Reduced-resolution synthetic-discriminant-function design by multiresolution wavelet analysis

Reduced-resolutionsynthetic-discriminant-function design bymultiresolution wavelet analysis

Paul C. Miller

Several approaches to the design of reduced-resolution synthetic discriminant functions 1SDF’s2 usingmultiresolution wavelet analysis 1MWA2 techniques are investigated. In the first approach, reduced-resolution approximations of a full-resolution SDF are obtained by MWA. In the second approach,reduced-resolution approximations of the training-image Fourier transforms are obtained byMWA, and areduced-resolution SDF is obtained directly by training on these. For both approaches, reduced-resolution MICE-SDF filters were designed with MWA and conventional down-sampling techniques.Simulations showed that filters designed by the second approach with MWA techniques permittedreductions in the number of filter pixels from 128 3 128 to 32 3 32, while still satisfying the designconstraints. In comparison, the performances of 32 3 32 filters designed by conventional down-sampling techniques were significantly degraded.Key words: Reduced-resolution filters, synthetic discriminant functions, multiresolution wavelet

analysis.

Introduction

An issue of importance with regards to the transitionof optical processing to systems1,2 1TOPS2 correlatorsis the rate at which the stored target reference filterscan be correlated against the input image. The rateat which this can occur depends on two factors: theframe rate at which the Fourier-plane spatial lightmodulator 1SLM2 can be driven and the speed ofmemory access and filter data transfer to the Fourier-plane SLM. These factors, along with the frame rateof the output detector and postprocessing electronics,limit the processing rate of optical correlators. Inthe case of the TOPS correlators the data are storedon a conventional memory chip and are seriallyaccessed and transported to the SLM, where they aredisplayed with varying degrees of parallelism. Moregenerally, memory access is regarded as a criticalproblem to almost any type of parallel processor.It is well acknowledged that the throughput of theprocessors is now a secondary issue with respect to

The author is with the Land, Space andOptoelectronics Division,Defence Science and Technology Organisation, P.O. Box 1500,Salisbury South Australia 5108, Australia.Received 19 January 1994; revised manuscript received 27

August 1994.0003-6935@95@[email protected] 1995 Optical Society of America.

overall processing speed and that the memory accessis now the main limitation. This is particularly thecase if the processing requires large amounts ofmemory fromwhich data have to be accessed continu-ally. Such an application is automatic target recogni-tion, the application that the TOPS correlators ad-dress. In this case an input image has to be templatematched against a relatively large database of storedreference filters in a search operation. For success-ful recognition and tracking it is anticipated thatthousands of reference filters have to be stored. Inaddition, access to the database has to occur at veryfast speeds, typically 1 ms for an entire 128 3 128filter.One approach to alleviating this problem is to

reduce the amount of data required to describe a filterwhilemaintaining its bandwidth 1or region of support2.The rationale behind this approach is essentially thesame as that behind the use of image-compressiontechniques for image data transfer, where the imagedata are compressed subject to a cost function, i.e.,minimization of the difference between the recon-structed image after transfer and the original. Inthis case the cost function is the difference betweenthe original and the reduced data filter. Reductionof the amount of data required to describe a filter hastwo main benefits. First, compressing the filter datameans the number of pixels on the SLM can be

10 February 1995 @ Vol. 34, No. 5 @ APPLIED OPTICS 865

reduced, leading to a faster frame rate and@or re-duced power requirements. Second, as a conse-quence of the first benefit, the time taken for filteraccess and transfer to the Fourier-plane SLM islessened. As the reduced-data filter still has thesame bandwidth as the original, its display on theFourier-plane SLM corresponds to a reduction infilter resolution with respect to the Fourier plane.Hence a drawback would be that the relative size of afilter pixel with respect to the optical Fourier-transform 1FT2 plane would be increased, leading toreduced manipulation of the Fourier plane and adegradation in the correlation peak.Previously, Kozaitis and Foor3 designed reduced-

resolution binary phase-only filters for a single train-ing image using the simple down-sampling technique,in which every other filter pixel in the horizontal andvertical directionswas discarded. Flannery andGus-tafson4 reported on reduced-resolution binary phase-only filters consisting of 3 3 3 super pixels designedby neural network optimization. Recently the au-thor5 reported on a procedure that enabled the designof optimal reduced-resolution filters with arbitrarilyconstrained values. In Ref. 5 an existing procedure6was extended to facilitate the design of reduced-resolution arbitrarily constrained filters that maxi-mized the correlation peak obtained for a singletraining image. Reduced-resolution phase-only fil-ters designed with the optimal technique were shownto achieve comparable correlation responses withdown-sampled filters, but with 4 to 16 times fewerpixels.In this study, the problem of reducing the resolution

