Space-variant simultaneous detectionof several objects by the use of multipleanamorphic fractional-Fourier-transform filters

Javier Garcıa, David Mendlovic, Zeev Zalevsky, and Adolf Lohmann

A fractional correlator that is based on the anamorphic fractional Fourier transform is defined. Thisnew, to our knowledge, correlator has been extended to work with multiple filters. The noveltyintroduced by the suggested system is the possibility of the simultaneous detection of several objects indifferent parts of the input scene 1when anamorphic optics are dealt with2, thereby permitting anindependent degree of space invariance in two perpendicular directions. Computer experiments aswell as experimental optical implementation are presented.Key words: Anamorphic systems, fractional Fourier transform, correlation, multiple filters, pattern

recognition, space variant systems. r 1996 Optical Society of America

1. Introduction

Many optical pattern-recognition systems are basedon the VanderLugt correlator.1 This setup providesa versatile linear configuration for performing real-time correlation or convolution operations. One ofthe milestones of this system is the shift-invariantproperty that it exhibits. This property is a directinheritance from the Fourier processing performedby the system. Nevertheless, this property maybecome a disadvantage in various practical situa-tions, for example, if the object to be detected is likelyto appear only in a certain area of the input image.The shift invariance of a conventional correlatorpermits the presence of false alarms, with a highprobability, in locations far from this possible loca-tion of the true target.Another typical case is produced when several

objects are to be detected simultaneously. The com-mon approach is the use of composite filters—alinear combination of the filters matched to the set oftargets,2 or their more sophisticated versions, suchas synthetic discriminant functions3 or minimum-

J. Garcıa is with the Departament Interuniversitari d’Optica,Universitat de Valencia, Calle Doctor Moliner, 50, 46100 Burjas-sot, Spain. D. Mendlovic and Z. Zalevsky are with the Faculty ofEngineering, Tel-Aviv University, 69978 Tel Aviv, Israel. A.Lohmann is with the Physikalisches Institut der Universitat,Rommel str. 1, 91058 Erlangen, Germany.Received 6 December 1995.0003-6935@96@203945-08$10.00@0r 1996 Optical Society of America

average correlation energy filters.4 In any case, aconventional correlator detects the presence of allthe targets over all the input scene.Although there have been several approaches for

obtaining a shift-variance property in the correlationscheme,5,6 the most used tool for this purpose inrecent times has been the fractional Fourier trans-form 1FRT2.7–13 This operation permits tuning of thefractional order from object space to Fourier space ina continuous way. The amount of shift invariance iscontrolled by the fractional order. Several recentcontributions show the abilities of a correlator basedon the FRT for performing a space-variant recogni-tion task.14–16 Recently, the FRT operation has alsobeen extended to the anamorphic case.17,18 Thisextension gives the possibility of independently vary-ing the space variance of the system in two perpen-dicular directions.In this paper a flexible system for obtaining the

fractional correlation is proposed. It is based on anadjustable-scale anamorphic fractional Fourier trans-former, followed in cascade by a second transformerthat, depending on the codification of the filter, canbe amorphic or anamorphic. The system is em-ployed for space-variant processing, i.e., implement-ing multiple targets to be detected in different zonesof the image.In Section 2 the FRT and fractional correlation in

the amorphic and anamorphic cases are sketched.Section 3 deals with the optical implementation of aFRT anamorphic correlator. In Sections 4 and 5computer simulations and optical experiments, re-

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spectively, are described and analyzed. Section 6outlines the conclusions.

2. Fractional Fourier Correlation

A. Amorphic Fractional Fourier Transform

The introduction of the FRT, with respect to optics,has been done in twoways. One deals with propaga-tion in a graded index medium,7,8 and the other dealswith lenses and free-space propagation.9 Both defi-nitions are fully equivalent19 and have been thestarting point for an increasing interest in space-variant signal processing. The basic definition of aFRT of the order P of a pattern f 1x2 is as follows:

