modified fringe-adjusted joint transform correlation to accommodate noise in the input scene

Modified fringe-adjustedjoint transform correlationto accommodate noise in the input scene

RuiKang K. Wang, Lin Shang, and Chris R. Chatwin

A modified fringe-adjusted joint transform correlator is proposed that is able to accommodate noise inthe input scene. The effect of noise in the input scene on the performance of the joint transformcorrelator is analyzed and quantified. When the target is embedded in a severely noise-corrupted inputscene, it is shown that the proposed modified fringe-adjusted filter joint transform correlator delivers abetter correlation performance and the capacity to accommodate this noise than does the fringe-adjusted filter-based correlator. When the power spectra of the input image and the reference imageare subtracted from the power spectrum of the joint-input image, it is found that the noise effect on theoutput plane is independent of the objects in the input scene and originates from the convolution of thereference image and noise in the input scene.Key words: Fringe-adjusted filter, joint transform correlation, power spectrum, scene noise. r 1996

Optical Society of America

1. Introduction

For optical pattern recognition, the optical jointtransform correlator1 1JTC2 is an extremely usefulalternative architecture to the VanderLugt configu-ration.2 A VanderLugt-type correlator requires apriori fabrication of the filter used in the correlationprocess, and this restricts its real-time optical imple-mentation. Furthermore, thematched spatial filtermust be accurately aligned with the Fourier trans-form of the input image. The JTC has the advan-tage that the reference images can be updated in realtime, and it does not require a priori filter fabrication.One of the main problems with the classical JTC isthat the presence of a strong zero-order peak in theoutput plane, which corresponds to the sum of theautocorrelations of the reference and the input im-ages, always overlaps the desired correlation sig-nals.3,4 For a single noise-free target input, forexample, the zero-order peak is at least 4 timesstronger than the first diffraction-order peaks 1corre-lation peaks2. This situation becomes more serious

The authors are with the Laser and Optical Engineering Group,the Department of Mechanical Engineering, The University ofGlasgow, Glasgow G1Z 8QQ, United Kingdom.Received 13 December 1994; revised manuscript received 10

July 1995.0003-6935@96@020286-11$06.00@0r 1996 Optical Society of America

286 APPLIED OPTICS @ Vol. 35, No. 2 @ 10 January 1996

in the presence of noise in the input scene. Afurther difficulty is that the classical JTC produces abroad correlation peak resulting in low accuracy inthe determination of the target position in the inputscene.Javidi and Kuo5 proposed a binary JTC in which

the joint power spectrum 1JPS2 is made binarythrough the use of a nonlinear hard-clipping tech-nique to truncate the values to 11 or 21, before theinverse Fourier-transform operation is performed.A binary JTC is found to be superior to a classicalJTC in terms of the correlation-peak intensity, corre-lation-peak width, and discrimination sensitivity.3,5,6However, a binary JTC involves computation-inten-sive Fourier-plane JPS binarization, which limitsthe system processing speed.6,7,8,9 Furthermore, thebinary JTC does not overcome the problem of theproduction of a large zero diffraction-order term atthe center of the output plane. Recently, Alam andKarim10 and Kuo11 proposed a power-spectrum-subtraction technique to remove the redundant partof the joint Fourier transform before producing thecorrelation output by means of the inverse Fourier-transform operation. This technique not only suc-cessfully removes the large zero-order term in theoutput plane, but also has been shown to dramati-cally improve the correlation-peak intensity.Recently, Alam and Karim12 introduced a fringe-

adjusted-filter-based JTC in which the JPS is multi-

plied by the fringe-adjusted filter 1FAF2 before theinverse Fourier transform is applied to yield thecorrelation-signal output. The FAF-based JTC wasshown to produce a better correlation performancethan either the classical or binary JTC’s for bothsingle- and multi-object input scenes10,12 under vary-ing illumination conditions that affect the gray-levelresponse of the input scenes.13 This technique ap-pears to be particularly attractive as it avoids theproblems associated with the other techniques. Alot of research has been reported with noise-freesingle- or multiple-object inputs; however this inves-tigation discovered that the FAF-based JTC is sensi-tive to noise in the input scene. Thus, to enable theFAF-based JTC to accommodate noise in the inputscene, we propose in this paper a modified JTC inwhich a modified fringe-adjusted filter 1MFAF2 isused. The performance of the MFAF-based JTC isinvestigated through the use of both noise-free andnoise-corrupted input images. The proposed schemeis found to yield better results with noisy inputimages than those of the FAF-based JTC.

