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Page 1: Gaussian–minimum average correlation energy filters

Gaussian-minimum average correlation energyfilters

David Casasent, Gopalan Ravichandran, and Srinivas Bollapragada

Correlation filters with sharp 8-function correlation peaks [such as phase-only filters and minimumaverage correlation energy (MACE) filters] do not recognize images on which they are not trained. Weshow that the MACE filter cannot always recognize intermediate images of true class objects (e.g., aspectviews or rotations midway between two training images). New Gaussian-MACE filters offer a solution tothis problem.

I. Introduction

Distortion-invariant pattern recognition is a classicproblem with many possible solutions. We considerthe synthetic discriminant function (SDF) class ofdistortion-invariant filter' used in an optical correla-tor. We restrict our choice to this filter since itreceives the most attention, has been widely tested,and allows full three-dimensional distortion invari-ance (not merely in-plane distortion invariance). Weconsider the newest filter of this type, the minimumaverage correlation energy (MACE) filter.2 It has beentested in noise"4 and extended to multiclass prob-lems.5 In Section II we review the basic MACE filter.We then (Section III) present the database to be usedand show (Section IV) that the MACE filter cannotalways recognize all distorted images of one class (i.e.,it can recognize images with 100 increments in rota-tion or aspect but cannot recognize all intermediateimages at 50 increments midway between two train-ing images). We expect similar results from all filtersthat produce sharp 6-function correlation peaks, e.g.,many phase-only filter designs.6 In Section V weintroduce the Gaussian-MACE (G-MACE) filter andshow (Section VI) that it can overcome this intraclass(within-class) recognition problem. A brief discussionand conclusion are then given in Section VII.

The authors are with the Center for Excellence in Optical DataProcessing, Department of Electrical and Computer Engineering,Carnegie Mellon University, Pittsburgh, Pennsylvania 15213.

Received 30 July 1990.0003-6935/91/355176-06$05.00/0.0 1991 Optical Society of America.

II. MACE Filter

We consider the MACE filter because it is the onedistortion-invariant filter that controls the responsein the full correlation plane while allowing the corre-lation peak value to be specified. (Prior SDF's onlycontrolled the response at the correlation peak or inthe vicinity of the correlation peak.') The MACE filterachieves full correlation-plane control by minimizingthe average correlation-plane energy (hence the nameMACE). It thus produces a sharp and easily detectedcorrelation plane peak with minimum sidelobes andfalse peaks. It also automatically performs linearpreprocessing (without the need for ad hoc edgeenhancement and other techniques).

The filter is synthesized in the frequency domain asH from the Fourier transform (FT) X, of a set oftraining images x (distorted versions of variousobjects). Uppercase letters denote FT's. We let Di be adiagonal matrix with elements equal to the square ofthe magnitude of the components of the FT X, of thetraining image i. The sum of the Di for all i is anotherdiagonal matrix D. To control the correlation peakvalue for each xi, we use a control vector u whoseelements ui specify the peak value for each i (typicallyUi = 1 for the true-class objects to be recognized andui = 0 for false-class objects to be rejected). Thus Hmust satisfy the peak constraint condition

H+X, = ui (1)

for all i, where + denotes a conjugate transpose. In amatrix-vector notation this becomes

H+X = u, (2)

where X is a matrix whose columns are X, To

5176 APPLIED OPTICS / Vol. 30, No. 35 / 10 December 1991

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minimize correlation-plane energy2 we write it as

E = H+DH (3)

in the FT plane to allow it to be described in terms ofH. The MACE filter is the function H that minimizesEq. (3) subject to the constraints in Eq. (2). TheLagrange multiplier solution2 is

H = D-X(X+D-'X)-'u.

This specifies the filter in the frequency domainwhere it is used. The operator D- in Eq. (4) performsthe automatic preprocessing. No iterations are usedin our H. (This was used in the original work.2 ) Notethat the MACE filter is not specified to be a linearcombination of the training set, as are SDF's.

