Mellin-Fourier correlation

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Mellin-Fourier correlation
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vi, 70 leaves : ill. ; 28 cm.
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Friday, Edward Carney
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Pattern recognition systems   ( lcsh )
Optical pattern recognition   ( lcsh )
Fourier transform optics   ( lcsh )
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Thesis:
Thesis (Ph. D.)--University of Florida, 1985.
Bibliography:
Includes bibliographical references (leaf 69).
Statement of Responsibility:
by Edward Carney Friday.
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Typescript.
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Vita.

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University of Florida
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Full Text










MELLIN-FOURIER CORRELATION


By

EDWARD CARNEY FRIDAY

























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


































Copyright 1985

by


Edward Carney Friday













ACKNOWLEDGMENTS

The author wishes to thank Dr. Henry Register for his

encouragement to pursue graduate studies at the University

of Florida. Dr. Roland Anderson has provided counseling and

essential critical review during the years of experimental

effort leading to publication. The Air Force Office of

Scientific Research provided initial funding for this

research through the basic research program conducted by the

Air Force Armament Laboratory at Eglin Air Force Base,

Florida. The final stages of the research would have been

impossible without the computing and word processing

resources provided by E G & G Special Projects,

Incorporated.


iii













TABLE OF CONTENTS



ACKNOWLEDGMENTS ...... ... ... ... ...... ..

ABSTRACT ..... ........... ........ .. ... ..

CHAPTERS


I INTRODUCTION .................. ............

Limitations of Classical Techniques ......
The Mellin-Fourier Correlation Process ....
Results Obtained by Other Workers .........
Areas Investigated in this Work ...........

II BACKGROUND ................................

Experimental Techniques ...................
Theory For Extending the Normal
Scale Invariance ..........................

III APPLYING OPTIMUM SAMPLING TECHNIQUES ......

Examples of Optimum Sampling ..............
Testing Theoretical Predictions ...........

IV ADDITIONAL EXPERIMENTS ....................

Higher Space Bandwidth ....................
Adding Noise .............. ...............


V CONCLUSIONS ............................ .

REFERENCES .........................................

BIOGRAPHICAL SKETCH ............ ... ............ ..


_l


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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


MELLIN-FOURIER CORRELATION

By

Edward Carney Friday

May 1985


Chairman: Roland C. Anderson
Major Department: Engineering Sciences


This dissertation evaluates the success of a pattern

recognition system which attempts to recognize an object

even though the object differs in size and orientation from

the object used as a reference. The criteria for

recognition were the peak value and shape of the cross

correlation function calculated in two dimensions. The

cross correlation was performed using Fourier transform

techniques to judge the utility of implementing the system

using many standard optical components. Size, or scale,

invariance was achieved using a logarithmic sampling

technique (the Mellin transform), and variations on

traditional sampling methods were shown to extend the scale







invariance predicted by previous researchers. The utility of

the sampling techniques developed was proven using images

acquired from infrared sensors carried aboard helicopters.

Limits of practical scale invariance were explored and

system design approaches were suggested from the results of

numerical experiments carried out in computer simulations.












CHAPTER I
INTRODUCTION

Limitations of Classical Techniques

Pattern recognition is often performed in autonomous

hardware by comparing imagery from an on-board sensor with

an ideal or reference image stored in a memory. Military

systems frequently are required to recognize patterns even

though they are slightly different from the reference for

which the system was designed. Signal processing techniques

to overcome certain types of distortions in sensed imagery

have been described in the literature (1). The most common

distortions are scale and orientation changes. Classical

matched filters are unreliable unless the reference and

sensed objects are very nearly the same. When the peak

signal to RMS noise ratio (SNR) of the correlation function

is used as a figure of merit for the holographic matched

filter technique, losses in SNR of 27 db can occur with as

little as 2% scale difference and 3.5 degrees of orientation

difference (2).


The Mellin-Fourier Correlation Process

Distortions such as scale changes and rotations of the

pattern of the sensed image may be overcome by using the

proper choice of geometrical transformations. Since most





2

scale invariant transformations are not invariant to

translation, the Fourier transform magnitude is sometimes

used as the first step in a pattern matching scheme to

achieve shift invariance before the scale invariant

geometrical transformation is performed. The scale

invariant transformation is achieved in practice by

resampling the original image. This resampling process

causes spatial distortion, e.g. warping, and is therefore

sometimes called space variant. Figure 1 shows the process

used to achieve Mellin-Fourier (scale and shift invariant)

correlation between a reference image and a sensed image.

The first step is to calculate the magnitude of the Fourier

transform of the reference and sensed images. The Mellin

transform is not invariant to object translation, so pattern

recognition using Mellin transforms is usually done using

the magnitude of the Fourier transform of an object. This

technique uses the shift invariance of the Fourier transform

to provide a centered image for later exponential

resampling. The magnitude of the Fourier transform is

equivalent to the diffraction pattern obtained when an image

is used as the aperture of an optical system. This pattern

contains no phase, i.e. position, information, and as a

result is invariant to the position of the pattern within

the system field of view.

The next step in the Mellin-Fourier correlation process

is to perform the geometrical transformation necessary to

































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create an image which has a useful relationship to scale

changes in the input image. The geometric transformation

used in this research maps the Fourier transform polar plane

into a rectangular plane. The orthogonal axes in the new

rectangular coordinate system are defined as the logarithm

of radial distance and the polar angle in the original

Fourier transform plane. These axes are denoted as In (R)

and 0, respectively. The quantity R is the radial distance

from the zero frequency, or d.c. term. The natural

logarithm of R is used in the transformation because a

multiplicative scale change occurring in the Fourier

transform plane causes a linear shift in the In (R) axis.

The quantity 9 is the angle measured counterclockwise from

the horizontal axis. When a Fourier transform image is

mapped using this geometric transformation, a new image

plane is created with each location in the In (R),9 plane

corresponding to one or more locations in the Fourier

transform plane. Figure 2 shows what happens in the ln(R),@

plane when a scale change occurs in the original image. The

top image in Figure 2 is the ln(R),O image generated by

a 10 x 10 reference square. The In(R),6 images below it are

sensed images with various scale changes present. As the

scale of the sensed image changes, the pattern in the

ln(R),0 image simply translates. Features in the ln(R),0










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image do not change size as they would in the Fourier

transform image. It is this scale-to-translation property

of the In(R),9 mapping that is exploited in the scale

invariant Mellin-Fourier correlator.

The final step in Mellin-Fourier correlation is a

standard cross correlation calculation performed with the

In(R),9 reference and sensed images as inputs. The

correlator in this work is implemented with a computer

simulation, and cross correlation is performed using

transform techniques rather than by direct calculation.

This technique requires fewer calculations than cross

correlation performed directly. The output of the

Mellin-Fourier correlator is a cross correlation image with

a peak value whose X coordinate reveals the amount of scale

change between the reference and sensed images. The Y

coordinate of the cross correlation peak gives the amount of

rotation difference between the reference and sensed images.