2. Analysis

A. Fringe-Adjusted JTC

A real-time fringe-adjusted JTC is shown in Fig. 1,12where the reference and input images are displayedsimultaneously in the input plane through the use ofa spatial light modulator 1SLM2. Assume thatr1x, y 1 y02 denotes the reference-image function andthat t1x, y 2 y02 represents the input-image function

in the input plane, separated by a distance 2y0 alongthe y axis. The input joint image function f 1x, y2 canbe expressed as

f 1x, y2 5 r1x, y 1 y02 1 t1x, y 2 y02. 112

Lens L1 in Fig. 1 performs the Fourier transform off 1x, y2, which is given by

F1u, v2 5 R1u, v2exp1 jvy02 1 T1u, v2exp12 jvy02, 122

where R1u, v2 and T1u, v2 are the Fourier transformsof r1x, y2 and t1x, y2, respectively; u and v are mutuallyindependent frequency-domain variables scaled by afactor 2p@lf, for which l is the wavelength of thecollimating light and f is the focal length of theFourier-transforming lenses L1 and L2. The jointpower spectrum, which is the intensity of the com-plex light distribution produced in the back focalplane of L1, is given by

0F1u, v2 02 5 0R1u, v2 02 1 0T1u, v2 02

1 R1u, v2T*1u, v2exp1 j2vy02

1 R*1u, v2T1u, v2exp12 j2vy02, 132

and it is detected with a square-law detector such asa CCD array or a liquid-crystal light valve. In aclassical JTC, the JPS, i.e., Eq. 132, is inverse Fouriertransformed by lens L2 to yield the correlationsignal. However, in a binary JTC, the JPS is firstbinarized by the application of a nonlinear hard-

Fig. 1. Schematic drawing of a fringe-adjusted JTC 1see Ref. 122.

10 January 1996 @ Vol. 35, No. 2 @ APPLIED OPTICS 287

clippingmask5 at the Fourier plane before taking theinverse Fourier transform of the JPS.Recently, a JTC based on an amplitude-modulated

filter 1AMF214 was proposed in which the AMF isdefined by

Hamf 1u, v2 51

0R1u, v2 02. 142

The JPS is multiplied by Hamf before the inverse-Fourier-transform operation is applied to producethe correlation output. This AMF-based JTC isfound to yield a better correlation performance thandoes a binary JTC. However, the fact that 0R1u, v2 022

may create one or more poles may contribute to otherserious problems. To overcome problems occurringwith poles in theAMF-based JTC,Alam and Karim12

proposed the fringe-adjusted JTC shown in Fig. 1, forwhich the FAF is defined by

Hfaf 1u, v2 5B1u, v2

A1u, v2 1 0R1u, v2 02, 152

where A1u, v2 and B1u, v2 are either constants orfunctions. When B1u, v2 is properly selected, onecan avoid having an optical gain greater than unity.With a small value of A1u, v2, the pole problem iseliminated, while at the same time it is possible toachieve high autocorrelation peaks. The FAF is areal-valued function because it involves only theintensity and has no phase terms; a FAF is thereforesuitable for optical implementation. The computa-tions required to produce the FAF may be completedbefore the input scene is introduced into the inputplane of the JTC. Thus, as long as there is norequirement to change the reference object, theinclusion of the filter does not have any significantdetremental affect on the processing speed of thesystem. However, an additional 1SLM2 is necessaryto display the FAF function, as is shown in Fig. 1.A major problem limiting the performance of a

FAF-based JTC is that it is sensitive to noise in theinput scene. When A1u, v2 in Eq. 152 is set to a smallvalue, the FAF greatly enhances the noise effect onthe JTC system; thus it reduces the performance forreal images in which noise is endemic.

