Real-time binary-amplitude phase-only filters

Ignacio Moreno, Esmail Ahouzi, Juan Campos, and Marıa J. Yzuel

A real-time binary-amplitude phase-only filter ~BAPOF! implemented in available phase-only modulatorsis presented. The BAPOF has an amplitude transmission equal to one only in a region of support, whilethe transmission is equal to zero in the complementary region. To implement zero transmission in aphase-only modulator we propose to add a linear phase to the region of support. In this way thecorrelation desired is obtained off axis. Computer simulations and experimental results obtained withthis technique are given. © 1997 Optical Society of America

Key words: Pattern recognition, binary-amplitude phase-only filters, spatial light modulators.

1. Introduction

Since their introduction by Horner and Gianino,1phase-only filters ~POF’s! have received considerableinterest for their high efficiency and their ability to beimplemented optically with real-time spatial lightmodulators ~SLM’s!. However, although optimal interms of light efficiency ~h!, POF’s are not optimal withrespect to other important parameters like the signal-to-noise ratio ~SNR!, peak-to-correlation energy ~PCE!,or discrimination capability ~DC!.2 Different ap-proaches have been suggested in the literature to im-prove the performance of POF’s. One of them is theintroduction of regions of support, i.e., regions thatindicate which frequencies of the POF have unit mag-nitude, in what is called binary-amplitude phase-onlyfilters ~BAPOF’s!.

Several methods of optimization of POF’s in terms ofdifferent parameters that evaluate the correlationhave been proposed in the literature. Kumar andBahri3 proposed an algorithm that generates the re-gion of support for POF’s that optimizes the SNR. InRef. 4, Kumar et al. reported the use of the PCE mea-surement to determine the POF with maximally sharpcorrelation. Refregier et al.5 proposed the generationof regions of support for POF’s that yield the optimaltrade-off among several parameters. And Ahouzi et

When this study was performed, the authors were with theGrupo de Optica, Departamento de Fisica, Universidad Autonomade Barcelona, 08193 Bellaterra, Spain. I. Moreno is now with theDepartamento Interuniversitario de Optica, Universitat de Valen-cia, 46100 Burjassot, Spain.

Received 14 November 1996; revised manuscript received 24March 1997.

0003-6935y97y297428-05$10.00y0© 1997 Optical Society of America

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al.6 have proposed an algorithm to design the region ofsupport that optimizes the DC of the POF betweensimilar objects using the phase-difference histogram-equalization method.

Another aspect to be considered in the generation ofa filter for pattern recognition in real applications is itscapacity to be implemented in real time by SLM’s.For instance, twisted nematic liquid-crystal SLM’s canproduce phase-only modulation under proper condi-tions of applied voltage and input polarization. Thiskind of modulation provides a useful device for gener-ating POF’s. However, no currently available updat-able SLM allows pure phase-only modulation whilealso permitting zero blocking of individual pixels.Consequently, it is not possible to implement BAPOF’sdirectly in real time with these SLM’s.

In this paper we propose a technique to generatereal-time BAPOF’s with a SLM that produces phase-only modulation. Blocking is accomplished by the ad-dition of a proper linear phase code to nonblockedfrequencies of the filter. The addition of appropriatephase codes may be used to separate different diffrac-tion orders.7 The linear phase code produces an off-axis shift in the correlation peak.8 With thistechnique we separate the optimized POF correlationregion from the correlation region given by undesiredfrequencies. We call this kind of filter a BAPOF inphase-only mode. With the BAPOF in phase mode, theregion of support is introduced in the phase information.

2. Binary-Amplitude Phase-Only Filters in Phase Mode

Let h~x, y! be the object to be recognized and let H~u,n! 5 uH~u, n!uexp@2if~u, n!# be its Fourier transform~FT!. The POF HPOF~u, n! is defined as1

HPOF~u, n! 5H*~u, n!

uH~u, n!u5 exp@2if~u, n!#. (1)

When a region of support R is designed to optimizesome parameters, a binary mask b~u, n! is obtained,and it is defined as

b~u, n! 5 H10

if ~u, n! [ Rif ~u, n! [y R. (2)

The BAPOF HBAPOF is defined as

HBAPOF~u, n! 5 b~u, n!HPOF~u, n!