This work is concerned with considerations in sampling the

ln(R),G image and the scale invariance expected of a system

using the sequence of operations described in Figure 1. In

addition, the effects of noise added to the reference and

sensed images on the shape and amplitude of the cross

correlation image are determined.









Results Obtained by Other Workers

Cassasent and Psaltis (3) have shown how space variant

processors can be used to perform scale invariant

correlation. With their technique, scale and rotation

changes in a sensed pattern were converted into a

translation by use of the logarithmic coordinate

transformation described above. Standard shift invariant

correlation methods, i.e., Fourier transform methods, were

then used for pattern recognition of the scaled function.

An analysis performed by Cassasent and Psaltis showed

that the spatial distortion introduced by the logarithmic

transformation resulted in increased space bandwidth

requirements. The logarithmic coordinate transformation is

equivalent to resampling an image at exponentially spaced

intervals, and the increase in space bandwidth (number of

samples) is caused by the oversampling which takes place at

small values of the input coordinate. Their analysis

considered the increased sampling requirements imposed by a

function which moved partially out of the sampled domain

when scaled by a factor A. This loss of information gave

rise to inaccuracies. Space bandwidth requirements were

said to increase by a factor of 5.3 when 100% scale change

(A=2) was accommodated (3).









Analysis done by Anderson and Callary (4) involved a

function which remained entirely within the sampled domain

when scaled by a factor A. In this case, there were no

inaccuracies introduced because the function was defined to

be zero outside the domain of interest, and no information

about the function was lost when scale changes occurred.

They showed that the increase in space bandwidth could be

minimized by careful choice of the constants used in the

logarithmic coordinate transformation. In their scheme,

oversampling at small values of radial distance was governed

by meeting the Nyquist criterion at the widest sampling

interval, and by oversampling only to the extent necessary

to accommodate the predicted scale change. Using this

scheme, space bandwidth requirements increased by a factor

of only 2.7 when accommodating a 100% scale change.

Variations of this geometrical transformation scheme

have been reported by other researchers (5). In those

studies, the location of the correlation peak in the

Mellin-Fourier correlation plane was used to identify the

actual scale change and rotation of the sensed image.

Then,a new reference image was synthesized at the proper

scale and rotation angle for application of traditional

correlation schemes.









Areas Investigated in this Work

This study evaluates the success of a pattern

recognition system which attempts to recognize objects in a

sensed image which differ in scale and rotation from the

reference image. The intent is to establish the limits of

use for the Mellin-Fourier correlation techniques rather

than to design a specific algorithm for pattern recognition.

Experience has shown that such algorithms are very dependent

upon a system application.

The approach used here was to calculate sampling

requirements based on a desired range of scale invariance,

and then to use a computer simulation of the correlation

process to calculate system performance. The scale of the

sensed image was then allowed to exceed the design limits of

the system to find the effect on the correlation peak. The

correlation coefficient and the shape of the correlation

peak were used as figures of merit to judge the success of

each trial.

This study applies the theory of Mellin-Fourier

correlation to imagery collected by a sensor carried aboard

a helicopter. The correlator modeled here is intended to

represent an optical implementation of a scale invariant






10

system. Such an optical signal processing system can be

modeled realistically by numerical experiments. The limited

dynamic range of optical systems is simulated by scaling all

imagery amplitudes between 0 and 255 each time a

transformation is performed. The response to high noise

levels is used to evaluate the immunity to background

clutter. Trials conducted with real sensed imagery provide

a realistic test for a scale invariant pattern recognition

system intended for tactical applications. These trials

show that the scale and rotation invariance predicted by

theory can be applied in practice. The space bandwidth of

the correlator is varied, and the effects on the correlation

images suggest design approaches for equipment using

Mellin-Fourier principles.

Experimental techniques for both zooming and

downsampling the reference image are developed to be able to

provide a sensed image of exactly the proper scale and

orientation for Mellin-Fourier correlation. In this way,

the Mellin-Fourier correlator is evaluated with respect to

small scale differences in the input. The effect of

different amplitude distributions of added noise is noted on

the peak value and shape of the correlation image.









The Mellin-Fourier image correlation process is

two-dimensional in nature, but insight into the In(R),Q

mapping operation and the correlation energy distribution is

gained when results are presented in the form of

one-dimensional plots. In this research, photographs taken

from the television monitor of an image processing computer

are used to portray the nature of the entire Mellin-Fourier

process, but one-dimensional plots are used to show details

of the energy distribution in the correlation image. Images

are characterized by a plot of the intensity values along a

line passing through the peak value of the image.

Additional experimental techniques, such as range

adaptive bandpass, are applied to extend the normal scale

invariance of the Mellin-Fourier type correlator. Anderson

and Callary (4) calculated only the value of the correlation

peak as a figure of merit for the Mellin-Fourier correlator.

This investigation calculates the entire correlation image

plane. Using this technique, the advantages of spatial

filtering are made clear.













CHAPTER II
BACKGROUND

In the course of this investigation, it was recognized

that use of the Mellin-Fourier correlation principle would

require careful application of the Discrete Fourier

Transform (DFT). The paragraphs that follow discuss several

aspects of the DFT and show how to properly apply it to

evaluate a pattern recognition system.



Experimental Techniques

Methods Used to Calculate Accurate DFTs

As mentioned earlier, in a pattern recognition system

which uses transform techniques to achieve invariance to

translation, scale, and rotation, the Fourier transform is

usually used initially to provide translation invariance.

Actual pattern recognition is performed on the magnitude of

the Fourier transform of an object rather than the object

itself. When the Fourier transform is computed, the

spectral resolution achieved may be insufficient for

reliable pattern recognition even though the Nyquist

criteria have been met. The two issues of spectral

resolution and Nyquist sampling must be considered

separately.





13

In a DFT algorithm, the input record consists of N data

points, each point representing T units, producing a record

length NT units long. The output of the DFT algorithm will

also contain N data points, each point representing a

spatial frequency interval

delta f = 1/NT

The discrete frequency spectrum is repeated periodically at

integer multiples of the sampling frequency fs = 1/T, with

the magnitude of the spectrum being symmetric about the

folding frequency

f = (N/2) delta f = 1/2T

This folding frequency is a direct result of sampling the

input signal, and gives rise to the Nyquist requirement of

sampling at twice the highest spatial frequency present in

the input signal. If sampling is done at less than the

Nyquist rate, an effect called aliasing causes inaccuracies

in the computed spectrum.

The DFT of an input record may be calculated with a

resolution of delta f even though the Nyquist condition has

not been met. However, the values computed for the spectrum

will not be accurate because of aliasing, and any further

use for pattern recognition may produce misleading results.






14

Alternatively, the number of data points, N, may be so small

that the widely spaced samples do not completely describe

the shape of the spectral curve, even though the Nyquist

condition has been met and the samples are numerically

accurate.

Thus, two considerations apply to all invariant pattern

recognition schemes which use digital transform techniques:

1. Spectral resolution must be sufficient to show

identifying features of the pattern.

2. Nyquist conditions must be met to assure

numerical accuracy of spectral samples.