B. Input-Noise Characterization

Let us assume that the input scene consists of areference object, with additive noise denoted byn1x, y2; thus the input scene can be expressed as

t81x, y2 5 t1x, y2 1 n1x, y2. 162

The input joint image function f 1x, y2 now becomes

f 1x, y2 5 r1x, y 1 y02 1 t1x, y 2 y02 1 n1x, y 2 y02. 172


Lens L1, shown in Fig. 1, performs the Fouriertransform of Eq. 172, which is given by

F1u, v2 5 R1u, v2exp1 jvy021 T1u, v2exp12 jvy02

1 N1u, v2exp12 jvy02, 182

where N1u, v2 denotes the Fourier transform of thenoise function n1x, y2. Thus the JPS at the backfocal plane of L1 is then written as

0F1u, v2 02 5 0R1u, v2 02 1 0T1u, v2 02

1 2 Re5R*1u, v2T1u, v26cos12vy02

1 2 Re5N1u, v2R*1u, v26cos12vy02

1 2 Re5N1u, v2T*1u, v26 1 0N1u, v2 02. 192

From Eq. 192 it can be seen that noise in the inputscene contributes to the last three terms of theoutput signal, which may alternatively affect thecorrelation-peak quality. A comparison of Eq. 192with Eq. 132 shows that noise in the input scene playsan important role in the JPS, which is recorded witha square-law detector.The FAF accentuates the higher-frequency values

when A1u, v2 is set to a small value; this accentuationenhances the relative magnitude of the last threeterms in Eq. 192 when the input scene is embedded innoise. Usually the Fourier transform of the refer-ence image concentrates most of the energy at lowspatial frequencies with little energy in the highfrequencies. This concentration at the low end re-sults in the power spectrum of the reference image,i.e., 0R1u, v2 02, having an extremely large dynamicrange. As 0R1u, v2 02 appears in the denominator ofthe FAF 3Eq. 1524, the dynamic-range problem isalleviated somewhat, but the system still is quitesensitive to noise in the input scene.

C. Modified Fringe-Adjusted JTC

To overcome the problems described in Subsections2.A. and 2.B., a modified fringe-adjusted JTC isproposed. The MFAF is defined by

Hmfaf 1u, v2 5B1u, v2

A1u, v2 1 0R1u, v2 0. 1102

Equation 1102 still retains the advantages of the FAF.When B1u, v2 is properly selected, an optical gaingreater than unity can be avoided. The pole prob-lem is also overcome when the value of A1u, v2 isselected to be small, while high autocorrelationpeaks are still achieved at the output plane. Clearlythe dynamic range of 0R1u, v2 0 is much smaller thanthat of 0R1u, v2 02, which is used in the FAF; thus thedistribution of energy in the JPS is better optimizedto cope with noise. AMFAF-based JTC is thereforeexpected to deliver better noise robustness than doesa JTC based on the FAF.The real-valued MFAF is more suitable for display

on a SLM than is the FAF, as the MFAF has a

smaller dynamic range. The computations re-quired to generate the MFAF may be completedsufficiently rapidly that its use does not limit theprocessing speed of the system. This is, of course,not true if the reference object has to be updated.The modified fringe-adjusted JPS is obtained by

multiplication of the filter function by the JPS.This multiplication is achieved in the same manneras for the FAF-based JTC12 illustrated in Fig. 1.Thus the modified fringe-adjusted JPS may be ex-pressed as

G1u, v2 5 Hmfaf 1u, v2 0F1u, v2 02

5B1u, v2

A1u, v2 1 0R1u, v2 03 0R1u, v2 02 1 0T1u, v2 02

1 2 Re5R*1u, v2T1u, v26cos12vy024, 1112

where the input image t1x, y2 is assumed to be thenoise-free case. If the input object is embedded inbackground noise n1x, y2, then, from Eq. 192, themodified fringe-adjusted JPS is given by

G1u, v2 5B1u, v2

A1u, v2 1 0R1u, v2 03 0R1u, v2 02 1 0T1u, v2 02

1 2 Re5R*1u, v2T1u, v26cos12vy02

1 2 Re5N1u, v2R*1u, v26cos12vy02

1 2 Re5N1u, v2T*1u, v26 1 0N1u, v2 024, 1122

when B1u, v2 5 1 and 0R1u, v2 0 : A1u, v2 and when thereference target is the same as the input target, i.e.,r1x, y2 5 t1x, y2, Eq. 1122 reduces to

G1u, v2 < 2 0R1u, v2 0 1 2 0R1u, v2 0 3cos12vy024

10N1u, v2 02

0R1u, v2 01 2 Re5N1u, v2exp12ifr26

3 31 1 cos12vy024, 1132

where fr denotes the phase of the Fourier transformof the reference image. Taking the inverse Fouriertransform of Eq. 1132 yields the output signal contain-ing the desired correlation patterns. It can be seenfrom equivalency 1132 that the second term is just likethat of the phase-only filter15 in the matched spatialfilter, as

2 0R1u, v2 0 3cos12vy0245 exp12ifr2R1u, v2exp1i2vy02

1 3exp12ifr2R1u, v2exp1i2vy024*.