Ill. DatabaseThe database we used in these tests consisted ofSynthavision solid models of two Soviet missilelaunchers. The ZSU-23 object was a top-down 90°image at two resolutions (64 x 64 and 32 x 32 pix-els). The SA-13 object used was at 64 x 64 pixelresolution with two different depression angles: 900(top-down) and 300 (from the horizontal). For each ofthese four object cases, we used a training set of up to36 images at 10° increments in rotation or the aspectview at the fixed depression angle. In tests we usedthese 36 images plus an additional 36 images atintermediate 5 rotations or aspects midway betweentwo adjacent original training images. We consideronly intraclass (one-class) problems since our presentattention is on achieving the recognition of most ofthe fixed 36 images and the 36 intermediate images.Figure 1 shows one view of each of these four objectcases. The number of gray levels present are limited,because they are caused only by the lighting effectsincluded in the model. Both missile launchers aresimilar in size. In all correlations the input was 64 x64 (the 32 x 32 image was padded with zeros), thefilter was 128 x 128, and the correlation plane was192 x 192.

IV. MACE Test Results

We formed a one-class MACE filter for each of thefour object classes and tested it versus the 36 originalimages and the 36 intermediate images within theassociated one true class. Each filter was synthesizedfrom only the 36 original images. The training imagesused from these 36 images were chosen by addingnew images and forming a new H filter until all 36original images gave a correlation peak above thethreshold T at the proper peak point. The image withthe largest energy (typically the broadside or 0 view)was used as the first training image. As the secondetc. image to be included in the filter, we used the onewith the lowest true correlation peak value with thepresent filter. We used ui = 1 for the true peak valueand thresholds T = 0.6, 0.65, 0.7, and 0.75.

We formed two MACE filters for each of the fourobject cases (with or without a constant training

(b)

(C)

(d)Fig. 1. Representative Synthavision image examples of the fourobject classes: (a) ZSU-23, 900, 32 x 32; (b) ZSU-23, 90°, 64 x 64;(c) SA-13, 900, 64 x 64; (d) SA-13, 30°, 32 x 32.

image included). When the constant image was in-cluded, it was an image with constant transmittance(the constant was chosen to make the energy of thisimage the same as that of the 0 or broadside objectview), and for it u = 0 was specified. This techniquewas noted elsewhere7 to be used in improving thesignal-to-noise ratio (SNR). However, the SNR defini-tion used (specified correlation peak values divided bythe mean of the noise at one point) was improper. Thevariance of the noise (,,2) should be used in the SNRdefinition, and the use of a constant training imagewill not improve the defined SNR (since u0 is unaf-fected by it). Our reason for the use of a constanttraining image is to improve the correlation planepeak-to-sidelobe ratio and to make the correlationindependent of the bias level of the input image. Wehave demonstrated that this occurs.8 It has been

10 December 1991 / Vol. 30, No. 35 / APPLIED OPTICS 5177

(4) (a)

Page 3: Gaussian–minimum average correlation energy filters

shown2 that the MACE filter performs linear prepro-cessing. However, the zero-mean filter constraint isnot a linear operation. The G-MACE filters we con-sider here have less discrimination than standardMACE filters, and so we also use the zero-mean filterconstraint to improve their discrimination. This con-straint is also needed to produce the zero-meaninputs needed with binary and ternary spatial lightmodulators9 with 0 and ± 1 transmittance values (toavoid the presence of an image of the input appearingin the correlation output when a phase-only filter isused).

We also include the filter energy Eh (the sum of thesquares of the pixel values in H) in our results. (Thisis the filter energy not the correlation-plane energy.)Our motivation for this is that a low Eh filter ispreferred for good performance in white Gaussiannoise. This is easily shown by considering an input fwith noise n as

g=f+n. (5)

The correlation peak value with a filter h is hTg, andthe detected intensity value is

c = I hT f 12 + I hTn I2 + 2h Tfh Tn. (6)

To consider the effect of noise we form the expectedvalue of Eq. (6), assuming that f and n (and h) areuncorrelated. If the noise is zero mean, E~n} = 0 andthe last term in Eq. (6) is zero. The real noise in theinput will be present on a bias (since g is real andpositive), and our use of a constant training imagethus removes the bias and makes this assumptionvalid. For white Gaussian noise the second term inEq. (6) becomes un2Eh, where cr is the standarddeviation of the noise and Eh = hh is the filterenergy. Thus we obtain

Ec) = IhTf 12 + an2Eh = 1 + °,nER, (7)

where we have assumed that h was designed toproduce a noise-free correlation peak value of 1. FromEq. (7) we see that a low filter energy Eh is preferred,as the correlation peak value obtained is then closerto the ideal value. A further analysis shows that Eh

also similarly affects the rest of the points in thecorrelation plane.

