Adaptive phase-input joint transform correlator

Adaptive phase-input joint transform correlator

Victor H. Diaz-Ramirez* and Vitaly KoberDepartment of Computer Science, CICESE, Km 107 carretera Tijuana-Ensenada, Ensenada B. C. Mexico, 22860

*Corresponding author: [email protected]

Received 9 April 2007; revised 4 July 2007; accepted 5 July 2007;posted 6 July 2007 (Doc. ID 81947); published 5 September 2007

An adaptive phase-input joint transform correlator for pattern recognition is presented. The input of thesystem is two phase-only images: input scene and reference. The reference image is generated with a newiterative algorithm using phase-only synthetic discriminant functions. The algorithm takes into accountcalibration lookup tables of all optoelectronics elements used in optodigital experiments. The designedadaptive phase-input joint transform correlator is able to reliably detect a target and its distortedversions embedded into a cluttered background. Computer simulations are provided and compared withthose of various existing joint transform correlators in terms of discrimination capability, tolerance toinput additive noise, and to small geometric image distortions. Experimental optodigital results are alsoprovided and discussed. © 2007 Optical Society of America

OCIS codes: 070.4550, 100.4550.

1. Introduction

The classical joint transform correlator [1] (JTC) is apopular processor for real-time pattern recognition.Its optical setup is less sensitive to misalignments [2]compared with that of the 4f correlator [3]. A maindrawback of the classical JTC is its poor performancewhen an input scene contains objects embedded intoa cluttered background. Note that its correlation out-put is very sensitive to small geometric distortions ofobjects (for instance, in-plane rotations and scalechanges). Several variants of the correlator have beenproposed to improve its performance in terms of var-ious quality measures [4]. Among these variants, thenonlinear [5,6], fringe-adjusted [7], and phase-input[8] JTCs are the most successful. These processorsyield better performance in terms of discriminationcapability (DC) than that of the classical JTC. How-ever, they are still sensitive to small geometric dis-tortions of objects to be recognized. Moreover, thesesystems have a poor tolerance to sensor noise andoften fail to distinguish a target and similar falseobjects. Many attempts have been made to incorpo-rate in the JTC the invariance to geometric distor-tions [9,10]. Synthetic discriminant function (SDF)filters [11,12] are attractive techniques for distortion-

invariant pattern recognition. The common way todesign correlation filters is to produce filters thatoptimize some performance criteria [4]. Some of thesemeasures can essentially be improved using an adap-tive approach to the filter design [13]. According tothis concept, we are looking for a filter optimized fora given observed scene, i.e., with a fixed set of pat-terns and a fixed background to be rejected, ratherthan a filter with average performance parametersover an assemblage of images (classical correlationfilters). Recently, an adaptive (JTC) (AJTC) fordistortion-invariant pattern recognition was pro-posed [14]. The AJTC yields superior performance interms of DC compared with that of the nonlinear andfringe-adjusted JTCs. A reference image for theAJTC is a real-valued bipolar image, which cannot bedirectly displayed on conventional amplitude spatiallight modulators (SLMs). For this reason, the op-todigital implementation of the AJTC requires eithertwo correlations or extensive pointwise postprocess-ing [14]. Conventional electrically addressed SLMssuch as twisted nematic liquid crystal displays(LCDs) can produce amplitude or phase-only modu-lations by configuring the polarization vector of anincident wavefront [15,16]. Phase-only modulationworks in a range of ��, ��. Use of the phase-onlymodulation regime has two advantages: any bipolarimage can be directly modulated on a phase-onlySLM, and no postprocessing is required to obtain the