5 Hexp@2if~u, n!#0

if ~u, n! [ Rif ~u, n! [y R. (3)

SLM’s that produce phase-only modulation can besuitable devices for implementing POF’s. However,if we want to implement binary phase-only filters,zero transmission is needed simultaneously withphase modulation. So it is necessary to have a mod-ulator that controls both phase and amplitude mod-ulations. Available SLM’s do not produce this kindof modulation, and the blocking technique cannot beapplied directly.

We propose a phase-only version of the BAPOF,defined as follows:

H9BAPOF~u, n! 5 exp@2if~u, n! 1 ib~u, n!~ux0 1 ny0!#,

(4)

where x0 and y0 are constants. A linear phase shiftin the direction determined by x0 and y0 is added tothe POF in those pixels that should not be blocked inthe BAPOF. On the other hand, those pixels thatshould be blocked keep the phase of the POF. As aresult the correlation peak corresponding to theBAPOF is obtained off axis in the final correlationplane.

Let cPOF~x, y! denote the correlation plane obtainedwhen the filter HPOF~u, n! is used. If s~x, y! is thescene to be analyzed and hPOF~x, y! is the impulseresponse of the POF, then cPOF~x, y! 5 s~x, y! phPOF~x, y!, where the asterisk denotes the convolu-tion operation. Let b~u, n! be a binary mask appliedto the POF and let b~x, y! be its FT. The correlationdistribution, cBAPOF~x, y!, obtained with the BAPOFgiven by Eq. ~3! is then

cBAPOF~x, y! 5 cPOF~x, y! p b~x, y!. (5)

Equation ~5! represents the distribution we want toreproduce using the BAPOF in phase mode given bythe Eq. ~4!. Taking into account that b~u, n! is abinary function that takes values of 0 or 1, we canrewrite H9BAPOF as

H9BAPOF~u, n! 5 $b~u, n!exp@i~ux0 1 ny0!#

1 @1 2 b~u, n!#%HPOF~u, n!. (6)

Let c9BAPOF~x, y! be the correlation plane whenH9BAPOF is used. It is given by

c9BAPOF~x, y! 5 cPOF~x, y!pFT$b~u, n!exp@i~ux0 1 ny0!#

1 @1 2 b~u, n!#%, (7)

where FT$. . .% denotes the FT operation. By usingEq. ~5! we obtain

c9BAPOF~x, y! 5 cBAPOF~x, y!pd~x 2 x0, y 2 y0!

1 cPOF~x, y!pr~x, y!, (8)

where r~x, y! is the FT of the complementary mask1 2 b~u, n!. The first term on the right-hand side ofin Eq. ~8! is the correlation distribution cBAPOF~x, y!centered at the coordinates ~x0, y0!. Consequently,those pixels of the filter that have an additional phasecode give rise to an optimized correlation peak shiftedfrom the center. On the other hand, the second termis the convolution of cPOF~x, y! with the FT of thecomplementary mask centered at the coordinates ~0,0!.

3. Results

We test the performance of the implementationmethod by using optimized BAPOF’s for different cri-teria. In the experiments we use a twisted nematicliquid-crystal TV from an Epson video projector~Model VP-100PS!. Previously we characterizedthis SLM and determined the configuration of theinput polarization and the positions of the potentiom-eters of control of the video projector that yield phase-mostly modulation.8

The scene we studied is shown in Fig. 1~a! andconsists of two similarly shaped letters, E and F.The target is the letter E. The phase-only distribu-tion of the POF matched to the letter E is shown inFig. 1~b!. Black areas correspond to a zero phaseand white areas to a 2p phase. The proposed tech-nique is applied to a BAPOF optimized for differentcriteria ~the SNR,3 a trade-off between the SNR andthe PCE,5 and the DC6!. As an example of a binarymask for a BAPOF, Fig. 1~c! shows one correspondingto the region of support that optimizes a trade-offbetween the SNR and the PCE.5 Figure 1~d! showsthe phase distribution corresponding to the BAPOFof Fig. 1~c!. A linear phase is added to those pixelsthat are not blocked in Fig. 1~c!, and we can see thatthe region to be blocked keeps the phase distributionof the POF.

