FINAL REPORT ON AUTOMATIC FOCUSING SYSTEM

• Document Type: CREST
• Collection: NGA Records (Formerly NIMA)
• Document Number (FOIA) /ESDN (CREST): CIA-RDP78B04747A002600020010-1
• Release Decision: RIPPUB
• Original Classification: K
• Document Page Count: 35
• Document Creation Date: December 28, 2016
• Document Release Date: May 25, 2011
• Sequence Number: 10
• Publication Date: November 1, 1965
• Content Type: REPORT

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FINAL REPORT
ON
AUTOMATIC FOCUSING SYSTEM
November 1965
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TABLE OF CONTENTS
PAGE
1.0
2.0
Introduction
Summary of Work Performed
2.1
Correlation System
2.2
Non-Linear Photoconducting Sensor
2.3
Sensor Development
2.4
optical Testing System
0 Detail Technical Discussion
5
3.1 Autocorrelation System for Autofocus
5
3.2 Non-Linear Photosensors for Autofocus
6
3.3 Photosensor Development
8
3.3.1 Non-Linearity in Photoconductors
8
3.3.2 "Binder" Panels
10
Sintered Panels
12
3.4 Testing and Evaluation
12
3.4.1 Photo Response
13
3.4.2 Current-Voltage Tests
14
3.4.3 Linearity Tests
14
3.4.4 Uniformity Tests
14
3.4.5 System Tests
15
4.0
Results
15
5.0
More Recent Developments
16
6.0
Conclusions
17
7.0
Recommendations
17
Appendix A
A-1
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LIST OF ILLUSTRATIONS
figure 1 Correlation Curve
2 Photo Response Measurement System
3 Photo Response of Binder Panel
4 Current-Voltage Characteristic
5 Linearity Plot
6 Autofocus Test System
7 Autofocus Test Results - Best Panels
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1.0 Introduction.
This report covers work done during the period
of March 11, 1965 to June,-,11,, 1965, and later tests
on the application and use of solid state sensors for
an Automatic Focusing System. The chief requirement for
this system was that it require no special focusing tar-
gets, but should work with any .Lnput imagery; that no
mechanical scanning motions be necessary; that it be cap-
able of developing an electronic servo signal for driving
the system into focus. Furthermore, this project being
of limited scope both in time and money, was directed
towards demonstrating basic feasibility rather than the
development of a breadboard or ;prototype apparatus. These
objectives were fully achieved.
2.0 Summary of Work Performed.
2.1 Correlation System.
Originally, the basic concept of an Automatic
Focusing was predicated on the utilization of an auto-
correlation technique. It relied on the fact that maxi-
mum autocorrelation is obtained, when an image is in sharp
focus. It was further planned to utilize a solid state
image correlator as the basic sensor. Such a correlator
had previously been developed by Newtek. After several
weeks of work, this approach had to be abandoned. While
the concept is perfectly valid, it turned out that the
configuration of the solid state correlator is such that
it does not perform a strict product correlation, i.e.
compute the integral,
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Rather, the output of the correlator is proportional to,
f-~ (Y,4 CL d X, a ~
( A - A r k I
seemed, initially, that the denominator terms
may be neglected. This, however, turned out not to be
so for the autofocus application.
2.2 Non-Linear Photoconducting Sensor.
Fortunately, all was not lost when the initial
scheme proved unworkable because during this initial ef-
fort it became apparent from our measurements that the
photoconducting materials used in the correlator exhibited
a relatively high degree of non-linearity. A second ap-
proach thus evolved which utilized this non-linear behavior.
This scheme is best illustrated with a photocon-
ductor having a square law response to light intensity.
For such a sensor the output signal is proportional to,
ff C, f- C~,~~] Ldx.dj
This is the same as the output of a perfect product cor-
relator. Actually, a square law response is not essential.
Any sensor which has an n-th power law where n is not
unity, is usable. Best sensitivity, however, is obtained
when n differs from unity by a substantial margin. In
the case of cadmium sulfide photoconductors, the power
law can be controlled in the sensor preparation by careful
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control of the dopants added to the photoconductor.
Sublinear behavior can also be obtained by properly
controlling the thickness of the photoconductive film,,
or by backing up this film with another semiconductor
film having a resistivity which is intermediate between
the light and dark resistivities of the photoconductor.