1142

This equation produces the two desired correlationpeaks, with the same performance as the phase-onlyfilter, but with a separation distance of 4y0 along they axis. This is a particularly attractive characteris-tic of the MFAF-based JTC.For the same condition, the FAF-based JTC is

expressed as

G1u, v2 < 2 1 2 cos12vy02 10N1u, v2 02

0R1u, v2 02

12

0R1u, v2 0Re5N1u, v2exp12ifr26

3 31 1 cos12vy024. 1152

The second term is very attractive because it isactually an inverse filter. However, it is evidentfrom equivalent 1152 that the noise effect in the lasttwo terms of that expression are greatly enhancedbecause of the very small value of 0R1u, v2 02 at highfrequencies 1usually 0R1u, v2 02 9 1.02. The MFAF-based JTC 3equivalent 11324 reduces the noise effect bya factor of 1@ 0R1u, v2 0 . Notice that 1@ 0R1u, v2 0 : 1.0for the higher-frequency components.From equivalent 1132 it can also be seen that a

zero-order and a noise term will be present in theoutput plane. It therefore would appear to be asimple matter to block the zero-order term throughthe use of an optical stop in the output plane.However, when the input image is corrupted bynoise, the zero-order term becomes more compli-cated, and the use of an optical stop may not beeffective; along with simulation results, a furtherexplanation of this issue is given in Subsection 2.D.Although the deleterious effect of noise on the JTC

has been reduced through the use of the MFAFrather than the FAF, the noise term, i.e., the last twoterms of expression 1132, may still result in poortarget detection when the input scene is embeddedin severe background noise. To further reduce thenoise effect in the output plane, and at the same timeeliminate the zero-order term, another architectureis suggested. One achieves this architecture bydisplaying the input scene at the input plane of theJTC in the absence of the reference image, thenrecording only the input-scene power spectrum,10which is expressed as

0I1u, v2 02 5 0T1u, v2exp1 jvy02 1 N1u, v2exp12 jvy02 02

5 0T1u, v2 02 1 0N1u, v2 02 1 2 Re5N1u, v2T*1u, v26,

1162

and finally displaying the reference image to produceonly its power spectrum, 0R1u, v2 02. When the input-scene-only power spectrum 3Eq. 11624 and the refer-ence-image power spectrum 0R1u, v2 02 are subtractedfrom the noise-corrupted JPS expressed by Eq. 192,the resultant modified JPS can be expressed as

P1u, v2 5 0F1u, v2 02 2 0R1u, v2 02 2 0I1u, v2 02

5 2 Re5T1u, v2R*1u, v26cos12vy02

1 2 Re5N1u, v2R*1u, v26cos12vy02, 1172


where the computation involving the reference-image power spectrum 0R1u, v2 02 may be completedbefore the JTC operation is performed. The subtrac-tion operation can be performed either optically16 orelectronically with the computer shown in Fig. 1.When Eq. 1172 is compared with Eq. 192, it can be seenthat the noise effect in the JTC is greatly reduced,although not completely eliminated; furthermore,the large zero-order term is completely removed.In Subsection 2.D., simulation results are used toillustrate how the subtraction method greatly im-proves the ability of the JTC to accommodate noisein the input scene.When the modified JPS 3Eq. 11724 is multiplied by

the MFAF, i.e., Eq. 1102, the final modified fringe-adjusted JPS becomes

P1u, v2 52B1u, v2

A1u, v2 1 0R1u, v2 0

3 3Re5T1u, v2R*1u, v26cos12vy02

1 Re5N1u, v2R*1u, v26cos12vy024. 1182

Lens L2 in Fig. 1 takes the inverse Fourier transformof Eq. 1182 to produce correlation patterns withreduced noise terms at the output plane.An alternative hybrid architecture for the imple-

mentation of the MFAF-based JTC is shown in Fig.2. Three separate CCD’s are used to capture the