0003-6935/07/266543-09$15.00/0© 2007 Optical Society of America

6543 APPLIED OPTICS � Vol. 46, No. 26 � 10 September 2007

correlator output. Actually, there are two ways tocarry out pattern recognition with a phase-only SLM:(i) design a filter taking into account characteristicsof a real SLM; (ii) design a conventional correlationfilter and then project its impulse or frequency re-sponse onto the coding domain of the modulator.Distortion-invariant pattern recognition with SDFfilters has been implemented on phase-only SLMs[17,18]. In this paper we propose a new adaptivephase-input JTC (APIJTC). A new iterative algo-rithm using introduced phase-only SDFs generates aphase-only reference image for the phase-input JTC.The algorithm guarantees a given value of the DC.The APIJTC is able to take into account calibrationlookup tables of all optoelectronics elements used inreal experiments. The obtained phase-only referenceimage contains information about a target and itsvariants to be detected and nondesired objects includ-ing a background to be rejected. The proposed API-JTC yields similar performance with respect torecognition capability to that of the AJTC. However,the APIJTC requires only one optical correlation andno postprocessing is needed. Finally, note that thefirst correlation of the JTC is a phase-only operationthat leads to maximum light efficiency at the firststep.

The paper is organized as follows: In Section 2 abrief review of conventional and phase-input JTCs isprovided. In Section 3 we present the design of aphase-only SDF-based reference image. This sectionis concluded with considerations for an experimentalimplementation of the system. In Section 4 computersimulation and experimental results obtained withthe APIJTC are provided and discussed. Section 5summarizes our conclusions.

2. Joint Transform Correlators

A. Classical Joint Transform Correlator

A block diagram of the classical JTC is shown in Fig.1. The input plane f�x, y� consists of a scene images̃�x, y� alongside a reference image r�x, y�, which areseparated by a distance, say 2�x:

f�x, y� � s̃�x � �x, y� � r�x � �x, y�. (1)

The scene s̃�x, y� contains input objects s�x, y� (de-sired and nondesired) embedded into a nonoverlap-ping background b̃�x, y� � w�x � x0, y � y0�b�x, y�.Here b�x, y� is a background image, �x0, y0� are un-known coordinates of the target in the input sceneand, w�x � x0, y � y0� is a binary function defined as

w�x � x0, y � y0� ��0, within the object area1, otherwise. (2)

The joint power spectrum of f�x, y� is given by

JPS��, v� � �S��, v��2 ��B̃��, v��2� �R��, v��2

� S��, v�B̃*��, v� � B̃��, v�S*��, v�� S��, v�R*��, v� � B̃��, v�R*��, v�� exp�i2�x�� R��, v�S*��, v�� R��, v�B̃*��, v��exp��i2�x��, (3)

where “*” denotes the complex conjugate, S��, v�,R��, v�, and B̃��, v� are the Fourier transforms ofs�x, y�, r�x, y�, and b̃�x, y�, respectively. Applying theinverse Fourier transform to Eq. (3), we obtain

c�x, y� � css�x, y� � cb̃b̃�x, y� � crr�x, y� � csb̃�x, y�� cb̃s�x, y� � csr�x � 2�x, y� � cb̃r�x � 2�x, y�� crs�x � 2�x, y� � crb̃�x � 2�x, y�. (4)

Here cab�x, y� � a�x, y� � b�x, y�, where “�” is thecorrelation operator. From Eq. (4), it can be seen thatthe cross-correlation signals of interest are placed atthe distances 2�x from the origin. Note that thecross-correlation terms containing b̃�x, y� severelyaffect the performance of the classical JTC withrespect to the DC. The DC is formally defined as theability of a system to distinguish a target amongother different objects. If a target is embedded in abackground that contains false objects, then the DCcan be written as [19]

DC � 1 ��CB�0, 0��2

�CT�0, 0��2, (5)

where |CB|2 is the maximum of intensity in the cor-relation plane over the area of the background to berejected and |CT|2 is the maximum intensity in thecorrelation plane over the area of the target position.The area of the target position is determined in closeproximity to the actual target location. The area ofthe background is complementary to the area of thetarget position. Negative values indicate that a testedfilter fails to recognize the target.