FP1x82 5 F P5 f 1x26 5 e2`

1`

f 1x2

3 exp1ip x2 1 x82

T 2exp12i2pxx8

S 2dx, 112

with

T 5 lf1 tan f, S 5 lf1 sin f, f 5 Pp

2,

and where lf1 is the global scaling factor for both theinput and the output. One can control the amountof shift invariance by changing parameter P. Avalue of P 5 0 results in the maximum spacevariance. In contrast, choosing a value of P 5 1makes the quadratic phase factor in Eq. 112 vanish.The transformation is then reduced to a conven-tional Fourier transform. It is worth noting thesimilarity between this expression and the analyti-cal expression for the Fresnel diffraction.20 In bothcases the output is obtained by the performance of aFourier transform of the input multiplied by aquadratic phase factor. The main difference be-tween these two transformations is that in the FRTcase the scaling factor for the quadratic phase factoris not the same as the scaling factor for the Fouriertransformation.The FRT can be accomplished optically by the use

of the setup depicted in Fig. 1 1see Ref. 92. In thissetup we can see how the FRT is obtained by thecascaded operation of a lens, free-space propagation,and a second lens. This splitting of the process canserve to define a method for performing an inexactFRT with an adjustable scale between the input andoutput.21The conventional correlation, mostly implemented

by means of the Fourier transform, has been ex-tended to the so-called fractional correlation.15 Thealgorithm for performing a fractional correlation isshown in Fig. 2. It consists of obtaining the productof the fractional transforms of the distributions to becorrelated and then rendering a last FRT to obtainthe final result. Analytically, the operation of frac-

3946 APPLIED OPTICS @ Vol. 35, No. 20 @ 10 July 1996

tional correlation of an input function, f 1x2, with areference pattern, g1x2, is defined as follows:

CP1,P2,P31x82 5 F P33F P15 f 1x26F P25g1x264, 122

where P1, P2, and P3 are the orders of the FRT’s to beperformed, which in principle are arbitrary. Forvarious reasons, detailed in Ref. 15, the most obviouschoice is

P1 5 P, P2 5 2P, P3 5 21, 132

with P ranging from 0 to 1. In this case, if the inputcoincides with the reference object, perfect phasematching between the object and reference FRT’s inthe fractional domain is obtained. The inverse Fou-rier transform will just focus the resulting planewave.

B. Anamorphic Fractional Fourier Transform

Recently, the FRT concept has been extended to theanamorphic case.17,18 This modification permits theuse of different fractional orders for two orthogonalaxes of a two-dimensional image. The main advan-tage of this extension is the possibility of varying theshift invariance of the processor according to thecharacteristics of the input image. A clear exampleoccurs when the detection of objects along a row isneeded. In the direction perpendicular to the row,there is no need to keep the shift invariance of aconventional Fourier-transform-based processor.Depending on the characteristics of the object to bedetected, the decrease of shift invariance may result

Fig. 1. Optical setup for performing a FRT operation.

Fig. 2. Algorithm for obtaining a generalized fractional correla-tion.

in a gain in the performance of the correlator, mainlyin peak sharpness and the signal-to-noise ratio.15,16The shift variance can also help to locate the object,because a detection peak is produced only when theinput object lies on the line along which the shiftinvariance is kept.The anamorphic FRT is defined as

FPx,Py1x8, y82 5 e2`

1`

f 1x, y2exp3ip1x2 1 x82

Tx1

y2 1 y82

Ty24

3 exp32i2p1xx8Sx1

yy8

Sy24dxdy, 142

with

Tx 5 lf1x tan fx, Sx 5 lf1x sin fx, fx 5 Px

p

2,

Ty 5 lf1y tan fy, Sy 5 lf1y sin fy, fy 5 Py

p

2,

where the subscripts x and y indicate the horizontaland vertical directions of the system, respectively.Because of the additivity of the FRT, the anamorphicFRT can be accomplished if two systems are cas-caded, with each one performing the appropriatetransform in one of the main axes and imaging in theother. Other setups providing higher compactnessor flexibility can be used as well.17,18The definition of the anamorphic fractional correla-

tion is a straightforward extension of Eq. 122.Analogous to the amorphic case, it is obtained byinverse Fourier transformation of the product be-tween the anamorphic FRT of the target and theinput image.The optical setup for obtaining an anamorphic