of SDF’s, where more than one training image isinvolved, is investigated. Since SDF’s were intro-duced to the optical processing community,7 they havebecome one of the major tools used for achievingdistortion-invariant target recognition. Many differ-ent types of SDF’s have been designed, with each typeaddressing some issue in addition to that of straight-forward target recognition, such as correlation peakshape,8 clutter,9 or a mixture of both.10 The reduc-tion of the number of pixels used to describe an SDFfor SLM reasons was discussed briefly in Ref. 10;however, to the author’s knowledge, no analyticapproach to the issue of SDF resolution has beenperformed. In the approach employed here, a mul-tiresolution wavelet analysis11,12 1MWA2 is used todesign reduced-resolution SDF’s. In Section 2 abrief overview of SDF design is presented, along witha description of more conventional techniques forreducing SDF resolution. Section 3 contains a de-scription of MWA, which is illustrated with a simpleexample. In Section 4 the design of reduced-resolu-tion SDF’s with wavelet analysis is described. Sec-tion 5 contains the results of simulations comparingthe performance of a minimum correlation envelopeenergy9 1MICE2 filter designed with the new proce-dure and that obtained with down sampling. Sec-tion 6 is a discussion of the simulation results.Finally, Section 7 contains a brief conclusion.

866 APPLIED OPTICS @ Vol. 34, No. 5 @ 10 February 1995

2. Brief Overview ofSynthetic-Discriminant-Function Design

Consider the case where there are N training images,each of which consists of n 3 n pixels. Let thed-dimensional vector

xi 5 3xi112, xi122, . . . , xi1d24T, 112

where d 5 n2, denote the ith training vector obtainedfrom the ith image by a lexicographical ordering of theimage rows. The Fourier transform of xi is

Xi 5 3Xi112, Xi122, . . . , Xi1d24T, 122

where the uppercase denotes a Fourier transform.In SDF design, for a set of training images denoted byxi, the following constraint must be satisfied in thefrequency domain:

H1Xi 5 ui, 132

where H is the frequency domain version of the SDF,the symbol 1 denotes the Hermitian transpose, and uiis the correlation-peak output value. This con-straint has to be satisifed for all training images.If there are N training images, then the correlation-peak output constraints for all the images can bewritten as

H1X 5 u, 142

where X is a d 3 N complex matrix with Xi as itscolumns and u is a vector with ui as its elements.In addition to the above constraints, it is also usuallydesirable that the expression H1AH be minimized,where A is a d 3 d matrix. The exact nature of Adepends on the application; for the minimum-vari-ance SDF,A is the noise correlationmatrixC, whereasfor the minimum average correlation energy filter, Ais a diagonal matrixD of the average energy spectrumof the training-set FT’s. The solution for the filterHthat satisfies the above constraints is13

H 5 A21X1X1A21X221u. 152

As stated in Section 1 we wish to reduce the filterresolution. There are a number of approaches toachieving this. In the conventional approach thetraining images xi are reduced in size to the nearestpower of 2, i.e., from 128 3 128 to 64 3 64, such thatthe training object is just contained within the image.From the reduced-size training image, reduced-sizeFT’s are generated that are then used to train areduced-size SDF according to Eqs. 142 and 152. Adrawback of this approach is that the size of the filteris determined by the largest object in the trainingimages. If the largest object is greater than 64 pixelsin size, either horizontally or vertically, then the filterhas to be 128 3 128 with the above approach. Thisproblem can be overcome with the down-samplingtechnique used by Kozaitis and Foor,3 in which thefull-resolution filter is simply down sampled. How-ever, both approaches ignore the fact that, in an

optical correlator, a continuous FT is generated andthat the SLM pixel has a finite size. Thus filterdesign should taken into account the variation of theoptical FT over the SLM pixel area.5 Because ofthese considerations, the reduced-resolution filter gen-erated with the down-sampling technique may not bethe filter at the reduced resolution that comes closestto satisfying the design constraints for the full-resolution filter. For proper design of such a filter ata reduced resolution, some form of functional analysistechnique is required that enables one to analyzefunctions at different resolutions. Such an analysistechnique is provided by multiresolution waveletanalysis 1MWA2, which is employed in this study todesign reduced-resolution SDF’s. In Section 3 a brieftreatment of MWA is given, followed by a simplifiedexample of a MWA.

3. Multiresolution Wavelet Analysis

A. Mathematical Theory

The following is a brief mathematical treatment ofMWA. The treatment is not rigorous; for a moredetailed description the reader is referred to Refs. 11and 12. As mentioned previously, MWA provides atool for analyzing a function at different resolutions.This is achieved by use of wavelets to construct basesof the function space, in this case the space of square-integrable two-dimensional 12-D2 functions L21R22.More specifically, there exist wavelets C1x, y2 [ L21R22such that the set of functions generated by binarydilations 1i.e., dilation by 2 j2 and dyadic translations1k@2 j, l@2 j2 of a specific wavelet form an orthonormalbasis of L21R22. These bases generalize the Haarbasis and are of the form

C2 j1x 2 22jk, y 2 22jl2 5 2 jC12 jx 2 k, 2 jy 2 l2, 162

where j, k, l [ Z 1the set of integers2, C1x, y2 is calledthe mother wavelet, and the coefficient 2 j appears inthe basis set in order to normalize the functions in theL21R22 norm. There can be many such bases witheach generated from a different mother wavelet. Fora specific value of j, the set of functions generated byall possible values of k and l form a subspace, denotedbyW2 j. In the case of orthogonal wavelets, L21R22 canbe decomposed as an orthogonal sum of the spacesW2 j:

