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Computer-generated holographic matched filters

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Computer-generated holographic matched filters
Creator:
Butler, Steven Frank
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Steven Frank Butler
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English

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Subjects / Keywords:
Dynamic range ( jstor )
Fast Fourier transformations ( jstor )
Fourier transformations ( jstor )
Image filters ( jstor )
Images ( jstor )
Images of transformations ( jstor )
Matched filters ( jstor )
Signals ( jstor )
Simulations ( jstor )
Squares ( jstor )

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University of Florida
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University of Florida
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Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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Full Text












COMPUTER-GENERATED HOLOGRAPHIC MATCHED FILTERS


By

STEVEN FRANK BUTLER




























A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY





UNIVERSITY OF FLORIDA


1985















ACKNOWLEDGEMENT

The author wishes to thank Dr. Henry Register and Mr. Jim

Kirkpatrick for their encouragement to continue graduate studies at

the University of Florida. Dr. Roland Anderson has tirelessly

provided counseling and guidance during the years of study,

experimentation, and writing. Dr. Ron Jones of the University of

North Carolina assisted greatly with the understanding of film non-

linearity. Dr. S.S. Ballard provided the scholastic background and

the interest in optics throughout the author's scholastic career at

the University of Florida. The Air Force Office of Scientific

Research and the Air Force Armament Laboratory funded the laboratory

support for this effort. The University of Florida provided academic

and administrative support for the author's entire period of graduate

studies.

















TABLE OF CONTENTS

Page

ACKNOWLEDGMENTS...................................................ii

LIST OF TABLES.................................. ........... .... ..

LIST OF FIGURES..................................................vi

ABSTRACT..................................... ..... .... .............. x

CHAPTER

I INTRODUCTION..... .............. .............................1

Machine Vision............................................2
Optical Computers ........................................5
Contribution........................................ .. 7

II BACKGROUND.................................................9

Communication Theory....................................9
Vander Lugt Filtering ................................... 20

III COMPUTER-GENERATED HOLOGRAMS (CGH).........................24

Continuous -Tone Holograms..............................25
Binary Holograms.......................................30
Sampling and Space-Bandwidth Requirements...............39

IV OPTIMIZATION OF OPTICAL MATCHED FILTERS...................... 63

Performance Criteria................................... 63
Frequency Emphasis.......................................65
Phase-Only-Filters......................................72
Phase-Modulation Materials..............................76

V PATTERN RECOGNITION TECHNIQUES .............................84

Deformation Invariant Optical Pattern Recognition........85
Synthetic Discriminant Functions.......................88

VI MATCHED FILTER LINEARITY..................................94

Measurement of Film Characteristics......................97
Models for Film Non-linearity..........................102
Computer Linearization of Filter Response...............112











CHAPTER Page

VII SIMULATIONS...............................................133

Techniques for Simulating Matched F,ilters............... 134
Simulation of a Continuous-Tone Hologram................. 145
Simulation of a Binary Hologram......................... 151
An Example Using an SDF as a Reference...................159

VIII OPTICAL IMPLEMENTATION...... ...............................170

Techniques for Optical Implementation.................170
Examples of CGH Matched Filters ......................179

IX SUMMARY..................................... ............. 191

Conclusions......................... .. ... ............194
Recommendation...........................................195

BIBLIOGRAPHY.....................................................197

BIOGRAPHICAL SKETCH...............................................201

















LIST OF TABLES


TABLE Page

7.1 Signal-to-noise ratio and efficiency for an ideal 146
auto-correlation of a square.

7.2 Signal-to-noise ratio and efficiency for a 157
continuous-tone CGH.

7.3 Signal-to-noise ratio and efficiency for an 165
A-K hologram of a square.

7.4 Signal-to-noise ratio and efficiency of an 169
A-K hologram of a SDF correlating with members
of the training set.
















LIST OF FIGURES


FIGURE Page

3.1 Brown and Lohmann CGH cell. 33

3.2 Complex plane showing four quadrature components. 36

3.3 Addressable amplitude and phase locations
using the GBCGH method. 38

3.4 Spectral content of an image hologram. 42

3.5 Spectral content of a Vander Lugt filter. 44

3.6 Spectral content of a Fourier Transform hologram. 50

3.7 Two dimensional spectrum of the Fourier Transform
hologram. 51

3.8 Two dimensional spectrum of the Vander Lugt filter. 53

3.9 Spectrum of a modified Vander Lugt filter. 55

3.10 Spectrum of the zero mean Vander Lugt filter. 58

3.11 Output of a 50% aliased Vander Lugt filter with
absorption hologram. 60

4.1 High-frequency emphasis of a square and a disk. 67

4.2 Phase-only filtering of a square and a disk. 74

5.1 Training set for the creation of a SDF. 91

5.2 SDF created from the images in Figure 5.1. 92

6.1 Typical H & D curve. 96

6.2 Computer output of the polynomial fit routine. 111

6.3 H & D plot for Agfa 10E75 photographic plates. 113

6.4 Amplitude transmission vs. exposure for Agfa
10E75 plates. 114









FIGURE Page
6.5 Computer output of the polynomial fit routine for
8E75 plates. 115

6.6 H & D plot for Agfa 8E75 photographic plates. 116

6.7 Amplitude transmission vs. exposure for Agfa
8E75 plates. 117

6.8 Image and plot of a linear gradient used for a
test input. 120

6.9 Image and plot of the output transmission on
film from the gradient input. 121

6.10 Image and plot of the pre-distorted gradient
used for an input. 122

6.11 Image and plot of the output transmission with
pre-distorted input. 123

6.12 Image and plot of a sinusoidal grating pattern
used for input. 125

6.13 Image and plot of the output transmission with
the sinusoidal input. 126

6.14 Output spectrum for a sinusoidal input. 128

6.15 Image and plot of a pre-distorted sinusoidal
grating used as an input. 129

6.16 Image and plot of the output transmission for the
pre-distorted sinusoidal input. 130

6.17 Output spectrum for a pre-distorted grating input. 131

7.1 Computer simulation of an ideal correlation. 136

7.2 Fourier transform of a square. 139

7.3 Fourier transform of a square with high-frequency
emphasis. 140

7.4 Ideal auto-correlation of a square with no
pre-emphasis. 141

7.5 Ideal correlation of a square with
high-frequency emphasis. 142

7.6 Ideal correlation of a square using
phase-only filtering. 143










FIGURE
7.7


7.8

7.9


7.10


7.11


Flow chart for the continuous-tone hologram
simulation.

Continuous-tone CGH of a square.

Continuous-tone CGH of a square with
high-frequency emphasis.

Continuous-tone CGH of a square with phase-
only filtering.

Auto-correlation of a square using a continuous-tone
CGH.


7.12 Auto-correlation of a square using a continuous-tone
CGH with high-frequency emphasis.

7.13 Auto-correlation of a square using a continuous-tonE
CGH with phase-only filtering.

7.14 Flow chart for the binary hologram simulation.

7.15 A-K binary hologram of a square.

7.16 A-K binary hologram using high-frequency
emphasis.

7.17 A-K binary hologram of a square with
phase-only filtering.

7.18 Auto-correlation of a square using an A-K binary
hologram with high-frequency emphasis.

7.19 Auto-correlation of a square using an A-K binary
hologram with phase-only filtering.

7.20 A-K binary hologram of the SDF using
high-frequency emphasis.

7.21 Correlation of a test image at 300 and the SDF
using an A-K hologram with high-frequency emphasis.

8.1 Photo of an interferometrically produced optical
matched filter.

8.2 Cathode-ray tube and camera produced by the
Matrix Corporation.

8.3 Cathode-ray tube imaged onto a translation table
produced by the Aerodyne Corp.

8.4 Electron-beam writing system at Honeywell Inc.


viii


Page

147

150


152


153


e


e










FIGURE Page
8.5 Magnified views of a binary hologram produced on the
Honeywell E-beam writer. 180

8.6 A-K CGH matched filters, using a square as a
reference produced on the Honeywell E-beam writer. 181

8.7 Reconstruction from an A-K CGH matched filter of a
square using no pre-emphasis. 183

8.8 Reconstruction from an A-K CGH matched filter of a
square using high-frequency emphasis. 184

8.9 Reconstruction from an A-K CGH matched filter of a
square using phase-only filtering. 185

8.10 A-K CGH matched filter of the letters "AFATL"
using a) high-frequency emphasis and
b) phase-only filtering. 186

8.11 Reconstruction from an A-K CGH matched filter of
the letters "AFATL" using high-frequency emphasis. 187

8.12 Reconstruction from an A-K CGH matched filter of
the letters "AFATL" using phase-only filtering. 188

8.13 A-K CGH matched filter of the SDF shown
in Figure 5.1. 189

8.14 Reconstruction of an A-K CGH matched filter of
an SDF. 190

















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

COMPUTER-GENERATED HOLOGRAPHIC MATCHED FILTERS

By

STEVEN FRANK BUTLER

December 1985

Chairman: Roland C. Anderson
Major Department: Engineering Sciences

This dissertation presents techniques for the use of computer-

generated holograms (CGH) for matched filtering. An overview of the

supporting technology is provided. Included are techniques for

modifying existing CGH algorithms to serve as matched filters in an

optical correlator. It shows that matched filters produced in this

fashion can be modified to improve the signal-to-noise and efficiency

over that possible with conventional holography. The effect and

performance of these modifications are demonstrated. In addition, a

correction of film non-linearity in continuous-tone filter production

is developed. Computer simulations provide quantitative and

qualitative demonstration of theoretical principles, with specific

examples validated in optical hardware. Conventional and synthetic

holograms, both bleached and unbleached, are compared.


















CHAPTER I

INTRODUCTION

Human vision is a remarkable combination of high resolution

sensors and a powerful processing machine. This combination permits

understanding of the world through sensing and interpretation of

visual images. The faculty of vision is so natural and common that

few pause to think how marvelous it is to acquire such clear and

precise information about objects simply by virtue of the luminous

signals that enter the eyes. Without consciousness of the complicated

process, objects are recognized by the characteristic qualities of the

radiations they emit. With the help of memory and previous

experience, the sources of these manifestations are perceived. This

process is known as sight, perception or understanding.

Images and photographs have long been used to identify and locate

objects. By photographing an area, perhaps from afar, a scene could

be given detailed study. This study might disclose the presence of

objects of interest and determine their spatial location. Images from

satellites show weather, agriculture, geology and global actions.

Special images may contain additional scientific information including

object spectral characteristics, velocity, temperature, and the like.

The traditional medium of these images has been photographic film.

It is capable of high resolution and is sensitive to visible and near-

visible wavelengths. Unfortunately, film based methods are slow due

to exposure, processing, and analysis time. This time lag is not a











problem for many applications and so film is still the primary medium

for reconnaissance. Electronic imagery (TV, radar, etc.) is used for

those applications that require faster interpretation. These images

can be viewed, like film, by people trained to interpret the

particular images. Because of the electronic nature of the images,

electronic hardware and computers are used for manipulation of the

images.

Machine Vision

For very high speed retrieval and interpretation, machines must be

designed around the specific tasks. Machine interpretation is also

necessary when a human is not available. Unmanned robots work in

hazardous areas and perform many jobs more efficiently without the

encumbrance of human intervention. However, to function and carry out

their assigned job, the robots must have information about their

surroundings. The ability to interpret imagery from self-contained

sensors is necessary for the proper function of a robot. This image

interpretation includes guidance, obstacle avoidance, target

recognition, tracking, and closed loop control of robot action. For

robot action without human intervention, machine intelligence must

have the ability to make decisions based on scene content. Computer

image processing and recognition refer to techniques that have evolved

in this field in which the computer receives and uses visual

information.

Image processing techniques prepare or preserve an image for

viewing. This includes enhancement, restoration, and reconstruction.

Image enhancement techniques are designed to improve image quality for

human viewing. For example, correction of a geometrically distorted











image produces an obvious improvement in quality to a human observer.

Image restoration techniques compensate an image, which has been

degraded in some fashion, to restore it as nearly as possible to its

undegraded state. For example, an image which is blurred due to

camera motion may be improved using motion restoration. To perform

the difficult task of image interpretation, extraneous noise must be

separated from the desired signals. This may occur in several stages

of enhancement where each stage reduces the extraneous noise and

preserves the information crucial to object recognition. Image

enhancement may include contrast transformation, frame subtraction,

and spatial filtering. The goal of image enhancement is to reduce the

image complexity so that feature analysis is simplified.1

Once the scene has been enhanced, the job of interpretation is

simplified. The interpreter must now decide what the remaining

features represent. The features present a pattern to the interpreter

to be recognized. This pattern recognition problem may be quite

difficult when a large number of features are necessary to

differentiate between two possibilities. Most people have to look

closely to see any difference between two twins. A computer might

have equal difficulty distinguishing a car from a house in low-

resolution image.

Recognition involves an interpretation of an image. This includes

scene matching and understanding. Scene matching determines which

region in an image is similar to a pictorial description of a region

of another scene. A reference region or template is provided and

systematically compared to each region in a larger image. Here the

computer attempts to match models of known objects, such as cars,










buildings, or trees, to the scene description and thus determine what

is there. The model objects would be described in memory as having

certain characteristics, and the program would attempt to match these

against various parts of the image. Scene understanding involves a

more general recognition problem describing physical objects in a

scene based on images. For example, a scene may be divided into

regions that match various objects stored in memory such as a house,

tree, and road. Once the scene is divided into known regions, the

interrelationship between these regions provides information about the

scene as a whole.

When it is necessary to recognize specific objects, correlation

techniques are often used.2 A reference image of the desired object

is stored and compared to the test image electronically. When the

correlation coefficient is over a specified threshold, the computer

interprets the image as containing the object. The correlation

procedure may also provide the location of the object in the scene and

enable tracking. The correlation coefficient may be used in decision

making to determine robot action. Because even a single object may

present itself in many ways, correlation procedures are complicated by

the immense reference file that must be maintained.3 Special

correlation techniques may provide invariance to specific changes, but

a wide range of object conditions (i.e., temperature, color, shape,

etc.) make correlation recognition a complicated computer task.4 The

best computer vision systems now available have very primitive

capabilities. Vision is difficult for a computer for a number of

reasons. The images received by a sensing device do not contain

sufficient information to construct an unambiguous description of the











scene. Depth information is lost and objects frequently overlap.

Vision requires a large amount of memory and many computations. For

an image of 1000 X 1000 picture elements, even the simplest operation

may require 108 operations. The human retina, with 108 cells

operating at roughly 100 hertz, performs at least 10 billion

operations a second. Thus, to recognize objects at a rate even

closely resembling human vision, very special processor technologies

must be considered. One promising technology has emerged in the form

of optical computing.

Optical Computers

Optical computers permit the manipulation of every element of an

image at the same time. This parallel processing technique involves

many additions and multiplications occurring simultaneously. Most

digital processors must perform one operation at a time. Even though

the digital processors are very fast, the number of total operations

required to recognize patterns in an image is very large. Using

optical Fourier transformers, an optical processor can operate on the

image and its Fourier transform simultaneously. This permits many

standard image processing techniques, such as spatial filtering and

correlation, to be performed at tremendous rates.

The Fourier transform is formed optically by use of a lens. The

usual case that is considered in optical computing is when the

illuminating source is located at infinity (by use of an auxiliary

collimating lens) and the image transparency is located at a distance

equal to focal length from the transforming lens. The distribution in

the output plane located a focal length behind the transforming lens

is the exact Fourier transform of the input distribution. The Fourier











transform contains all of the information contained in the original

image. However, the information is now arranged according to spatial

frequency rather than spatial location. The advantage of such an

arrangement is that objects or signals of interest may overlap with

noise in the image domain but exist isolated in the frequency domain.

This permits the possible separation of signal from noise in the

frequency plane when it would have been impossible in the image plane.

The image can be transformed into frequency space, frequency filtered

and then transformed back into image space with the noise removed.

The frequency filter may be low-pass, high-pass, or band-pass, chosen

to optimize the filtering of a specific signal. This frequency plane

filter is the heart of the analog optical computer.

The frequency plane filter can be constructed in many ways. Low-

pass and high-pass filters are accomplished using simple apertures

mounted on axis in the frequency plane. More complicated filters are

produced optically using holographic techniques. These filters may

also be produced using computer-generated holography (CGH). The

computer is used to model the desired filter response, mathematically

represent the holographic filter, and create a physical filter using a

writing device. One of the important advantages of computer-generated

holography is that the reference need not exist physically, but only

mathematically. This permits mathematical manipulation of the

reference prior to creation of the filter for purposes of

optimization.

The advantage of an analog optical processor is that it may

operate at very high speeds. In addition, the processor typically is

smaller, lighter, and consumes considerably less power than an











equivalent digital processor.5,6 When coupled with the ability to

manipulate and optimize the frequency plane filter, the optical

processor becomes a useful tool. With considerable justification,

there is great interest in the robotics community.

Contribution

This dissertation states that CGH matched filters should be used

in an optical correlator to recognize patterns in a complex scene, and

describes how to create that filter. The CGH matched filter is

superior to interferometric filters due to the ability to pre-process

the filter function and control the production of the hologram. The

use of optical elements for high speed pattern recognition was first

proposed 20 years ago.7 The concept of using computers to define and

generate holograms came only two years later.8 Since that time,

considerable effort has been devoted to exploring the potential of

these CGH elements for reconstruction holography. Most of this effort

was devoted to optimizing the methods for encoding and writing the

holograms.8-13 More recently, interest has grown in the area of

efficiency improvement.14 The efficiency of a hologram for optical

correlation must be high in order to utilize low power, light weight

diode lasers. In separate but parallel efforts in artificial

intelligence, researchers have studied the effects of image

enhancement on pattern recognition.15 Though research in the various

fields is proceeding, a unified approach to the interrelation of pre-

processing, holographic encoding and physical implementation is

lacking. Specifically, the research in CGH, to date, has only been

for display or reconstruction holography, not matched filtering. This










dissertation describes the steps necessary and possible to create

practical matched filters using CGH.

The approach presented here ties many areas of research together

as they apply to CGH matched filters. Modifications to existing

encoding schemes which provide real valued filter patterns for use in

an optical correlator are explained in Chapter III. In addition,

Chapter III defines the space-bandwidth-product (SBP) required for

holographic matched filtering rather than for display holography as is

presented in existing literature. This includes procedures for

minimizing the SBP required. Pre-processing methods which apply

specifically to matched filtering are presented along with rationale

for their use in Chapter IV. Techniques for the use of CGH matched

filters as a pattern recognizer are reviewed in Chapter V.

Linearization methods for writing on film are derived and evaluated in

Chapter VI.

These various considerations are not independent, but rather, are

interwoven in the production of CGH matched filters. These

interactions can be fully analyzed only with a complete model

incorporating all the parameters. Chapter VII describes such a model

created to analyze the pre-processing, encoding and writing techniques

used to produce optimal CGH matched filters. Now that the various

methods have been developed and the analytical tools demonstrated,

specific examples are presented and analyzed. Chapter VIII describes

approaches for physically producing a transparency including specific

examples taken from Chapter VII. Finally, conclusions based on the

analysis are offered in Chapter IX.

















CHAPTER II

BACKGROUND

The background technology is reviewed here to understand the

operation of an optical processor more fully. A number of different

types of optical processors are in use today. These include one-

dimensional signal processors, two-dimensional image processors and

multi-dimensional digital processors. Only two-dimensional image

processors used for matched filtering are described here. A matched

filter optimizes the signal-to-noise ratio at a specific point when

the characteristics of the input are known.16 Typically, the desired

pattern and the nature of the background or noise in the input image

are known. Specifically, the input consists of a known signal s(x,y)

and an additive noise n(x,y). The system is linear and space

invariant with impulse response h(x,y). The criterion of optimization

will be that the output signal-to-noise power ratio be a maximum.

This optimum system will be called a matched filter for reasons that

will become clear as the derivation proceeds.

Communication Theory

A system is any unit that converts an input function I(x,y) into

an output function O(x,y). The system is described by its impulse

response--its output when the input is an impulse or delta function.

A linear system is one in which the output depends linearly on the

input and superposition holds. That is, if the input doubles, so does

the output. More precisely stated, let 01 be the output when I1 is










the input and 02 be the output when 12 is the input. Then the system

is linear when, if the input is al1+bI2 the output is a01+b02. This

property of linearity leads to a vast simplification in the

mathematical description of phenomena and represents the foundation of

a mathematical structure known as linear system theory. When the

system is linear, the input and output may be decomposed into a linear

combination of elementary components.

Another mathematical tool of great use is the Fourier transform.

The Fourier transform is defined by

CO
F(u,v) = fff(x,y) exp -j2rr(ux+vy) dx dy = F {f(x,y)}. (2.1)
-00

The transform is a complex valued function of u and v, the spatial

frequencies in the image plane. The Fourier transform provides the

continuous coefficients of each frequency component of the image. The

Fourier transform is a reversible process, and the inverse Fourier

transform is defined by

00
f(x,y) = IfF(u,v) exp j27nux+vy) dx dy = F-1{F(u,v)}. (2.2)
-00

The transform and inverse transform are very similar, differing only

in the sign of the exponent appearing in the integrand. The magnitude

squared of the Fourier transform is called the power spectral density


Of = !F(u,v) 2 = F(u,v) F*(u,v). (2.3)


It is noteworthy that the phase information is lost from the Fourier

transform when the transform is squared and the image cannot, in

general, be reconstructed from the power spectral density. Several

useful properties of the Fourier transform are listed here.










Linearity Theorem


F { af1(x,y) + bf2(x,y)} = a F{f1(x,y)} + b F{f2(x,y)} (2.4)


The transform of the sum of two functions is simply the sum of

their individual transforms. The Fourier transform is a linear

operator or system.

Similarity Theorem


F {f(ax,by)} = F(u/a,v/b)/ab where F(u,v) = F {f(x,y)} (2.5)


Scale changes in the image domain results in an inverse scale

change in the frequency domain along with a change in the overall

amplitude of the spectrum.

Shift Theorem


F {f(x-a,y-b)} = F(u,v) exp [-j(ua+vb)] (2.6)


Translation of patterns in the image merely introduces a linear

phase shift in the frequency domain. The magnitude is invariant to

translation.


Parseval's Theorem


fJ/F(u,v) 12 du dv = If (x,y)12 dx dy (2.7)


The total energy in the images plane is exactly equal to the

energy in the frequency domain.

Convolution Theorem


F{f(x,y) g(x,y)} = ff F(u,v)F(uo-u,vo-V) du dv


(2.8)










The Fourier transform of the product of two images is the

convolution of their associated individual transforms. Also the

Fourier transform of the convolution of two images is the product of

the individual transforms.

Correlation Theorem


Rfg(x,y) = fff(x,y) f(x-xo,y-yo) dxo dyo (2.9)


The correlation is very similar to the convolution except that

neither function is inverted.


Autocorrelation (Wiener-Khintchine) Theorem


Off(u,v) = F {Rff(x,y)} (2.10)


This special case of the convolution theorem shows that the

autocorrelation and the power spectral density are Fourier transform

pairs.

Fourier integral Theorem


f(x,y) = F-1{ F{f(x,y)}} (2.11)


f(-x,-y) =F {F {f(x,y)}}


Successive transformation and inverse transformation yield that

function again. If the Fourier transform is applied twice

successively, the result is the original image inverted and perverted.

It is also useful to define here the impulse function. Also known

as the Dirac delta function, it describes a function which is infinite

at the origin, zero elsewhere, and contains a volume equal to unity.

One definition of the Dirac delta function is











6 (x) lim (a/ r ) exp -a2x2. (2.12)
a-u



The delta function possesses these fundamental properties:


6 (x) = 0 for x 0 (2.13)

Co
f 6 (x)dx = f6(x)dx = 1 (2.14)
CO CO

6 (x) = 6(-x) (2.15)


6 (ax) = (1/a) 6(x) a A 0 (2.16)


f f(x) 6(x-a)dx = f(a). (2.17)


The Fourier transform of the delta function is unity. This property

provides a useful tool when studying systems in which an output is

dependent on the input to the system. When an impulse is the input to

the system, the input spectrum is unity at all frequencies. The

spectrum of the output must then correspond to the gain or attenuation

of the system. This frequency response of the system is the Fourier

transform of the output when an impulse is the input. The output of

the system is the impulse response. Thus, the impulse response and

the frequency response of the system are Fourier transform pairs. To

determine the output of a system for a given input, multiply the

Fourier transform of the input by the frequency response of the system

and take the inverse Fourier transform of the result. The convolution

property shows an equivalent operation is to convolve the input with

the impulse response of the system.


O(u,v) = I(u,v) H(u,v)


(2.18)









o(x,y) = F 0(u,v)} =F-1 I(u,v) H(u,v)} (2.19)


= f i(xoYo) h(x-xo,y-yo) dxo dyo


= f(x,y) h(x,y)


where denotes convolution.

Consider the effect of an additive noise on the input of the

system. Although the exact form of the noise n(x,y) may not be known,

the noise statistics or power spectral density may be predictable.

Thus, the effect of the system on the input is determined by its

impulse response or frequency response. That is, when there is

knowledge of the input signal and noise, the output signal and noise

characteristics can be predicted. The relationship of the input and

output are expressed in the following diagram and equations. The

letters i and o indicate the input and output terms while the letters

s and n indicate the signal and noise portions.



Linear System
s(x,y) + n(x,y) h(x -- so(x,y) + no(x,y)
h(x,y)



i(x,y) = si(x,y) + ni(x,y) (2.20)


o(x,y) = So(x,y) + no(x,y) (2.21)


0(u,v) = I(u,v) H(u,v) (2.22)


So(u,v) = Si(u,v) H(u,v) (2.23)


No(u,v) = Ni(u,v) H(u,v)


-(2.24)










Now that the relationships between the input and output of a

linear system are known, such a system may be utilized to enhance the

input. For example, assume an image has been degraded by some

distorting function d(x,y). The original image was convolved with the

distorting function, and the spectral contents of the ideal image

Fi(u,v) were attenuated by the frequency response D(u,v) of the

distorting system. By multiplying the degraded image by the inverse

of the D(u,v), the original ideal image is obtained. Any distortion

which can be represented as a linear system might theoretically be

canceled out using the inverse filter. A photograph produced in a

camera with a frequency response which rolls off slowly could be

sharpened by Fourier transforming the image, multiplying by the

inverse filter, and then inverse transforming. In this case, the

inverse filter is one in which the low frequencies are attenuated and

the high frequencies are accentuated (high pass filter). Because the

high frequencies represent the edges in the image, the edges are

accentuated and the photo appears sharper.17 As indicated in the

following diagram, the image is distorted by the function D(u,v) but

in some cases can be restored by multiplying by 1/D(u,v).


fi(x,y)- Fi(uv) X D(u,v):> fd(x,y) = blurred photograph


fd(x,y)-;-Fd(u,v) X 1/D(u,v > f'd(xy) = enhanced photograph


The linear blur of a camera is another classic example. Consider

traveling through Europe on a train with your camera. Upon getting

home and receiving your trip pictures, you find that all of them are

streaked by the motion of the train past the scenes you photographed.

Each point in the scene streaked past the camera, causing a line to be











formed on the film rather than a sharp point. The impulse response is

a line, and the corresponding frequency response of the distorting

system is a sine function (sin u /u). To retrieve the European

photo collection, merely multiply the Fourier transform of the

pictures by u/sin u and re-image.

In the physical implementation of this process, there are several

practical problems. To multiply the image transform by the inverse

function, a transparency with the appropriate response is produced.

In general, a transparency can only attenuate the light striking it.

That is, the transparency can only represent non-negative real values

less than one. Herein lies the problem. The inverse response

required to correct a specific distortion may, in fact, be complex.

In some cases, a combination of two transparencies can be combined to

provide complex values. One transparency is used for amplitude or

attenuation, and another phase transparency or phase plate is used to

provide the appropriate phase shift at each point. A phase

transparency can be produced by bleaching film with an appropriate

latent image induced in the emulsion. Chu, Fienup, and Goodman18

demonstrated a technique in color film which consists of three

emulsions. One emulsion was used as an amplitude transparency and

another emulsion was used as a phase plate. The appropriate patterns

were determined by a computer and the film was given the proper

exposure using colored filters.

Even with a two-transparency system, not all distortions are

possible to remove. Note that in the linear blur case, the inverse

response is u/sin u. The denominator goes to zero for specific values

of u, and the response has a pole at those values. The filter cannot











represent those values, and the practical filter is merely an

approximation to the ideal filter. It is worth noting that when the

distorting response reduces a frequency component to zero or below

some noise threshold, that component cannot be recovered. That is,

information is usually lost during the distorting process and inverse

filtering cannot recover it.

It is desirable to remove noise from a corrupted image. Although

it is not always possible to remove all of the noise, the

relationships between the input and output of a.linear system are

known. A linear system is optimized when most of the noise is

removed. To optimize a system the input must be specified, the system

design restrictions known, and a criterion of optimization accepted.

The input may be a combination of known and random signals and noises.

The characteristics of the input such as the noise spectrum or

statistics must be available. The classes of systems are restricted

to those which are linear, space-invariant, and physically realizable.

The criterion of the optimization is dependent on the application.

The optimum filters include the least mean-square-error (Wiener)

filter and the matched filter. The Wiener filter minimizes the mean-

squared-error between the output of the filter and actual signal

input. The Wiener filter predicts the least mean-squared-error

estimate of the noise-corrupted input signal. Thus, the output of the

Wiener filter is an approximation to the input signal. The output of

the matched filter is not an approximation to the input signal but

rather a prediction of whether a specific input signal is present.

The matched filter does not preserve the input image. This is not the

objective. The objective is to distort the input image and filter the










noise so that at the sampling location (xo,yo), the output signal

level will be as large as possible with respect to the output noise.

The signal-to-noise ratio is useful in the evaluation of system

performance, particularly in linear systems. In the matched filter,

the criterion of optimization is that the output signal-to-noise power

be a maximum. The input consists of a known signal s(x,y) and an

additive random noise n(x,y). The system is linear and space

invariant with impulse response h(xo,yo). To optimize the system or

filter, maximize the expression


Ro = So2(oyo)/E{no2(x,y)} (2.25)


where E{no2(x,y)} =ff no2(x,y) dx dy


at some point (xo,Yo). The problem is then to find the system h(x,y)

that performs the maximization of the output signal-to-noise ratio.

The output signal so(x,y) is


so(x,y) = //si(xo,Yo)h(x-xo,y-yo) dxo dyo (2.26)


and the output noise no(x,y) power is


ff ino(x,y) 2 dx dy = f !No(u,v) 2 du dv

= If Ni(u,v)l2 IH(u,v)|2 du dv. (2.27)

The signal-to-noise output power ratio becomes


|ff si(xo,yo)h(x-xo,y-yo) dxo dyo 2 (2.28)
R =

!Ni(u,v) 2 IH(u,v) 2


Thus to complete the maximization with respect to h(x,y), the power

spectral density or some equivalent specification of the input noise









must be known. Once the input noise is specified, the filter function

h(x,y) is the only unknown. Equation (2.28) becomes


E{no2(xoo) aso2(xo,yo)} > 0


Ni2(u,v)H2u,v) du dv alff si(x,y)h(x-xo,y-yo) dxo dy0 2 > 0


where Ro max = 1/a


and the maximum signal-to-noise ratio at the output is obtained when

H(u,v) is chosen such that equality is attained. This occurs when


ff ni2(x,y) h(x-xo,y-yo) dxo dy = si(x,y).


(2.29)


(2.30)


Taking the Fourier transform of both sides and rearranging gives


S(-u,-v)
H(u,v) ,

INi(u,v) 12


exp -j(ux0+vy0)


Thus in an intuitive sense, the matched filter emphasizes the signal

frequencies but with a phase shift and, attenuates the noise

frequencies. This becomes clear when the additive noise is white. In

this case the noise power is constant at all frequencies and thus has

a power spectral density of


INi(u,v)12 = N/2


where N is a constant.


(2.32)


From equation 2.32 the form of the matched filter for the case of

white noise is


H(u,v) = Si(-u,-v)exp -j(uxo+vyo)


(2.33)


= S*i(u,v) exp -j(uxo+vyo)


(2.31)










or
h(x,y) = s(-x,-y). (2.34)


Equation 2.34 shows that the impulse response of the matched filter

(with white noise) is simply the signal image in reverse order

(inverted and perverted). Thus, the filter is said to be matched to

the signal. Filtering with a matched filter is equivalent to cross-

correlating with the expected signal or pattern. That is,


O(x,y) = Rhs(x,y)


= ff s(xo,yo)h(xo-x,yo-y) dxo dy (2.35)


Also, it can be seen that the frequency response of the matched filter

is equivalent to that of the signal but with the phase negated so that

the output of the filter is real. That is, the matched filter removes

the phase variations and provides a real valued output.19

Matched filters are used extensively in radar signal

processing, seismic data processing, and communications. These

filters are implemented using electronic circuitry and digital

computers. For image processing, the need to process large two-

dimensional arrays places a large burden on conventional filtering

techniques. For these applications, optical processing techniques

provide the highest throughput speeds for matched filtering. One such

optical processing technique was proposed by Vander Lugt7 in 1969.

Vander Lugt Filtering

If an image is placed on a transparent sheet and illuminated by a

plane wave of coherent light, its Fourier transform is formed using a

simple lens.19 Once the Fourier transform is formed, specific

frequency components in the image can be removed or attenuated. The











result may then be inverse Fourier transformed to recreate the

modified image. The aperture, which may be replaced by a complicated

filter, functions to perform specific filtering operations including

Wiener or matched filter. Unfortunately, there are certain

limitations to the functions which can be physically implemented. A

normal transparency merely attenuates the light passing through it.

Its transmission is real and non-negative. Thus, when a transparency

film is exposed to a waveform to be recorded, the phase information in

the waveform is lost. Two pieces of information, the real and

imaginary parts of the waveform, are recorded as only one value, their

magnitude. This loss of information can be corrected by taking

advantage of the redundancy in the wavefront and the use of additional

film space. Using the heterodyning technique proposed by Vander Lugt,

the complex waveform can be recorded on photographic film.

Vander Lugt proposed the use of holographic film to store the

filter response for a matched filter. A lens is used to Fourier

transform the reference and test images. Derivations of the Fourier

transforming capabilities of lenses can be found in the literature.10

The Fourier transform of the reference image is brought to focus on a

photographic film. Film is a nonlinear, time-integrating medium and

thus only the magnitude of the Fourier transform or power spectral

density is recorded. The power spectral density does not contain all

of the original image information. Only the autocorrelation of the

original image can be obtained upon inverse transformation. Neither

the power spectral density nor the autocorrelation uniquely describe

the original image. If a plane wave is mixed with the Fourier

transform of the reference image at the film plane, the film will










record the interference pattern caused by the summation of the two

fields. The result on the film then is


H(u,v) = 1 + IF(u,v) 2 + F(u,v)exp j2Tav + F*(u,v)exp -j2Trav, (2.35)


which contains a constant, the power spectral density, and two terms

due to a spatial carrier fringe formed due to interference with the

plane wave. The two spatially modulated terms contain the original

image and Fourier transform information. With this Fourier transform

recorded on the film, it is placed in the optical filter arrangement

and illuminated with the Fourier transform G(u,v) of the test image

g(x,y). The output of the film transparency is the product of its

transmittance and the illuminating Fourier transform.


O(u,v) = G(u,v) H(u,v) (2.36)


= G(u,v) + G(u,v)IF(u,v)l2


+ G(u,v)F(u,v)exp j2rav + G(u,v)F*(u,v)exp -j2rrav


The product of the transforms from the reference and test images is

then Fourier transformed by another lens to obtain the correlation of

the two images.


o(x,y) = g(x,y) + g(x,y)*h(x,y)*h*(x,y) (2.37)


+ g(x,y)*f(x,y)*6(x,y-a)


+ g(x,y)*f*(x,y)*6 (x,y+a)


The first two terms are formed on axis or at the origin of the output

plane. The third term is the convolution of the reference and test











images and is centered off axis. The last term is the correlation of

the reference and test images and is located off-axis opposite the

convolution. This optical arrangement provides the entire convolution

and correlation images at once while a digital processor must compute

one point at a time. In addition to the convolution and correlation

processes, additional image plane and frequency plane filtering may be

accomplished simultaneously in the same optical arrangement. The

convolution, correlation and any additional linear filtering are

accomplished with a single absorbing mask.