following: the joint transform power spectrum, thepower spectrum of the input scene, and the powerspectrum of the reference image. These power spec-tra are sent to the computer simultaneously toevaluate the MFAF-based JPS. Providing that theCCD’s have a large dynamic range, the power spec-tra of the input and reference images can be electroni-cally removed from the JPS by pixelwise substraction.The modified fringe-adjusted JPS is displayed onSLM2 1see Fig. 22, which is then optically addressedby a parallel laser beam to produce the desiredcorrelation patterns at the output plane by means ofa Fourier-transforming lens L2, also shown in Fig. 2.The computer, as shown in Fig. 2, can also bereplaced by a custom-designed high-speed micropro-cessor, so as to improve the processing speed of thesystem. Notice that the MFAF can be produceddirectly from the power spectrum of the referenceimage that is captured by CCD1 and then electroni-cally multiplied by the JPS; thus the reference imagecan be updated in real time, thereby making thissystemmore flexible.It should be noted that after the input-scene-only

spectrum and the reference-image power spectrumwere subtracted from the noise-corrupted JPS, thefinal modified fringe-adjusted JPS 3Eq. 11824, whichmust be encoded onto SLM2, may contain negativevalues. To overcome this problem, an additionalSLM operating in the binary-phase-only mode 1SLM3

Fig. 2. Schematic diagram of an alternative real-time fringe-adjusted JTC.

as shown in Fig. 22 must be added into the systemjust before SLM2. Thus, SLM3 applies the modula-tion

C1u, v2 5 511,

P1u, v2 $ 0,

21 otherwise1192

whereas SLM2, which is operated in the amplitudemode, applies

P81u, v2 5 5P1u, v2 P1u, v2 $ 0

2P1u, v2, otherwise, 1202

which guarantees that all the values encoded ontoSLM2 are positive. Therefore, if one is to achievethe required negative values obtained with Eq. 1182, a180° phase shift is made by the binary-phase-onlySLM3; this reverses the signs of the positive valuesencoded onto SLM2.

D. Multi-Object Modified Fringe-Adjusted JTC

If the input scene contains n objects as t11x 2 x1,y 2 y12, t21x 2 x2, y 2 y22, . . . , tn1x 2 xn, y 2 yn2 and alsocontains noise n1x, y 2 y02, the joint input image maybe expressed as

f 1x, y2 5 r1x, y 1 y02 1 oi51

n

ti1x 2 xi, y 2 yi2

1 n1x, y 2 y02, 1212

the corresponding JPS is given by

0F1u, v2 02

5 0R1u, v2 02 1 oi51

n

0Ti1u, v2 02

1 2 oi51

n

Re5Ti1u, v2R*1u, v26cos3uxi 1 v1 y0 1 yi24

1 oi51

n

ok51

n

Re5Ti1u, v2Tk*1u, v26

3 cos3uxi 1 v1 yi 2 yk24

1 2 Re5N1u, v2R*1u, v26cos12vy02 1 0N1u, v2 02

1 2 oi51

n

Re5N1u, v2Ti*1u, v26cos3ux0 1 v1 y0 2 yi24,

1222

where i fi k and Ti1u, v2 denotes the Fourier trans-form of ti1x, y2. The correlation output will containthe following terms: autocorrelations of the refer-ence and the input objects, cross correlations be-tween the reference and the input objects, and crosscorrelations between the different input objects andvarious noise terms. The last four terms in Eq. 1222may produce false alarms in the correlation plane;this possibility is especially likely for the noise

terms. Such false alarms can be avoided or reducedby the elimination of the cross-correlation termsbetween the different input objects and the reductionof the noise terms. These results can be achieved bythe subtraction of the input-scene-only power spec-trum and the reference-object power spectrum fromthe JPS expressed by Eq. 1222 and results in

P1u, v2 5 2 oi51

n

Re5Ti1u, v2R*1u, v26cos3uxi 1 v1 y0 1 yi24

1 2 Re5N1u, v2R*1u, v26cos12vy02. 1232

When this modified JPS 3Eq. 12324 is multiplied by theexpression for an MFAF, i.e., Eq. 1102, the finalmodified fringe-adjusted JPS for multiple input ob-jects is

G1u, v2 52B1u, v2

A1u, v2 1 0R1u, v2 0

3 3oi51

n

Re5Ti1u, v2R*1u, v26cos3uxi 1 v1 y0 1 yi241 Re5N1u, v2R*1u, v26cos12vy024. 1242

From Eqs. 1172 or 1182 and Eqs. 1232 or 1242, it is evidentthat noise in the output plane is independent of themultiple objects in the input scene; it comes onlyfrom the convolution of the reference object with thenoise. This very attractive result illustrates amethod for the reduction of the deleterious effect ofnoise on the performance of the JTC system.