FRT can be built as a cascade of an amorphic setup,which performs the FRT with the order that is thelower between Px and Py, and an anamorphic system,which renders a FRT in one axis and imaging in theperpendicular one.18 Nevertheless, in the case inwhich the transformer module is the first stage of acorrelator, an inexact FRT 1without the final qua-dratic phase factor2 may be obtained with a setupanalogous to the one described in Ref. 21. Insteadof the preparation of a full setup containing twolenses and free propagation, the object is illuminatedwith a converging beam. This permits the changein the convergence of the phase factor thatmultipliesthe object by its displacement along the optical axis.Matching between the distance object–filter and theconvergence of the beam can produce any desiredorder and scaling factor. Hence this approach ismore convenient for the experimenter, as the exactsizes of the input and filter transparencies are oftennot precisely determined. This is especially impor-tant when spatial light modulators 1SLM’s2 are usedfor implementing the filter. As the FRT is not exact,there will be a quadratic phase factor multiplyingthe output plane, meaning that the correlation plane

will be displaced along the optical axis. In the caseof an anamorphic FRT, the convergence of the beamat the output of the filter plane is different in the twomain axes. For the correlation to be focused, ananamorphic system is needed.A possible setup for performing the anamorphic

fractional correlation is depicted in Fig. 3. Theproper adjustment of distances ax and ay provides theFRT of the desired order in the filter plane. Thescale factor of the FRT is variable as a parameterindependent of the order. Because of the relativelyexpensive price, poor availability, and poor perfor-mance of cylindrical lenses, we made an effort toreduce the number of them in the optical setup. Inthe chosen configuration only three cylindrical lensesand one spherical lens are used. The price to bepaid for this simple setup is that the aspect ratio ofthe FRT 1the quotient between the x- and y-scaleratios2 cannot be adjusted. A modification can bemade to avoid this problem. It consists of insertingan additional anamorphic image-forming system1which provides differentmagnifications in both axes2,thereby creating a stretched image of the FRT plane.The output of this imaging system is then taken asthe input for the inverse transforming subsystem.Nevertheless, this additional complexity can beavoided in most practical cases. According to Figs.1 and 3, because the distance Z is equal for both axes,one may write the following:

Z 5 Zx 5 f1x sinPxp

25 Zy 5 f1y sin

Pyp

2. 152

Thus the aspect ratio 1AR2 between the two axes is

AR 5f1xf1y

5sin1Pyp@22sin1Pxp@22

. 162

3. Multiple Fractional Fourier Transform Filters

As mentioned in Section 2, one of our aims in thisstudy was to design a composite filter that is able torecognize object A, or a certain deformation of thisobject in region A, and object B, or a certaindeformation of it in regionB. To obtain this capa-bility we used the FRT. The shift variance of thistransformation is controlled by the fractional orderparameter, denoted by P. When there is a value ofP 5 1, the FRT becomes the conventional Fouriertransform, which is totally shift invariant. For thevalue P 5 0, the FRT domain is exactly the inputdomain, which means that the transform is maxi-

Fig. 3. Experimental setup for obtaining the anamorphic frac-tional correlation.

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mally shift variant. Thus, varying P from zero toone controls the amount of shift variance of thesystem 1see Ref. 22 for an analytical expression of theshift variance of the FRT2.Now, for implementing these results, we place

object A in regionA, which is assumed to be in theupper part of the input scene. If a FRT of 0.5, forinstance, were performed over this input, the ob-tained fractional spectrum would be mostly concen-trated in the upper region of the output plane 1regionA2. This effect can easily be understood because afractional order of 0.5 determines a transformationthat is not totally shift invariant. Thus, shiftingthe input object causes a shift of the fractionalspectrum. This shift does not occur in the conven-tional Fourier transform 1P 5 12, in which the shift ofthe input is expressed only by a linear phase factor inthe spectral plane. In contrast, placing object B inthe lower region of the input scene 1regionB 2 resultsin a fractional spectrum that is mostly concentratedin the lower part of the fractional spectrum. Thus,a simple summation of the fractional spectra for Aand B creates a joint spectrum that is basicallywithout overlap between each of the spectra 1Aand B2individually. Eventually the resultant filter is ableto recognize only objects A in regionA of the inputscene 1upper part2 and objects B in region B 1lowerpart2.So far we have performed a FRT of the order of 0.5