Windowing and Truncation

Practical application of the principles above involves

data windowing and truncation. These two effects are

manifest in the spatial domain and spatial frequency domain,

respectively. Windowing of input data must be performed to

minimize the effects on the Fourier transform of a finite

input record length. The DFT produces an output that is

always a combination of the actual data spectrum and the

window spectrum (6). The window is usually a rectangular

function of unity amplitude which is multiplied by the data.

The DFT results in the convolution of the spectrum of the

window with the spectrum from the pattern of interest.





15

the DFT is periodic, Nyquist conditions require that the

highest spatial frequency present in a record of N samples

be N/2 cycles per frame. The abrupt transitions at the

edges of a rectangular window cause very high spatial

frequencies to occur in the spectrum. These spectral

components can be significant at N/2 cycles and can cause

aliasing errors to occur. One way to reduce the components

at N/2 cycles is to make the edge transitions less abrupt by

using one of the well known smoothing windows such as the

Hamming, Hanning, or cosine functions. The need to

condition all data with a window before computing the DFT

stems directly from the Nyquist sampling condition. Proper

windowing assures numerical accuracy of all Fourier

transform samples computed using the DFT (6). The window

used in this research tapered all images to 1/2 the original

value at each edge, reducing the frequency components at the

edge of the Fourier transform plane.

In a way completely analogous to windowing, truncation

of Fourier transform records avoids errors in computing the

cross correlation of two functions. In one dimension, the

cross correlation of two records of length N results in 2N

valid data points. If transform techniques are used to

perform the cross correlation, each Fourier transform

computed must be truncated to one-half its original size

before multiplication and inverse transforming. Otherwise,





16

an effect called cyclic correlation causes errors in the

cross correlation plane (7). This effect is similar to

aliasing, but occurs in the cross correlation domain instead

of the spatial frequency domain.



Considerations for Geometric Transformations

The spatial frequency characteristics of the Fourier

transform magnitude of the input pattern must be considered

if system space bandwidth is to be used most effectively.

The symmetry of the Fourier transform magnitude may be used

to reduce space bandwidth requirements by taking all samples

from the upper half plane of the diffraction pattern. The

upper half plane of the diffraction pattern contains all of

the recoverable shape information about an object, even

though position information is lost. All available space

bandwidth (sampling capability) should be devoted to this

region when the geometric transformation is done. Only a

portion of the Fourier transform plane carries relevant

shape information, and a geometric transformation that

adapts to that frequency band and devotes full resolution to

the most relevant spatial frequencies is most likely to

provide the discrimination desired. In this way, invariance

to scale may be extended to cover more than the range

obtained with a transformation which maps over a fixed

frequency band.










Previous investigators have designed the polar

transformation to operate over a fixed spatial bandpass in

the Fourier transform plane, thereby limiting the inherent

scale invariance of the geometrical transform (8). Since

size information about the expected object is often

available, it may be used to select the most appropriate

area in the Fourier transform plane and thereby extend the

normal scale invariance of the Mellin-Fourier correlator.



Choosing a Figure of Merit

In evaluating a Mellin-Fourier correlator, a figure of

merit must be used which reflects the pattern recognition

ability of the system. Experimental considerations such as

the tendency of many sensors to drift or change the average

value of an image over time may cause the cross correlation

peak value to decrease even though the pattern match with a

reference image is nearly perfect. In cases where the

spatial frequency filter in the correlation plane includes

the center, or d.c., point, the correlation coefficient





18

should be normalized to the energy in the reference image

using the expression

C(X Y) = f[S(X',Y')'R(X+X',Y+Y')]dX'dY'
[ f[S(X ,Y')]2dX'dY'" f[R(X',Y')]2dX'dY']]11/2

where

C(X,Y) = correlation image plane

R(X,Y) = reference image

S(X,Y) = sensed image

X',Y' = coordinates used as dummy variables in

computing the cross correlation

This expression was used for calculation of the correlation

coefficient throughout this work.



Theory for Extending the Normal Scale Invariance

Use of the Mellin transform for pattern recognition

would seem to require more space bandwidth, i.e., more

samples, than required to obtain the original reference

pattern. A brief analysis of the resampling process will

show how the requirements for increased space bandwidth may

be minimized.

The theory developed to make full use of sampling

capability is best illustrated by an example in one

dimension. Figure 3 shows the relationship between sample

spacing in the Fourier transform plane, designated as X

space, and sample spacing in the In (R),9 plane, designated






19

as X' space. The samples in X' space occur at equally

spaced intervals of the variable X'. The coordinate

transformation that achieves scale invariance is represented

by a curve that follows the functional form

X'k = 1n (Xk / Xmin)

where

Xk = kth sample in X space

X'k = k sample in X' space

Xmin = minimum value of spatial

frequencies in Fourier plane

The sampling in the Fourier transform plane extends over an

interval between Xmax and Xmin, and the samples have a

non-uniform spacing along the X axis. The widest spacing in

X occurs at Xmax and the closest spacing occurs at Xmin.

Since the sample spacing in X is most dense at Xmin, it is

best to assure that this part of the sampling curve has

samples in X that are as far apart as possible if the system

bandwidth is to be most effectively used. At the same time,

the interval at Xmax must be no greater than the Nyquist

distance in order to preserve all the information present in

the original image. The slope of the logarithmic curve at

Xmax is determined by the Nyquist requirement, and the space

bandwidth of X' space is largely determined by the slope of

the sampling curve at Xmin'









When Mellin-Fourier correlation is done in two

dimensions, the X' axis in Figure 3 represents the In(R)

axis in the In(R),9 plane. The X axis represents the radial

frequency coordinate, or R, in the Fourier transform plane.

If the chosen region in the Fourier transform plane does not

change position and adapt to the new object spatial

frequencies as the object scale changes, information may

enter or leave the annulus defined by earlier size

estimates. The new information appearing in the In (R), 9

image will cause the cross correlation with a reference

pattern to be reduced. To prevent the reduction of the

cross correlation peak, the chosen region in the Fourier

transform plane must adapt by changing position when a scale

change occurs in the object. Specifically, system space

bandwidth may be conserved by using range adaptation in two

ways:

(1) using expected object range to select the annulus of

spatial frequencies in the Fourier transform plane over

which resampling will be done

and

(2) using the analysis developed for minimizing the slope of

the resampling curve at Xmin to modify the

transformation equations to map the spatial frequency

annulus into the full space bandwidth of the ln(R),9

plane.






21













SAMPLING IN A ND PA CE
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plane, X space, and in the ln(R),9 plane, X' space.






22

These steps will assure Nyquist sampling over the chosen

region in the Fourier transform plane, and will minimize

oversampling at the smallest value of the input coordinate.