3. Results

To investigate the performance of the proposedmodi-fied fringe-adjusted JTC, we considered the follow-ing three cases: 112 an input scene with a singlenoise-free object, 122 an input scene with a singleseverely noise-corrupted object, and 132 an inputscene containing multiple objects in a noisy back-ground. The results are compared with the FAF-based JTC. For both the MFAF- and FAF-basedJTC’s, A1u, v2 was taken to be 1 3 1026 to overcomethe pole problem, and B1u, v2 was set to unity. In allcases, the correlation-peak intensity was normalizedwith respect to the total energy of the output plane,17so that a perfect autocorrelation would use the fulldynamic range of the 256 gray levels 1i.e., 8 bits2 andall correlation outputs are fully resolved to yield ameaningful comparison of the performances of thedifferent filters investigated.

A. Input Scene with a Noise-Free Single Object

For the single-object input scene, a 30 3 48 pixel,noise-free image of a Bradley Armored PersonnelCarrier 1APC2 vehicle was used as the referenceimage, as shown in Fig. 31a2. The same vehicle, withthe same resolution, was taken as the target image.The two images were combined and zero padded toform a 1283 128 pixel-array joint image. This jointinput image was introduced to SLM1 1shown in Fig. 22


at the input plane, and then the power spectra of thejoint image, the input image, and the referenceimage were captured by CCD3, CCD2, and CCD1,respectively, as shown in Fig. 2. The computationof the MFAF or FAF, the power-spectrum subtrac-tion, and the multiplication operations are com-pleted by the computer. The final modified fringe-adjusted JPS is inverse Fourier transformed to yieldthe desired correlation output.The correlation output for the MFAF-based JTC is

shown in Fig. 41a2 and demonstrates that the output

1a21b2

Fig. 3. Bradley APC images used in the simulation: 1a2 anoise-free image, and 1b2 a noise-corrupted image with a signal–energy-to-noise–energy ratio of 0.21.

1a2

1b2

Fig. 4. Three-dimensional plots of the correlation output func-tions when the input scene is free of noise: 1a2 from the MFAF-based JTC, and 1b2 from the FAF-based JTC.


correlation peaks are extremely well defined, show-ing that the target can be detectedwithout ambiguity.For comparison, the correlation output for the FAF-based JTC is shown in Fig. 41b2. It can be seen fromFigs. 41a2 and 41b2 that the FAF-based JTC delivers abetter performance than the MFAF-based JTC whenthe input scene has a zero noise content. Theperformance is quantified by the results listed inTable 1 for the autocorrelation-peak intensity 1API2,the correlation–peak-to-secondary–peak ratio 1PSR218the correlation–peak-to-noise ratio 1PNR2, which isdefined by the ratio of the correlation-peak intensityto the average value of the noise intensity at theoutput plane,19,20 and the full width of the correlation-peak intensity at its half-maximum value.From the results in Table 1, it can be seen that the

correlation-peak quality of the FAF-based JTC ismuch better than that of the MFAF-based JTC; thisis because multiplication of the FAF with the JPSgreatly enhances the high-frequency components ofthe modified JPS. However, there is no doubt that,from the results in Table 1 and Fig. 41a2, the MFAF-based JTC delivers a good correlation performanceand very effectively detects the target in the inputscene. The FWHM metric is found to be 1 3 1,which is equal to that of the FAF-based JTC.

B. Input Scene with a Single Noise-Corrupted Object

An effective correlation system should be able toaccommodate noise in the input scene; this meansthat, if the target is embedded in a noisy back-ground, the correlator can still recognize the noise-corrupted target. In this subsection, the noise ro-bustness of the MFAF-based JTC is investigated andcompared with that of the FAF-based JTC. Thenoisy input scene is shown in Fig. 31b2, in which a30 3 48 pixel array of the Bradley APC vehicle isseverely corrupted by noise, the ratio of the targetsignal energy to the noise energy is 0.21. If onescrutinizes Fig. 31b2, it is evident that this is adifficult pattern-recognition problem.First, theMFAF-based JTC, with no subtraction of

the input-image or the reference-image power spec-tra from the joint-image power spectrum, is investi-gated. Figure 51a2 shows the three-dimensional 13D2plot of the correlation output from the MFAF-basedJTC. From Fig. 51a2 it can be seen that a large and