L21R22 5 · · · % W221 % W20 % W21 · · · , 172

Thus every function f 1x, y2 that is an element of L21R22has a unique decomposition of mutually orthogonalcomponents, as shown below:

f 1x, y2 5 · · · 1 g2211x, y2 1 g201x, y2 1 g211x, y2 1 · · · ,

182

where g2 j [ W2 j for all j [ Z. As each subspace isgenerated by differently scaled versions of the motherwavelet, the components g2 j can be considered as

different layers of detail of f 1x, y2 and are known asdifference functions.Using the above decomposition of L21R22, one can

generate the following nested sequence of closedsubspaces V2 j, j [ Z, of L21R22, defined by

V2 j 5 · · · % W2 j23 % W2 j22 % W2 j21, 192

where we restrict our interest to subspacesW2 j, whichare mutually orthogonal and where j [ Z. As wasthe case for the subspaces Wj, there exists a set offunctions f1x, y2, called scaling functions, such that,for a given j, the functions

f2 j1x 2 22jk, y 2 22jl2 5 2 jf12 jx 2 k, 2 jy 2 l2 1102

form an orthonormal basis of V2 j, where j, k, l [ Z, 2 j

is a normalizing coefficient, and each basis is gener-ated from a single scaling function. These subspaceshave a number of important properties, but only a fewrelevant ones will be discussed here 1for a morecomplete discussion, see Refs. 11 and 122. The first isthatV2 j can be interpreted as the subspace containingthe set of all approximations at the resolution 2 j offunctions inL21R22. Another important property, fromEq. 192, is that

· · · , V221 , V20 , V21 , · · · .

Hence the approximation of a function at a resolution2 j11 contains all the necessary information to com-pute the approximation of the function at a lowerresolution 2 j. Also, from Eq. 192, one has

V2 j11 5 V2 j % W2 j. 1112

Thus the approximation of f 1x, y2 at a resolution 2 j11

is

f2 j111x, y2 5 f2 j1x, y2 1 g2 j1x, y2, 1122

where f2 j1x, y2 and g2 j1x, y2 are the orthogonal projec-tions of the function onto V2 j and W2 j, respectively.Perhaps the most important property with respect tothis study is that the orthogonal projection of afunction f 1x, y2 in L21R22 onto V2 j gives the function inV2 j, which is most similar to f 1x, y2; i.e.,

;d2 j1x, y2 [ V2 j,

6d2 j1x, y2 2 f 1x, y26 $ 6 f2 j1x, y2 2 f 1x, y26, 1132

where d2 j denotes the set of all possible functions inthe space V2 j. Any set of vector spaces 1V2 j2j[Z thatsatisfies these properties is called a multiresolutionapproximation of L21R22.The orthogonal projection of a function f 1x, y2 in

L21R22 onto V2 j is given by

f2 j1x, y2 5 ok,l52`

`

7 f 1u, v2, f2 j1u 2 22jk, v 2 22jl28

3 f2 j1x 2 22jk, y 2 22jl2, 1142


where the following set of inner products,

7 f 1u, v2, f2 j1u 2 22jk, v 2 22jl28

5 ee f 1u, v2, f2 j1u 2 22jk, v 2 22jl2dudv, k, l [ Z,

1152

is termed the discrete approximation f2 j of f 1x, y2 atthe resolution 2 j. However, as mentioned previ-ously, f2 j can be obtained from f2 j11 as V2 j , V2 j11.The following equation gives the relationship be-tween the approximations f2 j and f2 j11:

7 f 1x, y2, f2 j1x 2 22jk, y 2 22jl28

5 on,m52`

`

7r1m 2 2k, n 2 2l8

3 7 f 1x, y2, f2 j111x 2 22j21m, y 2 22j21n28, 1162

where r1m, n2 is defined to be the discrete filter whoseimpulse reponse is given by

;m, n [ Z, r1m, n2 5 7f2 211u, v2, f1u 2 m, v 2 n28

1172

3see Appendix A for a detailed derivation of Eq. 11624.Thus Eq. 1162 shows that one can obtain f2 j from f2 j11

by convolving f2 j11 with r1m, n2 and down samplingby a factor of 2 in each direction. Hence one cancompute all the discrete approximations at resolu-tions below the original data resolution by iterativelyrepeating this process. A similar analysis can alsobe performed for g2 j1x, y2, which gives an iterativeequation that can be used to derive g2 j from f2 j11.However, a knowledge of this is not required for thepurposes of understanding this paper 1see Appendix Afor further details2.In summary, the MWA shows that the L21R22 func-

tion space can be decomposed into a sequence ofnested subspaces V2 j, each of which contains all thepossible functions at a resolution of j. The MWAapproximation of a function f 1x, y2 at a resolution j,f2 j1x, y2, is the function in the subspace V2 j, which ismost similar to f 1x, y2. Finally, the discrete approxi-mation of f2 j1x, y2, f2 j, can be derived from f2 j11 by aniterative relationship, given by Eq. 1162.