Adapting The Coordinate Transformation

The upper curve in Figure 3 represents sampling done

using a fixed set of transformation equations in which

Xmin = .10. The lower curve represents sampling done with

an adaptive transformation equation with Xmin = .50. If the

relevant information remains within a fixed domain when a

scale change occurs, and only translates, the lower curve

shows that sampling requirements in X space are less severe,

i.e., samples are further apart, than if the transformation

equations are left fixed and the upper curve is used to

generate sample coordinates in X space. The number of

samples in X' space necessary to meet the Nyquist conditions

in X and still keep oversampling to a minimum at Xmin

determines the space bandwidth requirements of the scale

invariant correlation system. The theory developed by

Anderson and Callary makes use of the assumption that no

information is allowed to leave the interval between Xmin

and Xmax when a scale change occurs (9). This means that

there are no "accuracy" problems such as the ones in the

analysis of Cassasent and Psaltis (10). This is the basis

for the savings in space bandwidth claimed in Reference 9.









Sample Adaptive Calulation

An example will show how range adaptation may be applied

to a specific pattern to derive a spatial frequency filter

for use in a Mellin-Fourier correlation scheme. The filter

is a window applied to the Fourier transform plane, defining

the area to be resampled. The width and boundary values of

the region are determined by the nature of the object to be

recognized and the expected range at which it will be

encountered. In the Fourier plane, the filter specification

will be two radial values determined from calculations of

expected object size in the sensor field of view. The

resampling scheme will then be applied to the region to

minimize oversampling in the Mellin-Fourier correlation.

For this example, it is assumed that other sensors have

provided approximate range as R meters, and that the image

and Fourier transform are made up of N x N arrays. If the

sensor field of view is F degrees, then the number of

pixels, P, occupied by a feature of size S is

P = (NS) / (R tan (F))

In a reference image, a square feature of size P produces a

first spectral lobe which goes to zero at N/P pixels from

the center pixel in the Fourier plane. The first zero

crossing defines a boundary value for the region chosen for

resampling. Two boundaries of the spatial frequency

bandpass may be established from estimated object feature






24

sizes in the image plane. The two cutoff frequencies will

then define an annulus of spatial bandpass in the Fourier

transform plane. For a sensor with a field of view of 3

degrees, the cutoff frequencies for a 6 meter object with .5

meter features at a range of 500 meters are

upper frequency = N/P = (R tan(F))/S

= (500/.5) tan (3)

= 52 cycles/frame



lower frequency = (500/6) tan (3)

= 4 cycles/frame

The full space bandwidth of the Mellin-Fourier correlation

resampling system may then be applied to an annulus between

4 and 52 pixels from the center or d.c. term in the Fourier

transform plane. The transformation equation necessary for

optimal resampling may be written in terms of radial

distances in the Fourier transform plane:

R' = In(R) In(Rmin)

where

Rmin < R < Rmax

This equation is used to generate sample coordinates in the

ln(R),9 plane where the value of Rmin is the lower spatial

frequency limit derived for the specific pattern, 4 pixels,

and the value of R extends to the upper frequency limit,

Rmax, of 52 pixels. The region in the Fourier plane that
is oversampled to construct the logarithmic data record does





25

not begin until a radius of 4 pixels has been reached in the

Fourier plane, thereby minimizing the oversampling that

occurs at small values of the Fourier radial coordinate.

Since no data of interest lie beyond a radius of 52 pixels

in the Fourier plane, the sampling that occurs at 52 pixels

from the center can be made to just meet the Nyquist rate.

The analysis above has shown how system space bandwidth

may be conserved by using range adaptation to define the

proper annulus in the Fourier plane and then adapting the

transformation equations to map the annulus into the ln(R),9

plane. When range adaptation is performed as a normal part

of signal processing, like proper windowing, the space

bandwidth requirements of a pattern recognition system may

be expressed in terms of spectral resolution in the Fourier

plane rather than sampling bandwidth in the ln(R),9 plane.

It may be argued that knowledge of range for the sensed

image would allow the sensed Fourier transform to be scaled

appropriately, thus making scale invariant schemes

unnecessary. In a practical system, however, range

information is usually only approximate, making the sensed

frequency annulus poorly defined. Range adaptation is

valuable because it makes a correlator more tolerant to

errors in size estimates for the sensed image. Evidence of

this tolerance is shown in the examples of the next chapter.













CHAPTER III
APPLYING OPTIMUM SAMPLING TECHNIQUES

Examples of Optimum Sampling

The Fourier transform patterns in Figures 4 and 5 have

been generated to illustrate the technique of designing a

transformation which maps the Fourier transform space into

another space with coordinates R and 9. The patterns in

Figures 4 and 5 represent the magnitude of the Fourier

transform of squares of sizes 10 X 10 and 20 X 20,

respectively. The Fourier transform image is represented by

a plot of intensity along the center of the image. Figures

6 and 7 are from the R,9 mapped images. Study of the size

of the features in Figures 6 and 7 shows that these features

change size and position with the object scale change, just

as they do in the Fourier transform image before mapping.

In this case the scale change encountered is a factor of 2.

Figures 8 and 9 are from the In (R),9 mapped images. Here

the features present in Figure 8 simply translate to a new

position in Figure 9; they do not change size. However, new

information is introduced from the right in Figure 9 which

was not present in Figure 8. Distortion is also introduced

in both figures because of the many-to-one mapping that

occurs at small values of radial distance in the Fourier

plane.





















FOURIER TRANSFORM MAGNITUDE
EFE'ENCvE 10 X 10 SQUARE



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- i
1 \
-t


0 20 40 80 80 100 12(
SPATIAL FREQUENCY (CYCLES/FRAME)

Figure 4. Amplitude along a line passing through
the center of the Fourier transform of a 10 X 10
square. Normalized from 0 to 255.


260

240

220
200
180
1 60

140

120
100
80
80

40

20
0


0


I


I























FOURIER TRANSFORM MAGNITUDE
SENSED 20 X 20 SQUARE


S 20 40 80 80 100 120
SPATIAL FREQUENCY (CYCLES/FRAME)

Figure 5. Amplitude along a line passing through
the center of the Fourier transform of a 20 X 20
square. Normalized from 0 to 255.


280

240

220

200

180

1 0

140

120

100

80

80

40

20

0























POLAR TRANSFORM
REFERENCE 10 X 10 SQUARE
280

240 -

220 -

200-

0 180-
I-
1 180 -
a-
140-
w 120 -
N

3 100-
0 80-

60 -

40-

20

0-
0 20 40 60 80 100 12C
RADIAL SPATIAL FREQUENCY (CYCLES/FRAME)

Figure 6. Amplitude along a line passing through
the center of the R,9 mapped image if a 10 X 10
square. Normalized from 0 to 255.






















POLAR TRANSFORM
SENSED 20 X 20 SQUARE


3 20 40 80 80 100 120
RADIAL SPATIAL FREQUENCY (CYCLES/FRAME)

Figure 7. Amplitude along a line passing through
the center of the R,9 mapped image of a 20 X 20
square. Normalized from 0 to 255.


280

240

220

200
180

160

140

120

100

80


40

20

0





















LOGARITHMIC


POLAR


TRANSFORM


REFERENCE 10 X 10 SQUARE


0 20 40 80 80 100 120
NORMALIZED LOG FREQUENCY (CYCLES/FRAME)

Figure 8. Amplitude along a line passing through
the center of ln(R),9 mapped image of a 10 X 10
square. Normalized from 0 to 255.