When used as a matched filter, the transparency multiplies the

expected pattern by its complex conjugate, thereby rendering an

entirely real field. This matched transparency exactly cancels all

the curvature of the incident wavefront. When an input other than the

expected signal is present, the wavefront curvature will not be

canceled by the transparency and the transmitted light will not be

brought to a bright focus. Thus the expected pattern will be detected

by a bright point of light in the correlation plane. If the pattern

occurs in the input plane but is shifted, the bright point of light in

the correlation plane will shift accordingly. This provides for the

detection of specific patterns in a larger image. The detection and

location of specific objects in large complicated images is a job well

suited for the high-speed processing capability of the Vander Lugt

filter.
















CHAPTER III

COMPUTER-GENERATED HOLOGRAMS

Vander Lugt described a technique by which the holographic matched

filter could be produced optically.7 At that time, no other

convenient method existed for the computation and creation of the

complicated filter function required. This limitation has faded away

with the availability of digital computers with large memories. Using

digital computers to determine the filter function and a computer-

driven writing device, a transparency with the appropriate filter

image can be produced. Using this technique, the computer determines

the appropriate value of the matched filter at each point and produces

a transparency with that absorption at each point. The resolution

required of the writing device depends on the application and, in some

cases, may be consistent with optically generated holograms.

Computer-generated holograms (CGH) have found applications in

optical information processing, interferometry, synthesis of novel

optical elements, laser scanning, and laser machining.20-23 CGHs can

implement computer-optimized pattern-recognition masks.24 The

computer writes the hologram by transferring the transmittance

function to an appropriate holographic medium. The computer drives

a plotter or scanner and writes the hologram one point at a time.

Typically, the primary limitation is writing resolution. A

conventional optical hologram may have a resolution of one-quarter of

a micron. A system using visible light to write holograms (plotters,









flying spot scanners, CRT's, etc.) cannot achieve resolutions much

better than several microns. Writing systems utilizing electron beams

are currently achieving better than 1-micron resolution. The electron

beam systems are typically binary and thus the transmittance function

must be quantized in some fashion into two levels, "on" or "off."

Binary holograms are attractive because binary computer-graphics

output devices are widely available and because problems with

nonlinearities in the display and recording medium are circumvented.12

When photographic emulsions are involved, granularity noise is

reduced.25

Continuous-Tone Holograms

When a hologram is produced optically or interferometrically, a

reference wave is superimposed with the wavefront to be recorded.

Typically, the reference wave is a tilted plane wave with constant

amplitude across the wavefront. The reference wave approaches at an

angle 9 relative to the direction of the wavefront to be recorded.

The resultant field is


E(x,y) = f(x,y) + Aexp(j2Tray) (3.1)


where a= sin 0
x
and the amplitude of the reference wave is 1. An interference pattern

is produced by the superposition of the waves. The fringe spacing is

dependent on the term a, known as the spatial carrier frequency, and

the details in the function f(x,y). A photographic film placed into

this field records not the field itself but rather the square

magnitude of the field. The pattern recorded on the film is then









h(x,y):= f(x,y) + A ej27a 12 (3.2)


= A2 + If(x,y) 2 + A f(x,y)ej2Tay + A f (x,y)e-j2aay.


The function recorded on the film contains a D.C. bias, A2, the base

band magnitude, If(x,y)12, and two terms heterodyned to plus and minus

a. These heterodyned terms contain the complex valued information

describing the input function f(x,y). If the spatial carrier

frequency is sufficiently high, the heterodyned terms are separable

and no aliasing exists. The original input function can be retrieved

with no distortion by re-illuminating the film with the reference beam

and spatially filtering the output to separate the various terms.

To make the hologram of the Fourier transform of an image, the

same procedure is applied. That is, the Fourier transform of the

image f(x,y) is used as the input to the hologram. Now


h(u,v) = A2 + F(u,v)2 + A F(u,v)eJ2,au + A F*(u,v)e-j27au (3.3)


where F(u,v) = Fourier Transform of f(x,y) = F {f(x,y)} and


A e-j27au = the off-axis reference wave used to provide the spatial

carrier for the hologram.


a = sin e = the filter spatial carrier frequency (9 = off-axis angle)
X

This filter contains the D.C. bias, A2; the power spectral density,

IF(uv)12; and two terms heterodyned to plus and minus a. These

heterodyned terms contain the complex valued information describing

the Fourier transform of the input f(x,y).

These optically generated holograms are formed

interferometrically by combining a plane wave with the wavefront to be










recorded.- The transmittance of the hologram is a real valued, non-

negative function of position on the plate. Recall that the input

F(u,v), which was used to create the hologram, is, in general,

complex. This conversion from a complex function to a pattern which

can be recorded on film is known as coding. The coding performed in

optical holography is a natural consequence of the action of the film.

Typically, the complex wavefront is coded to a real non-negative

function which can be recorded as transmittance values on film.

Equation 2.35 describes a way in which film (a square law detector)

would encode the complex input image in an optically generated

hologram.

Once produced, the hologram and its interference fringes may be

inspected by microscope. The hologram can be copied on another plate

by contact printing. The hologram consists of real valued positive

transmittance varying across the face of the photographic plate. To

record the hologram on a computer, the transmittance across the

surface of the plate is sampled. If the samples are many and the

transmittance determined with accuracy, the hologram can be accurately

reproduced from the recorded samples. In this way the hologram can be

represented with some accuracy using digital numbers stored on some

electronic media. An electronic device writes the physical hologram.

The computer can electronically record, modify an optically produced

hologram, and then rewrite the holographic pattern onto another plate.

The limitations to such a system include the ability to sample the

input hologram sufficiently often and accurately, the ability to store

the large number of sample values, and the ability to rewrite the

holographic pattern to film.









If the input wavefront is known, the optical step may be omitted

altogether. If the input wavefront can be accurately represented by

discrete samples stored electronically, the holographic pattern can be

computed. That is, the input is coded to create a function which can

be recorded on a transparency. In the case of the matched filter, the

Fourier transform of an image is recorded. The image is sampled and

stored on the computer, and equation 2.35 is used to determine the

holographic pattern. Note that the continuous variables are replaced

by discrete steps. At each sample point the actual value is

represented by a finite number. The value may be complex, but the

accuracy is limited by the sampling system. In any case the

holographic pattern is computed and written to the photographic plate.

The writing device is known as continuous-tone when the transmittance

of each point in the holographic plate can be controlled over a wide

range of transmittance values. That is, the transmittance varies

smoothly from clear to opaque, including gray scale values between.

These continuous-tone holograms most closely resemble the optically

generated holograms when the sampling is dense and many gray scale

values are available.

When continuous-tone holograms are written to the photographic

plate using equation 2.35 as the model, they include a D.C. term, a

square magnitude term, and the heterodyned terms due to the tilted

reference wave. Note that the first two terms are real valued and

that the sum of the last two terms is real valued. On the computer,

the film process is emulated using equation 2.35 or other coding

schemes for specific applications. The D.C. and square magnitude

terms need not be included in the computer-generated hologram as long










as the heterodyned terms are scaled and biased to provide results

between 0 and 1. The heterodyned terms contain the desired

information. Omission of the baseband terms has no adverse effect on

the hologram. The square magnitude term typically contains a large

dynamic range. Its omission from the coding algorithm helps reduce

the dynamic range of the hologram and, in most cases, improves the

hologram. Equation 3.3 can be replaced by the expressions


H(u,v) = 2!F(u,v)l + F(u,v)eJ2lau + F*(u,v)e-j2nau (3.4)


H(u,v) = A2 + F(u,v)eJ2Tau + F*(u,v)e-J27Tau (3.5)


where each of these expressions includes the reference information, is

real valued, and is non-negative.

The dynamic range in the hologram, defined as the largest value

divided by the smallest value, is limited by the writing device used

to create the hologram. Most films have dynamic ranges much less than

10,000. That is, the clearest portions of the film can transmit light

no better than 10,000 times better than the darkest portions. If the

coding scheme requires a dynamic range of over 10,000, the writing

device cannot faithfully reproduce the holographic pattern.

Unfortunately, the dynamic range of film is frequently much less than

10,000 and closer to 100. Additionally, the writing device also

limits the dynamic range. Most continuous-tone writing devices, which

are attached to computers, convert an integer value to an intensity on

a cathode-ray tube or flying spot scanner. Due to physical

limitations in the writing intensity, the dynamic range is usually

much less than 1000. Most commercially available computer-writing

devices are designed with a dynamic range of 256 or 8-bit writing









accuracy.. The resultant transmittance on the film will have one of

256 quantized levels determined by an integer value provided by the

computer. Quantization occurs when all values in a specified range

are assigned to a quantized value representative of that range. If

the quantization steps become large, the quantized level may be a poor

estimate of the actual values. The estimate is equivalent to the

actual pattern with an additive noise called quantization noise.

Quantization noise occurs in computer-generated holograms because the

computer-graphic devices have limited gray levels and a limited number

of addressable locations in their outputs. Quantizing the holographic

pattern into 256 gray scale levels introduces quantizing noise which

may be considerable when the dynamic range of the pattern is large.

To minimize the quantizing error, the coding scheme must produce a

result with a dynamic range compatible with the writing system.

Some writing systems are capable of only two quantized levels.

These binary devices are either on or off. Most metal etchers, ink

plotters, dot matrix printers, and lithographic printers are binary.

The mark they create is either completely on or completely off. To

represent the reference pattern on binary media accurately requires

specialized coding schemes.

Binary Holograms

Binary holograms are attractive because binary computer-graphics

output devices are widely available and because problems with

nonlinearities in the display and recording medium are circumvented.

When photographic emulsions are involved, granularity noise is

reduced. Using the normal definition of dynamic range, binary

holograms have a dynamic range of 1. The transmittance at each point










is completely on or completely off. All gray scale effects must be

created by grouping light and dark areas together and averaging over

an area large enough to provide the needed dynamic range. In this

case the dynamic range is the averaging area. Thus, dynamic range is

exchanged for increased area to represent each point. This is similar

to Pulse Code Modulation (PCM) in an electronic communication

systems.26 In PCM, each sample value is quantized to M levels. Then

each level is represented by a binary code requiring N=log2 M bits.

Rather than represent each point with a continuous variable with

sufficient dynamic range, N binary variables are used. Each variable

is either on or off, but N variables are required to provide

sufficient dynamic range. This exchanges dynamic range of the

variables for the number of variables required. In binary holograms,

the variables are not, in general, exponentially weighted as in PCM;

thus, M variables are required to represent M levels. It becomes very

important to code the hologram such that the number of variables M

needed to represent that dynamic range is reasonable.

One of the first practical binary coding schemes was introduced

when, in 1966, Brown and Lohmann8 devised a method for complex

spatial filtering using binary masks. They coded the Fourier

transform of an image f(x,y). When using this method, the complex

Fourier transform is sampled and represented at each point by an

amplitude and phase. To record a complex filter, both amplitude and

phase information are needed on the hologram. However, the hologram

transmittance is real-valued, non-negative, and in this case binary.

The amplitude can be recorded by opening or closing an appropriate

number of binary windows in the hologram, but the phase is not









correct. Brown and Lohmann proposed turning the hologram at an angle

to the incoming waveform. Thus, along the surface of the hologram, a

phase shift occurs. This phase shift is proportional to the position

along the hologram. Using this "tilted wave" technique, a phase shift

occurs as the aperture moves up and down the hologram causing the

total path length through that aperture to change.i The further the

detour through the aperture, the larger the phase shift. Phase shift

induced by this technique is known as detour phase. Thus, in the

Brown-Lohmann hologram, an aperture is moved up and down to create the

appropriate phase shift. The size of the aperture is varied to allow

the appropriate amount of light through. To synthesize the complex

filter function F(u,v), a continuous function is sampled. The cells

of a sizeA u by Av must be sufficiently small that the function F

will be effectively constant throughout the cell.


F(u,v) = F(nAu,mAv) = Fnm =Anmexp ienm (3.6)


where n and m are integers


For each cell in the hologram, the amplitude and phase are determined

by the size and position of an aperture as shown in Figure 3.1. From

each cell a complex light amplitude Fnm will emerge. The tilted wave

must approach at an angle steep enough to allow for a full wavelength

of detour phase within one cell. The dynamic range of the amplitude

and phase is limited by the number of resolvable points within the

cell. If a cell has only 4 by 4 resolvable points, the dynamic range

of the amplitude or phase can be no better than 4. The granularity in

the amplitude and phase may cause distortion in the reconstructed

















Phase Shift

I


Am


x = ndx


Figure 3.1 Brown and Lohmann CGH cell.


2litude


-4T


Y = M d y_ 1I I


I


It









image. -Many points are required to represent a transform with a large

dynamic range accurately.

Lee9 proposed a method in 1970 which helped relieve some of

the phase granularity. The Brown-Lohmann technique represented each

cell with an amplitude and phase component. The complex value for

each cell may be represented by a magnitude and phase or by the sum of

in-phase and out-of-phase terms. The Lee method represents each cell

with such a quadrature representation. For each cell the magnitude

and phase are converted to real and imaginary components. As in the

Brown-Lohmann method, the tilted wave is set to provide a wavelength

of delay across the cell. The cell is divided into four components

which represent the positive and negative real and imaginary axes.

Lee defined the functions as


IF(u,v)lexp[j 0(u,v)] = F1(u,v)-F2(u,v)+jF3(u,v)-jF4(u,v) (3.7)


where


F1(u,v)= IF(u,v) cos+(u,v) if cos((u,v) > 0
= 0 otherwise,


F2(u,v)= IF(u,v) sinp(u,v) if sini(u,v) > 0
= 0 otherwise,


F3(u,v)= IF(u,v)Icos4(u,v) if cost(u,v) > 0
= 0 otherwise,


F4(u,v)= IF(u,v) sin(u,v) if sinq(u,v) > 0
= 0 otherwise.


For any given complex value, two of the four components are zero.

Each of the components Fn(u,v) is real and non-negative and can be

recorded on film. The Lee hologram uses four apertures for each cell










shown in Figure 3.2. Each aperture is positioned to cause a quarter-

wave phase shift by increased path length (detour phase). The two

non-negative quadrature terms are weighted to vector sum to the

appropriate magnitude and phase for each pixel. The two appropriate

apertures are opened according to their weight. The Lee method uses

continuous-tone variables to represent the two non-zero components.

The phase is no longer quantized by the location of the aperture. The

phase is determined by the vector addition of the two non-zero

components. In a totally binary application of the Lee method, the

apertures are rectangles positioned to obtain the quarter-wave shift.

The area of each aperture is adjusted to determine the amplitude of

each component. Once again, in this binary case, the dynamic range is

limited by the number of resolution elements within one cell.

Burckhardt10 showed that while the Lee method decomposes the

complex-valued F(u,v) into four real and positive components, only

three components are required. Each cell can be represented by three

components 1200 apart. Any point on the complex plane can be

represented as a sum of any two of these three components. As in the

Lee method, two non-negative components are chosen to represent each

cell. Because only three instead of four components have to be

stored, the required memory size and plotter resolution are reduced.

Haskell11 describes a technique in which the hologram cell is divided

into N components equally spaced around the complex plane. It is

identical to the binary Lee (N=4) and the Burckhardt (N=3) where N may

take larger values. This Generalized Binary Computer-Generated

Hologram (GBCGH) uses N columns and K rows of subcells. Each subcell

can take a transmittance value of 1 or 0. The phase is delayed by 2/N



































U


Figure 3.2 Complex plane showing four quadrature components.


F1 F2 F3 F4









to provide N unit vectors. The K cells in each component are "opened"

or "closed" to provide the appropriate weights for each component.

The control over the amplitude and phase is not absolute with finite N

and K. The result at each cell is the vector sum of components with

integer length and fixed direction. Figure 3.3 shows that various

combinations of points turned on or off define an array of specific

points addressable in the complex plane. By increasing the number of

points N and K, the amplitude and phase can be more accurately

matched. When the total number of plotter dots is limited and more

subcells used for each cell, fewer cells can exist. Thus, with a

limited number of points, the hologram designer must choose between

space-bandwidth product (number of cells) and quantization noise.

The GBCGH allows more accurate determination of the amplitude and

phase of the cell by using more points. However, the complex sample

to be represented was taken at the center of the aperture. If N, the

number of points in the cell, is large, the outer pixel may have

noticeable error due to the offset in sample location. Allebach12

showed that the Lohmann hologram fell into a class of digital

holograms which sample the object spectrum at the center of each

hologram cell to determine the transmittance of the entire cell. The

Lee hologram fell into a class of digital holograms which sample the

object spectrum at the center of each aperture to determine its size.

He also described a new third class in which the object is sampled at

each resolvable spot to determine the transmittance at that spot.

Although the function to be recorded should be constant over the

entire cell, there is some phase shift across the cell dimensions. By

sampling the object spectrum at the center of each aperture rather






























Re


N=K=3


Figure 3.3


Addressable amplitude and phase locations
using the GBCGH method.









than at-the center of each hologram cell, some of the false images in

the reconstruction are removed. By sampling the object spectrum at

the center of each resolvable spot in the hologram, the hologram noise

is further reduced. Allebach described an encoding technique in this

last category known as the Allebach-Keegan (A-K) hologram.13 The A-K

hologram encodes the complex-valued object spectrum by quadrature

components as does the Lee hologram. Unlike the Lee hologram, the A-K

hologram compares subsamples within the aperture to an ordered dither

to determine whether each pixel is on or off. The input image is

padded to provide as many points in the FFT as there are resolvable

points. The FFT is decomposed into components spaced a quarter wave

apart (or more as in the GBCGH). Each point is then compared to a

threshold determined by the threshold matrix. The threshold values

are chosen to quantize the amplitude of each component. The threshold

values divide the range from zero to the spectrum maximum in steps

determined by the Max quantizer.27 The size of the dither matrix and

the corresponding points in the cell can increase as with the GBCGH

but the magnitude and phase are sampled at each pixel.

Sampling and Space-Bandwidth Requirements

To represent an image on a computer, the image must be sampled and

quantized into a set of numbers. To sample a continuous image or

function, the value of the function is determined at discrete points.

The values of a function f(x,y) are determined at regular intervals

separated by Ax and Ay. The continuous independent variables x and y

are replaced with discrete sample point denoted by mAx and ny .

Here AX and AY are the fixed sample intervals and m and n are

integers. The sampling rate is u=1/Ax in the x direction and v=1/Ay









in the y direction. To convert the continuous function f(x,y) to a

sampled version f(mAx,nAy), multiply f(x,y) with a grid of narrow unit

pulses at intervals of Ax and Ay. This grid of narrow unit pulses is

defined as


s(x,y) = Z
m_- n=-~


6 (x-m x,y-n y)


(3.8)


and the sampled image is


fs(mAx,nAy) = f(x,y) s(x,y).


(3.9)


The sampled version is the product of the continuous image and the

sampling function s(x,y). The spectrum of the sampled version can be

determined using the convolution theorem (equation 2.8).


Fs(u,v)= F(u,v) S(u,v)


where

and


S(u,v)=


(3.10)


F(u,v) is the Fourier transform of f(x,y)

S(u,v) is the Fourier transform of s(x,y)


6 (u-mAu,v-nAv)


where u = /Ax and v = / Ay


Thus Fs(u,v) = ff F(u-uo,v-Vo) Z 6 (uo-mAu,vo-nAv) duo dvo
m= co n= oo
(3.11)

Upon changing the order of summation and integration and invoking the

sampling property of the delta function (equation 2.17), this becomes


0O 00
F(u,v) = E Z F(u-mAu,v-nAv).
m= -o n= .


(3.12)










The spectrum of the sampled image consists of the spectrum of the

ideal image repeated over the frequency plane in a grid space (Au, A).

If Au and Av are sufficiently large and the ideal function f(x,y) is

bandlimited, no overlap occurs in the frequency plane. A continuous

image is obtained from the sampled version by spatial filtering to

choose only one order m,n of the sum in equation 3.12. If the image is

undersampled and the frequency components overlap, then no filtering

can separate the different orders and the image is "aliased." To

prevent aliasing, the ideal image must be bandlimited and sampled at a

rate Au >2fu and Av >2fv. The ideal image is restored perfectly when

the sampled version is filtered to pass only the 0,0 order and the

sampling period is chosen such that the image cutoff frequencies lie

within a rectangular region defined by one-half the sampling

frequency. This required sampling rate is known as the Nyquist

criterion. In the image, the sampling period must be equal to, or

smaller than, one-half the period of the finest detail within the

image. This finest detail represents one cycle of the highest spatial

frequency contained in the image. Sampling rates above and below this

criterion are oversampling and undersampling, respectively. To

prevent corruption of the reconstructed image, no overlap of the

desired frequency components can occur.

Frequency overlap is also a problem in holography. Recall that in

equation 3.2 the ideal function f(x,y) was heterodyned to a spatial

carrier frequency by mixing with an off-axis reference beam, i.e.,


h(x,y) = A2 + !f(x,y) 2 + A f(x,y)ej2,ay + A f (x,y)e-j2vay (3.13)


and that the spectrum (shown in Figure 3.4) of this recorded signal is




























-4B -3B -2B -B 0


H (u)




IF 2
2F (u)




B 2B 3B 4B FREQ


-a


Figure 3.4 Spectral content of an image hologram.









H(u,v) = IA12 + F(u,v)@F(u,v) + A F(u,v+a) + A F(u,v-a) (3.14)


where F(u,v) is the Fourier transform of f(x,y) and denotes

convolution.

The first term is a delta function at (0,0). The second term is

centered on axis (0,0) but has twice the width as the spectrum F(u,v).

The third and fourth terms are the Fourier transforms of the f(x,y)

but centered off axis at plus and minus a. To prevent frequency

overlap, the second term and the heterodyned terms must not overlap.

This requires that the spatial carrier frequency, a, used to

heterodyne the information must be sufficiently large. Specifically,

this carrier frequency must be larger than three times the one-sided

bandwidth of the information spectrum.

In the case of the Vander Lugt filter and the subsequent

correlation, the output of the holographic matched filter has the form


o(x,y) = g(x,y) + g(x,yf)f(x,yY)f*(x,y)


+g(x,y)f(x,y) 6 (x,y-a)


+g(x,y~@f (x,y) 6(x,y+a). (3.15)


The output, shown in Figure 3.5, contains a replica of the test image

g(x,y) centered on-axis along with a term consisting of the test image

convolved with the autocorrelation of the reference image f(x,y).

This term consumes a width of twice the filter size plus the test

image size. In addition to the on-axis terms, there are two

heterodyned terms centered at plus and minus a. These heterodyned

terms have a width equal to the sum of the widths of the test image

g(x,y) and reference image f(x,y). Again to prevent overlap of the



























AzI I\ 1


-7b -6b -5b -4b -3b -2b -b


gff
f


0 b 2b 3b 4b 5b 6b 7b


Figure 3.5 Spectral content of a Vander Lugt filter









information terms in the output, a spatial carrier of sufficiently

high frequency is required to separate the heterodyned terms from the

on-axis terms. Assuming as an example that the test image and the

reference image are the same size 2B. The output positions of the

various terms can be shown graphically. To prevent the information

terms from overlapping with the on-axis terms, the carrier frequency,

a, must be chosen to center the heterodyned terms at 5B or more. In

the general case, the reference image f(x,y) and g(x,y) may have

different sizes. Let 2Bf represent the size of the reference image

and 2Bg represent the size of the test image. Then the requirement on

the carrier frequency, a, to prevent aliasing is


a = 3Bf + 2Bg. (3.16)


Sampling and heterodyning cause aliasing when improperly

accomplished. The combination of the two in the CGH requires

specific attention to detail. To create a CGH from a continuous image

f(x,y), it must first be sampled and quantized. According to the

Nyquist criteria, there are two samples for the smallest details in

the image. The sampling rate is at least twice the highest spatial

frequency in the continuous image. If a limited number of sampling

points are available, the image should be low pass filtered to limit

the highest frequency in the continuous image to half the number of

sampling points. This can be accomplished in an electronic sensor by

blurring the optics before the detector. When using a television

camera to digitize a transparency or film, the camera must be blurred

to match the detail in the continuous image to the number of points in

the digitizer. The detail required in the reference and test images









is determined by the pattern or target to be recognized. To detect

the presence of a desired target while an unwanted object could appear

in the test scene, sufficient detail to discriminate the two must be

included. To pick out cars from a scene which contains both cars and

trucks, the resolution must be adequate to resolve the differences

between the two. This resolution is typically chosen in an ad hoc

fashion using the human eye to determine what resolution is required.

Computer techniques have been used to quantify the resolution

required, but the results are usually not different than what a human

would have decided by eye. Although beyond the scope of this

dissertation, the bandwidth and specific frequencies best suited to

discriminate between targets and clutter can be determined with large

computers operating on adequate training sets.

The resolution must be adequate for target recognition. However,

oversampling beyond that resolution required will drive the CGH to

impractical limits. The resolution in the test image must match that

in the reference image yet the test image usually represents a much

larger area and larger total number of points. If the image already

exists in digital form, the resolution can be reduced by averaging the

image to produce an unaliased image of the appropriate number of

points. If an image is blurred or averaged to reduce the highest

spatial frequency, the detail above that maximum frequency is lost.

That is, all frequency components above the maximum are zero and lost.

Sampling the image properly (Nyquist criteria) permits the perfect

reconstruction of the averaged image, not the original image.

It is worthwhile to define the concept of space-bandwidth product

(SBP) here. The bandwidth of an image is the width of the spatial










frequency content to the highest spectral component. The space is the

physical length over which the image exists. For example, a piece of

film may have a maximum resolution of 100 points/mm with an image

which occupies 1 cm along the length of the film. In this case the

SBP is 100 points/mm X 10 mm = 1000 points. This is in one dimension.

For a square image, the number of points is 1,000,000. The SBP is the

number of resolution points in an image. The maximum SBP capability

of the film may not be utilized by an image recorded on the film, and

the actual SBP of the stored image will depend on the image itself.

In general, the bandwidth will be determined by the finest detail in

the image and the area of the total image. The area of the smallest

detail divided into the total image area defines the SBP. When a

continuous image is sampled at the Nyquist rate, one sample per

resolution point in the image is required. Thus, the SBP of the image

sampled at the Nyquist rate matches that of the continuous image. The

SBP in the sampled image is a very practical detail because each

sample must be stored in the computer memory. The number of

resolution elements in a 4" X 5" holographic film may exceed 108. A

computer cannot practically store such a large number of values. With

a limited number of memory locations on the computer, the sampling

rate and SBP demand careful consideration.

A CGH is created using a digitized image. A continuous film image

may be sampled and quantized to create a non-aliased digital image.

Some imaging sensors output data in digital format with no further

digitizing required. Once the digital image is obtained, the image

values may be manipulated on a digital computer. If this digital

image is encoded on a continuous-tone CGH using equation 2.35 as a









model, a spatial carrier frequency on the Fourier transform of the

image must be induced. The image is encoded as f(mAx,nAy) with a SBP

of M x N where M and N are the number of points in the image in each

direction. If the Fast Fourier Transform (FFT) is applied to the

image, a digital representation of the Fourier transform of the image

is obtained. This transformed image F(mAu,nAv) contains the same

number of points as the image and obviously the same SBP. If the

image contained M points along the x direction, the highest spatial

frequency possible in this image would be M/2 cycles/frame. This

situation would exist when the pixels alternated between 0 and 1 at

every pixel. That is, the image consisted of {0,1,0,1, ...}. The

maximum frequency in the transform is M/2 cycles/frame in both the

positive and negative directions. The FFT algorithm provides the real

and imaginary weights of each frequency component ranging from -M/2+1

cycles/frame to +M/2 cycles/frame in one cycle/frame steps. This

provides M points in the u direction. The same is true for N points

in the v direction. Thus, the first point in the FFT matrix is

(-M/2+1,-N/2+1), the D.C. term is in the M/2 column and N/2 row, and

the last term in the FFT matrix is (M/2,N/2).

It is useful to point out that the FFT describes the frequency

components of the image f(x,y). The FFT pattern also contains

structure which can also be represented by a Fourier series. That is,

the FFT pattern or image has specific frequency components. Because

the image and the FFT are Fourier transform pairs, the image describes

the frequencies in the FFT pattern. For example, a spike in the image

implies the FFT will be sinusoidal. A spike in the FFT implies the

image is sinusoidal. The existence of a non-zero value on the outer









edge of-the image implies the FFT contains a frequency component at

the maximum frequency. A non-zero value on the corner of the image

implies the maximum frequency exits in the FFT pattern which is M/2 in

the x direction and N/2 in the y direction.

To record the complex Fourier transform as a hologram, the

function F(mAu,nAv) must be heterodyned to a spatial carrier frequency

so as to create a real non-negative pattern to record on film. To

prevent aliasing, the heterodyne frequency must be sufficiently high.

The frequency components in the hologram are shown in Figure 3.6 and

consist of the D.C. spike, the power spectral density of the function

F(u,v), and the two heterodyned terms. To record the function F(u,v)

on film without distortion from aliasing, the spatial carrier

frequency must be 3 times the highest frequency component of the FFT

pattern. This permits the power spectral density term to exist

adjacent to the heterodyned terms with no overlap. The frequencies in

the hologram extend to plus and minus 4B. Thus, the hologram has a

space-bandwidth product 4 times larger than the original image in the

heterodyne direction. When heterodyned in the v direction as implied

by equation 2.35, the resulting hologram matrix must be larger than

the original image by 4 times in the v direction and 2 times in the u

direction. The spectral content in two dimensions is shown in Figure

3.7. The space-bandwidth product is very large for this CGH to record

the information in H(u,v).

The requirement is even greater when the hologram is to be used as

a Vander Lugt filter. When used as a Vander Lugt filter, the CGH must

diffract the light sufficiently away from the origin and the

additional on-axis terms to prevent aliasing in the correlation plane.































I I


f I
f


-4B -3B

-a


-2B


-B


0 B 2B 3B 4B


Figure 3.6 Spectral content of a Fourier Transform hologram.


~:. --


l


































-B
-2B


2
: .l:,r


-3B I | f*
-4B L 2B






















Figure 3.7 Two-dimensional spectrum of the Fourier Transform
hologram.









The output of the Vander Lugt filter is shown in equation 2.37 and the

spectral contents are plotted in Figure 3.5. These spectral components

are shown in two dimensions in Figure 3.8. Here the space-bandwidth

product is 7 times larger than the image in the v direction and 3

times larger than the image in the u direction. To produce a

correlation image without stretching, the samples in the u and v

directions should have the same spacing. Usually for convenience, the

hologram contains the same number of points in both directions, giving

a pattern which is 7B by 7B. The FFT algorithm used on most computers

requires the number of points to be a power of 2. This requires that

the hologram be 8B by 8B. For example, if the original images to be

correlated contain 128 by 128 points, the required continuous-tone CGH

contains 1024 by 1024 points. In a binary hologram, each continuous

tone point or cell may require many binary points to record the entire

dynamic range of the image.

This illuminates the key problem with CGH-matched filters. The

space-bandwidth product becomes large for even small images. Yet it

is the ability of optical processors to handle large images with many

points that makes them so attractive. Holograms created with

interferometric techniques contain a large amount of information or a

large space-bandwidth product. However, these optically-generated

holograms lack the flexibility offered by CGH. Holographic filters

are produced by either optical or computer prior to their actual use.

The filter imparts its required transfer function to the test image

without any further computation of the hologram pattern. Even if the

task is difficult, production of the filter is a one-time job. The

more information stored on the hologram, the greater the potential





















gf




-g fef*

.


fIl2


g@ f*


Figure 3.8 Two-dimensional spectrum of the Vander Lugt filter.









processing capability of the Vander Lugt filter. To produce powerful

yet practical CGH filters, the space-bandwidth product and dynamic

range of the hologram must be understood and minimized within design

criteria.

One key to reducing the space-bandwidth product of the CGH is to

recognize that much of the spectrum is not useful information. The

terms in Figure 3.5 are described as the convolution of f and g, the

baseband terms ffti and the correlation of f and g. Only the

correlation term is useful for our purposes in the Vander Lugt filter,

but the other terms arrive as a by-product of the square law nature of

the film. The two heterodyned terms which result in the convolution

and correlation of f and g must come as a pair. That is, when the real

part of the heterodyned information is recorded, the plus and minus

frequencies exist. The real part, cos e, can be written as

exp(je)+exp(-je) using Euler's formula. The plus and minus exponents

give rise to the plus and minus frequency terms which become the

convolution and correlation terms. The convolution and correlation

terms are always present in a spatially modulated hologram.

A more efficient hologram is produced using equation 3.5. This

hologram consists of a D.C. term sufficiently large to produce only

non-negative values and the heterodyned terms.


H(u,v) = A2 + F(u,v)ej2'av + F*(u,v)e-j27av (3.17)


The output (shown in Figure 3.9) of the Vander Lugt filter using this

hologram is


O(u,v) = A2G(u,v) + F(u,v)G(u,v)ej2Tav + F*(u,v)G(u,v)e-j2rav (3.18)






































-5B -4B -3B -2B -B


gf


B 2B 3B 4B 5B


-a a


Spectrum of a modified Vander Lugt filter.


Figure 3.9









or

o(x,y) = A2g(x,y) + f(x,y)@g(x,y)~(x,y+a) + f(-x,-y)g(x,y)@S(x,y-a)


= A2g(x,y) + f(x,y)@g(x,y)@6(x,y+a) + Rfg(x,y)@S(x,y-a) (3.19)


which gives the spectrum shown in Figure 3.9 assuming Bf=Bg=B. Here

the spectrum extends to 5B rather than 7B and considerable space

saving is possible. However, the 5B is not a power of 2 and most

computer systems would still be forced to employ 8B points. The terms

in Figure 3.9 are the convolution term, the image term, and the

correlation term. The image term arises from the product of the D.C.

term with the test image g(x,y). In a normal absorption hologram, it

is not possible to eliminate the D.C. term. The image term takes up

the space from -B to B, forcing the spatial carrier frequency to 3B

and requiring 5B total space. If the absorption hologram is replaced

with a bleached hologram where the phase varies across the hologram,

the D.C. term may be eliminated.

As discussed in Chapter II, film may be bleached to produce a

phase modulation. This is accomplished at the expense of the

amplitude modulation. However, this phase hologram behaves much like

the original amplitude or absorption hologram. One advantage of the

bleaching process and the use of phase modulation is the opportunity

to eliminate the D.C. term (set it to zero) and reduce the space-

bandwidth product. Equation 3.17 is changed to


H(u,v) = F'(u,v)ej2;av + F*'(u,v)e-j2rav (3.20)


where the prime mark (') indicates the function has been modified by

the bleaching process. There is no D.C. term, so the output of the









Vander Lugt filter is

0(u,v) = F'(u,v)G(u,v)ej2rav + F*,(u,v)G(u,v)e-j2rav (3.21)

or

o(x,y) = f'(x,y)g(x,y)6(x,y+a) + f'(-x,-yg(x,y) (x,y-a) (3.22)


= f'(x,y)g(x,y)@6(x,y+a) + Rf,g(x,y)@(x,y-a)


which gives the spectrum shown in Figure 3.10, assuming Bf=Bg=B.

This phase hologram reduces the number of points to 4B, a power of 2.

This is the smallest possible size in a spatially modulated hologram.

As will be shown later, the phase modulation process may significantly

affect the information, and the correlation obtained may be a poor

approximation to the ideal correlation.

The Vander Lugt filter is typically used to detect the presence of

a small object in a large scene. This implies that Bf may be much

smaller than B In any case, the least theoretical hologram size

using equation 3.20 is still twice the size of the reference image and

test image combined in the y direction. For example, a large scene

consisting of 1024 by 1024 points is to be searched for an object that

would occupy 32 by 32 points in that scene. The smallest continuous-

tone hologram to perform that correlation would contain 2112 points in

the y direction (at least 1088 in the x direction). For most

practical applications, the absorption hologram illustrated in Figure

3.9 would be used. For the same example consisting of a 1024 by 1024

test scene and a 32 by 32 reference image, a square hologram would be

at least 2144 by 2144.

Another practical consideration provides some relief in the size

of the correlation plane. The correlation of two images creates a





























gf*


g f


-4B -3B -2B -B 0 B 2B 3B 4B
-a a


Figure 3.10 Spectrum of the zero mean Vander Lugt filter.









correlation image whose size is the sum of the individual image sizes.

Non-zero correlation values can exist when any points in the two

images overlap. However, the number of points which overlap becomes

very small near the outer edge of the correlation plane. In a

practical system, a threshold is set to determine correlations which

are "targets" (above threshold) or "background" (below background).