on axis y. On the other axis 1x2 we wish to obtainshift invariance so that object A, when it is located inregion A, can move along the x axis and yet berecognized. We wish to ensure the same propertyfor object B. Note that the shift variance is thesame all over the input image. This property isused to obtain different impulse responses in everydesired region of interest, permitting the detection ofseveral objects. Another way to exploit the shiftvariance is to design a system that provides adifferent FRT order 1and thus a different shift vari-ance2 in several regions of the image.22Let us now outline the procedure for preparing the

filter. Px denotes the fractional order performed onthe x axis, and Py denotes the fractional order on they axis. Object A is shifted to the center of regionA,and an anamorphic FRT is calculated with fractionalorders of the values Px 5 1 and Py 5 0.5. Thecomplex conjugate of this distribution is taken forencoding the filter. The use of this filter in ananamorphic FRT correlator can detect the presenceof object A in the center of region A, producing acorrelation peak located in the center of the output.For simplicity in the recognition process, it is conve-nient for the peak to be produced over the object.This is accomplished bymultiplication of the filter bya linear phase factor that diverts the correlationpeak to the object location. The linear phase factoris calculated directly according to the distance be-tween the center of the input image and the positionwhere the target has been displaced in the first stepof the filter preparation. The process is repeated for

3948 APPLIED OPTICS @ Vol. 35, No. 20 @ 10 July 1996

object B, with the corresponding displacement to thecenter of region B. The distributions obtained inthis way are added to render the final filter. Theresulting filter should be placed in the appropriatefractional Fourier domain in the adjustable anamor-phic FRT correlator. If instead of detecting object Ain regionA and object B in regionB, one prefers todetect a certain deformation of object A in regionAand a different deformation of object B in regionB ,the same approach can be applied.Let us assume that a one-dimensional, x-direction

scaling-invariance property of object A is required1while it is to be detected in region A2 and that aone-dimensional, y-direction scaling invariance ofobject B is required 1while the object is to be detectedin region B 2. According to Mendlovic et al.,23 toobtain a one-dimensional scale invariance, the loga-rithmic harmonic decomposition may be used.Thus here the proper logarithmic harmonic 1x scalinginvariant2 ofAis calculated. This harmonic is shiftedto the center of region A. In the same manner, aproper logarithmic harmonic 1 y scaling invariant2 ofB is calculated and shifted to the center of regionB.Then a FRT with fractional orders of the values ofPx 5 1 and Py 5 0.5 is performed over the sum. Acomplex conjugate of this function is obtained, andthe filter is placed in the proper fractional domain inthe adjustable anamorphic FRT correlator. Withrespect to the case in which no invariances areinvolved, the only difference in filter preparation isin the impulse responses that are selected. Theprocess for preparing the filter is identical. Notethat, by the use of this approach, any invariantproperty can be detected in regions A or B. Forinstance, it could be a rotation invariance of object Ain region A and a two-dimensional scale invarianceof B in region B . One is not restricted to anyspecific invariant property.

4. Optical Implementation

The construction of an anamorphic processor can bea complicated task. In addition to the usual difficul-ties with a conventional optical system, the use ofcylindrical lenses 1or in general spherocylindricallenses2 implies the existence of two independentsystems that share the same optical axis. In thecase of a correlator there is the additional problem1also present for conventional correlators2 of theimplementation of a spatially matched filter. His-torically, the first method employed for recording amatched filter was the use of an optically recordedhologram in high-resolution film. Thismethod, evenwhen it provides excellent results and flexibility inthe manipulation of the filter, is very seldom used inpresent times because real-time implementation isnot feasible. It is possible to use real-time opticallyrecorded holograms in specific photosensitivemateri-als such as photorefractive crystals, but the setupinvolves the use of two lasers 1one for recording andmaintaining the hologram and the other for readout2and the system results in a rather complex implemen-