280
240
220
200
180
1 0
140
120
100























LOGARITHMIC POLAR TRANSFORM
SENSED 20 X 20 SQUARE


0 20 40 80 80 100 120
NORMALIZED LOG FREQUENCY (CYCLES/FRAME)

Figure 9. Amplitude along a line passing through
the center of the ln(R),9 mapped image of a 20 X 20
square. Normalized from 0 to 255.


280

240

220

200

180

180

140

120

100

80

80

40

20

0









In producing both Figure 8 and Figure 9, the same

transformation parameters were used; i.e., the mapping was

governed by the same set of equations. For each figure,

pixel values in the In (R),9 space were derived from the

same calculated position in the Fourier transform space.

With a fixed set of transformation equations, a scale change

causes new information to enter the transformed image,

causing a decrease in the cross correlation peak value.

This difficulty may be overcome by adapting the

transformation equations to map a different region of the

Fourier transform plane into the new space. In Figures 8

and 9, range information was used to choose the size and

position of the annulus in the Fourier transform plane, and

the annulus tracked the region of critical object

information over a wider range of scale change. Of course,

spatial resolution in the transformed image will degrade as

the annulus chosen for the transformation becomes smaller

and smaller. This occurs at very close ranges, and is not

critical if the reference image is made at the closest

expected range. In a system with the ability for range

adaptive bandpass, range information is used to change the

transformation parameters to track the critical region of

spatial frequencies in the Fourier transform plane. The

cross correlation peak should remain high over a wider range






34

of scale change than when the peak is reduced due to scale

changes causing irrelevant information to enter the

transformed space.

Figures 10 through 13 show the results when the

principles discussed above are applied to real images. The

sequence of nine images that make up each figure show the

Mellin-Fourier correlation process outlined schematically in

Figure 1. The upper left pair of images labeled "REFERENCE"

and "SENSED" are infrared images of an armored vehicle. The

right image of the pair has been rotated about its center by

45 degrees. The second pair of images labeled "REFERENCE

FFT" and "SENSED FFT" are the magnitudes of the Fourier

transforms. The image pair labeled "LOG R, THETA

TRANSFORMS" are the ln(R),@ mappings done from the Fourier

transform images immediately above. The right In(R),@ image

shows that rotation in the sensed image produces vertical

translation of the pattern in the ln(R),9 image. The images

labeled "MELLIN TRANSFORMS" are the magnitudes of the

Fourier transforms of the images immediately above. Since

the only difference between the reference and sensed In(R),9

images is a vertical shift, the magnitude images labeled

"MELLIN TRANSFORMS" are nearly identical. The last step in

the Mellin-Fourier correlation process is shown as the

rightmost image labeled "CROSS CORRELATION." The real and

imaginary components of the In(R),9 images were used to

compute the cross correlation image plane. The dark spot in





































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43

the correlation image is enhancement performed by the image

processing computer to make the peak in the correlation

plane more visible. The vertical position of the

correlation peak reveals the correct rotation difference

between the reference and sensed images, i.e., 45 degrees,

and the amplitude, indicated by the peak intensity, shows

the degree of similarity between the reference and sensed

images.

Figure 10 shows that rotation of the input image by 45

degrees may be detected using standard space variant

processing techniques. In this example, no scale change was

involved, and pattern recognition was achieved with a

normalized correlation coefficient of .9. In Figure 11,

the sensed image was scaled by a factor of 2.0, and no

rotation was present. The images labeled "LOG (R), THETA

TRANSFORMS" show what happens when a scale change causes

information to leave the In (R),9 plane. The correlation

coefficient is reduced to .7, even though its position is

displaced from the center of the correlation plane by the

correct amount, indicating a scale change. Figure 12 shows

that combining a scale change with a rotation reduces the

correlation coefficient still further to .6, with the

appearance of noise at high spatial frequencies being

evident in the In (R),9 images. Application of range





44

adaptive bandpass in Figure 13 restores the correlation

coefficient to .89, with the rotation being correctly

sensed. However the ability to detect the amount of scale

change by the position of the correlation peak has been

given up by adapting the In (R),Q transform equations using

range information from an independent source.



Testing Theoretical Predictions

Predicted Scaling Limits

The success of the application of Mellin-Fourier

correlation to the tactical images shown in Figures 10

through 13 led to a careful analysis of the scale invariance

that should be expected of the Mellin-Fourier correlator

when used with a particular image frame size. The images

used in predicting the expected scale invariance were

centered in a frame of 256 X 256 and were scaled by sampling

in a polar coordinate system in which R was multiplied by a

scaling constant A. The value of A was used as an index to

denote the amount of scaling present in each trial. An

image of the Fourier transform of the reference scene, a

side view of an armored vehicle, was inspected to identify

the most critical features for pattern recognition. The

selection of "most critical" features in the Fourier

transform plane was subjective to the extent that some

spatial frequencies were identified with the reference image

window and were eliminated by specifying the lower limit of





45

the filter. Also, knowledge of imager characteristics as

well as inspection of large numbers of Fourier transform

images allowed an upper limit to be specified for the

spatial bandpass. Limiting the frequencies used for

transformation into the ln(R),8 plane allowed the

frequencies to be adjusted for the scale change present in

the sensed image. The adaptation of the filter according to

scale change is a capability that is often present in

tactical pattern recognition systems with radar altimeters

or laser rangers.

Calculations were done using the side view of the

armored vehicle as a reference image to verify that the

distinct features in the Fourier transform plane were due to

the spacing and number of the vehicle road wheels. The

window or frame size of the reference image was used to

calculate the location of the first zero crossing in the

Fourier transform plane. These calculations were patterned

after the example given in Chapter II, under "Sample

Adaptive Calculation." The filter limits calculated using

the methods described above resulted in an annulus in the

Fourier transform plane between 20 and 110 pixels. For

range adaptation, the lower and upper limits of this annulus

were scaled according to the factor A used for the sensed

image. The inverse relationship between scaling in the

image plane and scaling in the Fourier transform plane





46

caused the filter boundaries to be multiplied by 1/A

whenever the sensed image was scaled by A. The convention

used in this research was that values of A greater than 1.0

resulted in a smaller image but a larger Fourier transform.

Values of A less than 1.0 denoted a larger image and a

smaller Fourier transform.

The scaling limits predicted for the example discussed

above were determined from Nyquist sampling and spectral

resolution requirements. It was observed from inspection of

the Fourier transform of the reference image that this

imagery was not corrupted by aliasing effects, and that the

lower frequency limit of 20 pixels still provided an

adequate number of samples to define a spectral lobe when

scaling by a factor of A=.3 was performed. When scaling by

factors larger than A=1.5, it was observed that loss of high

frequency information pertaining to road wheel spacing,

e.g., the second lobe moving out of the sampled domain,

would violate the Nyquist condition. Therefore the

predicted range of scale invariance for the correlator

chosen for this example was from A=.3 to A=1.5.

Figure 14 shows the results obtained using the computer

simulation of the Mellin-Fourier correlator. The points

labeled with a square symbol were obtained from a correlator

with the spatial filter fixed at 20 to 110 pixels. The





47

filter did not adapt to scale changes in the sensed image.