B. Phase-Input Joint Transform Correlator

To reduce sidelobes in the JTC output owing to abackground, a phase-encoding of the JTC input planewas proposed [8]. This can be done by a monotonicmapping of intensity to phase distributions as fol-lows:

z � EXP�i� z�x, y� � Gmin

Gmax � Gmin�, (6)

Fig. 1. Block diagram of the classical joint transform correlator.

10 September 2007 � Vol. 46, No. 26 � APPLIED OPTICS 6544

where “EXP ” is a pointwise exponential operator,z�x, y� is an input image, and �Gmax, Gmin�, are maxi-mum and minimum values of z�x, y�. A block diagramof the phase-input JTC based on this mapping isshown in Fig. 2. Let f�x, y� be a bipolar input imagefor the JTC [see Eq. (1)] with a signal range of��1, 1�. Using Eq. (6) the phase-only image is given by

f�x, y� � EXP�i�s̃�x � �x, y�� EXP�i�r�x � �x, y��,(7)

where the exponential terms EXP�i�s̃�x, y�� andEXP�i�r�x, y�� are phase-only images of the scene andthe reference, respectively. The intensity of the out-put cross-correlation can be written as

c��x, �y� ��wr

EXPi��s�x, y� � b̃�x, y�

� r�x � �x, y � �y��dxdy�2

, (8)

where wr is the region of support of the reference. Ifs�x, y� � r�x, y� (autocorrelation) then at the actualtarget position �x0, y0� we have

ctar�x0, y0� ��wr

dxdy�2

� �E�2, (9)

where E is the area of the target, and outside of thetarget region, say at position ��bx, �by�, we obtain

cbg��bx, �by� ��wr

EXPi��b̃�x, y�

� r�x � �bx, y � �by��dxdy�2

. (10)

From Eqs. (9) and (10), it can be seen that the auto-correlation peak is always higher than the sidelobepeaks. Therefore, the phase-input JTC yields supe-rior performance in terms of the DC and the lightefficiency than that of the classical JTC, when theinput scene is noise-free. Moreover, the phase-inputJTC is not tolerant of different object distortions.

3. Adaptive Phase-Input Joint Transform Correlator

We wish to design a phase-input JTC that ensures ahigh correlation peak corresponding to the targetwhile suppressing possible false peaks. We are inter-ested in a JTC that with a high DC identifies a target(and its variants) in cluttered and noisy input scenes.In this case conventional JTCs may yield poor per-formance. With the help of adaptive SDF filters, agiven value of the DC can be achieved [13,14]. How-ever, adaptive SDF filters possess a signal range of��1, 1�, whereas conventional amplitude-only SLMs(used in classical JTCs architectures) have a dynamicrange of [0, 1]. Therefore, the optodigital implemen-tation of adaptive SDF filters in conventional JTCarchitectures requires either two correlations or exten-sive pointwise postprocessing. The proposed adaptivephase-only SDF filters can be directly implemented ina phase-input JTC.

A. Phase-Only Synthetic Discriminant Functions

A conventional SDF filter [11,12] is a linear combi-nation of training images: h�x, y� � a1s1�x, y� � · · ·� aKsK�x, y�, where si�x, y��i � 1, 2, . . . , K aretraining images, and ai�i � 1, 2, . . . , K are weight-ing coefficients, and they are chosen to satisfy theconstraints

ci � h�x, y� ● si�x, y�, (11)

where “●” denotes the inner product, and ci�i �1, 2, . . . , K are prespecified values in the correlationoutput at the origin for each training image. Inmatrix-vector notation the filter and the constraintscan be rewritten as h � Sa and c � S�h, respectively.Here S is a d � K matrix, where its ith column isgiven by the vector version of the training image si, ais a K � 1 vector of weighting coefficients ai, c is aK � 1 vector of prespecified values of the correlationpeaks for each training image, and the superscript“�” means a conjugate transpose. The �i, j�th elementof the matrix P � �S�S� is the value at the origin ofthe cross-correlation between the training images siand sj. If the matrix P is nonsingular, the solution ofthe linear equation system is given by

a � �S�S��1c, (12)

and the filter vector is

h � S�S�S��1c. (13)