Table 1. Quantified Results from an Input Scene with a Noise-FreeSingle Object

Typeof JTC

Auto-correlation-

PeakIntensity

API

Ratio of theCorrelationPeak to theSecondaryPeak PSR

Ratio of theCorrelationPeak to theNoise PNRa

Full Widthat Half-MaximumFWHM

MFAF based 11.90 24.95 839.43 1 3 1FAF based 54.25 100.54 6024.6 1 3 1

aThe PNR is defined by the ration of the correlation-peakintensity to the average value of the noise intensity at the outputplane.

broad zero-order term is present in the output plane;the correlation peaks are embedded in a severelynoise-corrupted background, which makes their de-tection a difficult task. It is evident from Fig. 51a2that the use of an optical stop to block the zero-orderterm would not be effective. For comparison, whenthe noisy image of Fig. 31b2 is input into the FAF-based JTC, the 3D plot of the output correlationfunction is shown in Fig. 51b2. Clearly the FAF-based JTC is extremely sensitive to noise in theinput scene, as the correlation peaks are completelylost in the output-plane noise. As one can see fromFig. 51a2, the MFAF-based JTC produces two correla-tion peaks at the output plane that, although theyare embedded in noise, are detectable.Figure 61a2 shows the result when the power

spectra of the input image and the reference imageare subtracted from the joint-image power spectrumby means of a 3D plot of the correlation outputfunction for the MFAF-based JTC. The MFAF-based JTC gives an excellent result. Comparedwith Fig. 51a2, it is evident that the subtractiontechnique is extremely useful in the reduction of thedeleterious effect of noise on the JTC. It can also beseen from Fig. 61a2 that the zero-order term iscompletely removed from the output plane and that

1a2

1b2

Fig. 5. Three-dimensional plots of the correlation output func-tions with no power-spectra subtraction when the input scene isnoise corrupted: 1a2 from the MFAF-based JTC, and 1b2 from theFAF-based JTC.

the correlation signal can be detected with a simplethreshold detector. For the same case, Fig. 61b2shows the 3D plot of the correlation output functionthrough the use of the FAF-based JTC. Two correla-tion peaks are evident in the output plane, but theyare seriously corrupted by noise, and this makestheir detection quite difficult. It can be concluded,from Figs. 61a2 and 61b2 that the MFAF-based JTC isfar more robust in response to noise than the FAF-based JTC. Furthermore, the results prove thatthe subtraction technique is effective in the reduc-tion of noise in the output correlation plane.The performances of both the MFAF- and FAF-

based JTC’s, Figs. 61a2 and 61b2, are quantified by theresults listed in Table 2. From the results listed inTable 2 it can be seen that, when the target isseverely corrupted by noise in the input scene, theautocorrelation-peak intensity 10.6912 produced bythe MFAF-based JTC is approximately 10% higherthan the same intensity 10.6342 produced by theFAF-based JTC. The autocorrelation–peak-to-sec-ondary–peak ratio and the correlation–peak-to-noiseratio from the MFAF-based JTC are found to bealmost twice those of the FAF-based JTC. Noticethat the value of the PSR from the FAF-based JTC is

1a2

1b2

Fig. 6. Three-dimensional plots of the correlation output func-tions with power-spectra subtraction from a noise-corrupted inputscene: 1a2 from the MFAF-based JTC, and 1b2 from the FAF-basedJTC.


1.82, which is less than 2.0; thus, the secondary-peakintensity is greater than half that of the correlation-peak intensity. This may lead to false alarms21,22 ifthe thresholding value of a detector used to detectthe correlation signal is set to PSR 5 2.0, whereasthe MFAF-based JTC with PSR 5 3.20 will detectthe target without ambiguity.

C. Multiple-Object Input Scene with Background Noise

When the input scene contains noise-free multipleobjects 3Fig. 7 1only a 128 3 192 pixel array is shown;this scene is zero padded to give a 256 3 256 pixelarray for the simulations24, the MFAF-based JTCdelivers a good performance, the correlation-peakintensity from the target object is extremely welldefined with only a small signal from the nontargetobjects. For brevity, this case is not reported herein.This subsection concentrates on the MFAF-basedJTC with a multiple-object input that is severelycorrupted by noise. The joint-image input consistsof a number of tanks with the reference imageseparated from the input scene. The input scenecontains several different tanks; however, the inputscene is severely corrupted by noise, as shown in Fig.8. Note that one of the objects in the input scene1the object located at the bottom left-hand-side posi-tion2 would be identical to the reference vehicle, i.e.,the Bradley APC vehicle, if the input scene were notcorrupted by noise.Figure 8 was used as the input to the MFAF-based

JTC, and the final correlation output is shown in Fig.

