• TABLE OF CONTENTS
HIDE
 Title Page
 Acknowledgement
 Table of Contents
 List of Tables
 List of Figures
 Abstract
 Introduction
 Background
 Computer-generated holograms...
 Optimization of optical matched...
 Pattern recognition techniques
 Matched filter linearity
 Simulations
 Optical implementation
 Summary
 Bibliography
 Biographical sketch
 Copyright






Title: Computer-generated holographic matched filters
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Title: Computer-generated holographic matched filters
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Language: English
Creator: Butler, Steven Frank
Publisher: Steven Frank Butler
Publication Date: 1985
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Table of Contents
    Title Page
        Page i
    Acknowledgement
        Page ii
    Table of Contents
        Page iii
        Page iv
    List of Tables
        Page v
    List of Figures
        Page vi
        Page vii
        Page viii
        Page ix
    Abstract
        Page x
    Introduction
        Page 1
        Page 2
        Page 3
        Page 4
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    Background
        Page 9
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    Computer-generated holograms (cgh)
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    Optimization of optical matched filters
        Page 63
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    Pattern recognition techniques
        Page 84
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    Matched filter linearity
        Page 94
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    Simulations
        Page 133
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    Optical implementation
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    Summary
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    Bibliography
        Page 197
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    Biographical sketch
        Page 201
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    Copyright
        Copyright
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




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