Table I shows the test results obtained for the twocases for each of the four object cases with thethreshold T = 0.6. We indicate the number of trainingimages NT required for the filter to recognize all 36

original images (NT is a maximum of 36), the numberof intermediate images N, recognized (N, is a maxi-mum of 36), plus Eh and the filter mean. These datashow that the MACE filter requires many trainingimages (NT is large) and that it does not recognizeintermediate images (N, is very small). Specifically, inno case are more than N, = 2 intermediate imagesrecognized (out of a maximum of 36), and generallyall 36 original images must be included in trainingand filter synthesis. In Test 1, NT = 26 is less becauseof the symmetries of the images that make severalviews similar at the lower resolution used. We expectthe Test 1 data set to be easier than the Test 2 set(because of its lower resolution, which makes theobject views more similar). The lower NT value (26versus 36) verifies this. We expect the Test 3 object tobe easier than the Test 2 object (since the SA-13 hasless detailed structure on the images used than doesthe ZSU-23). The larger N, = 1 versus 0 value verifiesthis. We expect the Test 4 object to be easier than theTest 3 object since aspect views are more similar witha 30° depression angle (since common object partsexist) than with the top-down 90° view. (In-planerotations represent more different distorted objectsthan do aspect views at a low depression angle.) Thelarger N, = 2 versus 1 value verifies this.

Filter energy Eh is also a measure of the difficulty ofa given problem. The lower Eh values for Test 1versus Test 2 and for Test 4 versus Test 3 verify this.The filter mean is very small (< 10-7), but the averagepixel energy EhI 1282 is also very small and compara-ble to the mean. Thus, although the filters with noconstant training image appear to be close to zeromean, this is not the case. Eh increases (indicating amore difficult problem) when a constant trainingimage is used. (Test 2 shows this, but the effect issmall for this case.)

We also obtained similar data for all four differentthresholds T. As the threshold T is increased (from0.6 to 0.75), we expect NT to increase and N todecrease and that the effect will be larger when aconstant training image is used (as it produces moredifferent images). For Test 1 we found NT = 27, andN. = 0 for T = 0.75, and for Tests 3 and 4 for T = 0.65and 0.75 we found N, = 0 and N, = 1, respectively (forthe case of no constant image) and N. = 0 (for the casewhen a constant training image was used). Thesedata verify these expected results. All our tests as-sume thresholded correlation peak detection. Wecould reduce T to include more true intraclass peaks.Our proposed new filters allow us to retain a large T

Table 1. MACE Test Data (T = 0.6) Showing a Large Number of Training Images NT and Failure to Recognize Most of the N, Intermediate Images

No Constant Training Image Constant Training Image

Test Object NT N1 Eh (X 10-') Mean (x 10-7) N, N1 Eh (x 10-4) Mean (x 10-7)

1 ZSU-23,900,32 x 32 26 0 0.39 0.44 26 0 0.39 02 ZSU-23,900,64 x 64 36 0 0.42 0.20 36 0 0.43 03 SA-13, 900, 64 x 64 36 1 0.15 0.22 36 1 0.15 04 SA-13,30, 64 x 64 36 2 0.10 0.11 36 2 0.10 0

5178 APPLIED OPTICS / Vol. 30, No. 35 / 10 December 1991

Page 4: Gaussian–minimum average correlation energy filters

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

' 2040 60 0 100 120

(b)Fig. 2. Typical MACE correlation function: (a) 3-D isometric; (b) cross-sectional scan.

and still achieve intraclass recognition of nontrainingobjects.