In the course of this effort we obtained panels
which had sublinear behavior and whose power law was
n - 1/2. These were then used in the system and a servo
signal was obtained which could be utilized to drive
the system into focus.
2.3 Sensor Development.
The key element in the autofocus system is the
photosensor which has the requisite power law behavior.
Many different sensor types may be applicable. Our work
was concentrated on the use of a photoconductive sensor.
This was further confined to CdS and CdSe which appear
to have most promise. The first phase of sensor develop-
ment concentrated on depositing thin layers of photocon-
ductor by blading onto a substrate a mixture of photocon-
ductive powder in a silicon rubber matrix. Panels pre-
pared by this method proved barely sufficient to demonstrate
the principle. Their main drawbacks were a lack of re-
producible behavior between panels prepared under substan-
tially identical conditions; different aging effects between
panels; noisy signal outputs. The physics of these "binder"
panels is not well understood and it is, therefore, dif-
ficult to predict what changes in the preparation method
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would improve their behavior. These "binder" panels
were sufficient, however, to demonstrate the soundness
of the basic principles underlying the system.
The next phase in the sensor development was di-
rected towards obtaining more reproducible and reliable
'behavior. To this end it was decided to prepare panels
using sintered photoconductive powder layers. Such panels
were successfully made and did, indeed, prove to be quite
superior to the binder panels. Final demonstration of
system feasibility was accomplished, using these sintered
thin film layers.
2.4 Optical Testing System.
In order to be able to test the autofocus system,
a special optical test rig is required. This consists
of an objective lens, beam splitter, sensor mounts and
several mirrors. In the course of this project such a
test bed was designed and built. It proved satisfactory
for our purposes. The output signals were measured and
plotted and, in some cases, displayed on an oscilloscope
and photographed.
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3.0 Detail Technical Discussion.
3.1 Autocorrelation System for Autofocus.
The basic principles involved in the autofocus sys-
tme, utilizing the autocorrelation approach, were described
in Proposal N614-22 and will not be repeated here. 25X1
This approach could not be used with the solid state sensor
of the "Photoconductive sandwich" type. The reason for
this was described in the interim progress report and
briefly mentioned in the introductory section above. The
following points should be noted:
1) several correlation panels were made and tested;
2) these panels, since they are not true product
correlators, could not be used for the autofocus
application, but they are nevertheless applicable
for the automatic registration of two images;
3) if, at some future time, true product correlator
panels could be prepared, they would be of use
for automatic.focusin.g.
The correlation panels which were made were of the
sandwich.type and were prepared by imbedding the photocon-
ductive powder in a silicone rubber binder. A typical
result obtained is shown in Figure 1. This is the correla-
tion curve of a single bar. Correlation curves of periodic
bar targets were also obtained with the experimental panels.
The results were in accordance with theory.
A fair amount of theoretical work was done in order
to=;predict the behavior of these correlator panels for
different target images. This wort is mostly mathematical
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z 10 -4 I& 22 z(,
CAAGIJ -t .oaf?
CoRREL~,-Ttow~ CURVE
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and will be found in Appensix A. This theory is still
applicable to the problem of automatic registry of images
by the use of these panels. The results of this math-
ematical analysis are as follows:
1) The correlation curve has a maximum when the
images are in register.
The sharpness of this correlation peak depends
on the amount of detaLl contained in the image.
With more detail, the peak will be tharper.
3) The sharpness of the peak also improves as the
average contrast in the image increases.
3.2 Non-Linear Photosensors for Autofocus.
The approach which was finally adopted for implement--
ation of the autofocus system was to utilize a non-linear
photosensor. In the progress report it was stated that
"it may be shown" that either a sublinear or superlinear
panel may be used. We wish to presently demonstrate this
fact, inasmuch as it forms the very basis of the system.
To facilitate the analysis, only a point image and Gaussian
spread function will be considered. The proof may, however,
be extended to any image and any realizable spread function.
If a point image is defocussed by a system having
a Gaussian spread function, then the resulting distribu-
tion of light intensity on the panel will be given by (one
dimension is treated for ease of manipulation),
L.
Z el-
~~ zrr
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If the panel has an n-th power :law response, then the
output signal will be,
where we had set,
171X
( 2rrr Q-
,1i -- (-LTr) C - i~ ,,~?~ -~,.
We now wish to study the shape of the output signal, S,
as a function of the width of the spread, Cam' , and the
power law, '7L . If we plot S as a function of 0
with /?L as a parameter., it is apparent that the curves
_7_
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have a.peak or trough at (5"= C> . For a superlinear
panel L > t) the output signal will peak when
for a sublinear one /n C %) the curve will be a minimum.