The correlation coefficient steadily decreases from .9 with

no scale change, and drops below the threshold of .7 if A is

less than .5. The points labeled with cross symbols were

obtained from a correlator in which the spatial filter

adapted to the scale change present in the sensed image.

The annulus defined in the Fourier transform plane for

the In (R),9 transformation was allowed to change size

according to the scaling factor A. This adaptation of the

spatial filter allowed the correlation coefficient to remain

fairly constant over the entire range of predicted scale

change. The correlation coefficient for the adaptive

correlator does not drop below .7 until A becomes smaller

than .1. This represents an extension of the scale

invariance for small values of A by a factor of 5. The

performance of the correlator when A was greater than 1.0

was similar for the fixed and for the adaptive filter. At

A=1.2 the upper frequency limit of the adaptive filter moved

to a value of 128, which was the edge of the available

bandwidth. Since the filter could no longer adapt, the

correlators behaved similarly. These results show that

adaptive filtering may be used to extend the normal scale

invariance of the Mellin-Fourier correlator to allow pattern

matches with high confidence (high correlation coefficient)

even when operating near the lower limits for the scaling

factor A.




















LIMITS OF CORRELATOR
EXTENDING SCALE INVARIANCE






| 0.8 / y
0.95 -



0.5
I- /I
z

_ 0875





0.87
9 O.8 -

0.55

0.5

0.45 ,
0.05 0.3 0.6 0.9 1.2 1.5 1.8
SCALE CHANGE IN SENSED IMAGE
D FIXED FILTER + RANGE ADAPTED

Figure 14. Scale invariance extended by adapting
the spatial filter.









Failure Modes

Data acquired while investigating the scaling limits of

the Mellin-Fourier correlator caused interest in information

that could be obtained from the correlation plane even after

the correlator had "failed" by giving a correlation

coefficient less than .7. The position of the peak value in

the correlation plane reveals the amount of scale change and

rotation difference sensed, and this information seems to be

reliable even when the value of the correlation peak is less

than that required for a confident pattern match. Figure 15

is a plot of the scale change detected by the X position of

the correlation peak versus the actual scale change present

in the sensed image. As before, the square symbols show

correlation done with a fixed filter, and the cross symbols

show correlation done with an adaptive filter. The fixed

filter gives incorrect values of scale change when the

scaling factor A goes below .5, as predicted by sampling

theory. The adaptive filter gives correct values of scale

change (within 20%) when A=.05. The rotation reported by

both correlators was within 10% of the correct value for the

entire range of scale changes tested. With larger values of

A, both correlators became insensitive to scale change,

probably because the spectral features become larger and

occupy more of the ln(R),9 plane, making small shifts in

cross correlation harder to detect. This result suggests





50

that reliable indications of scale and rotation may be

obtained from a Mellin-Fourier correlator using an adaptive

filter even when the designed scale change has been

exceeded. Correlator system design may be extended to cover

values of scale change not allowed by sampling theory if

information about scale change and rotation are required

rather than the degree of pattern match.























LIMITS OF
EXTENDING


CORRELATOR
SCALE INVARIANCE


0.3 0.6 0.9 1.2 1.5 1.8
ACTUAL SCALE CHANGE IN SENSED IMAGE
0 FIXED FILTER + RANGE ADAPTED


Figure 15. Detected scale change extended
the spatial filter.


by adapting


0.9

0.8
0.7
0.6

0.5
0.4


0.3
0.2-
0.1 -

0
0.05 --
0-05













CHAPTER IV
ADDITIONAL EXPERIMENTS

Higher Space Bandwidth

The tests shown in Figures 10 through 13 were repeated

with images having a space bandwidth of 256 X 256 pixels.

The larger image format was equivalent to sampling the

reference and sensed patterns at a higher frequency. To

create the Fourier transform patterns, the 128 X 128 input

images were simply centered in the 256 X 256 frames. No

additional information was added to the input image, but the

Fourier transform calculated over the 256 X 256 frame

extended to higher spatial frequencies. This added space

bandwidth allowed for more shift, i.e., more scale change,

in the In (R),9 images when scale changes caused distortions

in the Fourier transform plane. The annulus of spatial

frequencies chosen for logarithmic resampling was chosen

using the same criteria suggested in discussions of the

range adaptive technique; i.e., frequency boundaries were

established using approximate ranges supplied with the test

imagery.





53

Expected Benefits of Higher Space Bandwidth

The calculations of the upper and lower spatial

frequencies critical for pattern recognition may be done

without knowledge of the system space bandwidth available.

With the larger format frames however, the spatial frequency

boundaries could extend to larger distances, i.e., more

pixels, from the center or d.c. term. With the larger space

bandwidth available in a 256 X 256 format, it was expected

that it would be possible to vary scale over a wider range

while maintaining the value of the correlation coefficient.

Scale change was performed using a sample and hold

technique, with the sensed image remaining centered in the

frame for all scale changes. The sample and hold technique

was implemented using the same equations to generate sample

points that were used to rotate the reference image. The

sample coordinates for the sensed image had to be generated

in polar form in order to add the required rotation angle.

To generate a scaled image, the radial coordinate for the

sample points was simply multiplied by a scaling factor A.

The effect of this sampling scheme was to provide a sensed

image that was scaled or zoomed around the center of the

reference image as well as rotated by the desired angle.

When higher space bandwidth was used in the experiment shown

in Figure 12, with scaling and rotation present, the

correlation peak value changed from .6 to .7. This increase





54

was not considered significant since the correlation value

was so close to the threshold of .7 considered necessary for

pattern match. The threshold value of .7 for pattern match

was experimentally determined by correlation experiments

between both geometric shapes and real objects. In these

experiments, even dissimilar objects produced correlation

coefficients of .6. When range adaptation of the

transformation equations was performed as in Figure 13 but

using the higher space bandwidth, the correlation

coefficient changed from .89 (with the 128 x 128 frame) to

.92. This slight increase was not considered significant,

considering the price paid in a four-fold increase in the

number of sample points required.


Effects of Exceeding the Correlator Design

A detailed study of the effect of scale change on the

correlation peak value and curve shape was performed on the

images with the higher space bandwidth. The rotation angle

between the reference image and the sensed image was fixed

at 45 degrees to be sure that the location of the

correlation peak still revealed the correct rotation angle.

Scale factors of the sensed image were varied between .5 and

2.0, and Figure 16 shows a plot of the values of the

correlation images along a line passing through the peak

value. The two curves with the legends "A = .5" and

"A = 2.0" represent the extreme values over which the





















MELLIN-FOURIER CORRELATION
EFFECT OF SCALE CHANGE


0.2

0.1

0


0 A = 1.0


120 140 180
COLUMN OF CORRELATION IMAGE
+ A = 2.0 0 A = 0.5


Figure 16. Shape of the correlation curve for two
extreme values of scale change, A=.5 and A=2.0 .
Range adaptive spatial filter not applied. The curve
for A=1.0 represents a sensed image with no scale
change.