When the target fills the reference image and is entirely present in

the test image, the autocorrelation condition exists and the

correlation can be normalized to one. When the target begins to fall

off the edge of the test image, correlations will still occur.

However, the correlation value will fall from unity by the ratio of

the target area present in the test image to the target area in the

reference image. A practical rule of thumb might be to ignore the

correlations when half of the target falls outside the test image in

any direction. This reduces the correlation plane to the size of the

test image, offering some relief to the required hologram size. If

the outer edge of the correlation plane is ignored, it does not matter

if that edge is aliased. This reduces the sampling and heterodyning

requirements in the filter hologram especially when the reference

contains many points. When using the absorption hologram with 50%

aliasing (shown in Figure 3.11), the spatial frequency is


a = Bg + Bf (3.23)


and the number of points in the hologram in the v direction (SBPv) is


SBPv = 2Bg + 3/2 Bf. (3.24)


Phase encoding this hologram does not relieve the requirement on the































Overlap Area


gf


H I....


I N\


-4B -3B


Figure 3.11


-2B
-a


0 B 2B 3B 4B


Output of a 50% aliased Vander Lugt filter with
absorption hologram.


g@f*


^8S88S888S88/L!!N '"!.'ei8,8









carrier frequency or the total number of points. The edges of the

correlation plane will fall into the active correlation region if a or

SBPv is reduced from the values given in equation 3.23 and 3.24.

In review, the SBP of the hologram is determined by the

following criteria.

(1) The required resolution in the reference scene to recognize the

desired target.

(2) The size of the reference scene. This is not normally a

significant factor due to the small size of the reference compared to

the size of the test image.

(3) The size of the test scene. The potential advantage of optical

processing is to test a large scene for the presence of the reference

object. The test image must contain the same resolution as the

reference image but includes many times the image area. Thus, the SBP

of the test scene is very large and is the driving factor in the size

of the CGH-matched filter.

(4) Usually, aliasing can be tolerated at the edges. This depends

the threshold and expected intensity of false targets. When 50%

imposed aliasing can be tolerated, the SBP reduces to an even multiple

of two.

(5) The dynamic range in the reference scene. The hologram must

adequately represent the dynamic range in the reference scene. In the

case of binary holograms, many binary points may be required for

adequate representation of each hologram cell.

(6) Hologram type. The type of CGH produced determines the encoding

scheme and number of points required to represent the SBP and dynamic






62


range of the reference while preventing aliasing of the active

correlation plane.

(7) Incorporate D.C. elimination when possible to minimize on-axis

terms.

By following these guidelines it is possible to determine the minimum

possible SBP needed in the CGH.
















CHAPTER IV

OPTIMIZATION OF CGH-MATCHED FILTERS

The previous chapters describe the basic design techniques

employed to create CGH-matched filters. To determine the performance

of these filters, specific criterion must be established.

Performance Criteria

Because the matched filter is based on maximizing the signal-to-

noise ratio, that criteria is reasonable to apply to the result of the

CGH also. The matched filter created as a result of a CGH is only an

approximation of the ideal filter. The non-linearities of the film,

along with the sampling, heterodyning, and quantizing of the CGH

process, cause the correlation to be less than ideal. The noise is

not just caused by background in the input image but also by artifacts

from the hologram. The matched filter was intended to recognize a

specific target in a clutter background, yet, in some cases, the

target will vary in size and orientation. There is a tradeoff between

using high resolution to discriminate against false targets and too

much sensitivity for target size and orientation. When modifying the

frequency content of the scene to best distinguish target from

background, the signal-to-noise ratio may decrease from the ideal.

Another important property of the optical matched filter is the

efficiency or light throughput. In a practical system, the input

image is illuminated by a laser of limited size and power. Typically

the laser source could be an IR diode putting out 10 mW.6 Even if the









signal-to-noise ratio is large, the energy reaching the correlation

plane may be too small to measure. The efficiency of the hologram,

the ratio of the power in the correlation to the power in the input

test image, is an important criterion in evaluating a practical CGH-

matched filter. Mathematically, it is given as

ff Ig(x,y) f*(x,y) 2dx dy
= _____________________ (4.1)
H
ff Ig(x,y)I2dxdy

where H has been coined the Horner efficiency,28 f is the reference

scene, g is the test scene, and denotes an ideal correlation. The

correlation derived from a Vander Lugt-matched filter is not ideal.

To determine the Horner efficiency for a CGH-matched filter, equation

4.1 must include an accurate model of the encoding scheme. This

efficiency can be measured experimentally using a known input source

and calibrated detectors. Caulfield28 estimated that efficiencies for

certain matched filters could be as low as 10-6. Butler and Riggins29

used models of CGH filters to verify Caulfield's prediction and went

on to recommend techniques for improving the efficiency.

The matched filter is used to determine the presence of a target

in a large scene. A test scene is correlated with a reference, and

the correlation plane is thresholded to indicate the target location.

Occasionally, the Vander Lugt filter will generate correlation values

above the threshold in areas where no target exists. Accordingly, the

correlation of an actual target corrupted by noise may be lower than

the threshold. Due to the presence of noise, random and otherwise,

the performance of the filter must be measured in terms of the

probability of detection and the probability of false alarm. The

probability of detection, Pd, is defined as the probability that a










target will be indicated when there is, in fact, a target to be

detected. The probability of false alarm, Pfa, is defined as the

probability that a target will be indicated when there is, in fact, no

target to be detected. These two quantities are correlated by the

presence of noise. If the detection threshold at the correlation

plane is lowered, the probability of detection is increased, but the

probability of false alarm is also increased. As with the efficiency

measurements, determining Pd and Pfa for CGH-matched filters requires

accurate models or optical experiments.

Historically, efficiency was not a concern in laboratory

experiments because powerful lasers were available to overcome the

hologram loss. When attempts are made to improve the efficiency, the

signal to noise ratio may suffer. An efficient hologram is

impractical if the signal-to-noise ratio in the correlation plane is

so low that Pd goes down and Pfa goes up significantly. The

performance of matched filters are typically measured in terms of the

Pd and Pfa, but testing requires modeling the entire system and

providing realistic images. All of these measures must be considered

for the cases when the test target deviates from the reference.

Optimization criteria for optical matched filters depend on the

application. To improve the matched filter, modifications to the

filter design have been proposed. These modifications fall into areas

of frequency modification, phase filtering, and phase modulation.

Frequency Emphasis

High frequencies in an image correspond to the small details.

Most images contain large continuous areas bounded by sharp edges.

The large continuous areas contribute to the D.C. and low frequency










content-of the image, while the edges contribute to the high

frequencies. If the high frequencies are removed from the image

through spatial filtering, the sharp edges disappear, the large

continuous areas blend together smoothly, and the resultant image

appears soft or blurred. A low-pass image may not provide sufficient

resolution to discriminate between two similar objects. If the low

frequencies are removed from an image, the continuous areas become

dark with only the edges remaining. The image appears sharp with

well-defined edges and detail. This high-pass image provides, to the

human eye, the same or better discrimination of the original image.

That is, objects are identified and distinguished at least as well as

in the original image. For example, images containing a bright square

area and bright circular area are easily distinguished as a square and

circle. If the high frequencies are removed, both square and circle

appear as blobs with no distinct edges. However, if the low

frequencies are removed, the bright area in the center of the square

and circle disappears, leaving only a bright edge. Yet these bright

edges clearly indicate a square and a circle as shown in Figure 4.1.

Even if the square is not filled in, the edge clearly denotes the

square shape. The edge of the circular area still defines a circle.

The square and circle are easily distinguished in the high-pass

images. The information that distinguishes the square from the circle

is contained in the high frequencies.

The traditional matched filter, as outlined in Chapter II, is

created from the complex conjugate of the Fourier transform of the

reference image. Filtering with such a filter is equivalent to

correlating the reference image with a test image. Because most





































































Figure 4.1 High-frequency emphasis of a square and a disk.









scenes contain large continuous areas with edges, they contain a large

D.C. and low frequency component. Most images have spectra where the

magnitude tends to drop off drastically with increasing frequency.

The energy in the low frequencies may be several orders of magnitude

larger than the high frequencies. However, it is the high frequencies

which contain the useful information in separating the desired target

from false targets. A practical problem with holography is the

dynamic range to be recorded. Film cannot typically induce more than

two or three orders of magnitude of dynamic range. To record a

hologram of the Fourier transform, the film must accurately record the

entire dynamic range of the transformed image. If the dynamic range

of the transformed image is too large, the film cannot record the

Fourier transform linearly and the correlation is not ideal. The film

non-linearity will emphasize some frequencies and attenuate others.

The correlation signal-to-noise ratio will suffer if important

frequency components are attenuated. To reduce the dynamic range of

the transformed image and allow linear recording on the hologram, the

useless frequencies in the image should be eliminated. Because the

low frequencies contain most of the image energy but little of the

information, their omission considerably reduces the dynamic range

with little effect on the correlation except to reduce the overall

light through the hologram.

To determine which frequencies are important in target

discrimination involves considerably more work than can be considered

here. In general, a set of target images and a set of non-target

images can be compared on a large digital computer to determine which

frequencies appear most in the desired target. This requires a large










data base of true and false targets. Filtered images are correlated

and cross correlated to determine the most discriminating frequencies.

In practice, this process is too time consuming. Certain assumption

are reasonable in spatial filtering. It is reasonable to assume that

the reference and test images do not have much more detail than is

absolutely necessary to distinguish the true target. To reduce the

number of points needed in the digital imagery, the original sampling

process was accomplished by limiting the spatial frequencies to those

required to recognize the target. Thus, the appropriate filter to

eliminate unnecessary frequency components will have the form of a

high-pass filter. The nature of this high-pass filter is dependent on

the application of the matched filter.

The matched filter is created for a specific target. If the

target is present, the correlation is larger than for areas of the

image where the target is absent. If the target changes slightly from

the reference stored on the filter, the correlation drops. In a

practical application, small changes in the expected target are the

rule rather than the exception. If the target grows in size, rotates,

or changes its appearance slightly, the correlation may drop below the

threshold. This topic will be discussed further in Chapter V, but it

is necessary to point out that the invariance of the filter to small

changes in the target depends heavily on the frequencies used in the

correlation. Using the previous example, recall that the high-pass

images showing the edges allowed discrimination between the square and

circle. If the square were rotated slightly, the results would

change. The cross-correlation between a square and a slightly rotated

square depends on the frequencies used in the correlation. If only









low frequencies are used, considerable rotation can occur with little

effect on correlation. If high frequencies are used, the cross-

correlation drops quickly with rotation. Thus, a matched filter

created from a high-pass image to discriminate against out-of-class

targets will not correlate well on in-class targets with small

changes. That is, as more high frequency emphasis is applied to the

matched filter, the discrimination sensitivity is increased. The

probability of false alarm is increased, but the probability of

detection drops. The high frequency emphasis is then tied to the Pd

and Pfa which must be specified for a particular application.

There is another advantage to the frequency emphasis of matched

filters. As seen in equation 2.35, the transmission of the hologram

at each point depends on the magnitude of the reference image Fourier

transform. Yet the hologram transmission cannot be greater than 1.

Depending on the dynamic range of the film, the transmission out at

the edge of the hologram corresponding to the high frequencies is very

low or zero. As the magnitude drops off for high frequencies, so does

the transmission of light through the holographic filter, and hence,

filter efficiency is low. However, if the high frequencies are

emphasized (boosted), the transmission at those points in a positive

hologram is likewise emphasized. This creates an overall increase in

the hologram transmission. In an absorption hologram, the light which

is not transmitted is absorbed and lost to the system. The throughput

or efficiency is highly dependent on the total transmission of the

hologram. Thus, by emphasizing the high frequencies, the efficiency

of the Vander Lugt filter is increased. Because the maximum

transmission is limited to 1 and the dynamic range is limited on the









film, the greatest efficiency occurs when most of the frequencies have

equal weighting and the transmission is close to 1 across the entire

hologram. This implies that the throughput of the hologram will be

largest when the image transform is nearly white.

The following procedures determine the choice of frequency

emphasis.

(1) Specify the Pd and Pfa for the particular application.

(2) Choose a high-pass emphasis which satisfies the Pd and Pfa

requirements. Typical choices include gradient, exponential, and step

filters.

(3) Because the test image should be filtered in the same fashion as

the reference image, the frequency emphasis chosen should be squared

before inclusion in the hologram. This permits the pre-emphasis of

the test image without a separate stage of spatial filtering. That

is, the test image is spatially filtered for pre-emphasis with the

same hologram providing the correlation.

(4) The test image is typically much larger than the reference image

and can thus contain frequencies lower than any contained in the

reference. Since those frequency components can never contribute to

correlations, all frequencies below the lowest useful frequency in the

reference should be truncated to the value of the next smaller term.

(5) The frequency emphasis (squared) greatly reduces the dynamic

range of most scenes, simplifying the coding of the CGH-matched filter

and greatly improving the efficiency. The frequency-emphasized CGH

matched filter is created, as shown in Chapter III, but utilizes a

reference image whose frequency content is modified.


F'(u,v) = IP(u,v)!2 F(u,v)


(4.2)









where F' :is the modified image transform,

F is the original image transform,

and P(u,v) is the frequency emphasis chosen.

Phase-Only Filters

The preceding section describes techniques in which the high

frequencies are emphasized. This emphasis usually improves the

discrimination against false targets and increases hologram

efficiency. Frequency emphasis involves the multiplication of the

image transform by a filter function which attenuates or amplifies the

appropriate frequency components. The filter function adjusts the

spectral magnitude of the image. In the Fourier representation of

images, spectral magnitude and phase tend to play different roles and,

in some situations, many of the important features of a signal are

preserved even when only the phase is retained. Oppenheim15 showed

that when the magnitude portion of an image Fourier Transform is set

to an arbitrary constant and the phase left intact, the reconstructed

image closely resembles the original. Features of an image are

clearly identifiable in a phase-only image but not in a magnitude-only

image. Statistical arguments by Tescher30 and by Pearlman, and Gray31

have been applied to real-part, imaginary-part, and magnitude-phase

encoding of the discrete Fourier transform of random sequences. They

conclude that, for equivalent distortion, the phase angle must be

encoded with 1.37 bits more than the magnitude. Kermisch32 analyzed

image reconstructions from kinoforms, a phase-only hologram. He

developed an expansion of the phase-only reconstructed image I(x,y) in

the form










I(x,y) = A [Io'(x,y) + 1/8 Io'(x,y/@Ro'(X,Y)


+ 3/64 Io'(x,y)Ro'(x,y)@Ro'(x,y) + .] (43)


where Io'(x,y) is the normalized irradiance of the original object,

Ro'(x,y) is the two-dimensional autocorrelation function of Io'(x,y)

and denotes convolution. The first term represents the desired

image, and the higher terms represent the degradation. Kermisch

showed that the first term contributed 78% to the total radiance in

the image, giving a ratio of 1.8 bits.

The phase-only image typically emphasizes the edges as in the case

of the high-pass filtering as shown in Figure 4.1. This phase-only

filtering is closely related to the high-pass filter. Most images

have spectra where the magnitude tends to drop off with frequency. In

the phase-only image, the magnitude of each frequency component is set

to unity. This implies multiplying each pixel magnitude by its

reciprocal. The Fourier transform tends to fall off at high

frequencies for most images, giving a mound-shaped transform. Thus,

the phase-only process applied to a mound-shaped Fourier Transform is

high-pass filtering. The phase-only image has a high-frequency

emphasis which accentuates edges. The processing to obtain the phase-

only image is highly non-linear. Although the response 1/IF(u,v)l

generally emphasizes high frequencies over low frequencies, it will

have spectral details associated with it which could affect or

obliterate important features in the original. Oppenheim15 proposed

that if the Fourier transform is sufficiently smooth, then

intelligibility will be retained in the phase-only reconstruction.

That is, if the transform magnitude is smooth and falls off at high




































































Figure 4.2 Phase-only filtering of a square and a disk.









frequencies, then the principal effect of the whitening process is to

emphasize the high frequencies and therefore the edges in the image,

thereby retaining many of the recognizable features. In Figures 4.1

and 4.2 the phase-only filter emphasizes edges more strongly than a

gradient filter for the examples shown.

The advantage of using a phase-only image or high-pass image is

the increase in optical efficiency of the resultant matched filter.

As shown in equation 2.35, the transmission of each hologram element

depends on the magnitude of the reference image Fourier transform. As

the magnitude drops off for high frequencies, so does the transmission

of light through the holographic filter, and hence filter efficiency

is low. If the magnitude is set to unity (phase-only filter) for all

frequencies, the overall efficiency increases dramatically. The image

transform is white and thus the throughput of the absorption hologram

is highest. Horner14 shows that the maximum throughput efficiency of

an ideal autocorrelation of a 2-D rect function is only 44%, while the

autocorrelation using an phase-only filter achieves 100% efficiency.

The phase function, ((u,v) of an image Fourier transform is a

continuous function. To fabricate a phase-only filter for such an

image requires a linear process capable of faithfully reproducing the

whole range of values from 0 to 2 If the phase is quantized so as

to permit only two values, typically 0 and pi, such a filter is known

as a bi-phase filter.


H'(u,v) = sgn [cos 4(u,v)] = +1 if Re [H(u,v)] > 0 (4.4)
= -1 otherwise


where H(u,v) is the Fourier transform of the filter impulse response

h(x,y), the sgn operator gives the sign of the argument, and H'(u,v)









is the bi-phase transform. This bi-phase information is an

approximation to the phase-only information. In many cases,

reconstructions from this bi-phase information contain the same detail

as the ideal amplitude and phase information. This would indicate

that much of the information in an image is contained in the sign of

each pixel or where the zero-crossings occur.

In converting a complex wave, which contains continuous magnitude

and phase values, to binary values, much is thrown away. If the

reconstructions from the binary image transforms are similar to the

original image, then the bi-phase conversion reduces redundancy and

eliminates superfluous dynamic range. When this is accomplished in an

optical correlator without significant reduction in signal-to-noise

ratio, the CGH-matched filter is greatly simplified. Most important

is the ability to use binary light modulators. A number of electronic

spatial light modulators are commercially available. Of these

modulators, several can be used to phase-modulate a light wave. These

include deformable paddles, liquid crystals, and magneto-optical

modulators. These can be used as bi-polar phase modulators.33 If the

information in the reference image can be accurately represented using

only bi-phase information, binary phase modulators can be used as

real-time holographic filters. The ability to adapt the matched

filter in real time permits scanning the test image for various

targets with varying sizes and orientations. This technique is very

efficient because the light is phase shifted and not attenuated.

Phase-ModulationMaterials

Recall that spatially modulated holograms are needed for matched

filtering only because film cannot record a complex wavefront. Film










can record only real values. Film may be used to record, at baseband,

the magnitude of a wavefront, or it may be computer-encoded and phase-

modulated (bleached) to record the phase of a wavefront. Thus,

without using a spatially modulated hologram, the magnitude or phase

may be recorded. If only the phase information of the image is needed

to represent the reference image, a baseband hologram which records

the phase portion of the image transform can be used in the optical

correlator. This on-axis phase hologram, or kinoform, is recorded as

a relief pattern in which appropriate phase delays are induced in the

illuminating wavefront. To produce a Fourier transform kinoform, the

phase is restricted to a range from pi to + pi. The arctangent of

the ratio of the imaginary and real parts yields such a result. The

film is exposed to a pattern, whose intensity is proportional to the

desired phase, and bleached to create a relief pattern.34 These

kinoforms cannot record the amplitude variation of the image transform

and thus, the filter formed is a phase-only filter.

Several techniques have been proposed by which the phase could be

modified to introduce amplitude variation in the reconstructed

wavefront.35,36 Chu, Fienup, and Goodman18 used multiple layers of

film to represent both the phase and amplitude variation. Kodachrome

II, for color photography, contains three emulsions. The phase

variation was recorded on one emulsion and the amplitude on another.

The inventors named this technique Referenceless On-Axis Complex

Hologram (ROACH). To introduce amplitude variation to the

reconstructed wavefront, light must be removed from the wavefront,

resulting in a reduction in efficiency.









The reconstruction from the kinoform is formed on-axis and is a

phase-only image. When the phase values are uniformly distributed

between pi and + pi, the D.C. or average term is zero. However, if

the phase recording is unbalanced or the phase distribution is not

uniform, a D.C. term will exist in the hologram. When used as a

matched filter, the kinoform must be carefully phase-balanced to

prevent a large D.C. spike from occurring in the correlation plane.

Such a spike would be indistinguishable from an actual correlation.

If the phase hologram is produced using a "real time" holographic

device, the phase might be controlled using a feedback loop to

eliminate the D.C. term prior to correlation. To produce a

"permanent" hologram on film, the exposure and bleaching processes

must be carefully controlled.

Bleaching includes several processes which produce a phase

hologram from an exposed absorption hologram. The bleached hologram

retards the wavefront, causing a phase shift instead of attenuation.

The result is generally an increase in diffraction efficiency but

often with an accompanying decrease of signal-to-noise ratio.37 There

are three basic types of bleaches. The "direct" or "rehalogenizing"

method converts metallic silver back into silver halide which has a

different index than that of the gelatin. "Reversal" or

"complementary" bleaches dissolve the metallic silver from an unfixed

hologram, leaving the undeveloped silver halide which varies the index

of refraction. The third process creates a surface relief by

shrinking the exposed portions of the hologram by removing the

metallic silver. When the emulsion is bleached properly, the

attenuation of the transparency can be reduced to the point that phase










modulation due to index changes dominates any residual amplitude

modulation. Phase modulators prove to be more efficient in terms of

the portion of incident illumination that is diffracted to form the

desired correlation. A sinusoidal hologram using absorption or

amplitude modulation can theoretically diffract only 6.25% of the

incident energy into an image. Experimentally, the number is about

4%.38 A phase-modulated hologram transmits all of the light (ignoring

the emulsion, substrate, and reflection losses). A sinusoidal phase

hologram can diffract as much as 33.9% of the incident light into the

first order.

The bleaching process converts the real function F(u,v), recorded

in silver on the film, to a phase delay.


H(u,v) = exp j[ F(u,v) ] (4.5)


To produce a kinoform, the film is exposed to the phase function

e(u,v) of the image transform. Upon subsequent bleaching, the film

contains the response


H(u,v) = exp j[ 6(u,v) ]. (4.6)


The kinoform, produced in this fashion, records the phase-only

information of the image transform. The bleaching process is not

restricted to phase-only information. Rather, the absorption hologram

created from equation 2.35 can also be bleached.


H'(u,v) = exp j[ H(u,v) ] (4.7)


= exp j[1 + IF(u,v)12 + F(u,v)exp j2irav + F*(u,v)exp -j2rav]










where H'(u,v) is the bleached hologram response. The phase-only

information and the phase modulation obtained through bleaching are

entirely independent of one another. That is, a phase-modulated

hologram can be created from an image whose amplitude and phase are

intact or from an image whose amplitude or phase are modified or

removed. Considerable confusion continues to exist in the literature

in which a phase modulation process seems to imply, by default, phase-

only information. Cathey attempted to clarify this confusion in 1970

by defining specific terms for each case.39 The holographic process,

which is independent of the recorded information, was described as (1)

phase holography when only phase modulation was present, (2) amplitude

holography when only amplitude modulation was present, and (3) complex

holography when both amplitude and phase modulation were present. In

an equivalent fashion, the information to be recorded on the hologram

can be described as (1) phase-only information or (2) amplitude-only

information when either the amplitude or phase portion of the complex

waveform are deleted. Thus, for example, an amplitude hologram can be

created from phase-only information.

When an amplitude hologram is bleached, the density at each point

on the film is mapped to a phase delay. This mapping is linear when

the bleaching chemistry is correct. This new phase function on the

film is related to the original pattern on the film.


H(u,v) = exp j{F(u,v)} (4.8)


where H(u,v) is the complex function on the film after bleaching and

F(u,v) was the original transmission pattern recorded on the film.









The exponential expression in 4.5 can be expanded with a series

expression.


H(u,v)= 1 + jF(u,v) (1/2)F2(u,v) j(1/6)F3(u,v) +...


S [jF(u,v)]n (4.9)
n!

When reconstructed, this hologram can be expressed as a series of

convolutions.


h(x,y) = (x,y) + jf(x,y) (1/2)f(x,y)f(x,y) -


-j(1/6)f(x,y)@f(x,y)f(x,y) + ...


= jn f(n)(x,y) (4.10)
n!


where f(n)(x,y) = f(x,y)@f(x,y) .. f(x,y) n convolutions


and f(O)(x,y) = S(x,y)


f(1)(x,y) = f(x,y)


f(2)(x,y) = f(x,y)f(x,y)

and so on.

Thus, the phase modulation technique is very non-linear and the

resultant reconstruction is rich with harmonics. The reconstruction

from such a hologram is noisy due to the harmonic content. The higher

order correlations are broader, thus contributing less flux into the

reconstruction. Phase modulation in the form of bleached and

dichromated gelatin holograms have become the rule in display

holography due to the bright images. This fact indicates that the










noise is acceptable in many cases. In fact, the reconstruction of

such display holograms looks very good. Nevertheless, such an example

is deceiving because the repeated convolutions and correlations of

equation 4.10 become more detrimental for more complicated objects,

especially if the object has low contrast.32 The harmonics combine to

produce intermodulation terms within the bandpass of the desired

information, causing an increase in background noise. When used for

matched filtering, the decision to use phase modulation is a balance

between hologram efficiency and signal-to-noise ratio.

An interesting case occurs when a binary amplitude hologram is

converted to a phase modulation hologram. The bleaching process maps

an amplitude of zero and one to a phase shift of plus and minus pi.

This equates to an amplitude of plus and minus one. For this binary

mapping, the transfer function is 2x-1, which is a linear process. In

that sense, the binary hologram is inherently linear. The binary

hologram represents the continuous-tone amplitude hologram by opening

more or fewer binary "windows". Through the use of many "windows,"

the amplitude can be accurately represented by the appropriate

combination of binary values. The subsequent bleaching of the binary

hologram is a linear process and thus no additional harmonics are

contributed. This provides a means by which high efficiency holograms

may be produced without sacrificing signal-to-noise ratio due to non-

linearity. A sufficient number of points is necessary in the binary

hologram in order to minimize the non-linearity of the binary CGH

mapping. When a computer and writing device are available to produce

such binary holograms, subsequent bleaching or phase modulation

greatly improves the efficiency without any adverse effect on signal-






83


to-noise. This makes digital, phase-modulated holograms very

attractive for matched filtering.
















CHAPTER V

PATTERN RECOGNITION TECHNIQUES

Coherent optical correlators have been used as a means of

performing 2-D pattern recognition.40-43 An optical correlator system

could scan a large scene for the presence of a specific pattern. The

input image is correlated with the impulse response of the matched

filter to determine the presence and position of the reference

pattern. Because the Fourier transform is shift invariant (equation

2.6), correlation can occur anywhere in the input image and multiple

targets can be recognized simultaneously. However, other changes in

the input pattern do effect the correlation function. Rotation, scale

changes, and geometrical distortions due to viewing a 3-D scene from

various angles can lead to correlation degradation and a corresponding

loss in detectability.44 For example, to recognize a hammer in a box

of tools, the reference must be capable of correlating with the hammer

when it is laying in any orientation from 0 to 3600. The hammer could

lay on either side so that both orientations would need to be included

in the reference image. If we were not sure of the distance from the

camera to the hammer, we would not be sure of its size in the image.

The fundamental difficulty in achieving a practical recognition

system lies in correlation of the reference image with a real-time

image which differs in scale, aspect, contrast, and even content when

sensed in a different spectral band or at a different time than the

reference image. Matched filter pattern recognition systems, both










optical and digital, tend to suffer from two types of difficulties.

They tend to be too sensitive to differences likely to occur in the

desired pattern. These differences are termed "within-class

variations." Second, they tend to be too insensitive to differences

between real and false targets. These are "between-class variations."

While other deformations in the object condition are possible in

specific applications, translation, rotation, and scale are the most

common in pattern recognition whether it is accomplished optically or

digitally.

Deformation Invariant Optical Pattern Recognition

The basic operation performed in an optical processor is a two-

dimensional Fourier transform. Matched spatial filters are used to

perform correlations between an input image and a reference pattern.

While the reference pattern may exist in the input image, it may be

deformed by scale, rotation or geometrical distortion. The Fourier

transform is invariant to shift in two dimensions (see equation 2.6).

It is not however invariant to scale or rotation, and a dramatic loss

in signal-to-noise ratio (3 to 30 dB) occurs for small scale changes

(2%) or rotation (3.50).44

In some applications it is desirable to give up translation or

shift invariance in substitution for some other deformation

invariance. The technique described by Casasent and Psaltis45

involves a space variant coordinate transformation to convert the

deformation under consideration to a shift in the new coordinate

system. Because the optical processor performs two-dimensional

transforms, it is insensitive to shifts in two dimensions. Thus, two

separate invariances can be accommodated. Scale can be converted to a










shift in one direction and the normal shift can be left in the other

dimension. This would provide scale invariance, but the resultant

correlation would only yield the location of the target in only one

dimension (i.e. the x coordinate).

In another example, the scale can be converted to shift in one

dimension and rotation converted to shift in another dimension. Such

a two-dimensional optical correlator could provide correlations on

rotated and scaled objects but would no longer predict the location of

the object. The two-dimensional nature of the optical processor

allows the correlator to be invariant to both deformations. In order

to provide invariance to other deformations two at at time, a

coordinate transformation is needed to convert that deformation to a

coordinate shift. The Mellin Transform is an excellent example of

such a transformation used to provide scale and rotation invariance.

The Fourier transform is invariant to translation shift in two

dimensions. To provide invariance to other deformations, a coordinate

transformation is needed to convert each deformation to a shift. To

provide scale invariance a logarithmic transformation is used. The

logarithmic transformation converts a multiplicative scale change to

an additive shift. This shifted version will correlate with the

logarithmically transformed reference pattern. To provide rotation

invariance, a transformation is performed to map the angle to each

point in the image to a theta coordinate. If an object rotates in

the test image, it is translated along the theta coordinate. Usually

the two transformation are combined into the log r, theta

transformation. The test image as well as the reference image is

converted to polar form to provide r and theta values for each pixel.










The image is transformed into a coordinate system where one axis is

log r and the other axis is theta. In this system, scale changes

shift the object along the log r axis and rotation shifts the object

along the theta axis. Because this transform, known as the Mellin-

Fourier transform, is itself not shift invariant, it is normally

applied to the Fourier transform of the test image. This provides the

shift invariance but loses the location information in the test scene.

The cross correlation between the transformed test and reference

images no longer can provide the location of the object but does

determine the presence of the object, its size, and its rotation

relative to the reference pattern.

To perform the Mellin-Fourier transform for shift, scale, and

rotation invariance, the input image is first Fourier transformed and

the resultant power spectral density recorded. This magnitude array

is converted to polar coordinates and the linear radial frequency is

converted to a logarithmic radial coordinate. The new coordinate

space (log r,theta) is used for cross-correlation of the input image

with similarly transformed reference images. A high speed technique

is required to convert the image into log r, theta coordinates at a

speed compatible with the optical processor. This has been

demonstrated using holograms to perform geometrical

transformations.46-50 To do this, the coordinate transforming

hologram must influence the light from each point and introduce a

specific deflection to the light incorporating such modifications as

local stretch, rotation, and translation.

A practical correlator system might incorporate such an optical

transforming system or a sensor which collects data in the appropriate









format by the nature of its scan pattern. Whether accomplished by the

sensor scan or by a coordinate transformation, the logarithmic

coordinate transformation is equivalent to resampling an image at

logarithmically spaced intervals. An increase in space bandwidth

(number of samples) is caused by the oversampling which takes place at

small values of the input coordinate. This increased sampling at the

input is a cause for concern in a practical correlator design. In

such a system, the resolution required at the highest sampling rate

fixes the design of the entire system. This may cause the space-

bandwidth product required for adequate correlation to exceed the

capability of the sensor. However, Anderson and Callary51 showed

that previous researchers52 had overestimated the space-bandwidth

requirement and that practical Mellin-Fourier correlators were

possible.

Synthetic Discriminant Functions

Another technique for recognizing multiple orientations and

conditions is to cross-correlate with many different reference images

in parallel. The test image can be transformed by many lenses at

once, with each Fourier transform falling on an array of reference

filters chosen to give reasonable correlation to all conditions. By

the proper choice and location of the inverse transform lens, the

correlations of all the filters can coincide in one common plane.

This parallel setup has been extensively studied by Leib et al.53

They showed that with a memory bank of 23 views of a tank, an optical

correlator could obtain a 98% probability of detection and 1.4% false

alarm rate in scenes exhibiting both scale and rotation variations.










Unfortunately, this parallel technique is somewhat cumbersome to

implement due to alignment of the multiple lenses and filters.

To avoid the need for multiple lenses and filters, it is possible

to combine several reference images into one filter. The use of

multiple lenses and filters superimposes the outputs of the individual

correlators. Because the Fourier transform and correlation are

linear, the superposition at the output is equivalent to superimposing

the individual filter functions into one filter. Likewise, this is

equivalent to superimposing the reference images in the creation of

the filter. Rather than create separate filters from many images, a

single filter is created from a sum of the images. This simplifies

the optical hardware. Caulfield et a155 defines a "composite matched

filter" CMF as a single filter which is a linear sum of ordinary

matched filters, MF.


CMF = E wk MFk (5.1)
k


These filters can be implemented by either multiple exposure optical

holography or computer holography. In the optical hologram, the

weights in the linear combination are obtained by varying the exposure

time. The latter approach is to use computers to generate the CMF

off-line. In this way, the long-drawn-out creation of the CMF is

performed on a digital computer where time is not critical. This

takes advantage of the control, dynamic range, and flexibility of

digital processors.

Once the CMF function is determined, an optical filter is

produced, tested, and optimized. It is then inserted in an optical

correlator to take advantage of its real-time processing. To









implement the CMF optically, two techniques can be used: (1) transform

the digital image to optical image via a high resolution CRT or

digitally addressed camera and produce a Vander Lugt Filter in the

conventional holographic manner, or (2) retain the image in a digital

format and produce the filter through computer-generated hologram

techniques. This latter technique has the advantage of using the

dynamic range of the digital computer until the final product is

produced. That is, if the CMF function is displayed and transformed

optically, the display will limit the dynamic range. By producing a

computer-generated holographic filter, the dynamic range is retained

till a later stage. In addition, complex filter functions and

frequency pre-emphasis can be easily incorporated.