Of course, 0"- = C> , corresponds to the least spread
or best focus. Another way to see this is to differentiate
CI S
ccT
For, c7 , small the slope of the curve will be propor-
tional to ( I -'N-) . This is positive for t and
negative for ''1.7 1 . A point worth noting is that the
slope near d"` = 0 , is,
A'S _ l--_
Cr ( x`17-~ .~ '2 cr
This increases very fast with 'f'L . Thus, the more super-
linear the panel is, the better its sensitivity to focusing.
3.3 Photosensor Development.
3.1 Non-Linearity in Photoconductors.
When radiation of sufficient quantum energy is ab-
sorbed by a semiconductor or insulator, an increase of
electrical conductivity results. The effect of the radia-
tion, or light in our case, is either to increase the density
of free carriers or to decrease the resistance of barriers
in the material. When barriers are not present, the absorption
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of light ionizes host-crystal atoms, producing free elec-
trons and free holes, or it ionizes impurity atoms, pro-
ducing free charge carriers of only one sign. The presence
of barriers creates space charge layers which inhibit the
passage of current. The action of light in this case is
to decrease the barrier height. A figure of merit for a
photoconductor is given by the product TA. Here, T is the
lifetime of a free charge carrier, and g is the mobility of
the charge carrier. The mobility is a function of the
bulk material, whereas the lifetime is dependent upon the
imperfections. Usually a distinction is made between primary
and secondary photocurrents. A primary photocurrent is
produced directly by the ionization. The free carriers
travel toward the electrodes, where they are absorbed.
If there is an ohmic contact to the electrodes, current
may be supplied from there to the photoconductor, in re-
sponse to the primary current. These are secondary cur-
rents. The secondary current continues until the charge
carrier is re-absorbed by an impurity or by a carrier of
opposite sign. The presence of secondary currents gives
the maximum limit to the photosensitivity.
At a given light input, the photocurrent as a function J
of applied voltage may be divided into three regions. At
low voltage the current varies faster than linear, some-
times as the square of the voltage; at medium voltage,
the relation is linear; at high voltage the photocurrent
saturates. At constant voltage, the photocurrent as a
function of light input may usually be divided into two
regions, in each of which the photocurrent varies as some
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power of the light input. The low level region is generally
sufficient for our purposes. In many photoconductors, the
photocurrent varies linearly with light input. This is
characteristic of materials in which only one free carrier
is involved. In others, the current varies as the square
root of the light input. This is characteristic of materials
in which both free carriers are involved. In still other
materials, the variation of the photocurrent with light
input is some power between 0.5 and 1.0, especially 0.7
to 0.9. This is caused by traps,; the variation is sensi-
tive to the distribution of the traps. In several materials,
the variation is greater than linear; powers as high as 3
have been found experimentally. This results from an in-
crease in sensitivity with increasing light intensity, caused
by a decrease in the probability of capture of a free car-
rier by carriers of opposite`tsign.
3.3.2 "Binder" Panels.
In the first attempt at demonstrating system feas-
ibility, the photosensor used was a photoconductive layer
which was prepared by dispersing CdS powder in a silicone
rubber base. To get the proper consistency, a mixture of
t} grams rubber to one gram of powder was usually satisfactory.
Just before spreading this onto a NESA slide, a drop of
catalyst was added to the mixture. A "Doctor's blade"
device was used to obtain a layer of controlled thickness.
By controlling the doping of CdS, it was hoped to obtain
the desired non-linearity. Doping with copper in amounts
ranging from one part per million to a thousand parts per
million was tried. With the high concentration of dopants
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super linear response was expected. The results with all
the binder panels showed, however, a sublinearity of about
n - 0.5. This was nevertheless sufficient to show that the
method works in principle. It was demonstrated to the proj-
ect personnel that a minimum signal output is obtained for
sharpest focus.
Unfortunately, the "binder" panels had the following
drawbacks:
1) Noisy signal output.
2) Drift with age and temperature.
3) Non-uniform aging rates for panels prepared in
identical manner.
14) Sublinear response.
Non-uniformity of response at various areas on
the panel.
6) Low light to dark ratios.
7) Unreliable contacts between electrodes and photo-
conductor.
