AID-A1i51 711 RAYLEIGH CRITERIA RESOLUTION OF OPTICAL CORRELATIONS 1/1(U) ARMY MISSILE COMMAND REDSTONE ARSENAL AL RESEARCHDIRECTORATE D A GREGORY SEP 84 AMSMI/RR-84-i@-TR

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* U TECHNICAL REPORT RR-84-10

RAYLEIGH CRITERIA RESOLUTION OF OPTICAL CORRELATIONS

Don A. GregoryResearch DirectorateUS Army Missile Laboratory

SEPTEMBER 1984

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Rayleigh Criteria Resolution of Optical TechnicalCorrelations

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Don )L. Gregory1

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Commander, US Army Missile Command A U NM

Attn: AMSMI-RRRedstone Arsenal, AL 35898-5248

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This report describes an investigation into determining the resolv-

ability between correlations produced by Fourier transform matched filtersmade of very similar scenes. Knowledge of this type is needed to accuratelypredict the total number of matched filters needed to completely identifya scene or object regardless of its rotational position. Similar experimentsmay be done for scale and tilt as the variable. This report introduces theRayleigh criteria as the standard by which distinguishability may be measuredin a Vander we

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ACKNOWLEDGEMENT

The author wishes to acknowledge the able assistance of James C.Thompson, currently of Auburn Universit', for his devleopment of the plottingroutines used in this report.

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TABLE OF CONTENTS

PAGE

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

III* EXPERIMENTAL RESULTS .......... .**.* .. o.o.o... o.o*...*.... .o 4

IV OCLSO.........................

I. INTRODUCTION

Considerable interest has been generated in using Fourier transform

matched filters for object discrimination and tracking. In order to recognize

several objects or several orientations of the same object many matched

filters may be needed.' Interest has been shown in storing several of these

filters in a small area [1,21. This has been done by multiply exposing the

photographic plate or by using arrays produced by holographic elements orother means [3,4]. Arguments are presented here which investigate how closely

packed these filters can be and still have good resolution between them. Inthis instance close packing refers to double-exposing a photographic plate toan input scene at two slightly different rotation positions. Classic line-

shapes are used to model examples of experimental results and some conclusionsare drawn using the Rayleigh criteria as a guideline.

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II. THEORY

The Gaussian intensity distribution has the form

2 2II = Io e-2r /w (.)

where r is the radius from the center of the distribution and 2w is the full

line width at I =(I/e2 )lo. If another distribution of the same form is

located a small distance 61 away

12 - Io e-2(r-61) 2/w2 (2)

then the sum of these distributions is

I 11 + 12 10 [e-2r2/w2 + e -2(r-61)2/w2] (3)

LetI (r=O) . ,:

Pl I(r= c1 /2) (4)

Substituting from the above gives

(el /2w2 e3612/2w) ((5)i

P, 1/2 e+ e 5 ii

P1 is thus the ratio of the peak at r = 0 to the height of the valley betweenthe two distributions. Solving for 61 using the fact that the second term ismuch smaller than the first gives

61 w [2 Ln (2p) ]2. (6)

2

..........................................-.-.~...-....--.....-..............-.. . ..- ". ' " " " ' " " ' ' " -" " ' -" - " -" - " ' -' ." " " - ." ' ' " - ' " -" - , " " " "".". ." ".. . " 4 " " " " " ." "

' ': ' :

Using the Rayleigh criteria as a rough guideline, P1 = 1.23 and substitutingthis into Eq. (6) yields

61 1.34 w . (7)

The same sort of analysis may be applied to a Lorentzian distribution havingthe form

A + A

2 2 2 2 (8)r + (a/2) (r-6 2) + (/2)

where A is a constant equal to 10(o/2)2 , r is the radius and a the full

linewidth at half the maximum intensity. This produces a 62 of

62 o/2 4P2 -3 +416P2(P2-1)+l] 2 (9)

Then for P2 1.23 as a value of the intensity ratio

62= 1.03 a (10)

For a sinc 2 distributionF 12

.21 sinr 2 sin Q (r-6 3)

0 ok0r a(r-6-) 3(I)

where again r is the radius from the center of the distribution and

a = /r(o) (12)

where r(o) is the value of r at the fist zero of intensity. This distribution

yields

3 + sin2 63 (13)

8 sin 2 (a6 3/2)

which must be solved graphically or by iteration using P3 = 1.23. For smallvalues of 63 , note that a2 632 > > sin263 '

3* *~ * -* *-.-X*. ...- ,.