However the CMF is implemented, the weights must be chosen for

optimal performance in a specific application. Hester and

Casasent56,57 developed what is called the Synthetic Discriminant

Function (SDF) which is a CMF that gives the same correlation output

intensity for each pattern in the training set. The weights required

to provide a constant correlation output for each pattern are not

unique. Additional constraints can be placed upon the SDF to reduce

the response to specific unwanted targets, to reduce dynamic range, or

to incorporate other desirable features. Starting with a training set

(Figure 5.1) which adequately describes the conditions in which the

desired target could be found, the SDF is formed as a linear

combination of all of the training images (Figure 5.2). The weights

are determined using matrix techniques which typically requires

considerable time on a large computer.58-63 The weights are adjusted

to provide a correlation with each member of the training set as close




Full Text
COMPUTER-GENERATED HOLOGRAPHIC MATCHED FILTERS
By
STEVEN FRANK BUTLER
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1985

ACKNOWLEDGEMENT
The author wishes to thank Dr. Henry Register and Mr. Jim
Kirkpatrick for their encouragement to continue graduate studies at
the University of Florida. Dr. Roland Anderson has tirelessly
provided counseling and guidance during the years of study,
experimentation, and writing. Dr. Ron Jones of the University of
North Carolina assisted greatly with the understanding of film non¬
linearity. Dr. S.S. Ballard provided the scholastic background and
the interest in optics throughout the author's scholastic career at
the University of Florida. The Air Force Office of Scientific
Research and the Air Force Armament Laboratory funded the laboratory
support for this effort. The University of Florida provided academic
and administrative support for the author's entire period of graduate
studies.
ii

TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS ii
LIST OF TABLES v
LIST OF FIGURES vi
ABSTRACT . x
CHAPTER
I INTRODUCTION ...1
Machine Vision 2
Optical Computers 5
Contribution .7
II BACKGROUND . 9
Communication Theory 9
Vander Lugt Filtering 20
III COMPUTER-GENERATED HOLOGRAMS (CGH) ....24
Continuous -Tone Holograms 25
Binary Holograms 30
Sampling and Space-Bandwidth Requirements 39
IV OPTIMIZATION OF OPTICAL MATCHED FILTERS 63
Performance Criteria 63
Frequency Emphasis 65
Phase-Only-Filters 72
Phase-Modulation Materials 76
V PATTERN RECOGNITION TECHNIQUES 84
Deformation Invariant Optical Pattern Recognition 85
Synthetic Discriminant Functions 88
VI MATCHED FILTER LINEARITY 94
Measurement of Film Characteristics 97
Models for Film Non-linearity 102
Computer Linearization of Filter Response 112
iii

CHAPTER
Page
VII SIMULATIONS.. 133
Techniques for Simulating Matched Filters 134
Simulation of a Continuous-Tone Hologram 145
Simulation of a Binary Hologram 151
An Example Using an SDF as a Reference 159
VIII OPTICAL IMPLEMENTATION 170
Techniques for Optical Implementation 170
Examples of CGH Matched Filters 179
IX SUMMARY 191
Conclusions 194
Recommendation 195
BIBLIOGRAPHY 197
BIOGRAPHICAL SKETCH 201
iv

LIST OF TABLES
TABLE Page
7.1 Signal-to-noise ratio and efficiency for an ideal 146
auto-correlation of a square.
7.2 Signal-to-noise ratio and efficiency for a 157
continuous-tone CGH.
7.3 Signal-to-noise ratio and efficiency for an 165
A-K hologram of a square.
7.4 Signal-to-noise ratio and efficiency of an 169
A-K hologram of a SDF correlating with members
of the training set.
v

LIST OF FIGURES
FIGURE Page
3.1 Brown and Lohmann CGH cell. 33
3.2 Complex plane showing four quadrature components. 36
3.3 Addressable amplitude and phase locations
using the GBCGH method. 38
3.4 Spectral content of an image hologram. 42
3.5 Spectral content of a Vander Lugt filter. 44
3.6 Spectral content of a Fourier Transform hologram. 50
3.7 Two dimensional spectrum of the Fourier Transform
hologram. 51
3.8 Two dimensional spectrum of the Vander Lugt filter. 53
3.9 Spectrum of a modified Vander Lugt filter. 55
3.10 Spectrum of the zero mean Vander Lugt filter. 58
3.11 Output of a 50^ aliased Vander Lugt filter with
absorption hologram. 60
4.1 High-frequency emphasis of a square and a disk. 67
4.2 Phase-only filtering of a square and a disk. 74
5.1 Training set for the creation of a SDF. 91
5.2 SDF created from the images in Figure 5.1. 92
6.1 Typical H & D curve. 96
6.2 Computer output of the polynomial fit routine. 111
6.3 H & D plot for Agfa 10E75 photographic plates. 113
6.4 Amplitude transmission vs. exposure for Agfa
10E75 plates. 114

FIGURE
6.5
Computer output of the polynomial fit routine for
8E75 plates.
Page
115
6.6
H & D plot for Agfa 8E75 photographic plates.
116
6.7
Amplitude transmission vs. exposure for Agfa
8E75 plates.
117
6.8
Image and plot of a linear gradient used for a
test input.
120
6.9
Image and plot of the output transmission on
film from the gradient input.
121
6.10
Image and plot of the pre-distorted gradient
used for an input.
122
6.11
Image and plot of the output transmission with
pre-distorted input.
123
6.12
Image and plot of a sinusoidal grating pattern
used for input.
125
6.13
Image and plot of the output transmission with
the sinusoidal input.
126
6.14
Output spectrum for a sinusoidal input.
128
6.15
Image and plot of a pre-distorted sinusoidal
grating used as an input.
129
6.16
Image and plot of the output transmission for the
pre-distorted sinusoidal input.
130
6.17
Output spectrum for a pre-distorted grating input.
131
7.1
Computer simulation of an ideal correlation.
136
7.2
Fourier transform of a square.
139
7.3
Fourier transform of a square with high-frequency
emphasis.
140
7.4
Ideal auto-correlation of a square with no
pre-emphasis.
141
7.5
Ideal correlation of a square with
high-frequency emphasis.
142
7.6
Ideal correlation of a square using
phase-only filtering.
143
vii

FIGURE
7.7
Flow chart for the continuous-tone hologram
simulation.
Page
147
7.8
Continuous-tone CGH of a square.
150
7.9
Continuous-tone CGH of a square with
high-frequency emphasis.
152
7.10
Continuous-tone CGH of a square with phase-
only filtering.
153
7.11
Auto-correlation of a square using a continuous-tone
CGH.
154
7.12
Auto-correlation of a square using a continuous-tone
CGH with high-frequency emphasis.
155
7.13
Auto-correlation of a square using a continuous-tone
CGH with phase-only filtering.
156
7.14
Flow chart for the binary hologram simulation.
158
7.15
A-K binary hologram of a square.
160
7.16
A-K binary hologram using high-frequency
emphasis.
161
7.17
A-K binary hologram of a square with
phase-only filtering.
162
7.18
Auto-correlation of a square using an A-K binary
hologram with high-frequency emphasis.
163
7.19
Auto-correlation of a square using an A-K binary
hologram with phase-only filtering.
164
7.20
A-K binary hologram of the SDF using
high-frequency emphasis.
167
7.21
Correlation of a test image at 30° and the SDF
using an A-K hologram with high-frequency emphasis.
168
8.1
Photo of an interferometrically produced optical
matched filter.
171
8.2
Cathode-ray tube and camera produced by the
Matrix Corporation.
175
8.3
Cathode-ray tube imaged onto a translation table
produced by the Aerodyne Corp.
176
8.4
Electron-beam writing system at Honeywell Inc.
178
viii

FIGURE Page
8.5 Magnified views of a binary hologram produced on the
Honeywell E-beam writer. 180
8.6 A-K CGH matched filters, using a square as a
reference produced on the Honeywell E-beam writer. 181
8.7 Reconstruction from an A-K CGH matched filter of a
square using no pre-emphasis. 183
8.8 Reconstruction from an A-K CGH matched filter of a
square using high-frequency emphasis. 184
8.9 Reconstruction from an A-K CGH matched filter of a
square using phase-only filtering. 185
8.10 A-K CGH matched filter of the letters "AFATL"
using a) high-frequency emphasis and
b) phase-only filtering. 186
8.11 Reconstruction from an A-K CGH matched filter of
the letters "AFATL" using high-frequency emphasis. 187
8.12 Reconstruction from an A-K CGH matched filter of
the letters "AFATL" using phase-only filtering. 188
8.13 A-K CGH matched filter of the SDF shown
in Figure 5.1. 189
8.14 Reconstruction of an A-K CGH matched filter of
an SDF. 190
ix

Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
COMPUTER-GENERATED HOLOGRAPHIC MATCHED FILTERS
By
STEVEN FRANK BUTLER
December 1985
Chairman: Roland C. Anderson
Major Department: Engineering Sciences
This dissertation presents techniques for the use of computer¬
generated holograms (CGH) for matched filtering. An overview of the
supporting technology is provided. Included are techniques for
modifying existing CGH algorithms to serve as matched filters in an
optical correlator. It shows that matched filters produced in this
fashion can be modified to improve the signal-to-noise and efficiency
over that possible with conventional holography. The effect and
performance of these modifications are demonstrated. In addition, a
correction of film non-linearity in continuous-tone filter production
is developed. Computer simulations provide quantitative and
qualitative demonstration of theoretical principles, with specific
examples validated in optical hardware. Conventional and synthetic
holograms, both bleached and unbleached, are compared.
x

CHAPTER I
INTRODUCTION
Human vision is a remarkable combination of high resolution
sensors and a powerful processing machine. This combination permits
understanding of the world through sensing and interpretation of
visual images. The faculty of vision is so natural and common that
few pause to think how marvelous it is to acquire such clear and
precise information about objects simply by virtue of the luminous
signals that enter the eyes. Without consciousness of the complicated
process, objects are recognized by the characteristic qualities of the
radiations they emit. With the help of memory and previous
experience, the sources of these manifestations are perceived. This
process is known as sight, perception or understanding.
Images and photographs have long been used to identify and locate
objects. By photographing an area, perhaps from afar, a scene could
be given detailed study. This study might disclose the presence of
objects of interest and determine their spatial location. Images from
satellites show weather, agriculture, geology and global actions.
Special images may contain additional scientific information including
object spectral characteristics, velocity, temperature, and the like.
The traditional medium of these images has been photographic film.
It is capable of high resolution and is sensitive to visible and near-
visible wavelengths. Unfortunately, film based methods are slow due
to exposure, processing, and analysis time. This time lag is not a
1

2
problem for many applications and so film is still the primary medium
for reconnaissance. Electronic imagery (TV, radar, etc.) is used for
those applications that require faster interpretation. These images
can be viewed, like film, by people trained to interpret the
particular images. Because of the electronic nature of the images,
electronic hardware and computers are used for manipulation of the
images.
Machine Vision
For very high speed retrieval and interpretation, machines must be
designed around the specific tasks. Machine interpretation is also
necessary when a human is not available. Unmanned robots work in
hazardous areas and perform many jobs more efficiently without the
encumbrance of human intervention. However, to function and carry out
their assigned job, the robots must have information about their
surroundings. The ability to interpret imagery from self-contained
sensors is necessary for the proper function of a robot. This image
interpretation includes guidance, obstacle avoidance, target
recognition, tracking, and closed loop control of robot action. For
robot action without human intervention, machine intelligence must
have the ability to make decisions based on scene content. Computer
image processing and recognition refer to techniques that have evolved
in this field in which the computer receives and uses visual
information.
Image processing techniques prepare or preserve an image for
viewing. This includes enhancement, restoration, and reconstruction.
Image enhancement techniques are designed to improve image quality for
human viewing. For example, correction of a geometrically distorted

3
image produces an obvious improvement in quality to a human observer.
Image restoration techniques compensate an image, which has been
degraded in some fashion, to restore it as nearly as possible to its
undegraded state. For example, an image which is blurred due to
camera motion may be improved using motion restoration. To perform
the difficult task of image interpretation, extraneous noise must be
separated from the desired signals. This may occur in several stages
of enhancement where each stage reduces the extraneous noise and
preserves the information crucial to object recognition. Image
enhancement may include contrast transformation, frame subtraction,
and spatial filtering. The goal of image enhancement is to reduce the
image complexity so that feature analysis is simplified.^
Once the scene has been enhanced, the job of interpretation is
simplified. The interpreter must now decide what the remaining
features represent. The features present a pattern to the interpreter
to be recognized. This pattern recognition problem may be quite
difficult when a large number of features are necessary to
differentiate between two possibilities. Most people have to look
closely to see any difference between two twins. A computer might
have equal difficulty distinguishing a car from a house in low-
resolution image.
Recognition involves an interpretation of an image. This includes
scene matching and understanding. Scene matching determines which
region in an image is similar to a pictorial description of a region
of another scene. A reference region or template is provided and
systematically compared to each region in a larger image. Here the
computer attempts to match models of known objects, such as cars,

4
buildings, or trees, to the scene description and thus determine what
is there. The model objects would be described in memory as having
certain characteristics, and the program would attempt to match these
against various parts of the image. Scene understanding involves a
more general recognition problem describing physical objects in a
scene based on images. For example, a scene may be divided into
regions that match various objects stored in memory such as a house,
tree, and road. Once the scene is divided into known regions, the
interrelationship between these regions provides information about the
scene as a whole.
When it is necessary to recognize specific objects, correlation
techniques are often used.^ A reference image of the desired object
is stored and compared to the test image electronically. When the
correlation coefficient is over a specified threshold, the computer
interprets the image as containing the object. The correlation
procedure may also provide the location of the object in the scene and
enable tracking. The correlation coefficient may be used in decision
making to determine robot action. Because even a single object may
present itself in many ways, correlation procedures are complicated by
the immense reference file that must be maintained.3 Special
correlation techniques may provide invariance to specific changes, but
a wide range of object conditions (i.e., temperature, color, shape,
etc.) make correlation recognition a complicated computer task.1* The
best computer vision systems now available have very primitive
capabilities. Vision is difficult for a computer for a number of
reasons. The images received by a sensing device do not contain
sufficient information to construct an unambiguous description of the

5
scene. Depth information is lost and objects frequently overlap.
Vision requires a large amount of memory and many computations. For
an image of 1000 X 1000 picture elements, even the simplest operation
may require 10® operations. The human retina, with 10® cells
operating at roughly 100 hertz, performs at least 10 billion
operations a second. Thus, to recognize objects at a rate even
closely resembling human vision, very special processor technologies
must be considered. One promising technology has emerged in the form
of optical computing.
Optical Computers
Optical computers permit the manipulation of every element of an
image at the same time. This parallel processing technique involves
many additions and multiplications occurring simultaneously. Most
digital processors must perform one operation at a time. Even though
the digital processors are very fast, the number of total operations
required to recognize patterns in an image is very large. Using
optical Fourier transformers, an optical processor can operate on the
image and its Fourier transform simultaneously. This permits many
standard image processing techniques, such as spatial filtering and
correlation, to be performed at tremendous rates.
The Fourier transform is formed optically by use of a lens. The
usual case that is considered in optical computing is when the
illuminating source is located at infinity (by use of an auxiliary
collimating lens) and the image transparency is located at a distance
equal to focal length from the transforming lens. The distribution in
the output plane located a focal length behind the transforming lens
is the exact Fourier transform of the input distribution. The Fourier

6
transform contains all of the information contained in the original
image. However, the information is now arranged according to spatial
frequency rather than spatial location. The advantage of such an
arrangement is that objects or signals of interest may overlap with
noise in the image domain but exist isolated in the frequency domain.
This permits the possible separation of signal from noise in the
frequency plane when it would have been impossible in the image plane.
The image can be transformed into frequency space, frequency filtered
and then transformed back into image space with the noise removed.
The frequency filter may be low-pa3s, high-pass, or band-pass, chosen
to optimize the filtering of a specific signal. This frequency plane
filter is the heart of the analog optical computer.
The frequency plane filter can be constructed in many ways. Low-
pass and high-pass filters are accomplished using simple apertures
mounted on axis in the frequency plane. More complicated filters are
produced optically using holographic techniques. These filters may
also be produced using computer-generated holography (CGH). The
computer is used to model the desired filter response, mathematically
represent the holographic filter, and create a physical filter using a
writing device. One of the important advantages of computer-generated
holography is that the reference need not exist physically, but only
mathematically. This permits mathematical manipulation of the
reference prior to creation of the filter for purposes of
optimization.
The advantage of an analog optical processor is that it may
operate at very high speeds. In addition, the processor typically is
smaller, lighter, and consumes considerably less power than an

7
equivalent digital processor.^>6 When coupled with the ability to
manipulate and optimize the frequency plane filter, the optical
processor becomes a useful tool. With considerable justification,
there is great interest in the robotics community.
Contribution
This dissertation states that CGH matched filters should be used
in an optical correlator to recognize patterns in a complex scene, and
describes how to create that filter. The CGH matched filter is
superior to interferometric filters due to the ability to pre-process
the filter function and control the production of the hologram. The
use of optical elements for high speed pattern recognition was first
proposed 20 years ago.7 The concept of using computers to define and
generate holograms came only two years later.^ Since that time,
considerable effort has been devoted to exploring the potential of
these CGH elements for reconstruction holography. Most of this effort
was devoted to optimizing the methods for encoding and writing the
holograms.3 More recently, interest has grown in the area of
efficiency improvementJ^ The efficiency of a hologram for optical
correlation must be high in order to utilize low power, light weight
diode lasers. In separate but parallel efforts in artificial
intelligence, researchers have studied the effects of image
enhancement on pattern recognition.^ Though research in the various
fields is proceeding, a unified approach to the interrelation of pre¬
processing, holographic encoding and physical implementation is
lacking. Specifically, the research in CGH, to date, has only been
for display or reconstruction holography, not matched filtering. This

8
dissertation describes the steps necessary and possible to create
practical matched filters using CGH.
The approach presented here ties many areas of research together
as they apply to CGH matched filters. Modifications to existing
encoding schemes which provide real valued filter patterns for use in
an optical correlator are explained in Chapter III. In addition,
Chapter III defines the space-bandwidth-product (SBP) required for
holographic matched filtering rather than for display holography as is
presented in existing literature. This includes procedures for
minimizing the SBP required. Pre-processing methods which apply
specifically to matched filtering are presented along with rationale
for their use in Chapter IV. Techniques for the use of CGH matched
filters as a pattern recognizer are reviewed in Chapter V.
Linearization methods for writing on film are derived and evaluated in
Chapter VI.
These various considerations are not independent, but rather, are
interwoven in the production of CGH matched filters. These
interactions can be fully analyzed only with a complete model
incorporating all the parameters. Chapter VII describes such a model
created to analyze the pre-processing, encoding and writing techniques
used to produce optimal CGH matched filters. Now that the various
methods have been developed and the analytical tools demonstrated,
specific examples are presented and analyzed. Chapter VIII describes
approaches for physically producing a transparency including specific
examples taken from Chapter VII. Finally, conclusions based on the
analysis are offered in Chapter IX.

CHAPTER II
BACKGROUND
The background technology is reviewed here to understand the
operation of an optical processor more fully. A number of different
types of optical processors are in use today. These include one¬
dimensional signal processors, two-dimensional image processors and
multi-dimensional digital processors. Only two-dimensional image
processors used for matched filtering are described here. A matched
filter optimizes the signal-to-noise ratio at a specific point when
1 fi
the characteristics of the input are known. Typically, the desired
pattern and the nature of the background or noise in the input image
are known. Specifically, the input consists of a known signal s(x,y)
and an additive noise n(x,y). The system is linear and space
invariant with impulse response h(x,y). The criterion of optimization
will be that the output signal-to-noise power ratio be a maximum.
This optimum system will be called a matched filter for reasons that
will become clear as the derivation proceeds.
Communication Theory
A system is any unit that converts an input function I(x,y) into
an output function 0(x,y). The system is described by its impulse
response—its output when the input is an impulse or delta function.
A linear system is one in which the output depends linearly on the
input and superposition holds. That is, if the input doubles, so does
the output. More precisely stated, let 0^ be the output when 1-j is
9

10
the input and O2 be the output when I2 is the input. Then the system
is linear when, if the input is al^+bl2 the output is aO-j+bOg. This
property of linearity leads to a vast simplification in the
mathematical description of phenomena and represents the foundation of
a mathematical structure known as linear system theory. When the
system is linear, the input and output may be decomposed into a linear
combination of elementary components.
Another mathematical tool of great use is the Fourier transform.
The Fourier transform is defined by
CO
F(u,v) = //f(x,y) exp -j2ir(ux+vy) dx dy = f {f(x,y)}. (2.1)
— 00
The transform is a complex valued function of u and v, the spatial
frequencies in the image plane. The Fourier transform provides the
continuous coefficients of each frequency component of the image. The
Fourier transform is a reversible process, and the inverse Fourier
transform is defined by
00
f(x,y) = //F(u,v) exp j2n(ux+vy) dx dy = F-1(F(u,v)}. (2.2)
—CO
The transform and inverse transform are very similar, differing only
in the sign of the exponent appearing in the integrand. The magnitude
squared of the Fourier transform is called the power spectral density
Of = |F(u,v)!2 = F(u, v) F*(u, v). (2.3)
It is noteworthy that the phase information is lost from the Fourier
transform when the transform is squared and the image cannot, in
general, be reconstructed from the power spectral density. Several
useful properties of the Fourier transform are listed here.

11
Linearity Theorem
F { af ^ (x,y) + bf2(x,y)} = a F{f.,(x,y)} + b F{f2(x,y)} (2.4)
The transform of the sum of two functions is simply the sum of
their individual transforms. The Fourier transform is a linear
operator or system.
Similarity Theorem
F {f(ax,by)} = F(u/a,v/b)/ab where F(u,v) =F {f(x,y)J (2.5)
Scale changes in the image domain results in an inverse scale
change in the frequency domain along with a change in the overall
amplitude of the spectrum.
Shift Theorem
F {f(x-a,y-b)} = F(u,v) exp [-j(ua+vb)] (2.6)
Translation of patterns in the image merely introduces a linear
phase shift in the frequency domain. The magnitude is invariant to
translation.
Parseval's Theorem
//¡F(u,v)|2 du dv = ff |f(x,y)!2 dx dy (2.7)
The total energy in the images plane is exactly equal to the
energy in the frequency domain.
Convolution Theorem
F{f(x,y) g(x,y)} = ff F(u,v)F(uQ-u,vQ-v) du dv (2.8)

12
The Fourier transform of the product of two images is the
convolution of their associated individual transforms. Also the
Fourier transform of the convolution of two images is the product of
the individual transforms.
Correlation Theorem
Rfg(x,y) = ff f(x,y) f(x-x0,y-y0) dxQ dy0 (2.9)
The correlation is very similar to the convolution except that
neither function is inverted.
Autocorrelation (Wiener-Khintchine) Theorem
°ff(u,v) = F (Rff(x,y)} (2.10)
This special case of the convolution theorem shows that the
autocorrelation and the power spectral density are Fourier transform
pairs.
Fourier integral Theorem
f(x,y) = F-1 { F if(x,y)}} (2.11)
f(-x,-y) = F ( F Íf(x,y)}}
Successive transformation and inverse transformation yield that
function again. If the Fourier transform is applied twice
successively, the result is the original image inverted and perverted.
It is also useful to define here the impulse function. Also known
as the Dirac delta function, it describes a function which is infinite
at the origin, zero elsewhere, and contains a volume equal to unity.
One definition of the Dirac delta function is

13
6 (x) = lija (a/ TT ) exp -a2x2. (2.12)
The delta function possesses these fundamental properties:
6 (x) =
0 for x ¿ 0
(2.13)
CO CO
/ 6 (x)dx = /<$ (x)dx = 1
00 CO
(2.14)
6 (x) =
6 (-x)
(2.15)
ó (ax)
= (1/a) 6(x) a ¿ 0
(2.16)
/ f(x)
6 (x-a)dx = f(a).
(2.17)
The Fourier transform of the delta function is unity. This property
provides a useful tool when studying systems in which an output is
dependent on the input to the system. When an impulse is the input to
the system, the input spectrum is unity at all frequencies. The
spectrum of the output must then correspond to the gain or attenuation
of the system. This frequency response of the system is the Fourier
transform of the output when an impulse is the input. The output of
the system is the impulse response. Thus, the impulse response and
the frequency response of the system are Fourier transform pairs. To
determine the output of a system for a given input, multiply the
Fourier transform of the input by the frequency response of the system
and take the inverse Fourier transform of the result. The convolution
property shows an equivalent operation is to convolve the input with
the impulse response of the system.
0(u,v) = I(u,v) H(u,v)
(2.18)

14
o(x,y) = F *{0(u, v)} =F 1{I(u,v) H(u,v)} (2.19)
= ff i^xo^o) h(x-x0,y-y0) dxQ dyQ
= f(x,y) ® h(x,y)
where ® denotes convolution.
Consider the effect of an additive noise on the input of the
system. Although the exact form of the noise n(x,y) may not be known,
the noise statistics or power spectral density may be predictable.
Thus, the effect of the system on the input is determined by its
impulse response or frequency response. That is, when there is
knowledge of the input signal and noise, the output signal and noise
characteristics can be predicted. The relationship of the input and
output are expressed in the following diagram and equations. The
letters i and o indicate the input and output terms while the letters
s and n indicate the signal and noise portions.
s(x,y) + n(x,y) >
i(x,y) = s^Xjy) + ni(x,y)
o(x,y) = sQ(x,y) + nQ(x,y)
0(u,v) = I(u,v) H(u,v)
SQ(u,v) = Sj/u.v) H(u,v)
NQ(u,v) = N^Ujv) H(u,v)
(2.20)
(2.21)
(2.22)
(2.23)
« (2.24)
» sQ(x,y) + nQ(x,y)

15
Now that the relationships between the input and output of a
linear system are known, such a system may be utilized to enhance the
input. For example, assume an image has been degraded by some
distorting function d(x,y). The original image was convolved with the
distorting function, and the spectral contents of the ideal image
Fi(u,v) were attenuated by the frequency response D(u,v) of the
distorting system. By multiplying the degraded image by the inverse
of the D(u,v), the original ideal image is obtained. Any distortion
which can be represented as a linear system might theoretically be
canceled out using the inverse filter. A photograph produced in a
camera with a frequency response which rolls off slowly could be
sharpened by Fourier transforming the image, multiplying by the
inverse filter, and then inverse transforming. In this case, the
inverse filter is one in which the low frequencies are attenuated and
the high frequencies are accentuated (high pass filter). Because the
high frequencies represent the edges in the image, the edges are
accentuated and the photo appears sharper.^7 As indicated in the
following diagram, the image is distorted by the function D(u,v) but
in some cases can be restored by multiplying by 1/D(u,v).
fi(x,y>
Fi(u,v) X D(u,v) fd(x,y)
fd(x,y)
Fd(u,v) X 1/D(u,v)~^> f'd(x,y)
blurred photograph
enhanced photograph
The linear blur of a camera is another classic example. Consider
traveling through Europe on a train with your camera. Upon getting
home and receiving your trip pictures, you find that all of them are
streaked by the motion of the train past the scenes you photographed.
Each point in the scene streaked past the camera, causing a line to be

16
formed on the film rather than a sharp point. The impulse response is
a line, and the corresponding frequency response of the distorting
system is a sine function (sin u /u). To retrieve the European
photo collection, merely multiply the Fourier transform of the
pictures by u/sin u and re-image.
In the physical implementation of this process, there are several
practical problems. To multiply the image transform by the inverse
function, a transparency with the appropriate response is produced.
In general, a transparency can only attenuate the light striking it.
That is, the transparency can only represent non-negative real values
less than one. Herein lies the problem. The inverse response
required to correct a specific distortion may, in fact, be complex.
In some cases, a combination of two transparencies can be combined to
provide complex values. One transparency is used for amplitude or
attenuation, and another phase transparency or phase plate is used to
provide the appropriate phase shift at each point. A phase
transparency can be produced by bleaching film with an appropriate
latent image induced in the emulsion. Chu, Fienup, and Goodman^®
demonstrated a technique in color film which consists of three
emulsions. One emulsion was used as an amplitude transparency and
another emulsion was used as a phase plate. The appropriate patterns
were determined by a computer and the film was given the proper
exposure using colored filters.
Even with a two-transparency system, not all distortions are
possible to remove. Note that in the linear blur case, the inverse
response is u/sin u. The denominator goes to zero for specific values
of u, and the response has a pole at those values. The filter cannot

17
represent those values, and the practical filter is merely an
approximation to the ideal filter. It is worth noting that when the
distorting response reduces a frequency component to zero or below
some noise threshold, that component cannot be recovered. That is,
information is usually lost during the distorting process and inverse
filtering cannot recover it.
It is desirable to remove noise from a corrupted image. Although
it is not always possible to remove all of the noise, the
relationships between the input and output of a.linear system are
known. A linear system is optimized when most of the noise is
removed. To optimize a system the input must be specified, the system
design restrictions known, and a criterion of optimization accepted.
The input may be a combination of known and random signals and noises.
The characteristics of the input such as the noise spectrum or
statistics must be available. The classes of systems are restricted
to those which are linear, space-invariant, and physically realizable.
The criterion of the optimization is dependent on the application.
The optimum filters include the least mean-square-error (Wiener)
filter and the matched filter. The Wiener filter minimizes the mean-
squared-error between the output of the filter and actual signal
input. The Wiener filter predicts the least mean-squared-error
estimate of the noise-corrupted input signal. Thus, the output of the
Wiener filter is an approximation to the input signal. The output of
the matched filter is not an approximation to the input signal but
rather a prediction of whether a specific input signal is present.
The matched filter does not preserve the input image. This is not the
objective. The objective is to distort the input image and filter the

18
noise so that at the sampling location (x0,y0), the output signal
level will be as large as possible with respect to the output noise.
The signal-to-noise ratio is useful in the evaluation of system
performance, particularly in linear systems. In the matched filter,
the criterion of optimization is that the output signal-to-noise power
be a maximum. The input consists of a known signal s(x,y) and an
additive random noise n(x,y). The system is linear and space
invariant with impulse response h(xQ,y0). To optimize the system or
filter, maximize the expression
Ro = s02(xo»yo)/Eino2(x’y)} (2.25)
where E{nQ2(x,y)} =//nQ2(x,y) dx dy
at some point (x0,yQ). The problem is then to find the system h(x,y)
that performs the maximization of the output signal-to-noise ratio.
The output signal sQ(x,y) is
s0(x,y) = //si(x0,y0)h(x-x0,y-y0) dxQ dyQ (2.26)
and the output noise nQ(x,y) power is
ff !nQ(x,y) ¡2 dx dy = // ¡NQ(u,v)!2 du dv
= // ¡ Nj^u, v)! 2 ! H(u, v)'2 du dv. (2.27)
The signal-to-noise output power ratio becomes
!// s1(x0,y0)h(x-x0,y-y0) dxQ dyQ \2 (2.28)
â–  Ro =
¡N^u.v)!2 | H(u, v) j2
Thus to complete the maximization with respect to h(x,y), the power
spectral density or some equivalent specification of the input noise

19
must be known. Once the input noise is specified, the filter function
h(x,y) is the only unknown. Equation (2.28) becomes
E{n02(xQ,y0) - as02(x0,y0)} > 0 (2.29)
Ni2(u,v)H2(u,v) du dv - a \ff s^x^hix-x^y-y^ dxQ dyQ! 2 > 0
where Ro max = 1/a
and the maximum signal-to-noise ratio at the output is obtained when
H(u,v) is chosen such that equality is attained. This occurs when
// ni2(x,y) h(x-xQ,y-y0) dxQ dyQ = si(x,y). (2.30)
Taking the Fourier transform of both sides and rearranging gives
S(-u,-v)
H(u,v) = exp -j(uxQ+vy0) . (2.31)
¡N^u.v)!2
Thus in an intuitive sense, the matched filter emphasizes the signal
frequencies but with a phase shift and, attenuates the noise
frequencies. This becomes clear when the additive noise is white. In
this case the noise power is constant at all frequencies and thus has
a power spectral density of
|N^(u,v)|2 = N/2 where N is a constant. (2.32)
From equation 2.32 the form of the matched filter for the case of
white noise is
H(u,v) = S^-u.-viexp -j(uxQ+vy0) (2.33)
= S^ÍUjV) exp -j(ux0+vy0)

20
or
h(x,y) = s(-x,-y). (2.34)
Equation 2.34 shows that the impulse response of the matched filter
(with white noise) is simply the signal image in reverse order
(inverted and perverted). Thus, the filter is said to be matched to
the signal. Filtering with a matched filter is equivalent to cross-
correlating with the expected signal or pattern. That is,
0(x,y) = Rhs(x,y)
= // s(x0,y0)h(xQ-x,y0-y) dxQ dyQ . (2.35)
Also, it can be seen that the frequency response of the matched filter
is equivalent to that of the signal but with the phase negated so that
the output of the filter is real. That is, the matched filter removes
the phase variations and provides a real valued output.^
Matched filters are used extensively in radar signal,
processing, seismic data processing, and communications. These
filters are implemented using electronic circuitry and digital
computers. For image processing, the need to process large two-
dimensional arrays places a large burden on conventional filtering
techniques. For these applications, optical processing techniques
provide the highest throughput speeds for matched filtering. One such
optical processing technique was proposed by Vander Lugt^ in 1969.
Vander Lugt Filtering
If an image is placed on a transparent sheet and illuminated by a
plane wave of coherent light, its Fourier transform is formed using a
1 q
simple lens. Once the Fourier transform is formed, specific
frequency components in the image can be removed or attenuated. The

21
result may then be inverse Fourier transformed to recreate the
modified image. The aperture, which may be replaced by a complicated
filter, functions to perform specific filtering operations including
Wiener or matched filter. Unfortunately, there are certain
limitations to the functions which can be physically implemented. A
normal transparency merely attenuates the light passing through it.
Its transmission is real and non-negative. Thus, when a transparency
film is exposed to a waveform to be recorded, the phase information in
the waveform is lost. Two pieces of information, the real and
imaginary parts of the waveform, are recorded as only one value, their
magnitude. This loss of information can be corrected by taking
advantage of the redundancy in the wavefront and the use of additional
film space. Using the heterodyning technique proposed by Vander Lugt,
the complex waveform can be recorded on photographic film.
Vander Lugt proposed the use of holographic film to store the
filter response for a matched filter. A lens is used to Fourier
transform the reference and test images. Derivations of the Fourier
transforming capabilities of lenses can be found in the literature.^
The Fourier transform of the reference image is brought to focus on a
photographic film. Film is a nonlinear, time-integrating medium and
thus only the magnitude of the Fourier transform or power spectral
density is recorded. The power spectral density does not contain all
of the original image information. Only the autocorrelation of the
original image can be obtained upon inverse transformation. Neither
the power spectral density nor the autocorrelation uniquely describe
the original image. If a plane wave is mixed with the Fourier
transform of the reference image at the film plane, the film will

22
record the interference pattern caused by the summation of the two
fields. The result on the film then is
H(u,v) = 1 + |F(u,v)!2 + F(u, v)exp j2-rrav + F*(u,v)exp - j27rav, (2.35)
which contains a constant, the power spectral density, and two terms
due to a spatial carrier fringe formed due to interference with the
plane wave. The two spatially modulated terms contain the original
image and Fourier transform information. With this Fourier transform
recorded on the film, it is placed in the optical filter arrangement
and illuminated with the Fourier transform G(u,v) of the test image
g(x,y). The output of the film transparency is the product of its
transmittance and the illuminating Fourier transform.
0(u,v) = G(u,v) H(u,v) (2.36)
= G(u,v) + G(u,v)!F(u,v) ! 2
+ G(u,v)F(u,v)exp j27rav + G(u,v)F*(u,v)exp -j2iTav
The product of the transforms from the reference and test images is
then Fourier transformed by another lens to obtain the correlation of
the two images.
o(x,y) = g(x,y) + g(x,y)*h(x,y)*h*(x,y) (2.37)
+ g(x,y)*f(x,y) *S(x,y-a)
+ g(x,y)*f*(x,y)*6 (x,y+a)
The first two terms are formed on axis or at the origin of the output
plane. The third term is the convolution of the reference and test

23
images and is centered off axis. The last term is the correlation of
the reference and test images and is located off-axis opposite the
convolution. This optical arrangement provides the entire convolution
and correlation images at once while a digital processor must compute
one point at a time. In addition to the convolution and correlation
processes, additional image plane and frequency plane filtering may be
accomplished simultaneously in the same optical arrangement. The
convolution, correlation and any additional linear filtering are
accomplished with a single absorbing mask.
When used as a matched filter, the transparency multiplies the
expected pattern by its complex conjugate, thereby rendering an
entirely real field. This matched transparency exactly cancels all
the curvature of the incident wavefront. When an input other than the
expected signal is present, the wavefront curvature will not be
canceled by the transparency and the transmitted light will not be
brought to a bright focus. Thus the expected pattern will be detected
by a bright point of light in the correlation plane. If the pattern
occurs in the input plane but is shifted, the bright point of light in
the correlation plane will shift accordingly. This provides for the
detection of specific patterns in a larger image. The detection and
location of specific objects in large complicated images is a job well
suited for the high-speed processing capability of the Vander Lugt
filter

CHAPTER III
COMPUTER-GENERATED HOLOGRAMS
Vander Lugt described a technique by which the holographic matched
filter could be produced optically.? At that time, no other
convenient method existed for the computation and creation of the
complicated filter function required. This limitation has faded away
with the availability of digital computers with large memories. Using
digital computers to determine the filter function and a computer-
driven writing device, a transparency with the appropriate filter
image can be produced. Using this technique, the computer determines
the appropriate value of the matched filter at each point and produces
a transparency with that absorption at each point. The resolution
required of the writing device depends on the application and, in some
cases, may be consistent with optically generated holograms.
Computer-generated holograms (CGH) have found applications in
optical information processing, interferometry, synthesis of novel
optical elements, laser scanning, and laser machining.20-23 CGHs can
implement computer-optimized pattern-recognition masks.The
computer writes the hologram by transferring the transmittance
function to an appropriate holographic medium. The computer drives
a plotter or scanner and writes the hologram one point at a time.
Typically, the primary limitation is writing resolution. A
conventional optical hologram may have a resolution of one-quarter of
a micron. A system using visible light to write holograms (plotters,
24

25
flying spot scanners, CRT's, etc.) cannot achieve resolutions much
better than several microns. Writing systems utilizing electron beams
are currently achieving better than 1-micron resolution. The electron
beam systems are typically binary and thus the transmittance function
must be quantized in some fashion into two levels, "on" or "off."
Binary holograms are attractive because binary computer-graphics
output devices are widely available and because problems with
nonlinearities in the display and recording medium are circumvented.^
When photographic emulsions are involved, granularity noise is
reduced.25
Continuous-Tone Holograms
When a hologram is produced optically or interferometrically, a
reference wave is superimposed with the wavefront to be recorded.
Typically, the reference wave is a tilted plane wave with constant
amplitude across the wavefront. The reference wave approaches at an
angle 9 relative to the direction of the wavefront to be recorded.
The resultant field is
E(x,y) = f(x,y) + Aexp(j2uay) (3.1)
where a= sin 9
X
and the amplitude of the reference wave is 1. An interference pattern
is produced by the superposition of the waves. The fringe spacing is
dependent on the term a, known as the spatial carrier frequency, and
the details in the function f(x,y). A photographic film placed into
this field records not the field itself but rather the square
magnitude of the field. The pattern recorded on the film is then

26
h(x,y); = ¡f(x,y) + A eJ27rayi2 (3.2)
= + |f(x,y)!2 + A f(x,y)eJ2lTay + A f*(x,y)e“J27ray.
The function recorded on the film contains a D.C. bias, k^, the base
band magnitude, ¡f(x,y)¡2, and two terms heterodyned to plus and minus
a. These heterodyned terms contain the complex valued information
describing the input function f(x,y). If the spatial carrier
frequency is sufficiently high, the heterodyned terms are separable
and no aliasing exists. The original input function can be retrieved
with no distortion by re-illuminating the film with the reference beam
and spatially filtering the output to separate the various terms.
To make the hologram of the Fourier transform of an image, the
same procedure is applied. That is, the Fourier transform of the
image f(x,y) is used as the input to the hologram. Now
h(u,v) = A2 + F(u,v)2 + A F(u,v)eJ27,au + A F*(u, v)e“J27iau (3-3)
where F(u,v) = Fourier Transform of f(x,y) = F (f(x,y)} and
A e“j2lTau = the off-axis reference wave used to provide the spatial
carrier for the hologram.
a = sin 6 = the filter spatial carrier frequency (9 = off-axis angle)
X
This filter contains the D.C. bias, A2; the power spectral density,
¡ F(uiTv) l2; and two terms heterodyned to plus and minus a. These
heterodyned terms contain the complex valued information describing
the Fourier transform of the input f(x,y).
These optically generated holograms are formed
interferometrically by combining a plane wave with the wavefront to be

27
recorded.; The transmittance of the hologram is a real valued, non¬
negative function of position on the plate. Recall that the input
F(u,v), which was used to create the hologram, is, in general,
complex. This conversion from a complex function to a pattern which
can be recorded on film is known as coding. The coding performed in
optical holography is a natural consequence of the action of the film.
Typically, the complex wavefront is coded to a real non-negative
function which can be recorded as transmittance values on film.
Equation 2.35 describes a way in which film (a square law detector)
would encode the complex input image in an optically generated
hologram.
Once produced, the hologram and its interference fringes may be
inspected by microscope. The hologram can be copied on another plate
by contact printing. The hologram consists of real valued positive
transmittance varying across the face of the photographic plate. To
record the hologram on a computer, the transmittance across the
surface of the plate is sampled. If the samples are many and the
transmittance determined with accuracy, the hologram can be accurately
reproduced from the recorded samples. In this way the hologram can be
represented with some accuracy using digital numbers stored on some
electronic media. An electronic device writes the physical hologram.
The computer can electronically record, modify an optically produced
hologram, and then rewrite the holographic pattern onto another plate.
The limitations to such a system include the ability to sample the
input hologram sufficiently often and accurately, the ability to store
the large number of sample values, and the ability to rewrite the
holographic pattern to film.