In view of these problems, it was decided to accelerate
the development of the sintered photoconductive layers for
the autofocus application. Most of the problems with the
binder layers were thought to be due to the fact that each
powder particle was coated with a thin insulating layer of
rubber. Conduction across such "blocking" barriers may be
responsible for the effects that were observed.
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Sintered Panels.
The problem of "blocking" interparticle contacts is
avoided by the use of sintered photoconductive layers. These
are prepared by adding a "flux" to the photoconductive powder
spreading a layer of powder onto the NESA slide, and then
firing this layer for about six minutes at a temperature
of 590?C. It is generally believed that the CdS is slightly
solubld in the flux material at the sintering temperature.
This ensured good interparticle contact. Sintered panels
prepared in this manner were greatly superior to the binder
panels and enabled us to obtain good autofocus performance.
Still, there remain the following two problems to be solved
before the sintered panels can be used effectively in an
operational autofocus system.
1) Elimination of pinholes in the sintered layers
or alternately to perfect a method whereby a uni-
form electrical contact can be applied to a layer
with pinholes, but which does not short-circuit
the panel.
Perfecting the technique for adding the flux so
that the flux concentration is uniform over the
area of the panel.
Despite these two outstanding difficulties, we found
it possible to select some panels which were sufficiently
devoid of-these problems to allow their use for the tests
of the autofocus system.
3.4 Testing and Evaluation.
In the course of the program, an extensive series of
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tests were performed on the panels in order to assess their
quality. Approximately 50 panels were made and most of them
were tested for their photo response, uniformity and linear-
ity. Those which were found promising were then used in
the autofocus setup to test the whole system, using the given
panel. It is not the intent here to reproduce this extensive
data but rather to show a few typical examples and to describe
the test procedure.
3.4.1 Photo Response..
The setup used for measuring photo response is shown
in Figure 2. A light source is projected by a lens and beam
splitter onto the panel under test, and to a photomultiplier
tube. For a given bias,e, on the panel, the photomultiplier
current is measured by reading the voltage, Vi, across a
load resistor. Similarly, the current through the panel
is measured by reading the voltage, V. across load resistor
RL. Since the photomultiplier is linear with light input,
VI will be proportional to the light flux falling on the
panel and photomultiplier. By plotting V against VI as the
light intensity is varied, we obtain the photo response.
Figure 3 is a set of photo response curves for a typical
binder panel. One electrical contact was the tin oxide layer
and the other contact was pressure backed aluminum foil.
In Figure 3 the y-axis is proportional to the signal current
and the x-axis to the light input. Two points are worth
noting: The photo response is non-linear (sub linear), and
there is a difference in the photo response for forward bias
as against reverse bias. This is to be expected, inasmuch
as the aluminum to semiconductor contact forms a rectifying
junction. In order to obtain ohmic contacts, metals such
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9 Ve
RIZ SPo PS@ ~EArS~1I~ ME1r~'t S'( STEM
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O Fa 1~-K-+\R D Fi 1 A S
?C RCVFRSF B 1~S
aL o.4 ai 0.6
I LLVMt p lc~w~
/, o /. 2 1.s-
Fla's F4bTo RESPONSE OF BINDER Pr.OEL
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as gallium, indium or antimony would have to be used. For
some applications the rectifying junction may be of advantage.
3.4.2 Current-Voltage Tests.
To plot the current-voltage characteristic of the
panel, a setup is used which is similar to the one just de-
scribed. The light intensity is kept constant and the bias
voltage is varied. Figure 4 shows an i-v plot for a typical
binder panel. This is repeated for several light intensities.
For the curves in Figure 4, the light intensity was varied
by insertion of neutral density filters in the light path.
3.4.3 Linearity Tests.
Either the photo response data described in section
3.4.1, and shown in Figure 3, or the i-v curves of Figure
4, can be used to obtain the degree of non-linearity. This
is best done by plotting photo response on log-log paper
and measuring the slope. Such a plot is shown in Figure 5.
The data for these curves was obtained from the i-v character-
istic of Figure 4. For this binder panel, the power law was
approximately square root. The average slope was m = 0.55.
3.4.4 Uniformity Tests.
A test was made to check uniformity of response over
the surface of the panel. For this test the panel was mounted
in a microscope cross slide and a point source was imaged
onto it. The panel was moved in x and y and the signal cur-
rent monitored. Typical variations over the surface were
about a factor of 4, even though some selected panels showed
variations of less than 25% over a one square inch area.