B. Simple Multiresolution Wavelet Analysis Example

To illustrate the previous discussion more clearly, weconsider the following one-dimensional MWA. Al-though the analysis is carried out in one dimensionfor clarity, the results can be easily extended to twodimensions 1references to equations in the previoussubsection refer to their one-dimensional equivalents2.In this analysis the vector spaces V2 j and W2 j aregenerated by the following mother scaling and wave-let functions 1these particular functions are of rel-


evance for SLM reasons2:

f1x2 [ 51 for 0 # x , 1

0 otherwise, 1182

C1x2 [ 51 for 0 # x , 1⁄2

21 for 1⁄2 # x , 1

0 otherwise

, 1192

where the scaling function is the rectangular functionof the unit interval 0= 1, and the wavelet is the Haarfunction. Figure 11a2 shows the discrete approxima-tion of a function at a resolution j 5 0 1the signal isactually a slice through a 2-D MICE-SDF designed inSection 52. For a continuous approximation at thisresolution, Eq. 1142 shows that one must interpolatethe discrete approximation at j 5 0, given by Eq. 1152,with the scaling function described in relation 1182.The resulting continuous approximation is shown inFig. 11b2.To calculate the discrete approximation at a resolu-

tion of j5 21, one simply uses Eq. 1162. This involvescalculation of the discrete filter r1m2. From Eq. 1172,r1m2 is the discrete convolution of f1x2 and f2211x2. f1x2is described by relation 1182, while f2211x2 can beobtained with Eq. 1102 and is given by

f2211x2 [ 5221@2 for 0 # x , 2

0 otherwise, 1202

where the factor 221@2 arises as we are normalizing inL21R2 3it is 221 in L21R224. Figure 21a2 shows graphicalrepresentations of f1x2 and f2211x2. Their discreteconvolution results in r1m2 5 221@2 form equal to 0 and1 and zero for all other values ofm 3Fig. 21b24.For f221 the discrete approximation in Fig. 11a2 is

convolved with the filter r1m2 and down sampled by afactor of 2. Thus if we represent the discrete approxi-mation at j 5 0 by f0 5 5x0, x1, x2, x3, . . . , xn21, xn6, thediscrete approximation at j 5 21, shown in Fig. 11c2,is given by f221 5 31x0 1 x12@121@22, 1x2 1 x32@121@22, . . . ,1xn21 1 xn2@121@224. The continuous approximation isgiven by interpolation of the discrete approximationin Fig. 11c2 with f2211x2 and is shown in Fig. 11d2. Byiteratively performing this transform, one can obtainall the discrete approximations for j , 0 from theapproximation at j 5 0.

C. Multidimensional Function Space Interpretation

To design reduced-resolution SDF’s, one needs tounderstand the relationship between the variousfunction spaces and also how f2211x2 is derived fromf01x2 from a multidimensional function-space perspec-tive.In order to visualize this we can empirically derive

these relations from Eqs. 162 and 1102 by using thespecific wavelet and scaling functions defined inSubsection 3.B. For clarity we consider only thefirst two samples of the full-resolution signal f0 asdescribed in Subsection 3.B. That is, f0 equals x0 at

Fig. 1. Discrete approximations of a function 1the function is actually a slice through a 2-DMICE SDF2 at resolutions 1a2 j5 0 and 1c2 j5 21.Also shown are interpolations of 1a2 and 1c2with the scaling functions given by 1b2 relation 1182 and 1d2 relation 1202.

x 5 0 and x1 at x 5 1, and we ignore the subspacegenerated by the basis vectors f1x 2 k2 with k . 1 1Asall one-dimensional signals are composed of a concat-enated chain of such two-element signals, the analy-sis has to be repeated only on the other pairwiseadjoint elements of f02. Hence

f01x2 5 x0f1x2 1 x1f1x 2 12. 1212

Thus we can effectively treat the multiresolutionspace V0 as being generated by only two basis vectors,f1x2 and f1x 2 12, where f1x2 is once again given byrelation 1182. As there are only two basis vectors inV0, we have from Eq. 1112 that V221 andW221 have onlyone, f2211x2 and C2211x2, respectively. By calculatingthe inner products of these basis vectors, we can thusdetermine the relationships between the various sub-spaces 1see Appendix B for the derivation of theseinner products and a detailed diagram illustratingthe relationships between the basis vectors and sub-spaces2.

Figure 3 is a diagram showing the MWA functionspace for the above example, in which f01x2 is repre-sented as a vector f0. Also shown is the orthogonalprojection of f0 onto V221, f221. From the functionspace relationships shown in Fig. 3, the followingexpression can be derived for f221:

f2211x2 5 3x0@121@22 1 x1@121@224f2211x2, 1222

which is consistent with the result from the analysisin Subsection 3.B. Thus it has been empiricallyshown that, given a particular relationship betweenthe subspaces, the orthogonal projection of f0 ontoV221 is equivalent to Eq. 1162. For completeness thedifference function g221, which is the orthogonal projec-tion of f0 onto W221, is also shown. Similarly, anexpression for g221 can be derived from Fig. 3 and isgiven by

g2211x2 5 3x0@121@22 2 x1@121@224C2211x2. 1232


Adding f2211x2 and g2211x2 as described by Eqs. 1222 and1232 gives f01x2, as described by Eq. 1212, which isconsistent with Eq. 1122 of Subsection 3.A.Finally, consider the conventional down-sampling

approach described in Section 2 and used by Kozaitisand Foor3 to obtain a reduced-resolution approxima-tion of a function. In this approach, also shown inFig. 3, x1 is simply discarded, and the approximated

Fig. 2. Graphical representations of 1a2 f1x2 and f2211x2 and 1b2 r1m2.


function is given by

f 82211x2 5 x0f1x2 1 x0f1x 2 12

5 121@2x02f2211x2 1242

where the prime symbol denotes the conventionaldown-sampling technique. The difference functiong8221 in this case is composed of a component in V221,f221 2 f 8221, and a component in W221, g221. Theadditional component of the difference signal g8221 inV221 is consistent with relation 1132.