The SDF filter with different output correlationpeaks can be used for multiclass distortion-invariantpattern recognition. For simplicity, let us consider atwo-class recognition problem. Suppose that thereare N training images from the true-class si�x, y��i� 1, . . . , N and M training images from the false-class si�x, y��i � N � 1, . . . , N � M. The compositefilter h�x, y� is a linear combination of all training im-ages s1�x, y�, . . . , sN�x, y�, sN�1�x, y�, . . . , sN�M�x, y�.

Fig. 2. Block diagram of the phase-input JTC.


We can set the filter output ci � 1�i � 1, . . . , N forthe true-class objects and ci � 0�i � N � 1,N � 2, . . . , N � M for the false-class objects, i.e.,

c � �1 1 . . . 1 0 0 . . . 0�T. (14)

This is the common approach to the SDF filter design.Now we introduce new phase-only SDFs. Let us de-fine the phase-only SDF impulse response as follows:

hpo � EXPi�Sa. (15)

New constraints are given by

c � R�hpo, (16)

where R is a d � K matrix, where its ith column isgiven by the vector version of the phase-encodedtraining images si as

R � exp�i�s11� exp�i�s12� · · · exp�i�s1K�exp�i�s21� exp�i�s22� · · · exp�i�s2K�

É É Ì É

exp�i�sd1� exp�i�sd2� · · · exp�i�sdK��. (17)

An additional constraint is required to satisfy Eq.(15); that is,

a � ai�i � 1, . . . , K � �. (18)

For simplicity, let us consider a two-class patternrecognition problem. Assume that there are N train-ing images from the true-class si�x, y��i � 1, . . . ,N, and M training images from the false-classsi�x, y��i � N � 1, . . . , N � M, with K � N � M.The vector c can be determined as

c � ��t exp�i� 1�, . . . , �t exp�i� N�,�f exp�i� N�1�, . . . , �f exp�i� N�M��T, (19)

where the first N and last M exponential terms in Eq.(19) correspond to the filter output for the true-class

and for the false-class, respectively. Thus, |�t|2 is the

intensity value of the filter output at the positionsof the true-class phase-mapped images si�i �1, . . . , N and |�f|

2 is the filter intensity output atthe positions of the false-class phase-mapped imagessi�i � N � 1, . . . , N � M. Substituting Eqs. (17)and (19) in Eq. (16) and taking into account Eq. (18),weighting coefficients ai can be found by solving thefollowing nonlinear system of equations (NL):

�t cos�� 1� � �i�1

d

cos��j�1

K

ajsij�� si1�,É

�t cos�� N� � �i�1

d

cos��j�1

K

ajsij�� siN�,�f cos�� N�1��

i�1

d

cos��j�1

K

ajsij�� si�N�1��,É

�f cos�� N�M�� i�1

d

cos��j�1

K

ajsij�� si�N�M��,�t sin�� 1� � �

i�1

d

sin��j�1

K

ajsij�� si1�,É

�t sin�� N� � �i�1

d

sin��j�1

K

ajsij�� siN�,�f sin�� N�1��

i�1

d

sin��j�1

K

ajsij�� si�N�1��,É

�f sin�� N�M�� i�1

d

sin��j�1

K

ajsij�� si�N�M��. (20)

We can rewrite Eq. (20) in matrix-vector notation asfollows:

cre � Rre� COS�Sa � Rim

� SIN�Sa,

cim � Rim� COS�Sa � Rre

� SIN�Sa, (21)

Fig. 3. Block diagram of the proposed binary search algorithm.Fig. 4. Block diagram of the adaptive phase-input JTC designprocedure.