This non-uniformity could not be completely ascribed to the
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Vr o--,
1.5
to
30
\/ (v.LTS)
Ttk}kN S N\ SS%ab)
Ti
ut gv aT V , , r -~Q s. ~~. .~ ...
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7
J7._f +1 7y_
_tJ
-TH
1.5 I- - -
6
1 .1 w W1
1 111 1 11 Iii-H 11 iliAMME 111-11111 IN 11 if 114
L ~~ ? 10
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photoconductor. With the pressure backed aluminum foil
electrode, the electrical contact was not quite uniform,
and this may contribute greatly to the observed variations
in response. This is expected to improve dramatically by
using evaporated indium contacts.
3.4.5 System Tests.
Suitable panels were selected by checking linearity,
uniformity and sensitivity, as described above. These panels
were then used in the autofocus test system to evaluate their
performance. The test system is shown in Figure 6. A slide
is illuminated by a tungsten source and condenser lens. The
slide was mounted on a rack and pinion arrangement. It was
focused by the objective lens and beam splitter onto the
two panels. The output of the two panels (Sl and S2) could
be monitored separately. These two outputs were also fed
to a differential amplifier arrangement whose output S could
also be recorded. The procedure was to move the slide through
focus and record the three signals S, Sl, and S2. The signals
Sl and S2 show the response of the sensors to changes in
focus, while S is a signal suitable for use in a feedback
servo system to bring the slide back into focus automatically.
4.0 Results.
Using selected sintered panels, we were able to obtain
good autofocus performance. Figure 7 shows the final results
obtained. As such, they represent the major achievement
and success scored in this development. Figure 7a shows
the output signal from one panel, while the differential
output, S, is shown in Figure 7b. As can be seen from the
oscillograms, both the signal-to-noise ratio and sensitivity
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 E ^ E E ! F ! f 01 III I
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L r11*T
Soultct
St_j aF
o'bJcc.Tws
LENS
og
Pw- Dt ERE WTtr4L
V~u VF %V
S
ALT0 Focv S TEST Sf5'CEN\
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4fferential. Amplifier Output
Output of One Panel
J`~I.I~IVA
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to focus is adequate. The results of Figure 7 were obtained
with the best pair of panels.
In Figure "7 the vertical sensitivity is 200 millivolts
per cm. The horizontal sensitivity is 0.077 inches of object
motion per cm. The measured noise was about 20 mV. The
sensitivity of the discriminator signal, as can be seen on
the oscillogram, was 4.25 volts per inch of object motion.
The focusing accuracy should, therefore, be
noise voltage 0.020
6 Y, sensitivity 4.25
11 ~~ o
' = ~ 5 = T
ti
b ,
l20 p .
0.0047
This performance was btai:ried with continuous tone
imagery of moderate contrast. Even better performance was
gotten with a point source object.
5.0 More Recent Developments.
After the conclusion of work on this project, 25X1
has continued its development work towards attaining improved
panel performance and more reproducibility between panels.
The following techniques proved very successful:
1) Sequential sifting of powder to obtain smaller
grain sizes.
Use of unfluxed powder and addition of the flux
after deposition and before sintering.
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Improved settling technique whereby the settling
is done through a higher liquid column and per-
formed in several stages.
Evaporated metal electrodes.
In addition to these, work is in progress on another
method for preparing panels which holds great promise. This
involves a technique for spraying the photoconductor onto
a hot substrate and a subsequent firing at about 1200?F.
This looks so promising that we delayed this report in the
hope of being able to communicate first results obtained with
this method. These, however, have been delayed and will be
forthcoming in another three weeks.
5) Additional specialized test equipment was built
which greatly facilitates the evaluation of panel
performance.
6.0 Conclusions.
Using sublinear binder panels, it was possible to
demonstrate the basic system principles. With the advent
of the sintered panels enhanced performance was possible.
The operation of an autofocus system was demonstrated and
a signal suitable for driving a servo system was obtained.
This, we feel, fulfills all the expectations that were an-
ticipated at the outset of the project.
7.0 Recommendations.
In view of the fine results achieved thus far, it
is felt that the work be continued with the aim of attaining
an operational autofocus system to be incorporated into an
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existing projector. This work, it is recommended, be
divided into two phases: a sensor improvement phase
and an engineering design phase.
Under the sensor improvement phase, the following
performance objectives should be set:
1) Resolution of 20 linen per mm in the sensor
plane.