From these tests we see clear evidence that theMACE filter requires many (most) training images NTand does not generally recognize intermediate imagesN. We found this problem to become more acute asthe object resolution increases, as the object viewapproaches a 900 depression angle (top down), andwhen a constant image is used.5 The magnitude of alleffects clearly will be data dependent (i.e., with dif-ferent objects, fewer training images may be needed).However, the issue that intermediate nontrainingimages may not be recognized is of concern and hasnot been previously noted. It is due to the sharp(8-function) correlation response produced by theMACE filter. Figure 2 shows a typical MACE correla-tion plane (for the ZSU-13, 900, 32 x 32 image) andthe sharp peak (3-dB width of 1.0 pixels) and lowresponse elsewhere (maximum value of 0.135). Weexpect that similar effects will occur with all filterswith sharp correlation functions. Section V presentsthe new G-MACE filter solution we advance. Itspecifies that the MACE correlation peak be Gaus-sian. By varying the width d of the Gaussian correla-tion function, we expect to control the intraclassperformance of the filter. The preprocessing by D-lachieves the sharp standard MACE correlation shape.

This preprocessing is a form of edge enhancementthat whitens the spectrum of the object. In theG-MACE filter the amount of edge enhancement andspectrum preprocessing performed can be controlled.

V. G-MACE Filters

To improve intraclass performance (to increase N1)and reduce NT (thus reducing filter clutter), we addthe additional constraint to the MACE filter H thatthe output correlations have a specified shape func-tion f (or F for the FT of the correlation function);i.e., we minimize the mean-square error term

2, H*(u, v) Xi(u, v) - F(u, v) 12,i 0,0

(8)

where f = exp(-r2 /2d2 ) is a Gaussian amplitudefunction with r2 = x2 + y. The squared error in term(8) is

B = H+DH + F+F - H+AF - F+A+H, (9)

where F is now a column vector whose elements arethe FT of the desired Gaussian correlation function,A is a diagonal matrix whose elements are the sum ofthe components of X\, and D is a diagonal matrixwhose elements are the sum of the squares of the

Table 1. G-MACE Test Results Showing the Recognition of Nearly All N, = 36 Intermediate Images (T = 0.6)

No Constant Training Image Constant Training Image

Test Object Nr d N E, (x 10-4) Mean (x 10-7) Nr d N Eh (x 10-4) Mean (x 0-')

1 ZSU,90 0 32 x 32 9 5 35 0.04 4.66 9 5 35 0.05 02 ZSU, 900 64 x 64 36 6 36 0.16 2.22 36 6 33 0.17 03 SA 13, 900 64 x 64 28 8 35 0.02 3.83 28 8 34 0.02 04 SA 13, 300 64 x 64 15 8 36 0.01 1.83 16 8 36 0.01 0

10 December 1991 / Vol. 30, No. 35 / APPLIED OPTICS 5179

A N

(a)

140 160 180 200

Page 5: Gaussian–minimum average correlation energy filters

I

P..

(a) (b)Fig. 3. Typical G-MACE correlation function with d = 8: (a) 3-D isometric; (b) cross-sectional scan.

components of X,. By minimizing B we minimize thecorrelation-plane energy E. Thus the G-MACE filterminimizes

. = B - 2X,(HX, - u,) - -2XN(H XN - N), (10)

where minimizing the last N terms satisfies thecorrelation peak constraints in Eq. (1). Setting thegradient of Eq. (10) to zero yields

H = D-1AF + D-'XL, (11)

where L is the column vector of the unknownsXA . . N. Using Eq. (2), we solve Eq. (11) for L andobtain the Lagrange multiplier solution for the filteras

H = DW'X(X+D-'X)-lu + D-1 AF

- D-'X(X+D-'X)-X+D-AF. (12)

The output correlation intensity has the functionalshape exp(-r 2/d 2 ). We require that this exceed thethreshold T. This specifies the diameter of the correla-tion at a value T to be 2d[ln(1I/T)]12. As we increase Tthe diameter of the correlation output (at T) de-creases. As we increase d the diameter (at a fixed T)increases. We varied d from 2 to 8. (This resulted indiameters of <6.7% of the object size; thus theposition of the object was well localized.)