where Rre and Rim are d � K matrices containing thereal and imaginary parts of R, respectively. The vec-tors cre and cim are the real and imaginary parts of c,respectively. “COS ”, “SIN ” are pointwise cosineand sine operators. The system of equations in Eq.(21) has 2K equations with 2K � 2 unknown vari-ables. It can be solved with the use of numericalmethods such as the Levenberg–Marquardt algo-rithm [20]. Since prespecified values in the correla-tion output cannot be defined in a simple mannersuch as in Eq. (14), we introduce the following addi-tional objective function (correlation peaks ratio) thatis directly related to the DC:

CPR ��f�2

��t�2. (22)

Now we can state the problem to be solved. We wishto design a phase-only correlation filter with a givenvalue of the DC by minimizing the CPR subject to thesystem of equations in Eq. (21). The solution is basedon the Levenberg–Marquardt and binary search al-gorithms. A binary search algorithm [21] is a tech-nique for finding a particular value in a linear array,by ruling out half of the data at each step. The algo-rithm finds the median, makes a comparison to de-termine whether the desired value comes before orafter it, and then searches the remaining half in thesame manner. The computational complexity of thebinary search is bounded by O�log2�L��, where L isthe size of the input array. The block diagram of theproposed algorithm is shown in Fig. 3. It consists ofthe following steps:

1. Define a search interval for �t by setting its leftand right limits as Lt � 1 and Rt � d, respectively.

2. If Lt � Rt then the procedure is finished, or elsego to the next step.

3. Compute �t � �Lt � Rt��2. To guarantee a desiredvalue of DC, say DC̃, we calculate �f � ��1 � DC̃��t�1�2.

4. Solve the nonlinear system of equations in Eq.(21) using the Levenberg–Marquardt iterative algo-rithm. The initial solution is taken as ai � 1�d�si�i � 1, . . . , K and i � 0�i � 1, . . . , K, whereK � N � M.

5. If a solution is founded, then save all data fromthe solution in array SOL, set Lt � �t � 1, and go tostep 2; or else define a search interval for �f by settingits left and right limits as follows: Lf � 1, Rf � �f

subject to �f � ��1 � DC̃��t�1�2.6. If Lf � Rf, then go to step 9, or else compute

�f � �Lf � Rf��2.7. Solve the nonlinear system of equations in

Eq. (21) using the Levenberg–Marquardt algorithm.The initial solution is taken as ai � 1�d �si�i �1, . . . , K and i � 0�i � 1, . . . , K, where K �N � M.

8. If a solution is founded, then save all data fromthe solution in array SOL, set Rf � �f � 1, and go tostep 6; or else set Lf � �f � 1 and go to step 6.

9. If the array SOL is empty (no solutions found),then set Rt � �t � 1 and go to step 2; or else set Lt

� �t � 1 and go to step 2.

Finally, we take from the array SOL a pair ��t, �f�,which yields the lowest CPR. The weighting coeffi-

Fig. 5. Optical setup for implementation of the adaptive phase-input JTC.

Fig. 6. (a) Phase response of LCD1, (b) intensity response ofLCD2.

Fig. 7. Test images: (a) target, (b) nondesired object, (c) realbackground.

Fig. 8. (a) Input scene, (b) adaptive reference image.


cients corresponding to this pair form the desiredfilter with the help of Eq. (15).

B. Reference Image Design

Distortion-invariant pattern recognition can be im-plemented in a phase-input JTC using the phase-onlySDF filter (reference image) given in Eq. (15). How-ever, the obtained phase-only SDF filter may yield apoor performance because it is able to control onlycentral correlation outputs in response to trainingimages. This leads to the appearance of sidelobesbecause of the presence in the input scene of falseobjects and a cluttered background. This problem canbe solved by involving an adaptive approach to thefilter design [13]. For given objects to be recognizedand for false objects and a background to be rejected,a new algorithm to design an adaptive phase-onlySDF filter is proposed. At each iteration the algo-rithm suppresses the highest sidelobe peak and,therefore, it monotonically increases the value of theDC until a prespecified value is reached. The filterdesign procedure requires knowledge of a phase-encoded background, which can contains false objects