28
If the input wavefront is known, the optical step may be omitted
altogether. If the input wavefront can be accurately represented by
discrete samples stored electronically, the holographic pattern can be
computed. That is, the input is coded to create a function which can
be recorded on a transparency. In the case of the matched filter, the
Fourier transform of an image is recorded. The image is sampled and
stored on the computer, and equation 2.35 is used to determine the
holographic pattern. Note that the continuous variables are replaced
by discrete steps. At each sample point the actual value is
represented by a finite number. The value may be complex, but the
accuracy is limited by the sampling system. In any case the
holographic pattern is computed and written to the photographic plate.
The writing device is known as continuous-tone when the transmittance
of each point in the holographic plate can be controlled over a wide
range of transmittance values. That is, the transmittance varies
smoothly from clear to opaque, including gray scale values between.
These continuous-tone holograms most closely resemble the optically
generated holograms when the sampling is dense and many gray scale
values are available.
When continuous-tone holograms are written to the photographic
plate using equation 2.35 as the model, they include a D.C. term, a
square magnitude term, and the heterodyned terms due to the tilted
reference wave. Note that the first two terms are real valued and
that the sum of the last two terms is real valued. On the computer,
the film process is emulated using equation 2.35 or other coding
schemes for specific applications. The D.C. and square magnitude
terms need not be included in the computer-generated hologram as long

29
as the heterodyned terms are scaled and biased to provide results
between 0 and 1. The heterodyned terms contain the desired
information. Omission of the baseband terms has no adverse effect on
the hologram. The square magnitude term typically contains a large
dynamic range. Its omission from the coding algorithm helps reduce
the dynamic range of the hologram and, in most cases, improves the
hologram. Equation 3.3 can be replaced by the expressions
H(u,v) = 2¡F(u,v)¡ + F(u,v)ej2iTau + F*(u,v)e“J2uau (3.4)
H(u,v) = + F(u, v)e J^irau + p*(u?v)e-j2 uau (3.5)
where each of these expressions includes the reference information, is
real valued, and is non-negative.
The dynamic range in the hologram, defined as the largest value
divided by the smallest value, is limited by the writing device used
to create the hologram. Host films have dynamic ranges much less than
10,000. That is, the clearest portions of the film can transmit light
no better than 10,000 times better than the darkest portions. If the
coding scheme requires a dynamic range of over 10,000, the writing
device cannot faithfully reproduce the holographic pattern.
Unfortunately, the dynamic range of film is frequently much less than
10,000 and closer to 100. Additionally, the writing device also
limits the dynamic range. Most continuous-tone writing devices, which
are attached to computers, convert an integer value to an intensity on
a cathode-ray tube or flying spot scanner. Due to physical
limitations in the writing intensity, the dynamic range is usually
much less than 1000. Most commercially available computer-writing
devices are designed with a dynamic range of 256 or 8-bit writing

30
accuracy.: The resultant transmittance on the film will have one of
256 quantized levels determined by an integer value provided by the
computer. Quantization occurs when all values in a specified range
are assigned to a quantized value representative of that range. If
the quantization steps become large, the quantized level may be a poor
estimate of the actual values. The estimate is equivalent to the
actual pattern with an additive noise called quantization noise.
Quantization noise occurs in computer-generated holograms because the
computer-graphic devices have limited gray levels and a limited number
of addressable locations in their outputs. Quantizing the holographic
pattern into 256 gray scale levels introduces quantizing noise which
may be considerable when the dynamic range of the pattern is large.
To minimize the quantizing error, the coding scheme must produce a
result with a dynamic range compatible with the writing system.
Some writing systems are capable of only two quantized levels.
These binary devices are either on or off. Most metal etchers, ink
plotters, dot matrix printers, and lithographic printers are binary.
The mark they create is either completely on or completely off. To
represent the reference pattern on binary media accurately requires
specialized coding schemes.
Binary Holograms
Binary holograms are attractive because binary computer-graphics
output devices are widely available and because problems with
nonlinearities in the display and recording medium are circumvented.
When photographic emulsions are involved, granularity noise is
reduced. Using the normal definition of dynamic range, binary
holograms have a dynamic range of 1. The transmittance at each point

31
is complétely on or completely off. All gray scale effects must be
created by grouping light and dark areas together and averaging over
an area large enough to provide the needed dynamic range. In this
case the dynamic range is the averaging area. Thus, dynamic range is
exchanged for increased area to represent each point. This is similar
to Pulse Code Modulation (PCM) in an electronic communication
systems.26 in pcm, each sample value is quantized to M levels. Then
each level is represented by a binary code requiring N=log2 M bits.
Rather than represent each point with a continuous variable with
sufficient dynamic range, N binary variables are used. Each variable
is either on or off, but N variables are required to provide
sufficient dynamic range. This exchanges dynamic range of the
variables for the number of variables required. In binary holograms,
the variables are not, in general, exponentially weighted as in PCM;
thus, M variables are required to represent M levels. It becomes very
important to code the hologram such that the number of variables M
needed to represent that dynamic range is reasonable.
One of the first practical binary coding schemes was introduced
when, in 1966, Brown and Lohmann^ devised a method for complex
spatial filtering using binary masks. They coded the Fourier
transform of an image f(x,y). When using this method, the complex
Fourier transform is sampled and represented at each point by an
amplitude and phase. To record a complex filter, both amplitude and
phase information are needed on the hologram. However, the hologram
transmittance is real-valued, non-negative, and in this case binary.
The amplitude can be recorded by opening or closing an appropriate
number of binary windows in the hologram, but the phase is not

32
correct. . Brown and Lohmann proposed turning the hologram at an angle
to the incoming waveform. Thus, along the surface of the hologram, a
phase shift occurs. This phase shift is proportional to the position
along the hologram. Using this "tilted wave" technique, a phase shift
occurs as the aperture moves up and down the hologram causing the
total path length through that aperture to change.^ The further the
detour through the aperture, the larger the phase shift. Phase shift
induced by this technique is known as detour phase. Thus, in the
Brown-Lohmann hologram, an aperture is moved up and down to create the
appropriate phase shift. The size of the aperture is varied to allow
the appropriate amount of light through. To synthesize the complex
filter function F(u,v), a continuous function is sampled. The cells
of a size A u by A v must be sufficiently small that the function F
will be effectively constant throughout the cell.
F(u,v) = F(nAu,mAv) = Fnm sA^exp i0nm (3.6)
where n and m are integers
For each cell in the hologram, the amplitude and phase are determined
by the size and position of an aperture as shown in Figure 3.1. From
each cell a complex light amplitude Fnm will emerge. The tilted wave
must approach at an angle steep enough to allow for a full wavelength
of detour phase within one cell. The dynamic range of the amplitude
and phase is limited by the number of resolvable points within the
cell. If a cell has only 4 by 4 resolvable points, the dynamic range
of the amplitude or phase can be no better than 4. The granularity in
the amplitude and phase may cause distortion in the reconstructed

Figure 3.1
Brown and Lohmann CGH cell.

34
image. -Many points are required to represent a transform with a large
dynamic range accurately.
Lee9 proposed a method in 1970 which helped relieve some of
the phase granularity. The Brown-Lohmann technique represented each
cell with an amplitude and phase component. The complex value for
each cell may be represented by a magnitude and phase or by the sum of
in-phase and out-of-phase terms. The Lee method represents each cell
with such a quadrature representation. For each cell the magnitude
and phase are converted to real and imaginary components. As in the
Brown-Lohmann method, the tilted wave is set to provide a wavelength
of delay across the cell. The cell is divided into four components
which represent the positive and negative real and imaginary axes.
Lee defined the functions as
¡ F(u, v)¡exp[ j cj>(u,v)] = F-| (u, v)-F2(u,v)+jF3(u,v)-jF4(u,v) (3.7)
where
F1(u,v)=
F2(u,v)=
F3(u,v)=
F4(u,v)=
F(u,v)¡cos
<|>(u, v)
if cos(u,v)
0
otherwise,
F(u,v)¡sin
â– (u,v)
if simp(u,v)
0
otherwise,
i
/—\
c
<
COS if cos4>(u,v)
0
otherwise,
- !F(u,v) ¡
sin(u,v)
if sin(u,v)
0
otherwise.
For any given complex value, two of the four components are zero.
Each of the components Fn(u,v) is real and non-negative and can be
recorded on film. The Lee hologram uses four apertures for each cell

35
shown in Figure 3.2. Each aperture is positioned to cause a quarter-
wave phase shift by increased path length (detour phase). The two
non-negative quadrature terms are weighted to vector sum to the
appropriate magnitude and phase for each pixel. The two appropriate
apertures are opened according to their weight. The Lee method uses
continuous-tone variables to represent the two non-zero components.
The phase is no longer quantized by the location of the aperture. The
phase is determined by the vector addition of the two non-zero
components. In a totally binary application of the Lee method, the
apertures are rectangles positioned to obtain the quarter-wave shift.
The area of each aperture is adjusted to determine the amplitude of
each component. Once again, in this binary case, the dynamic range is
limited by the number of resolution elements within one cell.
Burckhardt^O showed that while the Lee method decomposes the
comp lex-valued F(u,v) into four real and positive components, only
three components are required. Each cell can be represented by three
components 120° apart. Any point on the complex plane can be
represented as a sum of any two of these three components. As in the
Lee method, two non-negative components are chosen to represent each
cell. Because only three instead of four components have to be
stored, the required memory size and plotter resolution are reduced.
Haskell^ describes a technique in which the hologram cell is divided
into N components equally spaced around the complex plane. It is
identical to the binary Lee (N=4) and the Burckhardt (N=3) where N may
take larger values. This Generalized Binary Computer-Generated
Hologram (GBCGH) uses N columns and K rows of subcells. Each subcell
can take a transmittance value of 1 or 0. The phase is delayed by 2/N

F1
F
4
Figure 3.2
Complex plane showing four quadrature components

37
to provide N unit vectors. The K cells in each component are "opened"
or "closed" to provide the appropriate weights for each component.
The control over the amplitude and phase is not absolute with finite N
and K. The result at each cell is the vector sum of components with
integer length and fixed direction. Figure 3.3 shows that various
combinations of points turned on or off define an array of specific
points addressable in the complex plane. By increasing the number of
points N and K, the amplitude and phase can be more accurately
matched. When the total number of plotter dots is limited and more
subcells used for each cell, fewer cells can exist. Thus, with a
limited number of points, the hologram designer must choose between
space-bandwidth product (number of cells) and quantization noise.
The GBCGH allows more accurate determination of the amplitude and
phase of the cell by using more points. However, the complex sample
to be represented was taken at the center of the aperture. If N, the
number of points in the cell, is large, the outer pixel may have
1 P
noticeable error due to the offset in sample location. Allebach
showed that the Lohmann hologram fell into a class of digital
holograms which sample the object spectrum at the center of each
hologram cell to determine the transmittance of the entire cell. The
Lee hologram fell into a class of digital holograms which sample the
object spectrum at the center of each aperture to determine its size.
He also described a new third class in which the object is sampled at
each resolvable spot to determine the transmittance at that spot.
Although the function to be recorded should be constant over the
entire cell, there is some phase shift across the cell dimensions. By
sampling the object spectrum at the center of each aperture rather

38
Figure 3*3 Addressable amplitude and phase locations
using the GBCGH method.

39
than at-the center of each hologram cell, some of the false images in
the reconstruction are removed. By sampling the object spectrum at
the center of each resolvable spot in the hologram, the hologram noise
is further reduced. Allebach described an encoding technique in this
last category known as the Allebach-Keegan (A-K) hologramJ3 The A-K
hologram encodes the complex-valued object spectrum by quadrature
components as does the Lee hologram. Unlike the Lee hologram, the A-K
hologram compares subsamples within the aperture to an ordered dither
to determine whether each pixel is on or off. The input image is
padded to provide as many points in the FFT as there are resolvable
points. The FFT is decomposed into components spaced a quarter wave
apart (or more as in the GBCGH). Each point is then compared to a
threshold determined by the threshold matrix. The threshold values
are chosen to quantize the amplitude of each component. The threshold
values divide the range from zero to the spectrum maximum in steps
determined by the Max quantizer.^7 The size of the dither matrix and
the corresponding points in the cell can increase as with the GBCGH
but the magnitude and phase are sampled at each pixel.
Sampling and Space-Bandwidth Requirements
To represent an image on a computer, the image must be sampled and
quantized into a set of numbers. To sample a continuous image or
function, the value of the function is determined at discrete points.
The values of a function f(x,y) are determined at regular intervals
separated by Ax and Ay. The continuous independent variables x and y
are replaced with discrete sample point denoted by mAx and nAy .
Here ax and Ay are the fixed sample intervals and m and n are
integers. The sampling rate is u=1 / A* in the x direction and v=1/Ay

40
in the y direction. To convert the continuous function f(x,y) to a
sampled version f(mAx,nAy), multiply f(x,y) with a grid of narrow unit
pulses at intervals of Ax and Ay. This grid of narrow unit pulses is
defined as
OO 00
s(x,y) = E E ó (x-m x,y-n y) (3.8)
mí» n=-°°
and the sampled image is
fs(mAx,nAy) = f(x,y) s(x,y). (3.9)
The sampled version is the product of the continuous image and the
sampling function s(x,y). The spectrum of the sampled version can be
determined using the convolution theorem (equation 2.8).
Fs(u,v)= F(u,v) ® S(u,v) (3.10)
where F(u,v) is the Fourier transform of f(x,y)
and S(u,v) is the Fourier transform of s(x,y)
CO CO
s(ufv) = E E 6 (u-mAu,v-nAv)
m= — o=n= -oo
where u = 1/Ax and v = 1/ Ay
00 CO
Thus Fs(u,v) = // F(u-u0,v-v0) E E ra= • co n= - oo
(3.11)
Upon changing the order of summation and integration and invoking the
sampling property of the delta function (equation 2.17), this becomes
CO oo
F(u,v) = £ i F(u-mAu, v-nAv).
ni— —oo n= * co
(3.12)

41
The spectrum of the sampled image consists of the spectrum of the
ideal image repeated over the frequency plane in a grid space (au, av).
If Au and Av are sufficiently large and the ideal function f(x,y) is
bandlimited, no overlap occurs in the frequency plane. A continuous
image is obtained from the sampled version by spatial filtering to
choose only one order m,n of the sum in equation 3.12. If the image is
undersampled and the frequency components overlap, then no filtering
can separate the different orders and the image is "aliased." To
prevent aliasing, the ideal image must be bandlimited and sampled at a
rate Au >2fu and Av >2fy. The ideal image is restored perfectly when
the sampled version is filtered to pass only the 0,0 order and the
sampling period is chosen such that the image cutoff frequencies lie
within a rectangular region defined by one-half the sampling
frequency. This required sampling rate is known as the Nyquist
criterion. In the image, the sampling period must be equal to, or
smaller than, one-half the period of the finest detail within the
image. This finest detail represents one cycle of the highest spatial
frequency contained in the image. Sampling rates above and below this
criterion are oversampling and undersampling, respectively. To
prevent corruption of the reconstructed image, no overlap of the
desired frequency components can occur.
Frequency overlap is also a problem in holography. Recall that in
equation 3.2 the ideal function f(x,y) was heterodyned to a spatial
carrier frequency by mixing with an off-axis reference beam, i.e.,
h(x,y) = A2 + ! f (x,y) ¡ 2 + A f (x,y)e^2lTay + A f *(x,y)e_J2lTay (3.13)
and that the spectrum (shown in Figure 3.4) of this recorded signal is

42
Figure 3.4 Spectral content of an image hologram.

43
H(u,v) = : |A¡2 + F(u, v)®F(u, v) + A F(u,v+a) + A F(u,v-a) (3.14)
where F(u,v) is the Fourier transform of f(x,y) and ® denotes
convolution.
The first term is a delta function at (0,0). The second term is
centered on axis (0,0) but has twice the width as the spectrum F(u,v).
The third and fourth terms are the Fourier transforms of the f(x,y)
but centered off axis at plus and minus a. To prevent frequency
overlap, the second term and the heterodyned terms must not overlap.
This requires that the spatial carrier frequency, a, used to
heterodyne the information must be sufficiently large. Specifically,
this carrier frequency must be larger than three times the one-sided
bandwidth of the information spectrum.
In the case of the Vander Lugt filter and the subsequent
correlation, the output of the holographic matched filter has the form
o(x,y) = g(x,y) + g(x,y$f (x,y)®f *(x,y)
+g(x,y)®f(x,y)® 6 (x,y-a)
+g(x,y)®f*(x,y)® 6(x,y+a). (3.15)
The output, shown in Figure 3.5, contains a replica of the test image
g(x,y) centered on-axis along with a term consisting of the test image
convolved with the autocorrelation of the reference image f(x,y).
This term consumes a width of twice the filter size plus the test
image size. In addition to the on-axis terms, there are two
heterodyned terms centered at plus and minus a. These heterodyned
terms have a width equal to the sum of the widths of the test image
g(x,y) and reference image f(x,y). Again to prevent overlap of the

- a
+ a
Figure 3.5 Spectral content of a Vander Lugt filter

45
information terms in the output, a spatial carrier of sufficiently
high frequency is required to separate the heterodyned terms from the
on-axis terms. Assuming as an example that the test image and the
reference image are the same size 2B. The output positions of the
various terms can be shown graphically. To prevent the information
terms from overlapping with the on-axis terms, the carrier frequency,
a, must be chosen to center the heterodyned terms at 5B or more. In
the general case, the reference image f(x,y) and g(x,y) may have
different sizes. Let 2Bf represent the size of the reference image
and 2Bg represent the size of the test image. Then the requirement on
the carrier frequency, a, to prevent aliasing is
a = 3Bf + 2Bg. (3.16)
Sampling and heterodyning cause aliasing when improperly
accomplished. The combination of the two in the CGH requires
specific attention to detail. To create a CGH from a continuous image
f(x,y), it must first be sampled and quantized. According to the
Nyquist criteria, there are two samples for the smallest details in
the image. The sampling rate is at least twice the highest spatial
frequency in the continuous image. If a limited number of sampling
points are available, the image should be low pass filtered to limit
the highest frequency in the continuous image to half the number of
sampling points. This can be accomplished in an electronic sensor by
blurring the optics before the detector. When using a television
camera to digitize a transparency or film, the camera must be blurred
to match the detail in the continuous image to the number of points in
the digitizer. The detail required in the reference and test images

46
is determined by the pattern or target to be recognized. To detect
the presence of a desired target while an unwanted object could appear
in the test scene, sufficient detail to discriminate the two must be
included. To pick out cars from a scene which contains both cars and
trucks, the resolution must be adequate to resolve the differences
between the two. This resolution is typically chosen in an ad hoc
fashion using the human eye to determine what resolution is required.
Computer techniques have been used to quantify the resolution
required, but the results are usually not different than what a human
would have decided by eye. Although beyond the scope of this
dissertation, the bandwidth and specific frequencies best suited to
discriminate between targets and clutter can be determined with large
computers operating on adequate training sets.
The resolution must be adequate for target recognition. However,
oversampling beyond that resolution required will drive the CGH to
impractical limits. The resolution in the test image must match that
in the reference image yet the test image usually represents a much
larger area and larger total number of points. If the image already
exists in digital form, the resolution can be reduced by averaging the
image to produce an unaliased image of the appropriate number of
points. If an image is blurred or averaged to reduce the highest
spatial frequency, the detail above that maximum frequency is lost.
That is, all frequency components above the maximum are zero and lost.
Sampling the image properly (Nyquist criteria) permits the perfect
reconstruction of the averaged image, not the original image.
It is worthwhile to define the concept of space-bandwidth product
(SBP) here. The bandwidth of an image is the width of the spatial

47
frequency content to the highest spectral component. The space is the
physical length over which the image exists. For example, a piece of
film may have a maximum resolution of 100 points/mm with an image
which occupies 1 cm along the length of the film. In this case the
SBP is 100 points/mm X 10 mm = 1000 points. This is in one dimension.
For a square image, the number of points is 1,000,000. The SBP is the
number of resolution points in an image. The maximum SBP capability
of the film may not be utilized by an image recorded on the film, and
the actual SBP of the stored image will depend on the image itself.
In general, the bandwidth will be determined by the finest detail in
the image and the area of the total image. The area of the smallest
detail divided into the total image area defines the SBP. When a
continuous image is sampled at the Nyquist rate, one sample per
resolution point in the image is required. Thus, the SBP of the image
sampled at the Nyquist rate matches that of the continuous image. The
SBP in the sampled image is a very practical detail because each
sample must be stored in the computer memory. The number of
resolution elements in a 4" X 5" holographic film may exceed 10®. A
computer cannot practically store such a large number of values. With
a limited number of memory locations on the computer, the sampling
rate and SBP demand careful consideration.
A CGH is created using a digitized image. A continuous film image
may be sampled and quantized to create a non-aliased digital image.
Some imaging sensors output data in digital format with no further
digitizing required. Once the digital image is obtained, the image
values may be manipulated on a digital computer. If this digital
image is encoded on a continuous-tone CGH using equation 2.35 as a

48
model, a;spatial carrier frequency on the Fourier transform of the
image must be induced. The image is encoded as f(mAx,nAy) with a SBP
of M x N where M and N are the number of points in the image in each
direction. If the Fast Fourier Transform (FFT) is applied to the
image, a digital representation of the Fourier transform of the image
is obtained. This transformed image F(m&i,nAv) contains the same
number of points as the image and obviously the same SBP. If the
image contained M points along the x direction, the highest spatial
frequency possible in this image would be M/2 cycles/frame. This
situation would exist when the pixels alternated between 0 and 1 at
every pixel. That is, the image consisted of {0,1,0,1, ...}. The
maximum frequency in the transform is M/2 cycles/frame in both the
positive and negative directions. The FFT algorithm provides the real
and imaginary weights of each frequency component ranging from -M/2+1
cycles/frame to +M/2 cycles/frame in one cycle/frame steps. This
provides M points in the u direction. The same is true for N points
in the v direction. Thus, the first point in the FFT matrix is
(-M/2+1 ,-N/2+1), the D.C. term is in the M/2 column and N/2 row, and
the last term in the FFT matrix is (M/2,N/2).
It is useful to point out that the FFT describes the frequency
components of the image f(x,y). The FFT pattern also contains
structure which can also be represented by a Fourier series. That is,
the FFT pattern or image has specific frequency components. Because
the image and the FFT are Fourier transform pairs, the image describes
the frequencies in the FFT pattern. For example, a spike in the image
implies the FFT will be sinusoidal. A spike in the FFT implies the
image is sinusoidal. The existence of a non-zero value on the outer

49
edge of-the image implies the FFT contains a frequency component at
the maximum frequency. A non-zero value on the corner of the image
implies the maximum frequency exits in the FFT pattern which is M/2 in
the x direction and N/2 in the y direction.
To record the complex Fourier transform as a hologram, the
function F(mAu,nAv) must be heterodyned to a spatial carrier frequency
so as to create a real non-negative pattern to record on film. To
prevent aliasing, the heterodyne frequency must be sufficiently high.
The frequency components in the hologram are shown in Figure 3.6 and
consist of the D.C. spike, the power spectral density of the function
F(u,v), and the two heterodyned terms. To record the function F(u,v)
on film without distortion from aliasing, the spatial carrier
frequency must be 3 times the highest frequency component of the FFT
pattern. This permits the power spectral density term to exist
adjacent to the heterodyned terms with no overlap. The frequencies in
the hologram extend to plus and minus 4B. Thus, the hologram has a
space-bandwidth product 4 times larger than the original image in the
heterodyne direction. When heterodyned in the v direction as implied
by equation 2.35, the resulting hologram matrix must be larger than
the original image by 4 times in the v direction and 2 times in the u
direction. The spectral content in two dimensions is shown in Figure
3*7. The space-bandwidth product is very large for this CGH to record
the information in H(u,v).
The requirement is even greater when the hologram is to be used as
a Vander Lugt filter. When used as a Vander Lugt filter, the CGH must
diffract the light sufficiently away from the origin and the
additional on-axis terms to prevent aliasing in the correlation plane.

50
-a
+ a
Figure 3.6 Spectral content of a Fourier Transform hologram.

51
Figure 3.7 Two-dimensional spectrum of the Fourier Transform
hologram.

52
The output of the Vander Lugt filter is shown in equation 2.37 and the
spectral contents are plotted in Figure 3.5. These spectral components
are shown in two dimensions in Figure 3.8. Here the space-bandwidth
product is 7 times larger than the image in the v direction and 3
times larger than the image in the u direction. To produce a
correlation image without stretching, the samples in the u and v
directions should have the same spacing. Usually for convenience, the
hologram contains the same number of points in both directions, giving
a pattern which is 7B by 7B. The FFT algorithm used on most computers
requires the number of points to be a power of 2. This requires that
the hologram be 8B by 8B. For example, if the original images to be
correlated contain 128 by 128 points, the required continuous-tone CGH
contains 1024 by 1024 points. In a binary hologram, each continuous
tone point or cell may require many binary points to record the entire
dynamic range of the image.
This illuminates the key problem with CGH-matched filters. The
space-bandwidth product becomes large for even small images. Yet it
is the ability of optical processors to handle large images with many
points that makes them so attractive. Holograms created with
interferometric techniques contain a large amount of information or a
large space-bandwidth product. However, these optically-generated
holograms lack the flexibility offered by CGH. Holographic filters
are produced by either optical or computer prior to their actual use.
The filter imparts its required transfer function to the test image
without any further computation of the hologram pattern. Even if the
task is difficult, production of the filter is a one-time job. The
more information stored on the hologram, the greater the potential

53
V
Figure 3.8 Two-dimensional spectrum of the Vander Lugt filter.

54
processing capability of the Vander Lugt filter. To produce powerful
yet practical CGH filters, the space-bandwidth product and dynamic
range of the hologram must be understood and minimized within design
criteria.
One key to reducing the space-bandwidth product of the CGH is to
recognize that much of the spectrum is not useful information. The
terms in Figure 3.5 are described as the convolution of f and g, the
baseband terms fOf’tSg, and the correlation of f and g. Only the
correlation term is useful for our purposes in the Vander Lugt filter,
but the other terras arrive as a by-product of the square law nature of
the film. The two heterodyned terms which result in the convolution
and correlation of f and g must come as a pair. That is, when the real
part of the heterodyned information is recorded, the plus and minus
frequencies exist. The real part, cos 0, can be written as
exp(j0)+exp(-j9) using Euler's formula. The plus and minus exponents
give rise to the plus and minus frequency terms which become the
convolution and correlation terms. The convolution and correlation
terms are always present in a spatially modulated hologram.
A more efficient hologram is produced using equation 3.5. This
hologram consists of a D.C. term sufficiently large to produce only
non-negative values and the heterodyned terms.
H(u,v) = A2 + F(u, v)e J2lTav + F*(u,v)e“J2lTav (3.17)
The output (shown in Figure 3.9) of the Vander Lugt filter using this
hologram is
0(u,v) = A2G(u, v) + F(u,v)G(u,v)eJ2lTav + F*(u,v)G(u, v)e-J2irav (3.18)

55
Figure 3.9 Spectrum of a modified Vander Lugt filter.

56
or
o(x,y) = A2g(x,y) + f (x,y)@)g(x,y)®5(x,y+a) + f(-x,-y)g(x,y)@$(x,y-a)
= A2g(x,y) + f(x,y)(5fe(x,y)®ó(x,y+a) + Rf g(x,y)®5(x,y-a) (3.19)
which gives the spectrum shown in Figure 3.9 assuming BfsBgsB. Here
the spectrum extends to 5B rather than 7B and considerable space
saving is possible. However, the 5B is not a power of 2 and most
computer systems would still be forced to employ 8B points. The terms
in Figure 3.9 are the convolution term, the image term, and the
correlation term. The image term arises from the product of the D.C.
term with the test image g(x,y). In a normal absorption hologram, it
is not possible to eliminate the D.C. term. The image term takes up
the space from -B to B, forcing the spatial carrier frequency to 3B
and requiring 5B total space. If the absorption hologram is replaced
with a bleached hologram where the phase varies across the hologram,
the D.C. term may be eliminated.
As discussed in Chapter II, film may be bleached to produce a
phase modulation. This is accomplished at the expense of the
amplitude modulation. However, this phase hologram behaves much like
the original amplitude or absorption hologram. One advantage of the
bleaching process and the use of phase modulation is the opportunity
to eliminate the D.C. term (set it to zero) and reduce the space-
bandwidth product. Equation 3.17 is changed to
H(u, v) = F'(u,v)eJarav + F**(u,v)e-j27rav (3.20)
where the prime mark (') indicates the function has been modified by
the bleaching process. There is no D.C. term, so the output of the

57
Vander Lugt filter is
0(u,v) = F'(u, v)G(u, v)e^2lTav + F*'(u, v)G(u,v)e-^2lTav (3.21)
or
o(x,y) = f’(x,yX?>g(x,y)Ste(x,y+a) + f'(-x,-y)®g(x,y)@fc (x,y-a) (3.22)
= f'(x,y)@g(x,y)®ó(x,y+a) + Rf lg(x,y)@S(x,y-a)
which gives the spectrum shown in Figure 3.10, assuming Bf=Bg=B.
This phase hologram reduces the number of points to 4B, a power of 2.
This is the smallest possible size in a spatially modulated hologram.
As will be shown later, the phase modulation process may significantly
affect the information, and the correlation obtained may be a poor
approximation to the ideal correlation.
The Vander Lugt filter is typically used to detect the presence of
a small object in a large scene. This implies that B^ may be much
smaller than B_. In any case, the least theoretical hologram size
©
using equation 3.20 is still twice the size of the reference image and
test image combined in the y direction. For example, a large scene
consisting of 1024 by 1024 points is to be searched for an object that
would occupy 32 by 32 points in that scene. The smallest continuous-
tone hologram to perform that correlation would contain 2112 points in
the y direction (at least 1088 in the x direction). For most
practical applications, the absorption hologram illustrated in Figure
3.9 would be used. For the same example consisting of a 1024 by 1024
test scene and a 32 by 32 reference image, a square hologram would be
at least 2144 by 2144.
Another practical consideration provides some relief in the size
of the correlation plane. The correlation of two images creates a

58
Figure 3.10 Spectrum of the zero mean Vander Lugt filter.

59
correlation image whose size is the sum of the individual image sizes.
Non-zero correlation values can exist when any points in the two
images overlap. However, the number of points which overlap becomes
very small near the outer edge of the correlation plane. In a
practical system, a threshold is set to determine correlations which
are "targets” (above threshold) or "background" (below background).
When the target fills the reference image and is entirely present in
the test image, the autocorrelation condition exists and the
correlation can be normalized to one. When the target begins to fall
off the edge of the test image, correlations will still occur.
However, the correlation value will fall from unity by the ratio of
the target area present in the test image to the target area in the
reference image. A practical rule of thumb might be to ignore the
correlations when half of the target falls outside the test image in
any direction. This reduces the correlation plane to the size of the
test image, offering some relief to the required hologram size. If
the outer edge of the correlation plane is ignored, it does not matter
if that edge is aliased. This reduces the sampling and heterodyning
requirements in the filter hologram especially when the reference
contains many points. When using the absorption hologram with 50%
aliasing (shown in Figure 3.11), the spatial frequency is
(3.23)
a
and the number of points in the hologram in the v direction (SBPV) is
SBPy = 2Bg + 3/2 Bf.
(3.24)
Phase encoding this hologram does not relieve the requirement on the

60
Figure 3.11 Output of a 50$ aliased Vander Lugt filter with
absorption hologram.

61
carrier.frequency or the total number of points. The edges of the
correlation plane will fall into the active correlation region if a or
SBPy is reduced from the values given in equation 3.23 and 3.24.
In review, the SBP of the hologram is determined by the
following criteria.
(1) The required resolution in the reference scene to recognize the
desired target.
(2) The size of the reference scene. This is not normally a
significant factor due to the small size of the reference compared to
the size of the test image.
(3) The size of the test scene. The potential advantage of optical
processing is to test a large scene for the presence of the reference
object. The test image must contain the same resolution as the
reference image but includes many times the image area. Thus, the SBP
of the test scene is very large and is the driving factor in the size
of the CGH-matched filter.
(4) Usually, aliasing can be tolerated at the edges. This depends
the threshold and expected intensity of false targets. When 50Í
imposed aliasing can be tolerated, the SBP reduces to an even multiple
of two.
(5) The dynamic range in the reference scene. The hologram must
adequately represent the dynamic range in the reference scene. In the
case of binary holograms, many binary points may be required for
adequate representation of each hologram cell.
(6) Hologram type. The type of CGH produced determines the encoding
scheme and number of points required to represent the SBP and dynamic

62
range of :the reference while preventing aliasing of the active
correlation plane.
(7) Incorporate D.C. elimination when possible to minimize on-axis
terms.
By following these guidelines it is possible to determine the minimum
possible SBP needed in the CGH.