4. Design of Synthetic Discriminant Functions UsingMultiresolution Wavelet Analysis

In Subsection 3.A, relation 1132 states that the MWAapproximation of a function at a reduced-resolution isclosest to the full-resolution signal in terms ofminimiz-ing the norm of the difference function between thetwo. This was further illustrated by an example inSubsection 3.C, where Fig. 3 shows the reduced-resolution approximation to be equivalent to theorthogonal projection of the full-resolution functiononto the reduced-resolution space. Given this prop-erty of MWA approximations, one would expect thatthe design of reduced-resolution SDF’s with MWAapproximations should result in reduced-resolutionSDF’s whose performances are closer to that of thefull-resolution SDF than those designed with anyother reduced-resolution approximation. This is thebasis of the approaches investigated here for thedesign of reduced-resolution SDF’s.In this study two different approaches to the design

of reduced-resolution SDF’s by MWA were investi-gated. Once again, for clarity, it is assumed thefull-resolution function space consists of only two

Fig. 3. Schematic diagram of simplified MWA example function space that shows the wavelet and down-sampling approximations, f221

and f 8221, respectively, of f0 in V221, and their respective difference functions, g221 and g8221 1some vectors are slightly displaced for clarity2.

Fig. 4. 1a2Graph showing the first approach to reduced-resolution SDF design, where the full-resolution filterH0 is approximated byH221

1wavelet2 and H8221 1down sampling2. 1b2 Graph showing the second approach, where the training images Xi0 are approximated by Xi221

1wavelet2 and X8i221 1down sampling2. The reduced-resolution training images are then used to train a reduced-resolution SDF 1somevectors are slightly displaced for clarity2.

dimensions. In the first approach, illustrated graphi-cally in Fig. 41a2, a full-resolution SDF H0 is obtainedby training on full-resolution FT’s Xi0 in V0. H0 isthen approximated by a reduced-resolution SDFH221,which is obtained by the orthogonal projection of H0onto V221. Also shown is the difference functionDH221 in W221. H8221 in Fig. 41a2 is the reduced-resolution approximation SDF obtained with downsampling. In the second approach, illustrated inFig. 41b2, the Xi0 are approximated by their ortho-gonal projections Xi221 onto the subspace V221. Thedifference functions DXi221 in W221 are also shown.

H221 is obtained by use of Xi221 as reduced-resolutiontraining images. As theXi221 are all containedwithinV221,H221 is also containedwithinV221. The reduced-resolution approximations X8i221 obtained by downsampling are also shown.For both approaches one needs only to determine

the filter r1m, n2. This requires both f2211x, y2 andf1x 2 m, y 2 n2. However, the choice of scaling func-tion in this case is governed by the fact that, inpractice, the filters are displayed on a 2-D pixelatedSLM. Thus, physically speaking, the basis vectors ofthe function space for this MWA are given by the


transmission of the SLM pixels. Therefore the scal-ing function is equal to the transmission of a SLMpixel, which, assuming a 100% fill factor, is given by

f1x, y2 [ 51 for 0 # x, y , 1

0 otherwise, 1252

which is simply the 2-D version of the rectangularfunction used in Subsection 3.B. The scaling func-tion at a resolution j 5 21 is therefore

f2211x, y2 [ 5221 for 0 # x, y , 2

0 otherwise. 1262

As stated in Section 3, r1m, n2 is the sampled convolu-tion of the above two functions and is given by

r1m, n2 5 5221 for m, n 5 0, 1

0 otherwise. 1272

Thus in the first approach to the design of reduced-resolution SDF’s, from Eq. 1162, one obtains the dis-crete approximation H221 by convolving r1m, n2 withH0 and down sampling by a factor of 2 in both x and ydirections. Similarly, the same goes for Xi0 and Xi221

in the second approach. For both approaches, whenH221 is represented in the full-resolution space, it hasto be interpolated with f2211x, y2, as illustrated inSubsection 3.B.