tation. Any complex filter can also be recorded as acomputer-generated hologram. This permits theuse of complex impulse responses, which are difficultto obtain in cases of optically recorded filters.Nevertheless, this method also cannot cope with areal-time implementation. The third main methodis based on the use of SLM’s. These devices candisplay amplitude or phase distributions. The filtercan be generated by the use of a carrier frequency orit can be generated directly if two SLM’s are used incontact 1one operating in an amplitude mode and theother in a phase-only mode2. A major drawback tothese devices is the low resolution they exhibit. Asa result, scaling the distribution when it is fed to theSLM cannot be arbitrary to avoid undesired aliasingand loss of information. Matching the scale of thefilter in the SLM with that of the optically obtaineddistribution must depend mainly on the opticalsystem.In the anamorphic FRT processor, if a filter is to be

placed at the FRT plane, the scale of the distributionis crucial for obtaining the desired results. For acase in which an input object recorded in photo-graphic film and a computer-generated filter areused, the scales can be matched during the recordingprocess. However, the process must have a highaccuracy. If SLM’s are to be inserted for either theinput transparency or the filter, the scale cannot becontrolled at will. Especially complicated is thecase in which the sizes of the input and output do notcoincide. Away to overcome this problem is to use avariable scale transformer. In the case of a conven-tional Fourier transform, this is a well-known tech-nique for both the amorphic and the anamorphicprocessor. For the case of the FRT, a variable scaletransformer can be also designed.21 It is based onthe setup depicted in Fig. 1, for which the order canbe varied if the focal lengths of the lenses and thedistance between the input and output are varied.Moreover, there are different combinations of thisorder parameter that produce the same FRT order,with the only difference occurring in the scalingfactors. When the input transparency is illumi-nated with a nonparallel beam, the displacement ofthe object along the optical axis varies the conver-gence of the beam illuminating the input. Thisvariation is fully equivalent to changing the focallength of the first lens in the setup shown in Fig. 1.The separation between the input and the outputmust be varied accordingly to keep the condition ofits being a fractional transform of the desired order.The second lens is removed from the setup. Theresult is a fractional transform with a variable scale,but with an additional quadratic phase factor in theoutput plane. If the transformer is the first stage ofa complete correlator, this implies a change only tothe position of the output correlation. In the pre-sent paper, this idea is extended to the case of ananamorphic correlator.

5. Results

A. Computer Simulations

Several computer simulations were performed todemonstrate filter performance. For the computersimulations the input image shown in Fig. 4 is used.The constructed filter is supposed to recognize anF18 airplane in the upper part of the input and aTornado airplane in the lower part. For such a filterto be constructed, the F18 was shifted to the center ofthe upper part of the image, and the Tornado wasshifted to the center of the lower part of the image.The procedure outlined in Section 4 was followed toobtain a filter with fractional orders with values ofPx 5 1 and Py 5 0.5. Thus the obtained filter shouldbe shift invariant in the x direction, and it shouldalso have a small amount of shift invariance in the yaxis. In the input scene illustrated in Fig. 4 the ypositions of the centers of the F18 airplanes in theupper part were separated by a few pixels. Thesame thing was done to the y positions of the centersof the Tornado airplanes in the lower part of thescene. Figure 5 is the obtained output plane. Onecan notice good correlation peaks that indicate theexistence of the F18 in the upper part and theTornado in the lower part of the image.The possibility of the use of arbitrary impulse

responses in different parts of the image is demon-strated in Fig. 6. In this case the purpose was toobtain an x-scale-invariant recognition of the F18airplanes in the upper part of the scene and ay-scale-invariant recognition of the same target inthe lower part. For the filter to be constructed, anx-scale-invariant logarithmic harmonic of an F18was calculated and shifted to the center of the upperregion of the image. Then a y-scale-invariant loga-rithmic harmonic of an F18 was calculated andshifted to the center of the lower region of the image.A multiple-FRT filter matched to these two targetswith the values Px 5 1 and Py 5 0.5 was prepared.Figure 7 is the output correlation plane. Distinct

Fig. 4. Input image used for computer simulations and opticalexperiments.

10 July 1996 @ Vol. 35, No. 20 @ APPLIED OPTICS 3949

correlation peaks demonstrate successful recognition.A threshold of 35% of the maximum intensity valueis enough to detect all true target peaks from thebackground, as well as to reject peaks correspondingto other objects.