CHAPTER IV
OPTIMIZATION OF CGH-MATCHED FILTERS
The previous chapters describe the basic design techniques
employed to create CGH-matched filters. To determine the performance
of these filters, specific criterion must be established.
Performance Criteria
Because the matched filter is based on maximizing the signal-to-
noise ratio, that.criteria is reasonable to apply to the result of the
CGH also. The matched filter created as a result of a CGH is only an
approximation of the ideal filter. The non-linearities of the film,
along with the sampling, heterodyning, and quantizing of the CGH
process, cause the correlation to be less than ideal. The noise is
not just caused by background in the input image but also by artifacts
from the hologram. The matched filter was intended to recognize a
specific target in a clutter background, yet, in some cases, the
target will vary in size and orientation. There is a tradeoff between
using high resolution to discriminate against false targets and too
much sensitivity for target size and orientation. When modifying the
frequency content of the scene to best distinguish target from
background, the signal-to-noise ratio may decrease from the ideal.
Another important property of the optical matched filter is the
efficiency or light throughput. In a practical system, the input
image is illuminated by a laser of limited size and power. Typically
the laser source could be an IR diode putting out 10 mW.^ Even if the
63

64
signal-to-noise ratio is large, the energy reaching the correlation
plane may be too small to measure. The efficiency of the hologram,
the ratio of the power in the correlation to the power in the input
test image, is an important criterion in evaluating a practical CGH-
matched filter. Mathematically, it is given as
// !g(x,y) ® f*(x,y)|2dx dy
n = <4-1)
H
// ¡g(x,y)¡2dxdy
where ^ has been coined the Horner efficiency,^® f is the reference
scene, g is the test scene, and ® denotes an ideal correlation. The
correlation derived from a Vander Lugt-matched filter is not ideal.
To determine the Horner efficiency for a CGH-matched filter, equation
4.1 must include an accurate model of the encoding scheme. This
efficiency can be measured experimentally using a known input source
and calibrated detectors. Caulfield2® estimated that efficiencies for
certain matched filters could be as low as 10"®. Butler and Riggins^
used models of CGH filters to verify Caulfield's prediction and went
on to recommend techniques for improving the efficiency.
The matched filter is used to determine the presence of a target
in a large scene. A test scene is correlated with a reference, and
the correlation plane is thresholded to indicate the target location.
Occasionally, the Vander Lugt filter will generate correlation values
above the threshold in areas where no target exists. Accordingly, the
correlation of an actual target corrupted by noise may be lower than
the threshold. Due to the presence of noise, random and otherwise,
the performance of the filter must be measured in terms of the
probability of detection and the probability of false alarm. The
probability of detection, P^i is defined as the probability that a

65
target will be indicated when there is, in fact, a target to be
detected. The probability of false alarm, Pfa> is defined as the
probability that a target will be indicated when there is, in fact, no
target to be detected. These two quantities are correlated by the
presence of noise. If the detection threshold at the correlation
plane is lowered, the probability of detection is increased, but the
probability of false alarm is also increased. As with the efficiency
measurements, determining Pd and Pfa for CGH-matched filters requires
accurate models or optical experiments.
Historically, efficiency was not a concern in laboratory
experiments because powerful lasers were available to overcome the
hologram loss. When attempts are made to improve the efficiency, the
signal to noise ratio may suffer. An efficient hologram is
impractical if the signal-to-noise ratio in the correlation plane is
so low that Pd goes down and Pfa goes up significantly. The
performance of matched filters are typically measured in terms of the
Pd and Pfa> but testing requires modeling the entire system and
providing realistic images. All of these measures must be considered
for the cases when the test target deviates from the reference.
Optimization criteria for optical matched filters depend on the
application. To improve the matched filter, modifications to the
filter design have been proposed. These modifications fall into areas
of frequency modification, phase filtering, and phase modulation.
Frequency Emphasis
High frequencies in an image correspond to the small details.
Most images contain large continuous areas bounded by sharp edges.
The large continuous areas contribute to the D.C. and low frequency

66
content-of the image, while the edges contribute to the high
frequencies. If the high frequencies are removed from the image
through spatial filtering, the sharp edges disappear, the large
continuous areas blend together smoothly, and the resultant image
appears soft or blurred. A low-pass image may not provide sufficient
resolution to discriminate between two similar objects. If the low
frequencies are removed from an image, the continuous areas become
dark with only the edges remaining. The image appears sharp with
well-defined edges and detail. This high-pass image provides, to the
human eye, the same or better discrimination of the original image.
That is, objects are identified and distinguished at least as well as
in the original image. For example, images containing a bright square
area and bright circular area are easily distinguished as a square and
circle. If the high frequencies are removed, both square and circle
appear as blobs with no distinct edges. However, if the low
frequencies are removed, the bright area in the center of the square
and circle disappears, leaving only a bright edge. Yet these bright
edges clearly indicate a square and a circle as shown in Figure 4.1.
Even if the square is not filled in, the edge clearly denotes the
square shape. The edge of the circular area still defines a circle.
The square and circle are easily distinguished in the high-pass
images. The information that distinguishes the square from the circle
is contained in the high frequencies.
The traditional matched filter, as outlined in Chapter II, is
created from the complex conjugate of the Fourier transform of the
reference image. Filtering with such a filter is equivalent to
correlating the reference image with a test image. Because most

67
Figure 4.1 High-frequeney emphasis of a square and a disk.

68
scenes contain large continuous areas with edges, they contain a large
D.C. and low frequency component. Most images have spectra where the
magnitude tends to drop off drastically with increasing frequency.
The energy in the low frequencies may be several orders of magnitude
larger than the high frequencies. However, it is the high frequencies
which contain the useful information in separating the desired target
from false targets. A practical problem with holography is the
dynamic range to be recorded. Film cannot typically induce more than
two or three orders of magnitude of dynamic range. To record a
hologram of the Fourier transform, the film must accurately record the
entire dynamic range of the transformed image. If the dynamic range
of the transformed image is too large, the film cannot record the
Fourier transform linearly and the correlation is not ideal. The film
non-linearity will emphasize some frequencies and attenuate others.
The correlation signal-to-noise ratio will suffer if important
frequency components are attenuated. To reduce the dynamic range of
the transformed image and allow linear recording on the hologram, the
useless frequencies in the image should be eliminated. Because the
low frequencies contain most of the image energy but little of the
information, their omission considerably reduces the dynamic range
with little effect on the correlation except to reduce the overall
light through the hologram.
To determine which frequencies are important in target
discrimination involves considerably more work than can be considered
here. In general, a set of target images and a set of non-target
images can be compared on a large digital computer to determine which
frequencies appear most in the desired target. This requires a large

69
data base of true and false targets. Filtered images are correlated
and cross correlated to determine the most discriminating frequencies.
In practice, this process is too time consuming. Certain assumption
are reasonable in spatial filtering. It is reasonable to assume that
the reference and test images do not have much more detail than is
absolutely necessary to distinguish the true target. To reduce the
number of points needed in the digital imagery, the original sampling
process was accomplished by limiting the spatial frequencies to those
required to recognize the target. Thus, the appropriate filter to
eliminate unnecessary frequency components will have the form of a
high-pass filter. The nature of this high-pass filter is dependent on
the application of the matched filter.
The matched filter is created for a specific target. If the
target is present, the correlation is larger than for areas of the
image where the target is absent. If the target changes slightly from
the reference stored on the filter, the correlation drops. In a
practical application, small changes in the expected target are the
rule rather than the exception. If the target grows in size, rotates,
or changes its appearance slightly, the correlation may drop below the
threshold. This topic will be discussed further in Chapter V, but it
is necessary to point out that the invariance of the filter to small
changes in the target depends heavily on the frequencies used in the
correlation. Using the previous example, recall that the high-pass
images showing the edges allowed discrimination between the square and
circle. If the square were rotated slightly, the results would
change. The cross-correlation between a square and a slightly rotated
square depends on the frequencies used in the correlation. If only

70
low frequencies are used, considerable rotation can occur with little
effect on correlation. If high frequencies are used, the cross¬
correlation drops quickly with rotation. Thus, a matched filter
created from a high-pass image to discriminate against out-of-class
targets will not correlate well on in-class targets with small
changes. That is, as more high frequency emphasis is applied to the
matched filter, the discrimination sensitivity is increased. The
probability of false alarm is increased, but the probability of
detection drops. The high frequency emphasis is then tied to the
and Pfa which must be specified for a particular application.
There is another advantage to the frequency emphasis of matched
filters. As seen in equation 2.35, the transmission of the hologram
at each point depends on the magnitude of the reference image Fourier
transform. Yet the hologram transmission cannot be greater than 1.
Depending on the dynamic range of the film, the transmission out at
the edge of the hologram corresponding to the high frequencies is very
low or zero. As the magnitude drops off for high frequencies, so does
the transmission of light through the holographic filter, and hence,
filter efficiency is low. However, if the high frequencies are
emphasized (boosted), the transmission at those points in a positive
hologram is likewise emphasized. This creates an overall increase in
the hologram transmission. In an absorption hologram, the light which
is not transmitted is absorbed and lost to the system. The throughput
or efficiency is highly dependent on the total transmission of the
hologram. Thus, by emphasizing the high frequencies, the efficiency
of the Vander Lugt filter is increased. Because the maximum
transmission is limited to 1 and the dynamic range is limited on the

71
film, the greatest efficiency occurs when most of the frequencies have
equal weighting and the transmission is close to 1 across the entire
hologram. This implies that the throughput of the hologram will be
largest when the image transform is nearly white.
The following procedures determine the choice of frequency
emphasis.
(1) Specify the and Pfa for the particular application.
(2) Choose a high-pass emphasis which satisfies the P requirements. Typical choices include gradient, exponential, and step
filters.
(3) Because the test image should be filtered in the same fashion as
the reference image, the frequency emphasis chosen should be squared
before inclusion in the hologram. This permits the pre-emphasis of
the test image without a separate stage of spatial filtering. That
is, the test image is spatially filtered for pre-empHasis with the
same hologram providing the correlation.
(4) The test image is typically much larger than the reference image
and can thus contain frequencies lower than any contained in the
reference. Since those frequency components can never contribute to
correlations, all frequencies below the lowest useful frequency in the
reference should be truncated to the value of the next smaller term.
(5) The frequency emphasis (squared) greatly reduces the dynamic
range of most scenes, simplifying the coding of the CGH-matched filter
and greatly improving the efficiency. The frequency-emphasized CGH
matched filter is created, as shown in Chapter III, but utilizes a
reference image whose frequency content is modified.
F’(u,v) = |P(u,v)i2 F(u,v)
(4.2)

72
where F? is the modified image transform,
F is the original image transform,
and P(u,v) is the frequency emphasis chosen.
Phase-Only Filters
The preceding section describes techniques in which the high
frequencies are emphasized. This emphasis usually improves the
discrimination against false targets and increases hologram
efficiency. Frequency emphasis involves the multiplication of the
image transform by a filter function which attenuates or amplifies the
appropriate frequency components. The filter function adjusts the
spectral magnitude of the image. In the Fourier representation of
images, spectral magnitude and phase tend to play different roles and,
in some situations, many of the important features of a signal are
preserved even when only the phase is retained. Oppenheim1^ showed
that when the magnitude portion of an image Fourier Transform is set
to an arbitrary constant and the phase left intact, the reconstructed
image closely resembles the original. Features of an image are
clearly identifiable in a phase-only image but not in a magnitude-only
image. Statistical arguments by Tescher30 and by Pearlman, and Gray31
have been applied to real-part, imaginary-part, and magnitude-phase
encoding of the discrete Fourier transform of random sequences. They
conclude that, for equivalent distortion, the phase angle must be
encoded with 1.37 bits more than the magnitude. Kermisch32 analyzed
image reconstructions from kinoforras, a phase-only hologram. He
developed an expansion of the phase-only reconstructed image I(x,y) in
the form

73
I(x,y) = A [I0'(x,y) + 1/8 Io'CxjyXSteo'^y)
+ 3/64 I0'(x,y)@R0'(x,y)@R0'(x,y) + . • .] (4.3)
where I0’(x,y) is the normalized irradiance of the original object,
V(x,y) is the two-dimensional autocorrelation function of I0'(x,y)
and ® denotes convolution. The first term represents the desired
image, and the higher terms represent the degradation. Kermisch
showed that the first term contributed to the total radiance in
the image, giving a ratio of 1.8 bits.
The phase-only image typically emphasizes the edges as in the case
of the high-pass filtering as shown in Figure 4.1. This phase-only
filtering is closely related to the high-pass filter. Most images
have spectra where the magnitude tends to drop off with frequency. In
the phase-only image, the magnitude of each frequency component is set
to unity. This implies multiplying each pixel magnitude by its
reciprocal. The Fourier transform tends to fall off at high
frequencies for most images, giving a mound-shaped transform. Thus,
the phase-only process applied to a mound-shaped Fourier Transform is
high-pass filtering. The phase-only image has a high-frequency
emphasis which accentuates edges. The processing to obtain the phase-
only image is highly non-linear. Although the response 1/¡F(u,v)¡
generally emphasizes high frequencies over low frequencies, it will
have spectral details associated with it which could affect or
obliterate important features in the original. 0ppenheim^5 proposed
that if the Fourier transform is sufficiently smooth, then
intelligibility will be retained in the phase-only reconstruction.
That is, if the transform magnitude is smooth and falls off at high

74
Figure 4.2
Phase-only filtering of a square and a disk,

75
frequencies, then the principal effect of the whitening process is to
emphasize the high frequencies and therefore the edges in the image,
thereby retaining many of the recognizable features. In Figures 4.1
and 4.2 the phase-only filter emphasizes edges more strongly than a
gradient filter for the examples shown.
The advantage of using a phase-only image or high-pass image is
the increase in optical efficiency of the resultant matched filter.
As shown in equation 2.35, the transmission of each hologram element
depends on the magnitude of the reference image Fourier transform. As
the magnitude drops off for high frequencies, so does the transmission
of light through the holographic filter, and hence filter efficiency
is low. If the magnitude is set to unity (phase-only filter) for all
frequencies, the overall efficiency increases dramatically. The image
transform is white and thus the throughput of the absorption hologram
is highest. Horner^ shows that the maximum throughput efficiency of
an ideal autocorrelation of a 2-D rect function is only 44$, while the
autocorrelation using an phase-only filter achieves 100$ efficiency.
The phase function, <|>(u,v) of an image Fourier transform is a
continuous function. To fabricate a phase-only filter for such an
image requires a linear process capable of faithfully reproducing the
whole range of values from 0 to 2 . If the phase is quantized so as
to permit only two values, typically 0 and pi, such a filter is known
as a bi-phase filter.
H’(u,v) = sgn [cos 0 (4.4)
= -1 otherwise
where H(u,v) is the Fourier transform of the filter impulse response
h(x,y), the sgn operator gives the sign of the argument, and H'(u,v)

76
is the bi-phase transform. This bi-phase information is an
approximation to the phase-only information. In many cases,
reconstructions from this bi-phase information contain the same detail
as the ideal amplitude and phase information. This would indicate
that much of the information in an image is contained in the sign of
each pixel or where the zero-crossings occur.
In converting a complex wave, which contains continuous magnitude
and phase values, to binary values, much is thrown away. If the
reconstructions from the binary image transforms are similar to the
original image, then the bi-phase conversion reduces redundancy and
eliminates superfluous dynamic range. When this is accomplished in an
optical correlator without significant reduction in signal-to-noise
ratio, the CGH-matched filter is greatly simplified. Most important
is the ability to use binary light modulators. A number of electronic
spatial light modulators are commercially available. Of these
modulators, several can be used to phase-modulate a light wave. These
include deformable paddles, liquid crystals, and magneto-optical
modulators. These can be used as bi-polar phase modulators.33 if the
information in the reference image can be accurately represented using
only bi-phase information, binary phase modulators can be used as
real-time holographic filters. The ability to adapt the matched
filter in real time permits scanning the test image for various
targets with varying sizes and orientations. This technique is very
efficient because the light is phase shifted and not attenuated.
Phase-Modulation Materials
Recall that spatially modulated holograms are needed for matched
filtering only because film cannot record a complex wavefront. Film

77
can record only real values. Film may be used to record, at baseband,
the magnitude of a wavefront, or it may be computer-encoded and phase-
modulated (bleached) to record the phase of a wavefront. Thus,
without using a spatially modulated hologram, the magnitude or phase
may be recorded. If only the phase information of the image is needed
to represent the reference image, a baseband hologram which records
the phase portion of the image transform can be used in the optical
correlator. This on-axis phase hologram, or kinoform, is recorded as
a relief pattern in which appropriate phase delays are induced in the
illuminating wavefront. To produce a Fourier transform kinoform, the
phase is restricted to a range from - pi to + pi. The arctangent of
the ratio of the imaginary and real parts yields such a result. The
film is exposed to a pattern, whose intensity is proportional to the
desired phase, and bleached to create a relief pattern.3^ These
kinoforms cannot record the amplitude variation of the image transform
and thus, the filter formed is a phase-only filter.
Several techniques have been proposed by which the phase could be
modified to introduce amplitude variation in the reconstructed
wavefront.35,36 QhU> Fienup, and Goodman^® used multiple layers of
film to represent both the phase and amplitude variation. Kodachrome
II, for color photography, contains three emulsions. The phase
variation was recorded on one emulsion and the amplitude on another.
The inventors named this technique Referenceless On-Axis Complex
Hologram (ROACH). To introduce amplitude variation to the
reconstructed wavefront, light must be removed from the wavefront,
resulting in a reduction in efficiency.

78
The reconstruction from the kinoform is formed on-axis and is a
phase-only image. When the phase values are uniformly distributed
between - pi and + pi, the D.C. or average term is zero. However, if
the phase recording is unbalanced or the phase distribution is not
uniform, a D.C. term will exist in the hologram. When used as a
matched filter, the kinoform must be carefully phase-balanced to
prevent a large D.C. spike from occurring in the correlation plane.
Such a spike would be indistinguishable from an actual correlation.
If the phase hologram is produced using a "real time" holographic
device, the phase might be controlled using a feedback loop to
eliminate the D.C. term prior to correlation. To produce a
"permanent" hologram on film, the exposure and bleaching processes
must be carefully controlled.
Bleaching includes several processes which produce a phase
hologram from an exposed absorption hologram. The bleached hologram
retards the wavefront, causing a phase shift instead of attenuation.
The result is generally an increase in diffraction efficiency but
often with an accompanying decrease of signal-to-noise ratio.37 There
are three basic types of bleaches. The "direct" or "rehalogenizing"
method converts metallic silver back into silver halide which has a
different index than that of the gelatin. "Reversal" or
"complementary" bleaches dissolve the metallic silver from an unfixed
hologram, leaving the undeveloped silver halide which varies the index
of refraction. The third process creates a surface relief by
shrinking the exposed portions of the hologram by removing the
metallic silver. When the emulsion is bleached properly, the
attenuation of the transparency can be reduced to the point that phase

79
modulation due to index changes dominates any residual amplitude
modulation. Phase modulators prove to be more efficient in terms of
the portion of incident illumination that is diffracted to form the
desired correlation. A sinusoidal hologram using absorption or
amplitude modulation can theoretically diffract only 6.25% of the
incident energy into an image. Experimentally, the number is about
4^.38 a phase-modulated hologram transmits all of the light (ignoring
the emulsion, substrate, and reflection losses). A sinusoidal phase
hologram can diffract as much as 33-9% of the incident light into the
first order.
The bleaching process converts the real function F(u,v), recorded
in silver on the film, to a phase delay.
H(u,v) = exp j[ F(u,v) ]
(4.5)
To produce a kinoform, the film is exposed to the phase function
0(u,v) of the image transform. Upon subsequent bleaching, the film
contains the response
(4.6)
H(u,v) = exp j[ 0(u,v) ].
The kinoform, produced in this fashion, records the phase-only
information of the image transform. The bleaching process is not
restricted to phase-only information. Rather, the absorption hologram
created from equation 2.35 can also be bleached.
H'(u,v) = exp j[ H(u,v) ]
(4.7)
= exp j[1 + lF(u,v)!2 + F(u,v)exp j2irav + F*(u,v)exp -j2nav]

80
where H'(ü,v) is the bleached hologram response. The phase-only
information and the phase modulation obtained through bleaching are
entirely independent of one another. That is, a phase-modulated
hologram can be created from an image whose amplitude and phase are
intact or from an image whose amplitude or phase are modified or
removed. Considerable confusion continues to exist in the literature
in which a phase modulation process seems to imply, by default, phase-
only information. Cathey attempted to clarify this confusion in 1970
by defining specific terms for each case.39 The holographic process,
which is independent of the recorded information, was described as (1)
phase holography when only phase modulation was present, (2) amplitude
holography when only amplitude modulation was present, and (3) complex
holography when both amplitude and phase modulation were present. In
an equivalent fashion, the information to be recorded on the hologram
can be described as (1) phase-only information or (2) amplitude-only
information when either the amplitude or phase portion of the complex
waveform are deleted. Thus, for example, an amplitude hologram can be
created from phase-only information.
When an amplitude hologram is bleached, the density at each point
on the film is mapped to a phase delay. This mapping is linear when
the bleaching chemistry is correct. This new phase function on the
film is related to the original pattern on the film.
H(u,v) = exp j{F(u,v)} (4.8)
where H(u,v) is the complex function on the film after bleaching and
F(u,v) was the original transmission pattern recorded on the film.

81
The exponential expression in 4.5 can be expanded with a series
expression.
H(u,v) = 1 + jF(u,v) - (1/2)F2(u,v) - j(1/6)f3(u,v) + ...
= ÃœF(u,v)3n (4.9)
n!
When reconstructed, this hologram can be expressed as a series of
convolutions.
h(x,y) = (x,y) + jf(x,y) - (1/2)f(x,y)®f (x,y) -
-j(1/6)f(x,yj@f(x,y)®f(x,y) + ...
=y jn f(n)(x,y) (4.10)
n!
where f(n)(x,y) = f(x,y)@f(x,y)® . . . ®f(x ,y) n convolutions
and f(°)(x,y) = 5(x,y)
f(1)(x,y) = f(x,y)
f^2^(x,y) = f (x,y)®f (x,y)
and so on.
Thus, the phase modulation technique is very non-linear and the
resultant reconstruction is rich with harmonics. The reconstruction
from such a hologram is noisy due to the harmonic content. The higher
order correlations are broader, thus contributing less flux into the
reconstruction. Phase modulation in the form of bleached and
dichromated gelatin holograms have become the rule in display
holography due to the bright images. This fact indicates that the

82
noise is Acceptable in many cases. In fact, the reconstruction of
such display holograms looks very good. Nevertheless, such an example
is deceiving because the repeated convolutions and correlations of
equation 4.10 become more detrimental for more complicated objects,
especially if the object has low contrast.32 The harmonics combine to
produce intermodulation terms within the bandpass of the desired
information, causing an increase in background noise. When used for
matched filtering, the decision to use phase modulation is a balance
between hologram efficiency and signal-to-noise ratio.
An interesting case occurs when a binary amplitude hologram is
converted to a phase modulation hologram. The bleaching process maps
an amplitude of zero and one to a phase shift of plus and minus pi.
This equates to an amplitude of plus and minus one. For this binary
mapping, the transfer function is 2x-1, which is a linear process. In
that sense, the binary hologram is inherently linear. The binary
hologram represents the continuous-tone amplitude hologram by opening
more or fewer binary "windows". Through the use of many "windows,"
the amplitude can be accurately represented by the appropriate
combination of binary values. The subsequent bleaching of the binary
hologram is a linear process and thus no additional harmonics are
contributed. This provides a means by which high efficiency holograms
may be produced without sacrificing signal-to-noise ratio due to non¬
linearity. A sufficient number of points is necessary in the binary
hologram in order to minimize the non-linearity of the binary CGH
mapping. When a computer and writing device are available to produce
such binary holograms, subsequent bleaching or phase modulation
greatly improves the efficiency without any adverse effect on signal-

83
to-noise.; This makes digital, phase-modulated holograms very
attractive for matched filtering.

CHAPTER V
PATTERN RECOGNITION TECHNIQUES
Coherent optical correlators have been used as a means of
performing 2-D pattern recognition.^0-^3 An optical correlator system
could scan a large scene for the presence of a specific pattern. The
input image is correlated with the impulse response of the matched
filter to determine the presence and position of the reference
pattern. Because the Fourier transform is shift invariant (equation
2.6), correlation can occur anywhere in the input image and multiple
targets can be recognized simultaneously. However, other changes in
the input pattern do effect the correlation function. Rotation, scale
changes, and geometrical distortions due to viewing a 3-D scene from
various angles can lead to correlation degradation and a corresponding
loss in detectability.1*1* For example, to recognize a hammer in a box
of tools, the reference must be capable of correlating with the hammer
when it is laying in any orientation from 0 to 360°. The hammer could
lay on either side so that both orientations would need to be included
in the reference image. If we were not sure of the distance from the
camera to the hammer, we would not be sure of its size in the image.
The fundamental difficulty in achieving a practical recognition
system lies in correlation of the reference image with a real-time
image which differs in scale, aspect, contrast, and even content when
sensed in a different spectral band or at a different time than the
reference image. Matched filter pattern recognition systems, both
84

85
optical'ánd digital, tend to suffer from two types of difficulties.
They tend to be too sensitive to differences likely to occur in the
desired pattern. These differences are termed "within-class
variations.n Second, they tend to be too insensitive to differences
between real and false targets. These are "between-class variations."
While other deformations in the object condition are possible in
specific applications, translation, rotation, and scale are the most
common in pattern recognition whether it is accomplished optically or
digitally.
Deformation Invariant Optical Pattern Recognition
The basic operation performed in an optical processor is a two-
dimensional Fourier transform. Matched spatial filters are used to
perform correlations between an input image and a reference pattern.
While the reference pattern may exist in the input image, it may be
deformed by scale, rotation or geometrical distortion. The Fourier
transform is invariant to shift in two dimensions (see equation 2.6).
It is not however invariant to scale or rotation, and a dramatic loss
in signal-to-noise ratio (3 to 30 dB) occurs for small scale changes
(2%) or rotation (3.5°).^
In some applications it is desirable to give up translation or
shift invariance in substitution for some other deformation
invariance. The technique described by Casasent and Psaltis1^
involves a space variant coordinate transformation to convert the
deformation under consideration to a shift in the new coordinate
system. Because the optical processor performs two-dimensional
transforms, it is insensitive to shifts in two dimensions. Thus, two
separate invariances can be accommodated. Scale can be converted to a

86
shift iñ one direction and the normal shift can be left in the other
dimension. This would provide scale invariance, but the resultant
correlation would only yield the location of the target in only one
dimension (i.e. the x coordinate).
In another example, the scale can be converted to shift in one
dimension and rotation converted to shift in another dimension. Such
a two-dimensional optical correlator could provide correlations on
rotated and scaled objects but would no longer predict the location of
the object. The two-dimensional nature of the optical processor
allows the correlator to be invariant to both deformations. In order
to provide invariance to other deformations two at at time, a
coordinate transformation is needed to convert that deformation to a
coordinate shift. The Mel 1 in Transform is an excellent example of
such a transformation used to provide scale and rotation invariance.
The Fourier transform is invariant to translation shift in two
dimensions. To provide invariance to other deformations, a coordinate
transformation is needed to convert each deformation to a shift. To
provide scale invariance a logarithmic transformation is used. The
logarithmic transformation converts a multiplicative scale change to
an additive shift. This shifted version will correlate with the
logarithmically transformed reference pattern. To provide rotation
invariance, a transformation is performed to map the angle to each
point in the image to a theta coordinate. If an object rotates in
the test image, it is translated along the theta coordinate. Usually
the two transformation are combined into the log r, theta
transformation. The test image as well as the reference image is
converted to polar form to provide r and theta values for each pixel.

87
The image is transformed into a coordinate system where one axis is
log r and the other axis is theta. In this system, scale changes
shift the object along the log r axis and rotation shifts the object
along the theta axis. Because this transform, known as the Mel 1in-
Fourier transform, is itself not shift invariant, it is normally
applied to the Fourier transform of the test image. This provides the
shift invariance but loses the location information in the test scene.
The cross correlation between the transformed test and reference
images no longer can provide the location of the object but does
determine the presence of the object, its size, and its rotation
relative to the reference pattern.
To perform the Mel 1 in-Fourier transform for shift, scale, and
rotation invariance, the input image is first Fourier transformed and
the resultant power spectral density recorded. This magnitude array
is converted to polar coordinates and the linear radial frequency is
converted to a logarithmic radial coordinate. The new coordinate
space (log r,theta) is used for cross-correlation of the input image
with similarly transformed reference images. A high speed technique
is required to convert the image into log r, theta coordinates at a
speed compatible with the optical processor. This has been
demonstrated using holograms to perform geometrical
transformations.46-50 jo this, the coordinate transforming
hologram must influence the light from each point and introduce a
specific deflection to the light incorporating such modifications as
local stretch, rotation, and translation.
A practical correlator system might incorporate such an optical
transforming system or a sensor which collects data in the appropriate

88
format by the nature of its scan pattern. Whether accomplished by the
sensor scan or by a coordinate transformation, the logarithmic
coordinate transformation is equivalent to resampling an image at
logarithmically spaced intervals. An increase in space bandwidth
(number of samples) is caused by the oversampling which takes place at
small values of the input coordinate. This increased sampling at the
input is a cause for concern in a practical correlator design. In
such a system, the resolution required at the highest sampling rate
fixes the design of the entire system. This may cause the space-
bandwidth product required for adequate correlation to exceed the
capability of the sensor. However, Anderson and Callary^ showed
that previous researchers^ had overestimated the space-bandwidth
requirement and that practical Mellin-Fourier correlators were
possible.
Synthetic Discriminant Functions
Another technique for recognizing multiple orientations and
conditions is to cross-correlate with many different reference images
in parallel. The test image can be transformed by many lenses at
once, with each Fourier transform falling on an array of reference
filters chosen to give reasonable correlation to all conditions. By
the proper choice and location of the inverse transform lens, the
correlations of all the filters can coincide in one common plane.
C-3 5ÍI
This parallel setup has been extensively studied by Leib et al. ’
They showed that with a memory bank of 23 views of a tank, an optical
correlator could obtain a 98? probability of detection and 1.4% false
alarm rate in scenes exhibiting both scale and rotation variations.

89
Unfortunately, this parallel technique is somewhat cumbersome to
implement due to alignment of the multiple lenses and filters.
To avoid the need for multiple lenses and filters, it is possible
to combine several reference images into one filter. The use of
multiple lenses and filters superimposes the outputs of the individual
correlators. Because the Fourier transform and correlation are
linear, the superposition at the output is equivalent to superimposing
the individual filter functions into one filter. Likewise, this is
equivalent to superimposing the reference images in the creation of
the filter. Rather than create separate filters from many images, a
single filter is created from a sum of the images. This simplifies
the optical hardware. Caulfield et al55 defines a "composite matched
filter" CMF as a single filter which is a linear sum of ordinary
matched filters, MF.
CMF = E wk MFk (5.1)
k
These filters can be implemented by either multiple exposure optical
holography or computer holography. In the optical hologram, the
weights in the linear combination are obtained by varying the exposure
time. The latter approach is to use computers to generate the CMF
off-line. In this way, the long-drawn-out creation of the CMF is
performed on a digital computer where time is not critical. This
takes advantage of the control, dynamic range, and flexibility of
digital processors.
Once the CMF function is determined, an optical filter is
produced, tested, and optimized. It is then inserted in an optical
correlator to take advantage of its real-time processing. To

90
implement the CMF optically, two techniques can be used: (1) transform
the digital image to optical image via a high resolution CRT or
digitally addressed camera and produce a Vander Lugt Filter in the
conventional holographic manner, or (2) retain the image in a digital
format and produce the filter through computer-generated hologram
techniques. This latter technique has the advantage of using the
dynamic range of the digital computer until the final product is
produced. That is, if the CMF function is displayed and transformed
optically, the display will limit the dynamic range. By producing a
computer-generated holographic filter, the dynamic range is retained
till a later stage. In addition, complex filter functions and
frequency pre-emphasis can be easily incorporated.
However the CMF is implemented, the weights must be chosen for
optimal performance in a specific application. Hester and
Casasent56,57 developed what is called the Synthetic Discriminant
Function (SDF) which is a CMF that gives the same correlation output
intensity for each pattern in the training set. The weights required
to provide a constant correlation output for each pattern are not
unique. Additional constraints can be placed upon the SDF to reduce
the response to specific unwanted targets, to reduce dynamic range, or
to incorporate other desirable features. Starting with a training set
(Figure 5.1) which adequately describes the conditions in which the
desired target could be found, the SDF is formed as a linear
combination of all of the training images (Figure 5.2). The weights
are determined using matrix techniques which typically requires
considerable time on a large computer.58-63 The weights are adjusted
to provide a correlation with each member of the training set as close

91
'/A
Figure 5.1 Training set for the creation of an SDF.

Figure 5.2 SDF created from the images in Figure 5.1.

93
as possible to one. That is, the quantity £ ‘ *'*gi“’'' is minimized
where f is the SDF and the g^'s are the images in the training set.
A constraint on the number of images in the training occurs when
the dynamic range is limited. To illustrate this, note that if the
medium on which the SDF will be reproduced has limited dynamic range,
small values can not be recorded on the same medium as the large
values. The images with larger weights will appear in the SDF of
limited dynamic range, while the images with smaller weights will be
lost in the noise. As more and more images are combined into the SDF,
the sums will become large but the small details in any one image will
be too small to appear in the recorded SDF image. This problem is
greatly simplified by leaving the SDF in the computer where the
dynamic range is not practically limited. That is, to create the
hologram pattern on the computer rather than producing the hologram
optically. As was shown in chapter 3, the dynamic range can be
reduced by eliminating unnecessary terms from the hologram. However,
when producing the hologram optically, the SDF image must be displayed
on a device with a definite limit to its dynamic range. This
restriction is quite severe and frequently prevents the use of SDFs in
optically-generated holographic matched filters. This can be somewhat
eliminated by the judicious choice of weights to reduce the dynamic
range to a minimum while maintaining adequate performance.

CHAPTER VI
MATCHED FILTER LINEARITY
Optical matched filters are almost always produced on some type of
film or photosensitive surface. The pattern recorded is typically the
Fourier transform of the reference scene or some pattern related to
the transform. The assumption of the previous chapters was that the
film responds as a square-law device. This implies that the
transmission of the film responds linearly with the irradiance or
exposure on the plate. However, photographic materials are rarely
linear, but rather, respond with a typical "s" curve response. The
study of the relationships between the irradiance of the light falling
on the film and the resulting blackening produced after development is
known as sensitometry. The sensitometry of any photographic material
is a crucial link in producing and optimizing a matched filter.
Sensitometry is based upon plotting the density (blackening) of a
photographic material as a function of exposure. The blackening of a
photographic emulsion is measured in terms of optical density. Light
striking a developed photographic negative is partially absorbed by
the metallic silver in the emulsion. Opacity is defined as the the
ratio of the irradiance of light incident on a film to the irradiance
of the light passing through the film. This ratio is always greater
than unity. The intensity transmittance is defined as the reciprocal
of the opacity and thus has a value less than 1. The density of a
photographic material is the logarithm (base 10) of the opacity.
94

95
Densities rarely exceed 3 for normal photographic materials (OD 3
implies only 0.1$ of the light passes through the film) but certain
holographic films are capable of higher densities. At the other
extreme, no materials have densities of zero. All materials have some
loss in the emulsion, substrate, and surfaces causing the unexposed
density or base fog to amount to as much as OD 0.5.
Hurter and Driffield performed the first successful work toward
finding a relationship between exposure irradiance and the resulting
density of a photographic emulsion. By exposing a photographic film
to varying intensities of light, the resultant density can be plotted
as a function of exposure. The Hurter and Driffield method of
plotting, which has come to be known as the H & D curve, is to plot
Optical Density against the logarithm of the exposure (log It), where
I is the irradiance (watts/cm^) and t is the time (seconds). The H &
D curve depends on the emulsion type, development, spectral content of
the exposing light, age and condition of the emulsion.
Figure 6.1 shows a typical H & D curve for a photographic
emulsion. At low exposures, (point A) there is an attenuation or fog
on the film even when the film is unexposed. This occurs due to the
losses of the emulsion and substrate, and surface reflections. The
base fog determines the lowest opacity possible with the film.
Typically, the base fog becomes considerably worse as the film ages,
especially if it is not stored in a cool place. The silver halide
crystals in the emulsion do not respond to light until a certain
number of photons strike the crystal. Due to this finite number of
photons which must strike the crystal before it responds and the
limited size of the crystals, the film is insensitive to light below a

Density
96
Figure 6.1 Typical H & D curve.