5. Simulation Results

To determine the effectiveness of both approachesdescribed in Section 5 and to compare MWA anddown-sampling techniques, the author designed vari-ous reduced resolution MICE-SDF’s. In the case ofthe MICE filter the matrix A in Eq. 152 is a diagonalmatrix T of size d 3 dwhose diagonal elements are

T1k, k2 5 max3D11k, k2, D21k, k2, . . . , DN1k, k24, 1282

where the diagonal elements Di1k, k2 of Di are theenergy spectra of the Xi 10FT02 of the training image2.The training images were 128 3 128 images of tenobliquely downward-looking views of a tank, witheach view differing by 10° in aspect angle, in therange from straight to side on 1the straight and theside-on views correspond to aspect angles of 0° and90°, respectively2. Figure 5 shows a number of differ-ent views of the tank.A full-resolution MICE-SDFH0 consisting of 128 3

128 pixels was designed from the ten training imagesby use of Eq. 152. Discrete approximations of reduced-resolution MICE-SDF’s, with j equal to 21 and 22and consisting of 64 3 64 and 32 3 32 pixels,respectively, were derived from H0 by the first ap-proach. Both MWA 1H12 j2, and down-sampling1H182 j2 techniques were used, where the 1 in H1 refersto the fact they were designed with the first approach.Similarly, reduced-resolution filters were designed bythe second approach with both MWA 1H22 j2 anddown-sampling 1H282 j2 techniques; 128 3 128 up-


sampled versions of the reduced-resolution filterswere obtained with the interpolation technique de-scribed in Section 3.The resulting filters were then correlated with each

of the ten training images, and a number of measure-ments were made of the correlation planes. Themeasurements made include the normalized correla-tion peak 1NCP2, which is the correlation-peak value1not necessarily the correlation-plane maximum2 di-vided by the filter energy, the standard peak-to-correlation energy 1PCE2, and the number of peaks,not including the correlation peak, with maximumvalues over 50% of the true correlation-peak maxi-mum 1NP502. Figures 61a2 and 61b2 show the variationof the NCP and the PCE, respectively, with thetraining image for the filter H0 and the filters H12 j

andH182 j with j equal to 21 and 22. Figure 7 showsthe corresponding measurements made on the corre-

Fig. 5. Gray-scale plots of tank training images at aspect anglesof 1a2 0°, 1b2 50°, and 1c2 90°.

Fig. 6. Graphs showing variation of 1a2 the NCP and 1b2 the PCEwith training-image number for filtersH0,H1221,H1222,H18221, andH18222.

lation planes obtained withH0 and alsoH22 j andH282 j

with j equal to 21 and 22. Finally, Table 1 showsthe variation in NP50 for all filters with the trainingimage.

6. Discussion

The study of Figs. 6 and 7 and Table 1 shows that, inall cases, filters designed with MWA gave superiorperformance to those designed with down-sampling

techniques. In particular, the performance of H2222

1Fig. 72 is far superior to H18222 and H28222 1Figs. 6 and7, respectively2. This is further illustrated in Fig. 8,which shows correlation-plane three-dimensional in-tensity plots with three training images 1numbers 0,5, and 9, which are shown in Fig. 52 for H2222 andH28222. Figures 81a2, 81c2, and 81e2, obtained withH2222, are characterized by well-defined sharp peakswhose maxima are well above the clutter with PCE’s


of 410, 204, and 227, respectively. In comparison, inFigs. 81b2, 81d2, and 81f 2, obtained with H28222, the truecorrelation peaks are buried in clutter with PCE’s of139, 8, and 21, respectively. Thus for H2222, targetidentification and location are easily achieved, whileforH28222 it is not possible.An interesting outcome of the simulations is that

the correlation-peak values obtained with the H12 j

filters 3Fig. 61a24, do not satisfy the design constraints,i.e., all peaks having the same value. Although thepeaks are sharp and well defined in most cases, theirmaximum value decreases with increasing training-image energy 1the training-image energies increasewith number2. This can be explained by consider-ation of the full-resolution case, where the filtersatisfies the constraints H0

1Xi0 5 ui. From Eq. 1122the full-resolution filter can be expressed as

H0 5 H1221 1 DH1221. 1292

The correlation-peak value for a training image Xi0and the filter approximation at a resolution j 5 21 is

H12211 Xi0 5 1H0

1 2 DH12211 2Xi0

5 ui 2 DH12211 Xi0, 1302

where we substitute Eq. 1292. Assuming that the Xi0are clustered together in V0, which is a reasonableassumption for real target imagery, the last term inEq. 1302. DH1221

1 Xi0, will be positive. Thus the cor-relation-peak value will decrease in proportion to themagnitude of the inner product. This depends di-rectly on the size of the Xi0 components in W221 andalso on 0Xi00. Furthermore, if we consider the case forj 5 22, the equivalent equation to Eq. 1302 is

H12221 Xi0 5 1H0

1 2 DH12211

2 DH12221 2Xi0

5 ui 2 DH12211 Xi0 2 DH1222

1 Xi0. 1312

Hence for j 5 22 there is an additional component,DH1222

1 Xi0 1which we shall also assume is positive2,that is subtracted from ui. As j decreases, moreterms have to be subtracted. Thus Eqs. 1302 and 1312show that the correlation-peak value will decrease for