VI. G-MACE Test Results

Table II summarizes the G-MACE test results for ourobject cases (two filters for each with and without theconstant training image used) for T = 0.6. Weincreased d from 0 (standard MACE), and for eachnew d we synthesized a G-MACE filter, using theminimum NT required to recognize all 36 original

images. Table II lists these new N values. All arelarge. Thus G-MACE significantly improves intra-class recognition (from the very small N = 0-2 val-ues for MACE in Table I). The results in Table II areas expected. Test 1 requires the fewest NT and thesmallest d and yields the lowest Eh for the ZSU-23object (because of the lower resolution compared withTest 2). In Test 2 we see that NT = 36 is still required,but with a larger d = 6 width all N = 36 intermediateimages are now recognized. For the 300 depressionangle, SA-13 (Test 4) versus the top-down 900 SA-13(Test 3), we find that fewer training images arerequired (NT= 15 versus 28) since the imagery ismore similar. The NT values for Test 3 are less thanthose for Test 2 since the SA-13 object has less detailthan the ZSU-23. The filters that a constant trainingimage show smaller differences (in Test 4, NT islarger, 16 versus 15; and in Test 3, N is less, 34versus 35). A lower NT value results in a lower(preferable) filter energy Eh (Test 1 versus 2 and Test3 versus 4). These G-MACE test results show thatG-MACE requires fewer training images and recog-nizes nearly all the intermediate images. Figure 3shows a typical G-MACE correlation peak with d = 8(as seen, compared with Fig. 2, it is broader).

VIl. Conclusion and Discussion

Two problems have been noted for distortion-invari-ant correlation filters that produce sharp correlationpeaks: the need for many training images and aninability to recognize intermediate images. Theseissues were quantified for the MACE filter, and a newG-MACE filter solution was advanced and shown toprovide preferable results. Only the intraclass (one-class) recognition problem was addressed in ourpresent work. As we improve recognition we expectdiscrimination (interclass or two-class problems) to

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become worse and that advanced solutions, such asinterference filters and symbolic filters, will be needed.It is important to reduce the training set size in allfilters,5 since this reduces correlation plane noise andfilter clutter and because many training images (e.g.,at 10 increments) are not always available.

The support of portions of this work by the NavalSurface Warfare Center is gratefully acknowledged.

References

1. D. Casasent, "Unified synthetic discriminant function computa-tional formulation," Appl. Opt. 23, 1620-1627 (1984).

2. A. Mahalanobis, B. V. K. Vijaya Kumar, and D. Casasent,"Minimum average correlation energy (MACE) filters," Appl.Opt. 26, 3633-3640 (1987).

3. D. Casasent, A. Mahalanobis, and D. Fetterly, "Advancedsymbolic and inference optical correlation filter results," inOptical Pattern Recognition, H. Liu, ed., Proc. Soc. Photo-Opt.Instrum. Eng. 1053,142-154 (1989).

4. B. V. K. Vijaya Kumar, D. Casasent, and A. Mahalanobis,

"Correlation filters for target detection in a Markov modelbackground clutter," Appl. Opt. 28, 3112-3119 (1989).

5. D. Casasent and G. Ravichandran, "Modified MACE filters fordistortion invariant recognition of mobile targets," in AirborneReconnaissance XIII, P. A. Henkel, F. R. LaGesse, and W. W.Schurter, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1156,177-187 (1989).

6. D. P. Casasent and A. G. Tescher, eds., Hybrid Image andSignal Processing II, Proc. Soc. Photo-Opt. Instrum. Eng.1297, 187-240 (1990), session on phase-only correlation filters.

7. H. H. Arsenault, "Improved composite filter for pattern recogni-tion in the presence of noise," J. Opt. Soc. Am. A 3(13), P89(1986). OSA Annual

8. A. Mahalanobis and D. Casasent, "Performance evaluation ofminimum average correlation energy filters," Appl. Opt. 30,561-572 (1991).

9. B. A. Kast, M. K. Giles, S. D. Lindell, and D. L. Flannery,"Implementation of ternary phase amplitude filters using amagnetooptic spatial light modulator," Appl. Opt. 28, 1044-1046 (1989).

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