97
certain threshold. At point B, sufficient numbers of photons are
available to convert the larger silver halide crystals to metallic
silver. This statistical event accounts for the ”toe" of the H & D
curve. When the number of photons arriving are larger than the
threshold, the conversion of the silver halides to metallic silver
depends on the projected area of each crystal. The crystal size is
carefully controlled during the production of the film. The
distribution of grain sizes provides a linear portion of the H & D
curve denoted part C in Figure 6.1. Because the crystal grain sizes
are limited in range, the film "saturates" at point D on the curve.
When all of the silver is converted, additional exposure can not
further reduce the transmission of the developed film. In fact, some
films will lose density with addition exposure (part E on the curve).
This effect, known as "solarization" is rarely used except by
photographic artists for special effects. This curve indicates that
the film response over a wide range of exposure is quite non-linear.
Measurement of Film Characteristics
In order to model the film accurately, it is important to collect
an adequate sample of data points. These samples must be gathered in
an experiment in which the conditions are representative. The
exposing energy and wavelength must be in a range consistent with the
eventual use of the film. Typically, a sensitometer and densitometer
are used to obtain the data points for the response of the film. The
densitometer exposes the film to a known irradiance for a fixed time
period. A Kodak step wedge is placed between the sensitometer light
source and the film to provide a wide range of exposures on the film
simultaneously

98
The.EGG sensitometer used in this lab provides the exposure with
an xenon flash lamp. The lamp has a consistent output irradiance and
the exposure time can be controlled electronically. The light from
the flash lamp is diffused to provide a uniform exposure across the
entire surface of the step wedge. This provides an accurate and
simple means by which the film is exposed to a known energy. This
method is used to calibrate most photographic films. Unfortunately,
several problems arise when attempting to use this sensitometer for
holographic films. The xenon lamp is rich with many spectral lines,
giving a brilliant white appearance. The film response to this white
light is not necessarily the same as the response to a single laser
spectral line. This limitation can be somewhat eliminated by
incorporating a spectral filter between the lamp and the film. When
the filter is properly chosen, the output of the sensitometer can
approximate that of the laser source.
A second limitation of the EG&G sensitometer is the extremely
short exposure time. The flash lamp emits light for a brief period
much less than a second. Considering that many holographic exposures
may last as long a minute, the time discrepancy between the use of the
film and the calibration of the film may be considerable. Reciprocity
is a term applied to the consistent response of film to the same
energy of light even when the exposure time is changed. When the
response does not depend on energy alone, but also exposure time, the
film suffers reciprocity failure. Reciprocity failure is common among
most films when attempting to use them at very long or very short
exposure times. It is important to calibrate the film with an
exposure time consistent with the intended use of the film.

99
Yet-another limitation of the EG&G sensitometer is the difficulty-
in measuring the energy of the exposing light. Although the
sensitometer provides a quick and consistent source, its flash is
difficult to measure with most radiometers due to their limited
ability to integrate the narrow pulse. It is therefore necessary to
use film calibrated by another source in order to calibrate the
sensitometer. This roundabout calibration lends itself to inaccuracy.
These limitations of the EG&G sensitometer suggest that another
technique is needed when accurate measurement of the film
characteristics is desired. To perform a more accurate measurement a
laser, beam expander, and step wedge were set up to expose the film in
a controlled fashion. It is interesting to note that even with the
limitations of the EG&G sensitometer, the results derived from the
sensitometer are quite consistent with those derived from the laser
setup. The inaccuracies range from 5% to 20%, depending on the films
used. Because the EG&G sensitometer is commonly available, easy to
use and fast, it is still used in this lab as a check of each new lot
of film and to test for development anomalies.
When a new film is received or a new developing technique is
tested, a rigorous film calibration is required. As described above,
the EG&G sensitometer is not always adequate for calibrated exposure
of the film. Instead a laser beam is spatially filtered and expanded
to provide a broad uniform beam. The irradiance of this beam is
measured using an NRC radiometer. This calibrated beam is then passed
through a Kodak step wedge to provide 21 discrete exposure levels.
Using the known irradiance of the original beam and the known
transmission of each step on the tablet, the irradiance at each point

100
on the photographic plate is determined. The resulting exposure
energy density in joules/cm^ at each point is computed as the product
of the irradiance (in watts/cm^) and the exposure time (seconds). The
exposure time is chosen based on an educated guess to give a
transmittance of 0.5 near the middle of the exposure range. It is
desirable to have as many of the 21 exposure levels provide points
within the dynamic range of the film. It may be necessary to repeat
the exposure using difference exposure times in order to bracket
properly the film response.
A number of holographic film types are used in this lab. These
films are mounted on glass substrates and are referred to as
photographic plates. These plates include Agfa 10E75, Agfa 8E75,
Kodak 120, and Kodak 649F. Full specifications are available from the
manufacturer but the general characteristics and uses of the plates
are outlined here. The Agfa emulsions are on a relatively thick
substrate. The 10E75 plate is 50 times more sensitive than the 8E75
plate but with an accompanying loss of resolution. The 10E75 plates
have sufficient resolution for two-beam holography, where the fringe
spacing is reasonably broad. However, for white-light holography, the
resolution of the 8E75 plates is necessary. Each of the Agfa plates
have thick emulsions of approximately 8 microns, permitting volume
holography. The Kodak 120 plates utilize a thinner substrate and
emulsion. The speed and resolution are between that of the Agfa 10E75
and 8E75 plates. The Kodak 120 plates are useful for two-beam
holography but lack the resolution and thickness for white-light
holography. The Kodak 649F is one of the first commercially available
holographic plates. It has an extremely thick emulsion (25 micron)

101
and is capable of high resolution. Although it is slow, the 649F
plate is a popular choice for white light holography. In addition,
most phase holograms using bleached halides or dichromated gels use
the 649F plates due to its thick emulsion.
Once the various plates have been exposed to a calibrated light
source, they are developed using a standardized developing technique.
In this case, development consists of 5 minutes in Kodak D—19 at
approximately 74°F with one-minute agitation intervals. The plate is
then placed in Kodak Stop Bath for one minute and Kodak Rapid Fix for
4 minutes. After a 30 minute wash, the plate is dipped in Kodak Photo
Flo and hung up to air dry. After processing, the intensity
transmission of each exposure step is measured. This measurement is
performed with a densitometer. A commercial densitometer is readily
available and easy to use. Unfortunately, the densitometer gives
results in terms of optical density which must be converted to
transmittance by
The commercial densitometer, as with the sensitoraeter, is designed
for use with photographic products which operate over a broad spectrum
of colors. Thus the white light in the densitometer may give results
inconsistent with the red laser light used for holographic
experiments. To determine whether this was the case or not, an
experiment to measure the transmission of the film was devised. The
irradiance of the collimated laser beam is measured. The beam is then
passed through each step of the developed plate and the exiting beam
irradiance measured. The power transmission is the ratio of the
transmitted irradiance to the incident irradiance. The amplitude
transmission is the square root of the power transmission. Using the

102
laser transmission experiment, extensive measurements were performed
on each of the photographic plates. Additional measurements were made
on the same plates using the densitometer set to the red color
position. In the red position, a red filter is placed over the light
source. Apparently, the red filter sufficiently matches the red laser
light. The results from the commercial densitometer very closely
match the results from the laser transmission experiment. The
agreement was consistent, with each of the plates indicating that the
commercial densitometer results could be trusted. This is
significant, due to the ease and speed with which this densitometer is
operated. The agreement has been so consistent that the laser setup
is used only as an occasional check of the densitometer.
The entire process of exposing the plates to a calibrated light
source, developing the plate, and measuring the resultant transmission
must be carefully accomplished. It is the film processing which
causes many to refer to holography as an art rather than a science.
There are many factors which affect the film response. Each of these
must be held constant for the calibrations to be meaningful. For
example, the base fog is highly dependent on the age of the film.
The film speed and contrast depend on the developer temperature and
freshness. Film and chemicals must be properly stored, and discarded
after their expiration date. Unless these conditions are held
constant, the film calibrations will be inconsistent and experimental
results inconclusive.
Models for Film Non-Linearity
When used in a coherent optical processor, the essential parameter
of any optical element is its amplitude transmittance, Ta(x,y). For a

103
photographic emulsion, this in turn is determined by the energy,
E(x,y) to which it has been exposed. Thus an emulsion is best
characterized experimentally by the transmittance versus exposure (Ta-
E) response curve. This is in contrast to the typical H & D curve
provided by most film manufacturers. The curves are, of course,
related. The amplitude transmission (ignoring any phase shift) is the
square root of the intensity transmission measured experimentally.
The exposure is the same as in the H & D curve but is not plotted
logarithmically. These curves also display an "s" shape due to the
saturation at low and high exposures. The Ta_E curves have the
opposite slope as the H & D curve because the transmission decreases
with exposure while the density increases with exposure.
For the matched filter to be recorded linearly, the recording
media must have transmittance directly proportional to exposure,
Ta=cE, for all values of E. Recall from the previous discussion of
optical matched filters that the hologram is created from the
interference of the Fourier transform wave and a reference wave. A
lens forms the Fourier transform of the reference image
S(u)=So(u)exp{i0(u)}, and a uniform reference beam R(u)=R0exp{iau} is
introduced at an angle <}> to the optical axis (a=ksin <{>). A
photographic film placed at the focal plane of the Fourier lens
receives an exposure
E(u) = |S(u) + R(u)¡2 t (6.1)
where t is the exposure time. Substituting for S(u) and R(u),
E(u) = [R0+S0(u)]t + tRoSo(u)[exp(i0)exp(-iau) + exp(-i0)exp(iau)]
= + Eac
(6.2)

104
where E¿j¿ = t[R0 + S0^
and Eac = tR0S0[exp(i9)exp(-iau) + exp(-i9)exp(iau)]
are the local average exposure and the varying components
respectively. Note that dc (direct current) and ac (alternating
current) subscripts apply here to the slowly varying terms and the
high spatial frequency terms respectively.
The ideal recording medium would have a transmittance directly
proportional to exposure, Ta = cE, for all values of E. For such a
linear material the transmittance would be
Ta(u) = c[Edc + Eac]. (6.3)
When such a transparency is illuminated with a collimated beam, light
is diffracted by the film. The first term in equation 6.3 is real and
gives rise to wavefronts propagating near the optical axis. The ac
term includes the factor exp(iau), a linear phase shift, which will
diffract light waves at angles plus and minus <{> to the optical axis.
Thus if a is chosen appropriately, or the recording angle properly
chosen, each of these wavefronts will be separated from the other and
from those which propagate on axis. One of the off-axis beams
contains the term S0(u) exp{-i9(u)} which is the complex conjugate of
the reference Fourier transform, and thus has the desired optimal
filter characteristic.
Unfortunately, photographic emulsions do not exhibit this ideal
linear response described above. Rather, they saturate at high and
low exposures. To understand the effect of this non-linearity, it is
important to describe the actual film response at all exposures and

105
model that response to predict the transmittance of the developed
emulsion. This model should be based on actual response curve
measurements performed on the film using a calibrated sensitometer and
densitometer. This model will assist in the proper choice of average
exposure and ratio of reference to signal exposures.
The Hurter-Driffield (H-D) curve has been used extensively to
predict photographic response. This plot of photographic density
versus log exposure demonstrates the key features of the film
saturation. A form of this curve could be described by
{ Ds when E > E3 (6.4)
D = { ylog(E) - log(Ebf) + Dbf Ebf < E < Es
{ Dbf E < Ebf
where s denotes saturation, bf denotes base fog, and y is the slope of
the linear portion of the curve. Modeled in this fashion, the film
exhibits a linear transmittance versus exposure only by producing a
positive print and developing to net y of -2. When this is the case,
the transmittance becomes
{ Tbf when E > Es (6.5)
Ta ={ c(E-Eg) + tbf Ebf ^ E < E3S
{ Tg E < Ebf
where the transmittances are those of the positive print and the
exposures refer to the original negative. Thus, when the exposure is
determined by equation 6.2, this model predicts linear recording will
occur when the signal and reference amplitudes satisfy
!Ro “ S0 max'^4 > Ebf and (6
'^o + ^0 max'2t < Es
where SQ max is the maximum signal amplitude in the Fourier plane.

106
This can be expressed in terms of the experimental parameters, E<-icinax
and K where
^dcmax = (^o^ + raax^ (6*7)
and K = R0^/S0 max^*
^dc max is maximum average exposure and K, frequently called the
beam balance ratio, is the ratio of the reference to signal beam
intensities. In terms of Es and Ebs these become
Edcmax = 1/2 (Ea + Ebf) (6.8)
and
K = C(E31/2 + Ebf1/2)/(E31/2 - Ebf1/2)]2
Under these conditions, the maximum amplitude of the diffracted signal
is given by 1/4(T3 _ Tbf). For all values of s(p) less than smax the
diffracted signal will be proportional to s(p), as desired.
The Hurter-Driffield model, although the most common in
photographic work, is not directly applicable to holography.
Transmittance, the ratio of transmitted amplitude to incident
amplitude, is the fundamental parameter in holographic filters.
Plotting the transmittance vs. exposure is more convenient than
inferring the information from the D-log(E) curve. The model as
expressed in equation 6.4 assumes the response is piece-wise linear
with two breakpoints. This neglects the smooth non-linearity which
exists throughout the film response. Although easier to analyze, the
piece-wise linear model does not predict the non-linear effects in the
regions near the breakpoints. To predict more accurately this non-

107
linear effect, a polynomial approximation can be made to fit an
experimentally measured Ta_E curve.
A polynomial representation provides a more accurate approximation
over a wider range of exposure than possible with a linear
representation. Such representations have been studied with
polynomials of degree three being the most common choice.64-66 Using
a third order polynomial, the transmittance can be expressed as
Ta = CQ + Ci ET + C2 Et2 + C3 Et3 (6-9)
where ET = E,jc + Eac. Since E be written
^a = Aq + A-| Eac + A2 Eac2 + ^3 EaG3 (6.10)
where the A¿ depend on the average local exposure E^q. Only the ac
portion of the exposing light causes fringes which will diffract light
in the resultant hologram.
Each power of Eac will diffract light to a unique angle or
diffractive order. The terms in equation 6.10 can be written
Ta = T0 + + T2 + T3 (6.11)
where Tq is a real quantity and T-] contains all those terms which
contribute to the first diffraction order. T2 and T3 contain only
terms which appear in the second and third diffracted orders. Thus T-j
can be separated spatially from all other contributions and contains
the filter term which is desired. This term when expanded can be
written
Ta = C(Ci + 2C2Er + 3C3Er2) + (2C2ES + 9C3ErEs +
3c3Es2)]Eac
(6.12)

108
where Er :=â–  R02 t and Es = So2 t are the reference and signal exposures
respectively. Written in this form, the first term in brackets does
not depend upon the signal strength. This term produces a linear
reproduction as in the ideal case. The second term in brackets
contains all contributions depending on the signal strength which
would produce a non-linear result. Thus, in this form, the first term
represents the signal coefficient and the second term the non-linear
noise coefficient. This provides a convenient method to determine the
signal-to-noise ratio for a given signal and reference exposure.
Several interesting cases are apparent from inspection. When Er
is large, the signal portion is dominant. The coefficient of Eac is
maximized when Er = Es. There is a trade-off between the need for
adequate signal and minimizing signal-to-noise ratio. This choice
depends upon factors such as the laser power available and the noise
level which can be tolerated. If Eac is maximized by setting Er = Es
and also set the sum Er + e3 = E', the exposure value at the
inflection point, the values of Er and Es are determined uniquely.
The result is Er = Es = -C2/6C3. By substituting these values into
equation 6.12, it is seen that the noise term is zero. This choice of
values is not very useful for matched filtering because the zero noise
condition holds only if E3 does not vary. No information can be
stored in this condition. If Es is allowed to vary, the noise
coefficient will increase. The worst case condition is when Es = Er/2
when the signal-to-noise ratio is 12.4. For many applications, this
noise is tolerable and this choice of Es and Er will be the preferred
choice

109
The expression in equation 6.12 is general for any t-E curve which
is described by a cubic polynomial. It is useful to explore
analytically the effect of varying reference and signal exposures and
to predict the signal and noise in the recorded pattern. However,
such analytical methods prove unwieldy when complicated signals are
involved. To analyze the more complicated patterns, the pattern and
film must be modeled on a computer. The computer provides the brute
force to expand the input pattern into the non-linear terms expected
from the film. By modeling the film on the computer, it is possible
to plot the experimental film response along with the predicted
response. This permits the comparison between the experimental data
and various order polynomial fits.
It has been recognized in experiments by this author, that cubic
polynomials have difficulty representing accurately the entire range
of the film response. In order to use the cubic polynomial to predict
the inflection point, only the experimental data points near that
point or in the linear region should be considered. Unfortunately,
this does not permit accurate modeling in the saturation regions.
When modeling an absorption hologram, where only the linear region is
utilized, the cubic polynomial will suffice. The cubic polynomial
loses accuracy when modeling holograms whose response extends into the
non-linear regions, such as phase holograms. This problem is greatly
relieved by using a 5th order polynomial. This gives a more accurate
fit to the t-E curve over the entire useful range of the film.
In order to model accurately the film response, a set of
experimental training points must be established. This is
accomplished by exposing the film to varying irradiances of light and

110
measuring the density of the film after developing. A Kodak step
wedge with densities varying in 21 discrete steps from 0D 0.05 to 0D
3.05 was placed before the test film. The step wedge and film were
then exposed to a known irradiance of laser light for a known duration
of time. The irradiance of the laser was attenuated by each step of
the wedge to give a wide range of exposures to the film. The exposure
for each section of the film is computed based on the laser
irradiance, step density, and exposure time. The film is then
processed in the darkroom using the standardized technique mentioned
previously. This processing plays a major role in the response of the
film and thus must be carefully controlled. The developed film is
placed on a densitometer to measure the resultant density of each
exposed portion. The set of exposure and density points are
incorporated into a polynomial fit routine to produce the
coefficients.
The output of the polynomial fit routine is shown in Figure 6.2.
All of the important parameters are recorded in order to reproduce the
same result in future exposures. The laser irradiance and exposure
time are recorded to verify that their combination would not suggest
reciprocity failure which occurs when exposure times are very short or
very long. It is advised that the exposure time for determining the
training set be near that to be used in experiments. This may require
that the experiment be run several times to determine what laser
irradiance is required. Unfortunately, it may dictate that more laser
power is needed than is available. In Figure 6.2, it is seen that the
exposure time was 120 seconds. This is longer than a normal
holographic exposure but no additional laser power was available to

Ill
DATE 11 FEB 85 TYPE OF FILM; 10E75 NAH
LOT #595906 DATE FILM RECEIVED 12 DEC 84
LIGHT INTENSITY 2000.00 ERGS/CM**2 .512 MICRONS
EXPOSURE TIME: 120 SEC.
DEVELOPING PROCEDURE
5 MIN. D-19 75 DEO. F.
1 MIN. STOP
4 MIN. FIXER
30 MIN. WASH
1 MIN. FOTO-FLO
STEP#
EXPOSURE
RESULTANT
(ERGS/CM**2)
DENSITY
1
1783.
7. 87
2
1262.
7. 86
3
893.
7. 86
4
632.
7. 83
5
448.
7. 64
6
317.
5. 67
7
224.
4. 50
8
159.
3. 58
9
112.
2. 65
10
80.
1. 82
11
56.
1. 10
12
40.
0. 64
13
28.
0. 36
14
20.
0. 24
15
14.
0. 17
16
10.
0. 12
17
7.
0. 10
18
5.
0. 09
19
4.
0. 09
20
3.
0. 09
21
2.
0. 09
THE POINT OF INFLECTION IS:
EXPOSURE = 32. ERGS/CM**2
AMPLITUDE TRANSMISSI0N= 0. 596
AMPLITUDE
TRANSMISSION
. 000
. 000
. 000
. 000
. 000
. 001
. 006
. 016
. 047
. 123
. 282
. 479
. 661
. 759
. 822
. 871
. 891
. 902
. 902
. 902
. 902
EXPOSURE FOR 0. D. =2
IS 89. ERGS/CM**2
SLOPE OF TANGENT LINE AT INFLECTION POINT IS -.014535
Y INTERCEPT IS 1.062
TAU vs. EXPOSURE IS APPROXIMATED BY A
FIFTH-ORDER LEAST SQUARES FIT
THE POLYNOMIAL IS OF THE FORM: THE COEFFICIENTS FOR
CO + C1*X + C2*X**2 + . . . + C5*X**5 EXPOSURE VS. TAU ARE
CO = 0. 903661728E+00
Cl = O. 132656097E-02
C2 = -. 568270683E-03
C3 = O. 838190317E-05
C4 = -. 422005542E-07
C5 = O. 553654900E-10
CO = 0. 14129B828E+03
Cl = -. 743000000E+03
C2 = O.263350000E+04
C3 = -. 5041OOOOOE+04
C4 = O.472500000E+04
C5 = -. 174050000E+04
THE COEFFICIENT OF CORRELATION IS.999964
Figure 6.2 Computer output of the polynomial fit routine

112
shorten,the exposure time. An independent test verified that a
shorter exposure on several points gave results consistent with the
120 second measurement. This indicates that the 120 second exposure
did not cause significant reciprocity failure. The film processing
procedure is outlined briefly in the printout with developing times
and temperatures recorded. The data from the measurement of the film
includes the step number, the exposure through that step, the density
on the developed film, and the associated amplitude transmission.
The computer tabulates several useful points from the data. The
exposure and transmission at the point of inflection are provided.
The best linear fit is determined by the slope and intercept of a line
tangent to the curve through the inflection point. Also, since most
phase holograms are exposed to OD 2.0, the appropriate exposure value
is provided. Finally, the printout provides the polynomial
coefficients for the transmission vs. exposure curve and its inverse
relationship along with the coefficient of correlation. The H & D
plot is found in Figure 6.3 for the AGFA 10E75 plates. The associated
transmission vs. exposure is plotted in Figure 6.4. The plot shows
the linear fit through the inflection point and the result of the 5th
order polynomial. Figures 6.5 through 6.7 are identical to Figures
6.2 through 6.4 except the results are for AGFA 8E75 plates.
Computer Linearization of Filter Response
The models presented here include a summation of terms
representing various powers or orders. The linear term or first order
represents the desired term where the transmission is linear with
exposure. If the linear term represents the signal and the higher
order terms represent noise, then the recording on the film is

I—
i J-J
LiJ
h-
lL
O
o
f-s
r-,
'•■i
O'
10®
1 1 i i'TTTT
1 TTTTT
T~r
¡ 1 ¡I
TTT
kV
i ¡v
í 0
"i—rrírm
10l
EXPU-jURE LhRGS/CM-2]
Figure 6.3 H & D plot for AGFA 10E75 photographic plates.
113

ÃœJ
Figure 6.4 Amplitude transmission vs. exposure for AGFA 10E75 plates
114

115
DATÉ 20 DEC 84 TYPE OF FILM: 8E7S HD
LOT #595707 DATE FILM RECEIVED 9 APR 84
LIGHT INTENSITY 10000.00 ERGS/CM**2 ,512 MICRONS
EXPOSURE TIME: 300 SEC.
DEVELOPING PROCEDURE
5 MIN. D-19 75 DEG. F.
1 MIN. STOP
4 MIN. FIXER
30 MIN. WASH
1 MIN. FQTO-FLO
STEP#
EXPOSURE
RESULTANT
AMPLITUDE
(ERGS/CM**2)
DENSITY
TRANSMISSION
1
8913.
4. 28
. 007
2
6310.
4. 27
. 007
3
4467.
4. 07
. 009
4
3162.
3. 69
. 014
5
2239.
3. 12
. 028
6
1585.
2. 77
. 041
7
1122.
2. 29
. 072
8
794.
1. 84
. 120
9
562.
1. 38
. 204
10
398.
0. 98
. 324
11
282.
0. 67
. 462
12
200.
0. 45
. 596
13
141.
0. 28
. 724
14
100.
0. 18
. 813
15
71.
0. 12
. 871
16
50.
0. 10
. 891
17
35.
0. 08
. 912
18
25.
0. 08
. 912
19
18.
0. 08
. 912
20
13.
0. 08
. 912
21
9.
0. 08
. 912
THE POINT OF INFLECTION IS: EXPOSURE FOR 0. D. =2
EXPOSURE = 163.ERGS/CM**2 IS 889. ERGS/CM**2
AMPLITUDE TRANSMISSION3 0.684
SLOPE OF TANGENT LINE AT INFLECTION POINT IS -.001774
Y INTERCEPT IS 0.973
TAU vs. EXPOSURE IS APPROXIMATED BY A
FIFTH-ORDER LEAST SQUARES FIT
THE POLYNOMIAL IS OF THE FORM: THE COEFFICIENTS FOR
CO + C1*X + C2*X**2 + . . . + C5*X**5 EXPOSURE VS. TAU ARE:
CO
=
0.
944401741E+00
CO
=
0.246046875E+04
Cl
=
120447576E-02
Cl
=
-.256770000E+05
C2
=
391120557E-05
C2
=
0.125204000E+06
C3
0.
106440439E-07
C3
=
-.284328000E+06
C4
=
869682104E-11
C4
=
0.295216000E+06
C 5
=
0.
233255255E-14
C 5
=
-.113680000E+06
THE COEFFICIENT OF CORRELATION IS. 999746
Figure 6.5 Computer output of the polynomial fit routine for
8E75 plates.

OPTICAL DEN
M ;
tí
Q
Ü i
f--j i
o
0
Q
rJ'
Q'
101
r i
ir is0
EXPOSURE LERGS/CM--23
m
10
vl
Figure 6.6 H & D plot for AGFA 8E75 photographic plates
116

ñ M P L I T U D E T R fi N S M I S S I 0 N
Cl”’
(D
f—i
o
r«.j
Figure 6.7 Amplitude transmission vs. exposure for AGFA 8E75 plates
117

118
optimized for signal-to-noise ratio when the higher order terms are
minimized or eliminated. As was shown in the previous section, the
choice of beam balance ratio in the conventional hologram determines
which part of the t-E curve is occupied by the information. When the
beam balance ratio is is high, the information resides in the linear
portion. The signal-to-noise ratio is high but there is little
variation in density on the film and the efficiency is low. When the
beam balance ratio is unity, the entire density range of the film is
used, giving high efficiency. However, since the information extends
into the saturation regions, the high-order non-linear terms are large
and the signal-to-noise ratio suffers.
Using the computer models for the film response, the non-linear
effects in conventional holography are predictable. It is impossible,
however, to affect the results, except by varying the average
irradiance and the beam balance ratio. To linearize the hologram, it
is necessary to produce the hologram as a CGH. The techniques
discussed in the previous chapters are used to compute the holographic
filter function. However, rather than writing the function directly
to film, the pattern is pre-distorted. Such a pre-distorted image,
when written to the film, will be distorted by the film so as to yield
the desired result.
In order to linearize the filter response over the entire dynamic
range of the film, it is necessary to model the film over that range.
The preceding sections describe the modeling process. Once the film
response has been determined experimentally, the computer is used to
determine the transfer function of the film. The transfer function is
described by the computer expression of the t-E curve. Whether that

119
expression is first order, third order, fifth order, or spline, the
inverse relationship can be applied to the desired pattern prior to
writing on the film. Thus by pre-distorting the pattern, the film can
be driven linearly to the extremes without saturation or non¬
linearity. This permits the film to operate over its entire dynamic
range while maintaining a linear response. Thus, the efficiency can
be maximized along with the signal-to-noise ratio. When using phase
holograms, the mapping from exposure to phase is quite non-linear.
The pre-distorting process permits accurate control of the phase
despite the inherent non-linearity of the bleaching process. This
ability to optimize many aspects of the hologram independently is an
overwhelming advantage of computer-generated holograms.
This linearization process was applied to Agfa 8E75 film to test
the signal-to-noise improvement. Figure 6.8 shows a linear gradient
provided as an input to an 8E75 photographic plate. Figure 6.9 shows
the resultant transmission on the plate. This figure demonstrates the
typical saturation effects noted in film. Note that the once linear
gradient is grossly distorted. This is the normal effect incurred in
conventional holography. However, if the pattern in 6.8 is pre¬
distorted, yielding Figure 6.10 and submitted as the input to the 8E75
plate, the output in Figure 6.11 is obtained. The pre-distorted
result in Figure 6.11 is superior to the normal exposure.
To analyze the extent of the distortion and the effect on the
signal-to-noise ratio, a suitable test pattern is utilized. The non¬
linearity manifests itself in the form of high frequency components
due to the powers expressed upon the input pattern. To measure the
extent of the non-linearity, the spectral energy in the perfect signal

Will IF l« I
Figure 6.8 Image and plot of a linear gradient used for a
input.

PWIIf Of LIU I
Figure 6.9 Image and plot of the output transmission on film
a gradient input.

122
Figure 6.10 image and plot of the pre-distorted gradient used for
an input.

123
Figure 6.11 Image and plot of the output transmission with pre¬
distorted input.

124
is compared to the spectral energy in the distorted energy. The
logical test input is a sine wave because it has a simple spectrum.
This technique for testing non-linearity is a long accepted method for
electrical systems. The output power when the sinusoidal spectral
components are removed, divided by the input power, is called the total
harmonic distortion (THD).
Amplifiers for speech and music are typically rated by their THD
specification. To measure THD in an audio amplifier, a sine wave is
input to the amplifier but is filtered from the output. Any remaining
power not at the input frequency is considered a result of harmonic
distortion. Applied to the analysis of film, a sinusoid input is
written to the film using the expression
s(x,y) = ( 1 + cos kx )/2 (6.13)
where k determines the spatial frequency. Note that the expression is
biased and scaled to provide an input between zero and one. This is
further scaled to utilize the entire dynamic range of the film. The
Fourier transform of equation 6.13 is
S(u,v) = 1/2 + 1/4 <$ (u-k) + 1/4 6 (u+k), (6.14)
and the power spectral density is
I(u,v) = 1/4 + 1/16 ó (u-k) + 1/l6fi(u+k). (6.15)
There is a constant term and two spectral terms. The spectral terms
in the intensity image are down from the constant term by 1/4. Now
that the input spectrum is known, the film distortion is applied to
the input and the effect on the output spectrum is analyzed. The
sinusoidal input is shown in Figure 6.12. The film distorts the
sinusoidal input and transmits the pattern shown in Figure 6.13. The
spectrum of the distorted pattern recorded on the Agfa 8E75 plate is

125
Figure 6.12 Image and plot of a sinusoidal grating pattern
used for an input.

126
Figure 6.13 Image and plot of the output transmission with
sinusoidal input.

127
shown in Figure 6.14. Note that the spectrum is rich with harmonics.
The input spectrum consisted of only the constant term and delta
functions at plus and minus 8 cycles. The distorted spectrum clearly
shows spectral components at the harmonic frequencies of plus and
minus 16, 24, 32, and so on. These harmonic components account for
the non-linear behavior and they represent noise in the recording.
The THD is determined for the film by calculating the power in the
output spectra with the constant and fundamental removed. This output
power, normalized by the input power, is the THD for the 8E75 when
exposed over its dynamic range. The THD for the 8E75 plate was 20$.
Now the input is pre-distorted and the resultant pattern used to
expose an 8E75 plate. The pre-distorted image is shown in Figure 6.15
and the resultant transmission pattern on the plate shown in Figure
6.16. Although not perfect, this modified output is close to the
desired result shown in Figure 6.12. The output spectrum is shown in
Figure 6.17. The harmonic components, although not zero, are largely
eliminated. The THD for the pre-distorted exposure was reduced to
4$, a factor of 5 improvement.
Significant improvement is possible by pre-distorting the desired
pattern. The non-linearity can be virtually eliminated by the use of
a high order polynomial or spline fit. Once linearized, the film can
be driven harder to take advantage of the entire dynamic range. This
provides increased efficiency and permits a greater dynamic range in
the input function to be recorded. Applied to continuous-tone CGH,
this allows the simultaneous optimization of efficiency and signal-to-
noise. Pre-distortion has no application to binary CGH because the

128
Figure 6.14 Output spectrum for a sinusoidal input.

129
Figure 6.15 Image and plot of the pre-distorted sinusoidal
grating used for an input.

Figure 6.16 Image and plot of the output transmission for pre¬
distorted sinusoidal input.

131
Figure 6.17 Output spectrum for a pre-distorted grating input

132
response :t.o a binary signal is inherently linear and no improvement is
possible with predistortion.
In display holography, non-linearity does not play an important
role. The harmonic terras are diffracted at broad angles and do not
significantly affect image quality. In fact, because of increased
efficiency, most display holograms are purposefully saturated. In a
hologram for use as a matched filter, the linearity is very important.
The harmonic terms contribute to noise and loss of efficiency. Even
though some efficiency is gained by driving the film into saturation,
light is diffracted to higher orders and this light is lost to the
system. In an optical correlator with limited light budget, this loss
may be intolerable. Such a system is the prime candidate for the
advantages offered by the CGH utilizing computer linearization.
Computer linearization is a vital step in optimizing the continuous-
tone CGH matched filter.

CHAPTER VII
SIMULATIONS
Hologram optimization has been described, based on such properties
as signal-to-noise ratio and efficiency. The improvements due to some
techniques are easily verified using analytical techniques. Other
optimization techniques are based on filter content and thus the
improvement is not determined until the actual filter response is
applied. The filter function is typically complicated and defies
analytical measurement. In those cases, it is possible to produce the
hologram physically and test its actual response in an optical
correlator. Though this truly represents the most realistic test
environment, it is not convenient to produce a new hologram for each
test case. In addition, with the many variables inherent in chemical
development of film, it is difficult to repeat the same experiment and
get the same result. With lengthy and careful efforts to standardize
each experiment, meaningful results may be possible. Such an effort
makes difficult the quick and exacting comparison necessary for
hologram optimization.
Computers have provided a compromise between the analytical
approach and the experimental approach. By simulating the analytical
effects in the hologram and submitting digitized images to the
simulation, a simulated experiment can take place. This simulation
allows as much realism as desired, according to the extent of the
simulation software. The advantages of digital simulation include the
133

134
ease with which an experiment can be run, the ability to vary
parameters, and repeatability. Assuming a sufficiently powerful
computer is available, the cost of these digital experiments is small
compared to their hardware implementation. A large number of computer
experiments can be performed in the same time it takes to produce and
process one optical hologram.
In the case of a computer-generated hologram, the digitized images
and the hologram are already on the computer. It is a simple matter
then to evaluate the hologram on the computer using simulation
techniques. This step would naturally occur after the computation of
the CGH but before physically writing the pattern to film. If the
holographic pattern does not perform as expected in the simulation, it
need not be produced until all problems are corrected. The computer
allows many experiments to be used to optimize the CGH for its
specific application and to predict the performance before investing
the time and expense of writing the pattern on film. This chapter
describes these simulation techniques and shows specific examples and
results for common CGH types.
Techniques for Simulating Matched Filters
Many of the digital processing techniques are described in Chapter
III, but the simulations described here go beyond the creation of the
hologram. The theory introduced in Chapter II shows that the
correlation of two images can be accomplished by inverse transforming
the product of the transforms of the two images. This approach is
used in an optical correlator because of the ease with which Fourier
transforms are accomplished using lenses. In each simulation, the
effects are limited to those possible in a Vander Lugt optical

135
correlator in which no active devices are used. That is,
transmittance values cannot exceed one. All pre-processing is
performed on the hologram, only to preserve the real-time capability
of the optical correlator.
Figure 7.1 shows a flow chart of a simulation of an ideal
correlation. In this case a hologram is not used to perform the
correlation. Rather, the image transforms are multiplied directly in
their complex form. Although this simulation could be used to
represent the effects associated with an on-axis hologram capable of
simultaneous amplitude and phase modulation, no hologram noise, film
noise, or dynamic range limitations are included. This ideal
correlation serves as a standard for latter comparison with off-axis
hologram-derived correlations. There is an interaction between the
pre-processing and the hologram that will affect the influence of the
pre-emphasis. However, the ideal correlation simulation is useful for
determining correlation properties of the images and pre-processing
techniques independent of the influence of the hologram.
In Figure 7.1, the digitized reference image is stored in a matrix
^ij. Likewise, the digitized test image is stored in a matrix gjj.
The correlation image will contain more points than each of the input
images. Specifically,1 the correlation image contains a number of
points in each direction equal to the sum of the number of points of
the two images in those direction.j In general, the two images are
equal in size and the correlation is twice the size of the images in
each direction. In order for the FFT routine to perform the
appropriate transformation resulting in a 2N by 2N image, the input
images need to be padded to create two images of size 2N by 2N.