with unknown locations. The first step is to carry outthe joint transform correlation between the phase-encoded background and a basic phase-only SDF fil-ter given in Eq. (15), which is initially trained onlywith the target. Next, the maximum of the filter out-put is set as the origin, and around the origin we forma new object to be rejected from the background. Theobject has the region of support equal to the union ofshapes of all used objects (desired and nondesired).The created object is added to the false class of ob-jects. Now, the two-class recognition problem is uti-lized to design a new phase-only SDF filter; that is,the true class contains only the target and the falseclass consists of the false-class objects. The describediterative procedure is repeated until a given value ofthe DC is obtained. Finally, note that if other objectsto be rejected are known, they can be directly in-cluded into the false class and used for the design ofan adaptive phase-only SDF filter. A block diagram ofthe procedure is shown in Fig. 4.

C. Optodigital Implementation

An optical setup for the implementation of the adap-tive phase-input JTC is shown in Fig. 5. The jointpower spectrum of the classical JTC contains infor-mation about an input scene and a reference object.In the case of the phase-input JTC, the unused areaof the SLM forms in the joint power spectrum anadditional high energy term. A simple solution to theproblem is based on the modulation property of theFourier transform. We encode the unused area ofthe SLM with a periodic phase mask. As a conse-quence, a delta-like spectrum of the mask is dif-fracted to a given unused high frequency area,where it can be easily filtered [8].

Fig. 9. Computer simulation intensity correlation planes obtained for the scene in Fig. 8(a) with (a) CJTC, (b) BJTC, (c) FAJTC, (d) PIJTC,(e) MACEmap, and (f) SDFproj.

Fig. 10. Computer simulation intensity correlation plane ob-tained for the scene in Fig. 8(a) with (a) AJTC and (b) APIJTC.

Table 1. Performance of JTCs: 95% Confidence DC

DC DC

CJTC 0.057 � 0.006 MACEmap 0.26 � 0.07BJTC �0.39 � 0.04 SDFproj 0.53 � 0.02FAJTC 0.27 � 0.05 AJTC 0.89 � 0.006PIJTC 0.38 � 0.004 APIJTC 0.85 � 0.006


4. Results

A. Calibration of Devices

Real experiments were carried out using the opticalsetup in Fig. 5. First we characterized optoelectronicsdevices such as a twisted nematic LCD of 832 �624 pixels and a monochrome CCD camera of 640� 480 pixels. LCD1 and LCD2 are red and blue chan-nels from a Toshiba projector (TLP4101), respec-tively. The display model is a Sony LCX016ALC7.The CCD camera model is a MTI (CCDC72). Theexperimental lookup tables of the SLMs are shown inFig. 6: (a) the phase response of the LCD1, (b) theintensity response of the LCD2. These plots containinformation about a dynamic range of signals as wellas a nonlinearity introduced by the SLMs and theCCD camera. It can be seen from Fig. 6(a) that agray-scale range of [0, 119] corresponds to a phaseshift of �0, ��, whereas a dynamic range of [120, 255]corresponds to a phase shift of ��, 7�4��. In a similarmanner, one can observe from Fig. 6(b) that a gray-scale signal range of the LCD2 is [0, 48]. Note thatthis plot is nonlinear due to quantization effects,and it is well approximated by the kth-law of non-linearity: output � ��input�2�1�k, when k � 0.7. Thisinformation is used in the design procedure of theAPIJTC.