Table 2. Quantified Results from an Input Scene with aNoise-Corrupted Single Object

Typeof JTC

Auto-correlation-

PeakIntensity

API


Ratio of theCorrelationPeak to theNoise PNR



Fig. 7. Noise-free multiple-object input scene used for the simu-lation.


91a2. For comparison Fig. 91b2 shows the correlationoutput from the FAF-based JTC. From Figs. 9 itcan be seen that the MFAF-based JTC delivers abetter ability to accommodate noise in the inputscene than does the FAF-based JTC. Table 3 liststhe quantified results from Figs. 91a2 and 91b2. It canbe seen from this table that the target correlation-peak intensity produced by the MFAF-based JTC isapproximately 20% higher than that produced by the

Fig. 8. Noise-corrupted multiple-object input scene used in thesimulation.

1a2

1b2

Fig. 9. Three-dimensional plots of the correlation output func-tions with power-spectra subtraction from a noise-corruptedmulti-object input scene: 1a2 from the MFAF-based JTC, and 1b2from the FAF-based JTC.

FAF-based JTC. The correlation-peak intensityfrom the target object is 3.29 times that of thesecondary-peak intensity; hence the MFAF-basedJTC can detect the target from the noise-corruptedmultiple-object input scene, without ambiguity,through the use of a thresholding detector that is setto half the maximum correlation-peak height. ThePSR value for the FAF-based JTC is only 1.80, whichmeans that there are several secondary peaks higherthan half the value of the maximum correlationpeak, and these peaks may cause false alarms whenthe target is being detected from the input sceneshown in Fig. 8.The PNR metric quantifies the noise robustness of

the correlator at the output plane. From Table 3one can see that theMFAF-based JTC gives a PNR of72.66, which is approximately 2.5 times greater thanthat for the FAF-based JTC 129.802. Thus, theMFAF-based JTC delivers a far greater capacity toaccommodate noise in the input scene when thereare multiple objects than does the FAF-based JTC.In all cases, the full width of the correlation-peak

intensity at its half-maximum is found to be 1 3 1 forboth the MFAF- and the FAF-based JTC’s, eitherwith the single-object input or with the multiple-object input. Hence the position of the target objectwithin the input scene can be located with highaccuracy.

4. Conclusions

A modified fringe-adjusted joint transform correla-tion 1JTC2 filter that can accommodate noise in theinput scene has been proposed. The effect of noisein the input scene on the JTC has been analyzed andquantified. If the joint power spectrum 1JPS2 ismodified by the subtraction of the power spectra ofthe input image and of the reference image from theJPS, the noise effect in the output plane is indepen-dent of the objects in the input scene. Output planenoise results only from the convolution of the refer-ence image with noise in the input scene. Anarchitecture to implement the proposed modifiedfringe-adjusted JTC in real time has been suggested.The proposedmodified fringe-adjusted filter 1MFAF2

based JTC is found to yield an unambiguous, in-tense, high-fidelity correlation peak for single- andmultiple-object input scenes with either a noise-freeor a severely noise-corrupted input scene. When

Table 3. Quantified Results from an Input Scene with MultipleNoise-Corrupted Objects

Typeof JTC

Auto-correlation-

PeakIntensity

API


Ratio of theCorrelationPeak to theNoise PNR



compared with the fringe-adjusted filter 1FAF2 basedJTC, it has been shown that the MFAF-based JTCdelivers a better capacity to accommodate noise inthe input scene. Furthermore, when the targetobject is severely corrupted by noise, the correlation-peak intensity from the target image when theMFAF-based JTC is used is at least 10% higher thanthat for the FAF-based JTC; this is true for eithersingle- ormultiple-object input scenes. It is interest-ing that, for the noise-free case, the FAF-based JTCgives better correlation results than does the MFAFsystem; this result highlights the importance ofassessing the filter-noise performance.Future work is planned that will concentrate on

the optimization of the functions A1u, v2 and B1u, v2for the MFAF and on the optimization of the JTC toenhance its ability to accommodate noise in theinput scene.

R. K. Wang and L. Shang were supported by theUniversity of Glasgow Postgraduate ScholarshipScheme and the Overseas Research StudentsAwardsScheme at the time this research was completed.

The authors can be reached by means of e-mail [email protected].

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