56

Mellin-Fourier correlation scheme was exercised. The shape

of the correlation curve for no scale change is shown in the

curve labeled "A = 1.0." Symbols are included in the plots

to define the curves when they cross each other.

The Mellin-Fourier correlator in this example was

designed to cover a range of scale change from A = 2.0

to A = .5 without the use of range adaptive techniques.

Figure 16 shows that the value of the correlation peak at

these extreme scale changes drops just below the threshold

value of .7. The range of scale change over which the

system could operate in this example was not extended using

range adaptive techniques. The use of higher space

bandwidth has increased the range of scale invariance of the

correlator only by the amount that was predicted from

sampling theory.



Adding Noise

Effects of Adding Uniform Noise

A further investigation of the Mellin-Fourier

correlation scheme was performed by adding noise to the

sensed image and noting the effect on the shape and

magnitude of the correlation peak. The intent of this part

of the investigation was to simulate the imagery collected

by a sensor that might be corrupted by various noise

sources. The image used as a reference was the same as the





57

reference image in tests involving changes in scale and

rotation, i.e., one which contained no added noise. A

random variable function from a computer statistical library

was used which was capable of generating a pseudo random

number uniformly distributed between 0 and 1.0. The

function is considered pseudo random because the seed, or

initializing integer, completely determines the string of

random digits generated by the function each time it is

used. However, the probability distribution of the digits

produced by the function is the desired uniform

distribution. For initial investigations into the

sensitivity of the Mellin-Fourier correlator to added noise,

the sensed image was treated as a random variable to which

another random variable with the desired statistical

distribution was added. The addition of noise took the

mathematical form

NOISY SENSED IMAGE = SENSED IMAGE + 255 X NOISE X RAN (SEED)

where

NOISE = value from 0 to 1.0 providing a

measure of the amount of added noise

RAN (SEED) = random variable uniformly

distributed from 0 to 1.0

The mathematical form above produced a sensed image with an

altered amplitude distribution between values determined by

the noise measuring variable NOISE. The total amplitude

distribution of the sensed image was the combination of

uniformly distributed noise and the original distribution.





58

The effect of varying the value of NOISE was to extend the

area of the image histogram over which the added noise had

its effect. One meaningful measure of this type of noise is

the width of the histogram region that is altered. The

degradation of the correlation peak when the sensed image

was transformed by noise of uniform amplitude distribution

is shown in Figure 17. As with the evaluations involving

scale change, the correlation image was represented by a

horizontal scan through the correlation peak. The three

curves shown represent altered histogram widths of of 25,

50, and 128 counts; each region centered around the image

mean value of 59 counts. Figure 17 shows that introduction

of noise causes the cross correlation curve to decrease

smoothly past the threshold amplitude considered necessary

for pattern match.


Effect of Adding Gaussian Noise

A more physically significant Gaussian distribution was

used in a second series of trials with the same sensed

image. The Gaussian distribution was achieved by making use

of the Central Limit theorem applied to the uniformly

distributed random function described above. The

mathematical form for the transformation was

K = 48

Y = E [ (RAN (SEED) .5) / 2 ]

K = 1




















MELLIN-FOURIER CORRELATION
EFFECT OF UNIFORMLY DISTRIBUTED NOISE


O0 70 90 110 130 150 170 190
COLUMN OF CORRELATION IMAGE

Figure 17. Uniformly distributed noise added to
sensed image. Added noise was centered around the
image mean value, and had the three histogram widths
shown.





60

A new random variable, Y', with the desired mean of U

and standard deviation V was obtained using the

transformation

Y' = U + V Y

The results of trials conducted with Gaussian distributed

random noise are shown in Figure 18. The three curves

represent the shape of the correlation curve after the

addition of noise having standard deviations of 0, 10.0, and

20.0. The horizontal scale of Figure 18 was changed to see

better that the correlation curve still has a discernible

peak which is located at the proper scale and rotation

coordinates even though the amplitude of the correlation

peak has fallen below the threshold value of .7. This

characteristic of the correlation image suggests that the

ability of the Mellin-Fourier correlator to detect scale and

rotation changes may be preserved even when noise levels

have reduced the confidence in pattern matches between

reference and sensed images.



Effect of Optimal Spatial Filter

During the course of the investigation into noise

sensitivity, the spatial filter used to eliminate low

frequencies in the Fourier transform plane was fixed for the

reference and sensed images to pass all frequencies between

radii of 10 and 128 pixels. The filter was not adapted to

the scale change present in the sensed image. When the






61
















MELLIN-FOURIER CORRELATION
EFFECT OF GAUSSIAN NOISE


0.56 -

0.54 --
100 120 140


0 NO NOISE


COLUMN OF CORRELATION IMAGE
+ STD. DEV.=10 0 STD. DEV.=20


Figure 18. Gaussian distributed noise added to
sensed image. Added noise was zero mean, with the
three standard deviations shown.


0.8

0.78

0.78

0.74

0.72

0.7

0.88

0.668

0.84

0.82

0.8


180





62

sensed spatial filter was optimized for the scale change,

Gaussian noise was again introduced into the sensed image to

determine the effect on the correlation curve. Figure 19

shows that use of a filter optimized for the scale change in

the sensed image can restore the correlation peak to values

required for full confidence in a pattern match. The shape

of the correlation curve in Figure 19 is significant because

the secondary peak, present in Figure 18 when the filter was

not optimized, is reduced. The secondary peak in Figure 18

represents a partial match caused by energy present in the

window of the In (R),9 image. The location of the secondary

peak is at the center of the correlation plane, as expected,

and represents a match between the reference and sensed

windows. Figure 19 also shows that the level of Gaussian

noise necessary to reduce the correlation peak below .7 is

higher when the sensed filter is optimized for scale change.



Demonstration of Insensitivity to Window

The results shown in Figure 19 caused concern that most

of the amplitude observed in the correlation peaks of

Figures 16 through 19 was caused by the window, or frame

size, of the sensed image. The scaling algorithm used to

produce precisely scaled sensed imagery also served to

scale precisely the image window by the same factor. Since

the position of the secondary peak did not change with the






63














MELLIN-FOURIER CORRELATION
GAUSSIAN NOISE WITH OPTIMUM FILTER


120


D NO NOISE


124 128 132 138 140 144 148
COLUMN OF CORRELATION IMAGE
+ STD. DEV=20 0 STD. DEV.=40


Figure 19. Gaussian distributed noise added to the
sensed image. Added noise was zero mean, with the
three standard deviations shown. Spatial filter
optimized to the scale change in the sensed image.





64

scaling factor of the sensed image, it was considered likely

that correlation amplitude was generated by the In (R),9

window size, which was independent of the object scaling

factor. An experiment was performed to demonstrate that the

correlation peak was indicating the correct scale change and

degree of similarity between the reference image and the

sensed image. The Mellin-Fourier system was made to perform

a correlation between the usual reference image and an

incorrect sensed image. The incorrect sensed image was

scaled by a factor of A = .7 and the filter bandpass was

optimized for this scale change. The resulting correlation

peak was not expected to indicate the proper scale factor

nor was it expected to have sufficient amplitude to indicate

a pattern match. The amplitude of the correlation peak was

.662, well below the threshold for match, and the X position

of the correlation peak indicated a scale factor of .85.