Table 1. NP50 versus Training Image for All Filters

Training-ImageNumber H0 H1221 H1222 H18221 H18222 H2221 H2222 H28221 H28222

0 0 0 0 0 1 0 0 0 11 0 0 0 0 14 0 0 0 442 0 0 0 0 194 0 0 0 1633 0 0 0 0 20 0 0 0 194 0 0 0 0 43 0 0 0 415 0 0 0 0 1024 0 0 0 9286 0 0 0 0 1544 0 0 0 11087 0 0 0 0 204 0 0 0 2068 0 0 3 0 98 0 0 0 1029 0 0 0 0 142 0 0 0 144


decreasing j and that, for the same value of j, thecorrelation-peak values will be smaller for largervalues of 0Xi00. Both of these results are consistentwith the simulation results.In the case of theH22 j filters the constraints on the

correlation-peak values are satisfied 3Fig. 71a24. Thisis because the SDF’s are trained on Xi2 J. Thus theH22 j filters, which are also in V2 j, are designeddirectly so that the design constraints are satisfied forthe Xi2 J. For the case of j 5 21,

H22211 Xi221 5 ui. 1322

However, in the simulation, H2221 is correlated withthe Xi0. From Eq. 1112, each Xi0 can be decomposedinto Xi221 and its difference function DXi221:

Xi0 5 Xi221 1 DXi221. 1332

Thus the correlation-peak value of the reduced-resolution filter with the full-resolution training im-age is given by

H22211 Xi0 5 H2221

1 1Xi221 1 DXi2212. 1342

As DXi221 is orthogonal to V221, which contains H2221,

H22211

DXi221 5 0. 1352

Thus DXi221 does not contribute to the correlation inEq. 1342; i.e.,

H22211 Xi0 5 H2221

1 Xi221 5 ui. 1362

A similar case can be advanced for j 5 22 by theobservation thatDXi221 andDXi222 are both orthogonalto V222, which containsH2222.In the case of down sampling, the results are more

or less similar for both approaches with neithersatisfying the constraints. For all filters there is adecrease in the NCP with a reduction in filter resolu-tion. This is to be expected; however, the decrease ismuch more significant for filters designed with downsampling techniques. In the case of the H22 j filters3Fig. 71a24 the ratio of their NCP to theH0 NCP is,0.72and 0.18 for resolutions of j equal to 21 and 22,respectively. This can be explained by the fact thatthe magnitude of Xi2 J decreases with decreas-ing j.As mentioned in the introduction, in Ref. 5 a

heuristic technique for the design of arbitrarily con-strained reduced-resolution filters was described.In Ref. 5 only a single training image was involved,and the technique was optimal in the sense that itmaximized the correlation-peak magnitude obtainedwith an arbitrarily constrained filter at a given resolu-tion. The technique involved generation of a reduced-resolution representation of the single training-imageFT by partitioning of the full-resolution FT into anarray of reduced-resolution elements and integrationof the complex amplitude over the area of eachelement 3see Eq. 122 of Ref. 54. The arbitrarily

Fig. 7. Same as Fig. 6 but forH0,H2221,H2222,H28221, andH28222.

constrained filter was then matched to this reduced-resolution FT. The technique was motivated by theobservation that this is, in essence, what physicallyoccurs in the optical correlator; the reduced-resolu-tion element corresponds to a SLM pixel, and thecontribution of the FT that passes through a particu-lar pixel to the correlation peak is the integration ofthe complex amplitude over this pixel area. Compar-ing this technique to the MWA approach adoptedhere, we see that the integration of the full-resolution

FT over a reduced-resolution element 1SLM pixel2 isequivalent to the use of the following rectangularbasis functions:

f2 j1x 2 22jk, y 2 22jl2 5 f12 jx 2 k, 2 jy 2 l2, 1372

where f1x, y2 is given by relation 1252. In Eq. 1372 thenormalization factor 2 j in Eq. 1102 is dropped. Theabove basis functions correspond to havingr1m, n2 5 1 for 1m, n2 5 10, 12, as opposed to 221 for


Fig. 8. Three-dimensional intensity plots of the correlation plane obtained with training images 0 31a2 and 1b24, 5 31c2 and 1d24 and 9 31e2 and 1f 24as inputs. The filters used areH2222 31a2, 1c2, and 1e24 andH28222 31b, 1d2, and 1f 24.

the MWA. Thus the approximation used in theprevious heuristic technique is shown to be rigorouslycorrect, apart from a normalization factor. The im-portant point is that both representations are basi-cally equivalent as they are averaged versions offull-resolution FT’s, as opposed to the pick-one, miss-one approach of conventional down sampling.

7. Conclusion

In this study several approaches to the design ofreduced-resolution SDF’s have been investigated.


The best results, which satisfied the design con-straints, were obtained with MWA techniques togenerate reduced-resolution approximations of thetraining images, which were then used to train thefilter. Simulations performed with reduced-resolu-tion MICE filters showed that the use of MWA tech-niques gave much better results than those obtainedwith simple down-sampling techniques and permit-ted a reduction by a factor of 4 in the number of pixelsneeded to describe a filter. Although there is anaccompanying decrease in the NCP with reduced-

resolution filters, this study suggests that 128 3 128filters are not required for typical applications.In previous research performed by the author, the

correlation peak obtained for an arbitrarily con-strained filter matched to the reduced-resolution ap-proximation of a single training-image FT was maxi-mized. A heuristic reduced-resolutionapproximation, based on what physically occurs in acorrelator, and which by coincidence differs only by anormalization factor from the MWA approximationsused here, was employed. Future research will ad-dress the integration of the previous research, withthe MWA approaches used here, to design reduced-resolution smart arbitrarily constrained filters thatare trained on more than one image. This shouldpermit filters to be written onto SLM’s with a reducednumber of pixels 1i.e., 64 3 64 or 32 3 322, therebypermitting faster frame rates and@or reduced powerrequirements for the Fourier-plane SLM.