B. Optical Results

The performance of the suggested system was testedexperimentally in the laboratory. The binary maskfor the computer-generated hologram was plotteddirectly onto a DuPont scanner film by the useof a Dolev PS Scitex machine. It was thenreduced times 20 by the use of a high-resolutioncamera. The hologram was a 128 3 128 pixelLohmann’s encoding mask.24 The input-scene sizewas 4.2mm3 4.2mm, whereas the filter size was 8.5mm 3 12 mm. The aspect ratio between the axeswas calculated according to Eq. 162 3AR 5sin10.5p@22@sin1p@22 5 8.5@124. At the output plane,

Fig. 5. Numerical calculation of the correlation, showing thedetection of an F18 target in the upper part of the image and of aTornado target in the lower part.

Fig. 6. Input image used for computer simulations.

3950 APPLIED OPTICS @ Vol. 35, No. 20 @ 10 July 1996

a CCD camera connected with a Matrox image-LCframe grabber was used to grab the correlator out-puts. The input scene is illustrated in Fig. 4. Theexperimentally obtained output is shown in Fig. 8,which shows the intensity at the output correlationplane. The experimentally obtained results matchthe computer simulations illustrated in Fig. 5.For adjusting the system, a gauge plate is prepared.

It consists of the FRTwith fractional orders of valuesof Px 5 21 and Py 5 20.5, with four dots arranged inthe corners of a square. This actually representsthe addition of four zone plates. When the gaugeplate is placed in the input plane of the suggestedoptical system, the arrangement of the four dots isreproduced in the filter plane. In addition, a maskwith this arrangement of four dots is prepared toadjust the scaling of the system. This mask isplaced in the filter plane, and the system scaling isadjusted so that the four dots reproduced by thegauge plate coincide with the four dots of the mask.This arrangement is produced by the same methodas are the computer-generated filters, hence it hasthe same aspect ratio and scaling factor.The optical setup is constructed according to the

sketch illustrated in Fig. 3. The simplified systemdoes not permit the independent adjustment ofx- and y-scaling factors. Because of this, the aspectratio must be predefined. The setup is adjustedwith the following procedure: First an x-axis cylin-drical lens is inserted into the system and thefocusing plane of the illuminating point source islocated. This plane is the filter plane. The adjust-ing gauge plate is inserted between the lens and thefilter plane. The mask with the four dots is placedin the filter plane. By moving the gauge plate alongthe optical axis, one should be able to adjust the xsize of the optical intensity pattern to the x size of themask 1determined by the horizontal separation of the

Fig. 7. Numerical calculation of the correlation, showing thedetection of an F18 target with invariance to the one-dimensionalscale in two axes.

Fig. 8. Experimental results for multiple anamorphic FRT correlations with the image in Fig. 4.

four dots2. Then the y-axis cylindrical lens is added.Its position is determined so that, in the filter plane,the best focus is obtained. After the adjustment,the filter mask and the input object can be placed inthe setup. The second part of the system, consist-ing of a cylindrical and a spherical lens, render thefinal output in the CCD sensor.

6. Conclusions

In this paper several advancements in the use ofshift-variant pattern-recognition correlators havebeen introduced and demonstrated. An anamor-phic fractional correlator has been proposed and wastested in computer simulations and in optical experi-ments. With respect to the amorphic FRT correla-tor, the possibility of having a different degree ofshift variance on two orthogonal axes comprises themain advantage. The setup proposed for the experi-mental demonstrations permits matching the scalesbetween the optically obtained FRT distribution forthe input object and for the filter mask 1both havingan arbitrary size2. The shift-variant property of theproposed correlation scheme has been exploited forproducing a correlator that is able to detect differentobjects in different regions of the input image. Thisprocedure can also be used with different impulseresponses. In particular, when single terms of or-thogonal expansions are recorded in the filter, differ-ent invariances can be achieved in different zones ofthe input image. Computer simulations and opticalexperiments have been performed to demonstratethe above-mentioned characteristics.

Z. Zalevsky acknowledges the Eshcol Fellowshipgranted by the Israeli Ministry of Science and Arts.J. Garcıa acknowledges a grant from the Universitatde Valencia. This research was partially supported

by the Comision Interministerial de Ciencia y Tecno-logia, Spain, under project TAP93-0667-103-03.

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