V W P. 7 ..

III. EXPERIMENTAL RESULTS

Figure I shows three distributions compared with experimental data taken

using visible (Helium-Neon) real-time optical correlation methods. The inputscene is fed via an RCA television camera (model number TC 1005) into a

Videotek Monitor (model VM-12PR) which is used as the input to a Hughes ..

liquid crystal light valve (the same one used in Ref. 4) which produces the

coherent image that is fed into a real-time Vander Lugt type correlator. - -

Experiments of this type are described in detail elsewhere [5]. All threedistributions have been arbitrarily fit to experimental data at r - 0 and r =

2.940 using w - 2.80, a - 2.07, and a = .775 deg-1 . This gives 61 = 3.80,

62 - 2.13, and 63 - 4.0% Note that these widths should depend upon the

spatial frequency of the input scene as well as other factors and thus serveonly as examples here. In this particular experiment the input scene was alow resolution black and white aerial photograph of Huntsville, Alabama. Thephotograph was placed on a rotatable stage and matched filters made at the

desired rotation angles. Experimental data in Figure 1 is the correlation

intensity for a filter made at 00 rotation of the input scene. After develop-

ment of the film plate (Kodak 649F) it was replaced in the correlator and the

input scene rotated slowly as the corresponding correlation intensity wasmeasured using an RCA CCD camera fed into a television monitor then digitizedby a Colorado Video Analyzer (model 321). This signal was then recorded by a

Hewlett-Packard model 680 strip chart recorder. Figure 2 is representative of

actual data taken as the input scene was rotated through the position (0

rotation) where the matched filter was made.

In order to investigate multiply stored filters, a single plate was

exposed twice with different rotation angles of the input scene. After deve-

lopment using standard techniques for Kodak 649F plates, the filter was rein-

serted into the correlator and data taken as before. Figure 3 shows the

actual data and a plot of Equation 8 using the Rayleigh criteria. The expo-

sures were made using the same criteria for 62 of 2.130 of scene rotation.

The agreement is reasonably good. Some variation is expected due to the dif-

ficulty in obtaining equally intense correlations for the two filters. This

depends upon the exposure times used, film response, and several other

variables. In the theoretical analysis the calculations could easily have

been done using distributions having different peak heights but then the value

of 6 would depend upon these heights which can't be known with any certainty

in advance of doing the experiment. In the calculations presented, it was

also assumed that individual correlation intensities should be added directly

rather than adding the corresponding fields then squaring for the intensity.

This has been justified experimentally by storing one filter made of 00 scene

rotation and another filter at 900 scene rotation then adding the resulting

correlation intensities algebraically. This distribution was then compared

with data similar to Figure 3. The results were found to be essentially

equal. This indicates that there is little if any phase addition contribution

to the sum of the two correlation intensities.

* - -.. .....

"...'' .' *.. .',s.e, '. .... .... . . . ............................................. .......... .. . . .,..

IV. CONCLUSIONS

In this brief report a method of analyzing closely spaced optical matchedfilters has been presented along with experimental data which tends to supportthe findings. The use of the Rayleigh criteria may not be strictly correct in

that this criteria is for a Bessel function intensity distribution only but it

serves as an initial guideline [6]. Experimental data seems to suggest that theseparation criteria will become more stringent as laboratory methods improve.

5

.1h . d . . .

13

12 '

9..Gaussian

Lorentzian

27" 51nc

6 ~5 ** * * Experimental

2 1 23

SCENE ROTATION (DEGREES)

Figure 1. Correlation intensity vs angle of rotation of input scene

6

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References

1. P. Meuller, Appi. Opt., Vol. 8, No. 2, 1969 (297).

2. K. Lieb and J. Mendelsohn, Proc. SPIE, Vol. 317, 1981, (148).

3. K. Lieb, R. Bondurant, S. Hsiao, R. Wohiers, and R. Herold, Appi.

Opt., Vol. 17, No. 18, 1978, (2892).

4. H. K. Liu and J. H. Duthie, Appi. Opt., Vol. 21, No. 18, 1982, (3278).

5. J. Smith, B. Guenther, and C. Christensen ,J. Appi. Photo. Engr.,Vol. 5, No. 4, 1979, (236).

6. E. Hecht and A. Zajac, Optics, Addison-Wesley, Reading, Massachusetts,1974, p. 352.

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