2) Autofocus sensitivity of 30 microns.
3) Signal-to-noise ratio of 50 to 1.
4) Super-linear response with a square or higher
power law.
Light to Dark impedance ratio of 500:1.
6) Rise time of 10 milliseconds.
7) Fall time of 5 milliseconds.
8) Uniformity of response of + 5% over any 16 square
millimeters, and + 20% over 600 square mm.
9) Drift to be less than 15% per hour.
In order to achieve the above performance goals, the
following techniques should be explored:
A. Sintering Technique.
1) Multiple layer sintering to eliminate pinholes.
2) Grazing angle contact evaporation onto the
sintered layer to prevent low impedance paths.
3) Sintering in a nitrogen atmosphere at elevated
pressures.
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4) Spray application of flux material.
Spray Application Method.
1) Application of CdS and CdSe powders by spraying
the powdered suspension onto the substrate.
2) Application of photoconductor by spraying liquid
mixture of chemicals containing Cd and S ions, as for
example CdC1 and Thiourea.
C. Evaporation of CdS by a vapor phase reaction.
Under the engineering design phase the following object-
ives should be attainable:
1) Design amplifier, stabilizer, discriminator, and
servo circuits to stabilize and compensate for any resid-
ual drifts until these are reduced to less than be-
tween pairs of panels.
2) Design servo system to have a response of 10 cps.
3) Modify projector mechanically as needed to incor-
porate the autofocus system and provide for manual
override.
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Appendix A
For the solid state correlator the output signal is
proportional to
where T is the amount of misregistry. It would be of interest
to evaluate this output signal for a typical image. Prefer-
ably, we should like to describe the typical image in such
a way that the description may fit a wide range of imagery,
at least in a statistical sense.
To do so, let us assume that the image is made up of
rectangular objects of size n x m. Let us further assume
that there are a whole range of such rectangles, each of
different dimensions, and that each such rectangular spot
has a gray shade u. Furthermore, we presume that the shade
of gray is independent (statistically) of the size patch.
The distribution of gray levels is given by p(u); i.e. the
percentage of area having gray shades lying between u and
'A -t Q U is P t"~ pLt . We also have to assume some dis-
tribution function for the patch sizes. A convenient dis-
tribution which is close to reality is, __
.e 4L
where eh, is the average patch length in the x-direction
and .`m in the y-direction. Actually, since the n and m
are statistically independent we can work with one dimension,
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save ourselves some mathematical complexity, and then gen-
eralize to two dimensions. We thus take,
-n
e
c ?
If we now have some function f(x) which consists of a series
of these patches of varying gray shade and the same function
shifted by an amount i. then some patches having the same
shade of gray will overlap. All those patches which are
larger than T will, in fact, overlap partially. Let us now
calculate this percentage overlap, G.
The percentage of any one overlapping patch will be
The total amount of overlap will be the percentage overlap
for each patch and summed over all patches larger than T.
Thus,
j )_' "71- ]
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We now have to evaluate the signal
This may be written as,
-gyp-W
-tr -l- LAX
where the bar denotes the ensemble average and where we
abbreviated,
For a statistical image of the sort being considered, the
integration is, of course, an averaging operation.
There are two choices now: either v = w or v w.
When a pulse overlaps (for n 7 %) v - f (x) &- w = f (x+T).
Otherwise, v and w are unrelated. Hence, in order to obtain
Z we can treat the two cases separately. For v - w,
Z0 -= -i p , -ti--\ . ..t,r
`2..
-Er
Hence, Ta
't-UCH Cr
A-
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What is left to be done is to evaluate Z for the case
when v and w are different and thus statistically independ-
ent. This is given by
-1T-f- VLr
For any arbitrary distribution of gray levels p(v), this
is a most complex integration. For some simple cases,
however, this can be worked out. Let us take a uniform
distribution of gray shades,
Then,
_L_j j -r (Ar j -,r j W
Cy ip %Ar
This integral can be evaluated and shown to be,
0. 2 C.
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Hence, the complete expression now becomes,
0-25
e
/t'i
-f- 0. 2 C (/
2 C
For two dimensions, this becomes,
In this manner we can predict what the correlator signal
will be for an image whose average patch size is n x m
and whose range of gray tones is from zero to C. This
curve peaks at T r 0. The sharpness of the peak depends
on and ii, as well as on C.
A-5
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