B. Computer Simulation Results

In this section, computer simulation results obtainedwith the APIJTC are presented. These results arecompared with those obtained with the classical JTC(CJTC), the binary JTC (BJTC), the fringe-adjustedJTC (FAJTC), the adaptive JTC (AJTC), the conven-tional phase-input JTC (PIJTC), and two phase-inputJTCs. The latter JTCs are generated by the mappingof the impulse response of common SDF filters to aphase distribution. The first phase-input JTC is re-ferred to as “MACEmap” and it is synthesized by themapping [see Eq. (6)] of the impulse response of theMACE filter [22] to a phase distribution. The secondJTC is based on a projection algorithm proposed byMontes-Usategui, et. al. [18]. We slightly modifiedthis algorithm to implement a SDF filter in a phase-input JTC. It is referred to as “SDFproj.” The modi-fied design procedure consists of the following steps:

1. Let z0 be a column vector of d elements. Designa conventional SDF filter x � S�S�S��1c and projectit to phase distribution using Eq. (6), i.e., k � 0, zk

� EXP�i�x�.2. Make k � k � 1. Design a generalized SDF

filter [23]

hgk � x � �Id � S�S�S��1S��zk�1, (23)

where Id is the d � d identity matrix.3. Project hg

k onto the unitary circle in the com-plex plane employing the minimum Euclidean dis-tance principle [24]; that is, compute for i � 1 to d:zk�i� � argmin∀ s�D��hg

k�i� � s�2�, where D representsthe phase-only coding domain and s is an arbitraryvalue in this domain.

4. If the difference between the phase-only andgeneralized SDF filters is less than a prespecifiedvalue, then exit, or else go to step 2.

The target is the butterfly shown in Fig. 7(a). Figure7(b) shows a false object with a similar shape but withdifferent gray-scale content. Figure 7(c) shows a spa-tially inhomogeneous real background. The size of allimages used in our experiments is 128 � 128 pixels.The signal range is [0, 1] with 256 quantization lev-els. We use a test input scene shown in Fig. 8(a). Thescene contains the target and false object embeddedinto a real background at unknown coordinates. Therecognition performance of the proposed APIJTC iscompared with that of the CJTC, BJTC, FAJTC,

Fig. 11. Input scene corrupted by additive white noise with thestandard deviation of �n � 0.31.

Table 2. Noise Tolerance: 95% Confidence DC of Tested JTCs for Additive White Gaussian Noise

�n � 0.1 �n � 0.2 �n � 0.28 �n � 0.31 �n � 0.37

CJTC 0.04 � 0.005 0.04 � 0.05 0.03 � 0.004 0.03 � 0.005 0.02 � 0.005FAJTC 0.29 � 0.058 0.33 � 0.062 0.33 � 0.067 0.32 � 0.073 0.33 � 0.076PIJTC 0.38 � 0.005 0.37 � 0.005 0.37 � 0.005 0.37 � 0.005 0.37 � 0.005MACEmap 0.23 � 0.065 0.13 � 0.053 0.15 � 0.049 0.08 � 0.036 0.07 � 0.038SDFproj 0.46 � 0.024 0.46 � 0.025 0.45 � 0.026 0.43 � 0.027 0.42 � 0.029AJTC 0.89 � 0.005 0.89 � 0.008 0.88 � 0.01 0.88 � 0.01 0.87 � 0.01APIJTC 0.82 � 0.01 0.82 � 0.01 0.81 � 0.01 0.81 � 0.009 0.80 � 0.01