Appendix A

To compute f2 j from f2 j11, we observe thatf2 j1x 2 22jk, y 2 22jl2 for any k, l [ Z is a member ofV2 j, which is included in V2 j11. It can thus be ex-panded in the orthonormal basis of V2 j11:

f2 j1x 2 22jk, y 2 22jl2

5 on,m52`

1`

7f2 j1u 2 22jk, v 2 22jl2,

f2 j111u 2 22j21m, v 2 22j21n28

3 f2 j111x 2 22j21m, y 2 22j21n2. 1A12

By changing variables in the inner-product integral,one can show

7f2 j1u 2 22jk, v 2 22jl2, f2 j111u 2 22j21m, v 2 22j21n28

5 7f221(_((u, v2, f3u 2 1m 2 2k2, v 2 1n 2 2l248. 1A22

Substituting Eq. 1A22 into 1A12 and taking the innerproduct of f 1x, y2with both sides gives

7 f 1x, y2, f2 j1x 2 22jk, y 2 22jl28

5 on,m52`

1`

7f2211u, v2, f3u 2 1m 2 2k2, v 2 1n 2 2l248

3 7 f 1x, y2, f2 j111x 2 22j21m, y 2 22j21n28. 1A32

We define r1m, n2 to be the discrete filter whoseimpulse response is given by

;m, n [ Z, r1m, n2 5 7f2211u, v2, f1u 2 m, v 2 n28,

1A42

and substituting Eq. 1A42 into Eq. 1A32 gives

7 f 1x, y2, f2 j1x 2 22jk, y 2 22jl28

5 on,m52`

1`

7r1m 2 2k, n 2 2l8

3 7 f 1x, y2, f2 j111x 2 22j21m, y 2 22j21n28, 1A52

which is the relationship described by Eq. 1162.Similarly, an analysis can be performed to show thatthe difference g2 j can be iteratively derived from f2 j11

with a filter whose impulse response is given by

;m, n [ Z, s1m, n2 5 7C2211x, y2, f1x 2 m, y 2 n28.

1A62

Appendix B

To empirically determine the relationship betweenV221, W221, and V0, we have to determine their basisvectorsf1x2,f1x 2 12,f2211x2, andC2211x2. f1x2,f1x 2 12,and f2211x2 are given by relations 1182, 1192, and 1202,respectively, while from Eq. 162 and relation 1192,C2211x2 is given by

C2211x2 [ 5221@2 for 0 # x , 1

2221@2 for 1 # x , 2

0 otherwise

. 1B12

FromEq. 1A12, to express the single basis vector f221 ofV221 in terms of the basis vectors of V0, we have todetermine the inner products of f221 with the basisvectors of V0. These inner products were calculatedwhen we determined r1m2 in Subsection 3.B; they aregiven by 7f1x2, f2211x28 5 7f1x 2 12, f2211x28 5 [email protected] we have

f2211x2 5 221@2f1x2 1 221@2f1x 2 12. 1B22

Likewise, to express the single basis vector C2211x2 ofW221 in terms of the basis vectors of V0, we have todetermine the inner products of C2211x2 with the basis

Fig. 9. Graphical representation of f1x2 and C2211x2.


vectors of V0. Figure 9 is a graphical illustration off1x2 and C2211x2. It is clear from Figure 9 that7f1x2, C2211x28 5 221@2, while 7f1x 2 12, C2211x28 5 [email protected] we have

C2211x2 5 221@2f1x2 2 221@2f1x 2 12, 1B32

which corresponds to a difference filter of s1m2 equal to221@2 and 2221@2 for m equal to 0 and 1, respec-tively, and zero for all other values ofm.These relationships are graphically illustrated in

Fig. 10, which shows that f221 is rotated by 45° withrespect to both basis vectors of V0. This reflects Eq.1B22 1i.e., cos21 221@2 5 45°2 and is to be expected, asthe reduced resolution approximation in V221 musthave equal components of f1x2 and f1x 2 12. Alsoshown is C221, which is rotated by 45° to f1x2 and by135° to f1x 2 12 1i.e., cos21 2221@2 5 135°2. HenceC2211W2212 is orthogonal to f2211V2212, which is consis-tent with Eq. 1112. By plotting f01x2 3Eq. 12124 as a

Fig. 10. Schematic diagram showing the relationships betweenthe vector spaces V0, V221 and W221 and the basis vectors f1x2,f1x 2 12, f2211x2, and C2211x2.


vector in the function space of Fig. 10, we obtainFig. 3.

I thank the referees and my colleague Chris Wood-ruff for their comments and suggestions.

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Reduced-resolution synthetic-discriminant-function design by multiresolution wavelet analysis