136
Display
&
'Analysis'
Figure 7.1 Computer simulation of an ideal correlation.

137
The padding consists of placing the N by N image into a bed of zeros.
That is, the matrix values in the resultant image, f'ij, are all zero
except the N by N center portion which contains the original image.
This padded image, when Fourier transformed, provides a matrix which
is always smooth over two pixels in both directions. A standard
Cooley-Tukey Fast Fourier Transform (FFT) algorithm is used to provide
F^j, the Fourier transform of the reference image.
The normal correlation is performed by taking the product of Fij
and Gij point by point. This product is performed using real or
double precision real numbers and provides more than adequate dynamic
range. In addition to a normal correlation, the simulation software
includes a pre-processing option. This permits the reference image to
be modified using a frequency emphasis or phase-only filter. In the
frequency emphasis option, the frequency plane values are multiplied
by a real valued coefficient based on the emphasis desired. In the
experiments shown in this report, the frequency emphasis applied is a
gradient filter. That is, the weight of each frequency component is
equal to the radius of that component. Because the reference image
must account for the filter effect for both the reference and the test
image, the actual filter applied is the square of the desired filter
response. In this case the gradient squared is known as a Laplacian
filter or radius-squared weighting.
Another pre-processing technique possible in the ideal correlation
simulation is the phase-only filter. In this option, the magnitude at
each location is set to one while the angle of each complex value is
left intact. This is accomplished by dividing the real and imaginary
components by the magnitude. This normalizes the pattern such that

138
the transmission is one at every location except when the magnitude is
zero. When the magnitude is zero, both components are set to zero.
The Fourier transform of a test square is shown in Figure 7.2. The
same transform with high-frequency emphasized is shown in Figure 7.3.
Note in Figure 4.1 that the edges are sharply accentuated in the
modified image while broad areas are dark. Despite the modified
frequency, the image is still easily recognized. The Fourier
transform of a phase-only-filtered image has minimum dynamic range,
but the image is distorted by the extreme edge enhancement typical of
phase-only filtering. When all pre-emphasis is completed, the
reference and test patterns are normalized to possess continuous
values between 0 and 1 inclusive. Normalization is performed to
simulate the action of a transparency as would be used in an optical
correlator and allows the determination of light efficiency.
Once the reference transform is modified according to the desired
pre-processing technique, and the point by point product taken with
the test image transform, the result is transformed again. Note that
the inverse is not taken because the lenses which are simulated can
only take forward Fourier transforms. Recall that the only difference
between a forward and inverse transform is that the result will be
inverted and perverted (see equation 2.11). It is of no consequence
in this case that the correlation is upside-down. Figures 7.4 through
7.6 show the auto-correlation of a square with no pre-emphasis,
high-frequency emphasis, and phase-only filtering applied. Note that
the correlation spikes with pre-emphasis are considerably sharper.
This is to be expected as the correlation length is inversely related
to the high-frequency content of the images.

139
Figure 7.2 Fourier transform of a reference square.

MMOMM
Figure 7.3 Fourier transform of a square with high-
frequency emphasis.

Figure 7.4 Ideal auto-correlation of a square with no pre-emphasis.

142
Figure 7.5 Ideal auto-correlation of a square with high-frequency-
emphasis .

143
Ideal auto-correlation of a square using
phase-only filtering.
Figure 7.6

144
Once :the correlation image is obtained, an analysis program
evaluates the signal-to-noise ratio and the efficiency of the
correlation process. The signal-to-noise ratio is determined by the
average of the correlation peak with its nearest six neighbors divided
by the average of the entire correlation image.
Signal-to-Noise = Correlation Peak (7.1)
Correlation Plane Average
The efficiency is determined by dividing the energy in the correlation
peak by the energy in the input test image. This efficiency is always
less than or equal to 1 because the reference pattern is normalized
and can only attenuate the test image energy. In the case of the
phase-modulated reference, the efficiency can actually reach 1 if all
of the diffracted light reaches the correlation peak. In this case,
no attenuation occurs at each pixel but rather a phase shift occurs.
Total efficiency can be divided into two effects, medium
efficiency and correlation efficiency. The medium efficiency depends
on the attenuation by the reference pattern and can be computed by
dividing the total light out of the reference pattern by the input
energy.
Medium Efficiency = Energy Leaving Hologram (7.2)
Energy in Test Image
The medium efficiency goes to 1 for a phase-modulated hologram. The
correlation efficiency is the energy in the correlation spike divided
by the total energy output from the hologram. This efficiency term
depends on how much of the output light is diffracted to the
correlation spike rather than to other orders.

145
Correlation Efficiency = Energy in Correlation Peak (7.3)
Energy Leaving Hologram
The term Horner efficiency was originally applied to that which is here
called the medium efficiency.^ More recently, the Horner efficiency
has been re-defined to describe the total efficiency including the
correlation efficiency due to diffraction.^9 This new definition
seems to describe more accurately Horner's original intent that the
efficiency represent the entire correlation process including the
transparency and diffraction effects.
Total Efficiency = Energy in Correlation Peak (7.4)
Energy in Test Image
The signal-to-noise and efficiency for the ideal auto-correlation of a
square are tabulated in Table 7.1 for cases where no pre-processing,
high-frequency emphasis, and phase-only filtering are applied.
Simulation of a Continuous-Tone Hologram
The ideal correlation is useful for understanding the various
pre-processing effects. In fact, for the on-axis hologram this ideal
correlation model accurately predicts the actual results. However,
holographic materials cannot record complex values, so off-axis
techniques are required. The spatial modulation or other mapping from
complex to real valued functions has a pronounced effect on the action
of the reference filter. This step must be included to reflect the
action of the hologram in an optical correlator. Figure 7.7 shows a
block diagram of the simulation of an optical correlator using a
continuous-tone hologram. Although quite similar to the ideal

146
Table 7.1
No Pre-emphasis
High-Frequency
Emphasis
Phase-Only
Filtering
Signal-to-noise ratio and efficiency of an ideal
auto-correlation of a square.
Signal-to- Medium Total
Noise Ratio Efficiency Efficiency
661 44.9$ 44.7$
967 0.1$ 0.1$
1631 100.0$ 97.8$

147
Reference Image Test Image
Figure 7.7 Flow chart for the continuous-tone hologram
simulation.

148
correlation model, this model includes a step which creates the
computer-generated continuous-tone hologram.
This hologram, when used as the reference filter, provides a
correlation similar to an ideal correlation. However, the differences
can be quite pronounced. For example, the output not only includes
the correlation, but also the convolution and other terms. These
terms may, in some cases, overlap and cause degradation of the signal-
to-noise ratio. In all cases, light will be lost to the convolution
term and the on-axis term with a resulting loss of total efficiency.
Obviously, the hologram has a significant effect on the result of the
optical correlator and must be adequately modeled to obtain reasonable
predictions of correlator performance.
The hologram produces not only the correlation plane but also
other terms. As was discussed in Chapter III, the space-bandwidth
requirement for the hologram is dependent on the spatial carrier
frequency and the number of points in the reference image. Thus,
based on these factors, the reference image must be padded in a field
of zeros of the appropriate space-bandwidth. This requires greater
padding than is needed in the ideal correlation and additional
computing power for analyzing the same images.
In addition, there are modifications to the hologram which are
possible when produced via computer generation. These modifications,
discussed in the previous chapters, are modeled here to predict their
effect and usefulness. These hologram modifications include the use
of non-linear pre-distortion of the filter function to remove the
distorting film response. The pre-distortion of the filter function
would occur in the computer generation of the CGH. However, since the

149
film is-also modeled in this simulation, both the pre-distortion and
film distortion must be included. To represent the overall effect,
the pre-distortion must be the same as the one to be used in the
production of the CGH. This pre-distortion model needs only to
provide the accuracy desired in the CGH. However, the model of the
film distortion must be as accurate as possible to provide an accurate
simulation of the overall transfer function. Thus, while a third or
fifth order fit may suffice for the pre-distortion, the model for the
film distortion may require a higher order or a spline fit.
The type of modulation utilized in the hologram is also modeled.
Recall that the pre-emphasis or pha3e-only filtering of the
information used to create the hologram is independent of the
modulation technique incorporated in the physical hologram. That is,
the hologram may be produced as an amplitude hologram with constant
phase but variable transmission, or produced as a phase hologram with
constant amplitude but variable phase. The proper choice of hologram
is modeled in the simulation by mapping the hologram pattern to
transmission or phase. This modified amplitude or phase pattern is
used as the reference and is multiplied by the Fourier transform of
the test image. The following steps of the correlation and analysis
are identical with the ideal correlation simulation with the exception
that the correlation peak is no longer centered on-axis.
Figure 7.8 shows the continuous-tone hologram of a square,
produced by the simulation. The carrier frequency was maximized in
this case as half the sampling frequency to produce the greatest
separation of the various output terms. The Fourier transform shown
in Figure 7.2, a sine function, is spatially modulated by a sinusoidal

150
Figure 7.8 Continuous-tone CGH of a square.

151
term diffracting the correlation and convolution terms off-axis.
Figures 7.9 and 7.10 show the same hologram but with high-frequency
emphasis and phase-only filtering. Figure 7.11 shows the auto¬
correlation of a square using a continuous-tone hologram. Figures
7.12 and 7.13 show the same auto-correlation with high-frequency
emphasis and phase-only filtering. Table 7.2 lists the performance of
the continuous-tone hologram when the various pre-processing
techniques are applied.
Simulation of a Binary Hologram
The next case to be considered is the binary hologram. The binary
hologram differs considerably from the continuous-tone hologram in
that no gray-scale values are permitted. This requires that the
dynamic.range in the hologram be obtained through the use of
additional points. This additional space-bandwidth requirement
impacts the simulation and testing as well as the generation of the
CGH. Figure 7.1*1 shows a block diagram of the binary hologram
simulation. The structure is the same as the ideal and continuous-
tone correlation with only small changes. The pre-distortion and
distortion of the film is unnecessary for binary holograms since the
two points can always describe a straight line and thus are always
linear. It is meaningless to include the film response to the binary
signal. The programming of the binary hologram simulation differs
from the continuous-tone case in that the variables used to describe
the hologram pixel values are not continuous, but rather, assume only
integer values. When an amplitude hologram is selected, the
transmission assumes values of zero or one. When a phase hologram is

152
Figure 7.9 Continuous-tone CGH of a square with
high-frequency emphasis.

153
Figure 7.10 Continuous-tone CGH of a square with phase-only
filtering.

154
Figure 7.11 Auto-correlation of a square using a
continuous-tone CGH.

Figure 7.12 Auto-correlation of a square using a continuous-tone CGH
with high-frequency emphasis.

156
Figure 7.13 Auto-correlation of a square using a continuous-tone CGH
with phase-only filtering.

Table 7.2 Signal-to-noise ratio and efficiency for continuous-tone
CGH.
Signal-to- Medium Total
Noise Ratio Efficiency Efficiency
Absorption
Hologram
No Pre-emphasis
661
30.6%
2.56%
High-Frequency
Emphasis
902
88.2%
0.04?
Phase-Only
Filtering
1630
37.5%
5.87%
Phase
Hologram
No Pre-emphasis
661
100.0?
CO
•
High-Frequency
Emphasis
902
100.0%
1.70?
Phase-Only
Filtering
1630
100.0?
15.78?

158
Reference Image Test Image
Figure 7.14 Flow chart for the binary hologram simulation.

159
selected,; the transmission is set to one but the phase assumes values
of zero or pi.
The pre-processing and modulation options are the same as in the
previous cases but there is an additional choice in the type of binary
hologram. As was discussed in Chapter III, many mappings are possible
to convert the complex filter function to a real binary pattern.
Presently, Lohmann, Lee, and Allebach-Keegan (A-K) type holograms are
available to the CGH algorithm and the simulation. Figures 7.15,
7.16, and 7.17 show the A-K hologram of the square using no pre¬
processing, frequency emphasis and phase-only filtering. Figures 7.18
and 7.19 show the auto-correlation of the square using the A-K hologram
with frequency emphasis and phase-only filtering. Table 7.3 shows the
signal-to-noise and the efficiency for the auto-correlation of the
square using the A-K hologram.
An Example Using an SDF as a Reference
The auto-correlation of the square is theoretically interesting
and provides a common tool by which various techniques can be
compared. Additionally, the auto-correlation of a square is a problem
which can be solved analytically for many of the types of holograms,
lending credibility to the simulation results. However, the real
power of the simulations occur when they are applied to more
complicated imagery. Actual images with complicated shapes and
patterns are impossible to correlate analytically but rather must be
correlated by a computer. Figure 5.1 shows various images from a
training set which was used to create the Synthetic Discriminant
Function (SDF) shown in Figure 5.2. The SDF was created from 36 views
of the object rotated 10° between each view. The SDF should therefore

160
Figure 7.15 A-K binary hologram of a square

Figure 7.16
A-K binary hologram using high-frequenoy emphasis.

162
Figure 7.17 A-K binary hologram of a square with
phase-only filtering.

163
Figure 7.18 Auto-correlation of a square using an A-K binary-
hologram with high-frequency emphasis.

164
Figure 7.19 Auto-correlation of a square using an A-K
binary hologram with phase-only filtering.

165
Table 7.3 Signal-to-noise ratio and efficiency for an
of a square.
Signal-to- Medium
Noise Ratio Efficiency
Absorption
Hologram
No Pre-emphasis 66 15.6?
High-Frequency
Emphasis 287 42.5?
Phase-Only
Filtering 615 34.6?
Phase
Hologram
No Pre-emphasis 66
100.0?
High-Frequency
Emphasis 287 100.0?
Phase-Only
Filtering 615 100.0?
A-K hologram
Total
Efficiency
1.97?
9.17?
7.22?
7.88?
36.68?
28.87?

166
give reasonable cross-correlations with each of the 36 views. An A-K
hologram of the SDF using high-frequency emphasis is shown in Figure
7.20. The cross-correlation of the SDF with the 40° view is shown in
Figure 7.21. The signal-to-noise and efficiencies for several
representative cases are shown in Table 7.4.

167
Figure 7.20 A-K binary hologram of the SDF using
high-frequency emphasis.

168
Figure 7.21
Correlation of a test image at 30° and the SDF using an
A-K hologram with high-frequency emphasis.

169
Table 7.4 Signal-to-noise and efficiency of an A-K hologram of an SDF
correlating with members of the training set.
Signal-:to- Medium Total
Noise Ratio Efficiency Efficiency
Absorption
Hologram
No Pre-emphasis
30°
2.1
7.4$
0.03$
90°
2.1
8.1$
0.03$
130°
2.1
9.1$
0.04$
330°
High-Frequency
2.1
8.2$
0.04$
Emphasis
30°
7.2
34.0$
0.50$
90°
7.7
35.5$
0.52$
130°
8.0
37.6$
0.62$
330°
7.2
34.1$
0.51$
Phase-Only
Filtering
30°
13.4
27.7$
7.62$
90°
13.5
27.7$
6.85$
130°
14.9
27.2$
7.27$
330°
13.6
27.5$
6.42$
Phase
Hologram
No Pre-emphasis
30°
2.1
100.0$
0.13%
90°
2.1
100.0$
0.13%
130°
2.1
100.0$
0.16$
330°
2.1
100.0$
0.14$
High-Frequency
Emphasis
30°
7.2
100.0$
2.00$
90°
7.7
100.0$
2.11$
130°
8.0
100.0$
2.49$
330°
7.2
100.0$
2.02$
Phase-Only
Filtering
30°
13.4
100.0$
3.05$
90°
13.5
100.0$
2.74$
130°
14.9
100.0$
2.91$
330°
13.6
100.0$
2.57$

CHAPTER VIII
OPTICAL IMPLEMENTATION
As shown in the previous chapters, tremendous control over the
matched filter hologram is possible when produced using computer-
generation techniques. Figure 8.1 shows an interferometrically
produced matched filter demonstrating the typical saturation in the
dc and low frequency terms. The interferometric holograms include a
magnitude squared term which produces a wide dynamic range signal to
be recorded on the film along with the desired information. The non¬
linearity of the film limits the dynamic range of the information to
be recorded interferometrically. In the matched filter shown in
Figure 8.1, the useful, unsaturated information exists only in the
higher harmonics shown enlarged. When produced as a computer¬
generated hologram, the matched filter need not include the magnitude
squared term and permits a wider dynamic range to be recorded. The
digital representations of the CGH are easily tested, analyzed,
and modified to obtain the optimum pattern. To produce the
hologram, a writing device is needed to convert the digital
representation into a physical transparency for use in an optical
correlator.
Techniques for Optical Implementation
Certainly one of the major shortcomings of computer-generated
holograms has been the limited space-bandwidth product which could be
accommodated. This has been limited in part by the computational
170

Photo of an interferometrically produced optical matched
filter.
Figure 8.1

172
facilities required for encoding the holograms, but more importantly,
the plotting devices have been the major bottleneck. The typical
procedure for fabricating a computer generated hologram is to have the
digitized interference pattern, which has been calculated and encoded
by computer, drawn to a large scale by a computer-driven plotter. The
drawing is then reduced photographically onto high-resolution film to
the desired final size. Unfortunately, errors are introduced in the
plotting and photo-reduction processes. In addition, optical plotting
devices are limited in spatial resolution and space-bandwidth product,
typically to 10^ pixels. The resolution is limited by the number of
discreet points which can be placed on the paper and the accuracy of
the copy process. This printing technique is strictly binary and
requires binary mapping techniques as described in Chapter III.
To produce continuous-tone holograms, a gray-scale writing device
is needed. With such a device, a pattern can be produced containing
transmittance values ranging continuously from zero to one. One of
the earliest examples is the rotating drum scanner. Such scanners
have been used in the newspaper industry for years to send pictures
electronically over phone lines. In the rotating drum scanner, a
photographic film is wrapped about a drum rotating at a fixed speed.
As the drum rotates, an incoherent light source is focused to a spot
on the film and moves sideways along the length of the film. In this
fashion, the light source scans a helix along the drum, providing a
raster scan on the film. The light is intensity modulated according
to the pattern to be written to the film. These drum writers are
commonly available from a number of companies, along with interfaces
for most common computers.

173
The disadvantages of such a system include spatial accuracy and
film format limitations. The mechanical drum and light source drive
have inherent position noise which limit the ultimate accuracy. The
visible light source can, in general, be focused to a spot 10
micrometers in diameter. To produce a hologram on a glass plate or a
film format not compatible with the drum fixture, a second copying
process becomes necessary as with the plotters and printers. This
step is tedious and introduces further noise.
To avoid this copy process, the plate or film can be mounted on a
flat surface and the light source scanned in both horizontal and
vertical directions. This is usually accomplished using a fixed light
source and scanning mirrors. A computer controls the source intensity
and the mirror deflection. Such a system can quickly write onto
standard film backs, including glass plates. Such two dimensional
scanners are not currently available from commercial sources but have
been produced in several labs including that of Sing Lee at the
University of California in San Diego and Roland Anderson at the
University of Florida. These scanner writers produce position
accuracies similar to that possible with the drum writers but provide
greater writing speed and flexibility.
Another technique which is becoming popular is to use commercial
image display systems coupled with a camera. Systems such as the
DeAnza and Eyecom convert the digital information from a computer to
a raster-scan television image. This image, viewable on a normal
television monitor, is easy to see and manipulate with the computer.
The screen is then photographed with a camera mounted before the
cathode-ray tube (CRT). Several cameras, including one produced by

174
the Matrix Corporation, are specifically designed for this task. The
Matrix camera contains a flat field CRT, a lens system, and a number
of camera backs to accommodate most film formats, including glass
plates. The CRT exhibits excellent positional stability and gray
scale accuracy sufficient for 8-bit resolution. The Matrix camera,
shown in Figure 8.2, is versatile, convenient and fast, but is
currently limited to spot sizes of around 25 micrometers. This
accuracy could probably be made comparable with the drum and scanner
systems with modification to the camera optics. A more important
limitation to the video based systems is the space-bandwidth product.
The standard video format permits only computer images 512 by 512
points or smaller. This problem is overcome by a new digital Matrix
camera capable of 4096 by 4096 points. However, this larger format
precludes the use of common and convenient television equipment to
view the entire image.
Another device, which combines the CRT imaging system with a
scanning capability, is the Aerodyne Holowriter shown in Figure 8.3.
This device, created for the sole purpose of producing holograms by
Caulfield et al^l, uses a CRT imaged onto film to produce an image as
in the matrix camera. However, the Holowriter translates the
photographic plate such that many CRT fields can be placed together,
side-by-side. In this way, the convenience of CRT imaging is combined
with a capability to produce holograms of enormous space-bandwidth-
products by interweaving frames. The Caulfield system is capable of
10 micrometer resolution with 6-bit amplitude accuracy. The system is
also capable of blurring pixels together in order to gain better
amplitude resolution at the expense of position resolution.

175
produced by the Matrix Corp
Figure
8.2 Cathode-ray
tube and camera

176
Figure 8.3 Cathode-ray tube imaged onto a translation table produced
by the Aerodyne Corp.

177
Each of the systems thus described is capable of continuous-tone
response and uses visible sources to expose photographic films or
plates. The use of visible sources causes a basic limitation in the
ultimate spot size. A diffraction-limited system might be capable of
1 micrometer resolution at visible wavelengths but, as is seen in each
of these systems, 10 micrometers or larger spot sizes are typical. To
produce CGHs of finer detail, shorter wavelengths are necessary. Some
improvement is obtained by writing with ultraviolet light such as in
the GCA Corp. 4600 pattern generator, which uses an excimer laser.
However, for significant improvement an electron-beam is used.
The electron-beam writing system pictured in Figure 8.4 was
produced by Honeywell for large scale integrated circuits.'67 This
fine structure capability lends itself readily to the production of
high resolution holograms. This system is capable of 0.5 micrometer
resolution over several millimeters of work space. The primary
advantage of the E-beam system is the fine detail size. With 0.5
micrometer resolution, high spatial frequencies are possible, thus
providing large angular outputs from an optical correlator. This
permits the incorporation of high space-bandwidth products in the
hologram while minimizing the size of the optical correlator. As with
each of the other systems, phase modulation holograms can be produced
with the Honeywell E-beam system by etching various depths into the
glass. At this time, the E-beam device is strictly binary. When the
resist is sufficiently exposed, the etchant removes the metal
entirely. Thus, to maintain a large dynamic range, spatial resolution
may need to be discarded in exchange for amplitude resolution using
the techniques described in Chapter III.

178
FOCUSED
ELECTRON BEAM
HOLOGRAM
SUBSTRATE
ELECTRON SOURCE
DEMAGNIFICATION LENS
DEFLECTION COILS
COMPUTER CONTROLLED X-Y TABLE
Figure 8.4 Electron-beam writing system at Honeywell Inc.

179
The E-beam systems are still too expensive for most labs to own.
Presently, Honeywell, General Dynamics, MIT, the Office of Naval
Research and several integrated circuit companies in California are
producing E-beam holograms for CGH researchers. Each of these systems
caters primarily to the integrated circuit industry, and hologram
producers must wait in line. Turn-around times of over a month are
typical, with cost per hologram exceeding $1000. Obviously, the E-beam
holograms, while superior in their spatial resolution, must be
supplemented with in-house holograms of lesser quality until research
and testing indicate that a hologram is ready for production.
Examples of CGH Matched Filters
The following section does not attempt to quantitatively verify
the simulations. That work is needed and is continuing. However, the
figures shown here indicate that indeed the appropriate patterns can
be placed on film or glass plates, and that their reconstructions
verify qualitatively results predicted by the appropriate simulations.
In each case, the holograms shown here were produced using the
Allebach-Keegan (A-K) algorithm and written to chrome-on-glass plates
using the Honeywell E-beam writer. These CGH matched filters are
amplitude-modulated holograms and do not provide the high efficiency
possible with phase-modulation relief holograms. However, the
holograms do include examples of various preprocessing techniques,
namely frequency emphasis and phase-only filtering.
Figure 8.5 shows magnified views of one of the E-beara written
holograms indicating the fine detail possible. Figure 8.6 contains
A-K CGH matched filters using no pre-processing, frequency emphasis
and phase-only filtering. These photographs are taken of the actual

180
Figure 8.5
Magnified views of a binary hologram produced on the
Honeywell E-beam writer.

181
(b)
(c)
Figure 8.6 A-K CGH matched filters, using a square as a reference,
produced on the Honeywell E-beam writer. The holograms shown measure
1 milli meter along a side and include a) no pre-processing, b) high-
frequency emphasis, and c) phase-only filtering.

182
holograms produced from the patterns shown in Chapter VII. Figures 8.7
through 8.9 show the reconstruction from each of the holograms in
Figure 8.6. In each case, the hologram was placed in on-axis
collimated light to produce the pattern in transmission. Figure 8.10
shows holograms of the letters "AFATL" using high-frequency emphasis
and phase-only filtering. Figure 8.11 and 8.12 show the appropriate
reconstructions of those holograms. The final hologram in Figure 8.13
is of a A-K CGH matched filter of an SDF using phase-only filtering.
The reconstruction is shown in Figure 8.14 and should be compared to
the original pattern shown in Figure 5.2.

183
Figure 8.7 Reconstruction from an A-K CGH matched filter of a square
using no pre-emphasis.

184
Figure 8.8 Reconstruction from an A-K CGH matched filter of a square
using high-frequency emphasis.

185
Figure 8.9 Reconstruction from an A-K CGH matched filter of a square
using phase-only filtering.

186
Figure 8.10 A-K CGH matched filter of the letters."AFATL" using
a) high-frequency emphasis and b) phase-only filtering.

187
Figure 8.11 Reconstruction from an A-K CGH matched filter of the
letters "AFATL" using high-frequency emphasis.

188
Figure 8.12 Reconstruction from an A-K CGH matched filter of the
letters "AFATL" using phase-only filtering.

189
Figure 8.13 A-K CGH matched filter of the SDF shown in Figure 5.1

190
Figure 8.14 Reconstruction of s.n
A-K CGH matched filter of an SDF.

CHAPTER IX
SUMMARY
This dissertation presents techniques for the use of computer¬
generated holograms for matched filtering. Due to the enormity of the
information presented, an executive summary is provided here.
Chapters I and II review the appropriate background to introduce the
reader to the power of optical matched filtering. This power is
particularly useful when a fixed detection algorithm, such as Fourier
transformation or correlation, is to be used consistently at an
extremely high rate. The parallelism of the optical processor
provides high speed processing of large SBP images. Thus, as input
arrays provide large images at high speeds, optical correlators will
serve as efficient processors to perform specific tasks.
Chapter III described the recording of the reference signal on a
hologram. The continuous-tone hologram closely matches an actual
interferometric hologram. This hologram can be improved by the
incorporation of equations 3.4 and 3.5. This reduces the dynamic
range and yields a recording containing more information. The binary
techniques of the B-L, Lee, Haskell, and A-K holograms are presented
to indicate the current evolution of binary holograms. The optimum
combination appears to be a combination of the GBCGH and the A-K. By
increasing the intra-cell sampling to an odd valued N, as in the
GBCGH, the hologram can represent large dynamic ranges. In addition,
191

192
the hologram should be sampled at each pixel, as in the A-K, to reduce
or eliminate false images.
Chapter III also describes the SBP required of the CGH to perform
the optical correlation. The SBP drives the size, complexity and
expense of the CGH. When used as a matched filter, the requirements
are more stringent than when used for reconstruction holography.
However, when the techniques presented at the end of the chapter are
incorporated, the SBP requirements can be minimized.
Chapter IV describes preprocessing techniques useful in optimizing
the CGH matched filter. The high-frequency emphasis and phase-only
filtering increase the discrimination against false targets but
sensitize the filter to scale and rotation. The phase-only filter,
although easy to implement, is not unique in its ability to improve
signal-to-noise and efficiency. Rather, it is a form of high
frequency emphasis which may or may not be optimum. It does minimize
the dynamic range but may not be the appropriate choice for a specific
application. The rules describing the application of high frequency
emphasis should be applied to each case. The exception is when an on-
axis hologram is to be phase modulated so as to require the reference
to contain no amplitude information.
The phase-modulation process applied to the physical hologram
should not be confused with the phase-only filtering step which is
applied to the reference information. That is, a phase-modulated
hologram can utilize the normal reference information, frequency
emphasis or phase-only filtering. In general, the phase modulation
provides considerably higher efficiency with an accompanying loss of
signal-to-noise ratio due to the non-linearity. When using binary

193
holograms, there is no non-linearity and phase modulation should
always be incorporated. A binary, phase-modulated, on-axis hologram
with the dc term eliminated, provides the least SBP and highest
efficiency. For this reason, it is recommended for future work in
programmable holograms. However, the bi-phase conversion required for
this hologram may not provide sufficient signal-to-noise ratio for all
applications.
Chapter V describes two approaches for handling scale, rotation
and other deformations. Presently, the Mellin-Fourier approach is
hard to implement. The SDF approach is easy to implement but
typically yields low (<10) signal-to-noise ratios.
Chapter VI describes the techniques for minimizing the THD of
holographic films. This process is important for optimizing
continuous-tone holograms. The inability to perform this
linearization is a severe limitation of conventional matched filter
holograms. This chapter outlines the steps necessary for proper
modeling of the film. Using a polynomial model and correcting for the
non-linearity, a 100 times improvement was realized using 8E75
holographic plates. It is probable that further improvement could be
obtained using a spline model. Another improvement is possible by
whitening the modulation transfer function (MTF). The MTF, like the
film linearity, effects the film response. It can be measured and
whitened to prevent undesired frequency emphasis. The effect on
spatially modulated holograms is small due to the small frequency
variations about the carrier and has been ignored in this
dissertation. Future work may wish to include this effect by
performing a second frequency emphasis upon the encoded CGH pattern.

194
The effect would be more pronounced with an on-axis hologram where a
wide spread of spatial frequencies is possible. However, on-axis
holograms for matched filtering must be phase modulated, complicating
the correction to the MTF.
Chapter VII describes the simulations used to demonstrate the
various techniques. The ideal correlation is performed using real
valued variables with no hologram encoding. This provides an
essentially infinite dynamic range correlation with no CGH encoding
noise and serves as a standard by which holographic techniques can
be compared.
The continuous-tone simulation provides the best approximation to
the performance of a conventional hologram. By eliminating the
magnitude terms, the dynamic range is reduced and the hologram is
greatly improved. The binary simulation shows that equivalent
performance is possible using binary techniques. The example of an
SDF matched filter demonstrated a practical example. This example
shows that although correlations are obtained, the signal-to-noise and
efficiencies are low for SDF filters. For some applications, this may
preclude the use of SDFs and require sequential single filter steps.
Chapter VIII demonstrated the implementation of the holograms and
shows reconstruction of several examples.
Conclusions
By accounting for the changes in SBP, CGH techniques proposed by
Lohmann, Lee, Allebach and others for reconstruction holography were
used to create matched filters. These holograms, when used for
matched filtering, require a larger space-bandwidth product to
separate the correlation terms. Modifications to the conventional

195
matched'filter produced in this fashion improve the signal-to-noise
and efficiency over that possible with conventional holography. Pre¬
distortion of the holographic pattern for minimizing the total
harmonic distortion (THD) created by film non-linearity provided a
factor of five improvement. Each of these effects, along with various
CGH types, were encoded in powerful computer simulations to test and
demonstrate the benefits of the optimization techniques. The
simulation showed that in all cases, high-frequency or phase-only
filtering of the reference transform provided improved signal-to-noise
ratios and efficiencies. Although phase-only filtering provided the
lowest dynamic range, it did not in general provide results superior
in overall performance. A-K CGH matched filters produced using E-beam
lithography performed as predicted by the simulations.
Recommendations
This dissertation shows that CGH matched filters can be produced
and perform satisfactorily when pre-processing is utilized. However,
further study is recommended in several areas. The A-K hologram can
be improved by utilizing larger cell sizes and by making the cell
dimension, N, an odd number as in the GBCGH. This would provide
better representation of the complex value and render more pixels
open, improving efficiency. Techniques for real-time dc elimination
could permit programmable on-axis holograms, and thus, sequential
single filters.
A better algorithm is needed to choose the appropriate frequency
weighting for a given probability of detection and false-alarm rate.
The non-linear analysis would benefit from the incorporation of a
spline model for the film. Corrections to the MTF should be modeled

196
and included in the simulation to verify the belief that its effect is
negligible. More simulation runs on a wide variety of images would
better determine the interaction of image parameters with hologram
performance.
Optical processors will provide low cost and high speed pattern
recognition for specific tasks.68 It must be noted, however, that
Vander Lugt-type optical pattern recognizers are very limited in their
application. The correlation process is hard-wired into the optical
computer and the pattern recognition algorithm cannot be modified.
This is in contrast to the slower digital processors which are
software-controlled to perform a wide variety of image functions. The
Vander Lugt correlator will fill a near-term need for a cheap
processor of moderate performance in a controlled environment. This
works well for machine vision application where the in-class and out-
of-class patterns are well known. Performance of such a hard-wired
system drops when the target is in a hostile environment.
The next generation of optical processors must be able to adapt
and learn. Such a device must support a wide variety of algorithms at
a speed compatible with the application. Future research in this area
should address the need for optical processors which not only perform
Fourier transforms and correlation, but many other functions as well.
This could be accomplished using real-time spatial light modulators as
holographic elements. The hologram might perform as a holographic
lens for Fourier transforming or it might perform a coordinate
transform depending on the command of a digital, high level, image
understanding algorithm. Such a flexible processor should prove
successful for the next generation of pattern or target recognizers.

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BIOGRAPHICAL SKETCH
Steven Frank Butler was born on December 18, 1955, in Valparaiso,
Florida. He attended Choctawhatchee High School in Fort Walton Beach,
Florida, where he took great interest in the growing field of lasers.
In 1973 he entered the University of Florida on a mathematics
scholarship but later changed his major to physics to pursue optics.
He graduated with honors in 1976 and entered the University of Florida
graduate school in electrical engineering where he studied
communication theory. He received his M.S. in 1978 after completing
his thesis on "Atmospheric Sounding Using a Frequency Modulated
Radar." During these years at the University of Florida, he was an
active member of the Amateur Radio Club, Tau Beta Pi, Sigma Pi Sigma,
Eta Kappa Nu, and student government.
Upon graduation, Mr. Butler became involved with infrared
measurements for the Air Force at Eglin AFB, Florida. Until 1982, he
was involved in the design and use of infrared sensors and the
analysis of infrared signatures. In 1982, he transferred to the
Armament Laboratory Optics Research Group at Eglin to pursue optical
processing research for target recognition. Having taken classes
during his work at Eglin, Mr. Butler went back to the University of
Florida campus from 1983 to 1984 for full-time study in aerospace
engineering. Since his return to Eglin, Mr. Butler has led a team of
engineers in the development of holographic matched filters for use in
guided missiles.
201

202
Mr. Butler is active in amateur radio and participates in many
sports including water skiing, sailing and tennis. He attends Trinity
Methodist Church where he serves on the administrative board.

I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is
fully adequate, in scope and quality, as a dissertation for the Doctor
of Philosophy.
Roland C. Anderson, Chairman
Professor of Engineering Sciences
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is
fully adequate, in scope and quality, as a dissertation for the Doctor
of Philosophy.
Charles E. Taylor '
Professor of Engineering Sciences
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is
fully adequate, in scope and quality, as a dissertation for the Doctor
of Philosophy.
^//h 'e( M tis2-*re]/
Ulrich H. Kurzweg v
Professor of Engineering Sciences
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is
fully adequate, in scope and quality, as a dissertation for the Doctor
of Philosophy.
lyC^rrxJ-^ j/A-
es E. Milton
rofessor of Engineering Sciences

I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is
fully adequate, in scope and quality, as a dissertation for the Doctor
of Philosophy.
J‘ lift ( C c-l W'(-
Stanley S. Ballard
Distinquished Service Professor of Physics
This dissertation was submitted to the Graduate Faculty of the College
of Engineering and to the Graduate School, and was accepted as partial
fulfillment of the requirements for the degree of Doctor of Philosophy.
December, 1985
Dean, Graduate School

Page 2 of 2
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