AJTC, PIJTC, MACEmap, and SDFproj. The APIJTCis designed with the help of the proposed iterativealgorithm. After the first iteration the value of the DCis negative. After only three iterations in the designprocess, the APIJTC reaches DC � 0.87. The phase-only adaptive reference is shown in Fig. 8(b). Thecorrelation intensity planes obtained with the testedJTCs are shown in Figs. 9 and 10. Next, 30 statisticaltrials for different positions of the target were carriedout. With 95% confidence, the DC values are shown inTable 1. We see that the AJTC and APIJTC yield thebest results. Since the AJTC design algorithm is notsubject to any coding domain restrictions, it performsslightly better than the APIJTC. The BJTC fails todetect the target, whereas the performance of theCJTC is poor. Note that the SDFproj, the AJTC, andAPIJTC are designed iteratively. The SDFproj algo-rithm synthesizes the filter with the best recognitionperformance �DC � 0.39� in the input scene in 28iterations. The AJTC algorithm designs the filterwith a DC value of 0.88 in 12 iterations. To achieve asimilar recognition performance as the AJTC, theproposed APIJTC requires only three iterations. Nextwe tested the tolerance of the tested JTCs to additivenoise. An example of the input scene corrupted byzero-mean white Gaussian noise with a standard de-viation of �n � 0.31 is shown in Fig. 11. We performed150 statistical trials for different positions of the tar-get and realizations of additive noise. With 95% con-fidence, DC values are shown in Table 2. One canobserve that the proposed correlator possesses a verygood robustness to input noise. Next, we investi-gated the tolerance of the APIJTC to small geomet-ric distortions of the target. Several methods wereproposed to improve pattern recognition in the pres-ence of such distortions. These methods can bebroadly classified into two groups. The first class isconcerned with 2D scaling and rotation distortions.Such methods include space-variant transformsand circular harmonic functions. The second classuses training images that are sufficiently descrip-tive and representative of the expected distortions.The proposed method belongs to the second approach.First, the input scene shown in Fig. 8(a) with an em-bedded rotated object is used. The APIJTC is trainedwith only two versions of the target rotated by 0° and

5°. After two iterations in the design process, theAPIJTC yields DC � 0.86. The obtained reference isshown in Fig. 12(a). We tested the APIJTC when thetarget was rotated by 0°, 2°, 4°, 6°, 8°, and 10°. Thetolerance of the APIJTC to the rotations is shown inFig. 13(a). Tolerance of the APIJTC to scale distor-tions of the target is investigated. In this case, theAPIJTC was trained with three versions of the objectscaled by factors of 0.9, 1, and 1.1. After two itera-tions in the design process, the obtained JTC yieldsDC � 0.87. The adaptive reference image is shown inFig. 12(b). The performance of the APIJTC for recog-nition of the target scaled by factors of 0.9, 0.95, 1,1.05, and 1.1 in the input scene shown in Fig. 8(a) ispresented in Fig. 13(b). We see that the distortedtarget can be clearly detected.

C. Experimental Results

Experimental results were obtained with the help ofthe optical setup shown in Fig. 5. The input image ofthe correlator is shown in Fig. 14. It contains thephase-encoded input scene in Fig. 8(a), the phase-only adaptive reference in Fig. 8(b), and the unusedarea of the LCD encoded with a phase modulationmask. According to the lookup tables of the SLMsshown in Fig. 6, the positive part of the input imageof the correlator was mapped into a gray-scale rangeof [0–119], whereas the negative part of the inputimage was mapped into a gray-scale range of [120–255]. The obtained intensity correlation plane isshown in Fig. 15. The target can be easily detected

Fig. 12. Adaptive reference image invariant to (a) rotation dis-tortions and (b) scale distortions.

Fig. 13. Tolerance of the APIJTC to geometric distortions. (a) DCversus rotation degree, (b) DC versus scale factor.

Fig. 14. Joint input image used for real experiment.


with one optical correlation and without any postpro-cessing.

5. Conclusions

A new adaptive phase-input JTC was proposed toimprove the recognition of a target embedded into aknown cluttered background. A new iterative algo-rithm for the design of an adaptive phase-input JTCbased on phase-only synthetic discriminant functionswas suggested. It was shown that the proposed algo-rithm with a few training iterations helps us to takecontrol over the whole correlation plane. The com-puter simulation results demonstrated superiority inthe performance of the proposed JTC for pattern rec-ognition compared with that of the phase-input JTC,and modified phase-only versions of SDF and MACEfilters. The suggested adaptive phase-input JTC pos-sesses a high scene adaptivity and good robustness toadditive noise and small geometric distortions. Ex-perimental results were also presented. Very goodaccordance between computer simulation and exper-imental results was obtained.

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Fig. 15. Optical intensity correlation plane obtained with theAPIJTC.


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Adaptive phase-input joint transform correlator