These results indicate that the amplitude and position of

the correlation peak are reliable indicators of pattern

match and scale factor, and are not completely determined by

image window size.













CHAPTER V
CONCLUSIONS

The numerical experiments described in this paper have

shown how several issues can affect the evaluation of the

space bandwidth requirements of a scale invariant

correlation system. Spectral resolution and Nyquist

sampling were seen to be the two main determinants of the

space bandwidth required to achieve correlator invariance

over a given range of scale change. Other things, such as

windowing and truncation of the Fourier transform plane,

were shown to have an effect on the validity of the cross

correlation image.

Calculations done under the assumptions used by Anderson

and Callary (4) were used to see how savings in system space

bandwidth could be realized by careful choice of geometric

transformation parameters. Further refinements of the

choice of transformation parameters were applied in the

development of the technique of range adaptive bandpass.

The series of examples presented in the section

discussing optimum sampling for scale invariance showed how

each of the principles described in the preceding sections





66

could be used successfully to restore the amplitude of the

cross correlation peak in a scale invariant correlator.

The theoretical scaling limits of a Mellin-Fourier

correlator were calculated and a computer simulation was

used to verify that the theoretical benefits of range

adaptive filtering could be applied in practice. Using only

the techniques of range adaptive sampling and careful choice

of transformation parameters, the scale invariance of the

Mellin-Fourier correlator was extended by a factor of 5 for

scaling factors less than 1.0 without increasing the space

bandwidth of the system. The performance of the

Mellin-Fourier correlator which used a range adaptive filter

was similar to the fixed filter correlator when the scaling

factor A was greater than 1.0.

Design approaches were suggested for systems required

only to detect scale and rotation changes in a sensed image

rather than to perform a detailed measure of pattern match.

Range adaptive filtering techniques were seen to provide

accurate measures of scale and rotation changes even though

scale changes in the sensed image had violated Nyquist

sampling conditions.

The techniques developed for Mellin-Fourier correlation

were tested in an investigation of the benefits of higher

space bandwidth. The effect of the larger format 256 X 256






67

frame was to increase the system space bandwidth by a factor

of four. When standard scale invariant techniques were

applied to the large format frame in the same way as with

the 128 X 128 image, the larger space bandwidth produced a

modest improvement in the correlation peak; not a

significant improvement considering the large price paid in

increasing the system space bandwidth. When range adaptive

techniques were applied to the Mellin-Fourier correlator

using the larger frame size, the correlation peak value

still did not show a significant increase. These

experiments suggest that signal processing techniques such

as range adaptive transformations and spatial filtering

contribute more to the success of the Mellin-Fourier

correlator than an increase in correlator space bandwidth.

The experiments further suggest that the theory for

predicting performance of Mellin-Fourier correlator may be

based on the well-established principles of Nyquist sampling

and spectral resolution.

Investigations into the sensitivity of the

Mellin-Fourier correlator to noise measured the influence of

both uniform and Gaussian noise distributions on the value

and position of the correlation peak. Higher space

bandwidths were investigated and sensitivity to added noise

was evaluated using the space bandwidth available in a

256 X 256 frame. Optimal choice of the spatial filter used





68

in processing the sensed image made the Mellin-Fourier

correlator less sensitive to Gaussian noise. Experiments

with incorrectly matched library images verified that the

correlator was not simply responding to the scaled window

through which the sensed image was viewed.













REFERENCES


1. Cassasent, D., & Psaltis, D. (1978). Deformation
Invariant, Space Variant Optical Processing. In E. Wolf
(Ed.), Progress in Optics, 16, (pp. 291-356). New York:
North-Holland Publishing Company.

2. Cassasent, D., & Psaltis, D. (1976, July). Position,
Rotation, and Scale Invariant Optical Correlation.
Applied Optics, 15(7), 1795.

3. Cassasent, D., & Psaltis, D. (1977, June). Space
Bandwidth Product and Accuracy of the Optical Mellin
Transform. Applied Optics, 16(6), 1472.

4. Anderson, R.C., & Callary, P.R. (1981, April). Space
Bandwidth Product for the Mellin Transform. Applied
Optics, 20(8), 1272.

5. Horev, H. (1980,December). Picture Correlation Model for
Automatic Machine Recognition, 7. Unpublished master's
thesis, Air Force Institute of Technology,
Wright-Patterson Air Force Base, OH.

6. Childers, D., & Durling, A. (1975). Digital Filtering
and Signal Processing. (pp. 290-314). New York: West
Publishing Company.

7. Childers, D., & Durling, A. (1975). Digital Filtering
and Signal Processing. (p. 302). New York: West
Publishing Company.

8. Anderson, R.C., & Callary, P.R. (1981, April). Space
Bandwidth Product for the Mellin Transform. Applied
Optics, 20(8), 1272.


9. Anderson, R.C., & Callary, P.R. (1981, April). Space
Bandwidth Product for the Mellin Transform. Applied
Optics, 20(8), 1272.

10. Cassasent, D., & Psaltis, D. (1977, June). Space
Bandwidth Product and Accuracy of the Optical Mellin
Transform. Applied Optics, 16(6), 1472.













BIOGRAPHICAL SKETCH

Mr. Friday graduated from the University of Alabama in

1973 with a Master of Science in Electrical Engineering.

From 1974 to 1978 he served as a system engineer for an

airborne infrared measurement system operated by the U. S.

Air Force at Eglin Air Force Base, Florida. He was selected

by the Air Force to attend the University of Florida for

long-term, full-time study where he was admitted to

candidacy for the PhD degree. In 1978 he joined the Air

Force Armament Laboratory where he was program manager for

projects involving computer generation of synthetic infrared

images. From 1981 to 1983, he received funding from the Air

Force Office of Scientific Research (AFOSR) to do basic

research in pattern recognition using Mellin transforms. He

is now employed by E G & G Special Projects Inc., in Las

Vegas, Nevada, and is continuing studies of computer

simulations of electro-optical pattern recognition

algorithms.







I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.

Rol nd C. Anderson
Professor, Department of Engineering
Sciences

I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.

7/1< 7/e. Kfr7__
Ulrich H. Kurzweg
Professor, Department of Engineering
Sciences

I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.


Alex E. Green
Graduate Reasearch Professor, Department
of Physics

I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.


Stanley S. Ballard
Distinguished Service Professor,
Department of Physics

I certify that I have read this study and that in my
opinion it conforms to acce pble standards of scholarly
presentation and is fully c e and quality,
as a dissertation for egre0 Do ot of Philosophy.


Engineer


n, Department of


ces









I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.


Richard L. Fearn
Professor, Department of Engineering
Sciences












This dissertation was submitted to the Graduate Faculty
of the College of Engineering and to the Graduate School
and was accepted as partial fulfillment of the
requirements for the degree of Doctor of Philosophy.

May 1985 14A a -_____4X
Herbert Bevis
Dean, College of Engineering

Madelyn Lockhart
Dean, Graduate School







































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