Figure 2~a! shows the correlation plane obtained bycomputer simulation when the optimized POF interms of the SNR3 is used with the blocking tech-nique. We can see that the peaks are wide, which isa characteristic of this kind of optimization. Figure2~b! shows the computer-simulated correlation planewhen the phase-only version of the BAPOF is used.We can see that the distribution of Fig. 2~a! is repro-duced in Fig. 2~b! but is shifted from the center. Inthe center, other peaks appear as a result of thenonblocked pixels. Figure 2~c! shows the intensitydistribution of the correlation plane obtained withthe liquid-crystal TV. Figure 2~d! shows a three-dimensional plot of the distribution of Fig. 2~c!. Wecan see that there is very good agreement with thesimulated results @Fig. 2~b!#. The main differencebetween Figs. 2~b! and 2~d! is in the peaks that ap-pear centered. In the optical experiment the scene

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Fig. 1. ~a! Input scene. ~b! Phase-only distribution corresponding to the POF matched to the letter E. ~c! Binary mask correspondingto the BAPOF that optimizes a trade-off between the SNR and the PCE. ~d! Phase-only distribution corresponding to the BAPOF in phasemode.

is reproduced in the correlation plane centered at theorigin because of the nonlinearities of the SLM.8This results in some noise superimposed on the cen-tral peaks, which are those that do not give the de-sired correlation.

Figures 3~a! and 3~b! show computer-simulationand experimental results of the regions of interest inthe correlation plane corresponding to optimization ofthe trade-off between the SNR and the PCE.5 Wecan see that the peaks are much narrower than thosein Figs. 2~b! and 2~d!. Figures 4~a! and 4~b! corre-spond to the case of optimization of the DC by use ofthe phase-difference histogram method.6 For thecase of the POF, the cross-correlation peak corre-

7430 APPLIED OPTICS y Vol. 36, No. 29 y 10 October 1997

sponding to the letter F has an intensity equal to 64%of the autocorrelation peak. With the application ofthe binary mask designed with the phase-differencehistogram method the cross-correlation peak de-creases to 43% of the autocorrelation peak. Goodagreement between the simulated and experimentalresults is also obtained in this case.

4. Conclusions

In this study we have proposed a method to imple-ment BAPOF’s in devices that produce phase-onlymodulation. The method consists of the addition of

Fig. 2. Intensity distribution in the correlation plane when the BAPOF that optimizes the SNR is used: ~a! Computer simulation for theBAPOF. ~b! Computer simulation for the BAPOF in phase mode. ~c! Experimental results for the BAPOF in phase mode by use of aliquid-crystal SLM. ~d! Three-dimensional representation of ~c!. The desired correlation peaks are marked with arrows.

a linear phase code to the POF distribution in thosepixels that should not be blocked. The result is anew phase-only distribution that can be implementedin a phase-mostly SLM and reproduces, in an off-axis

direction, the intensity distribution of the correlationplane obtained with the BAPOF. We have demon-strated by computer simulations and experimentsthe validity of the proposed technique.

Fig. 3. Intensity distribution in the correlation plane when the BAPOF that optimizes a trade-off between the SNR and the PCE is used:~a! Computer simulation for the BAPOF in phase mode. ~b! Experimental results for the BAPOF in phase mode by use of a liquid-crystalSLM.

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Fig. 4. Intensity distribution in the correlation plane when the BAPOF that optimizes the DC is used: ~a! Computer simulation for theBAPOF in phase mode. ~b! Experimental results for the BAPOF in phase mode by use of a liquid-crystal SLM.

This work was financed by the Comision Intermin-isterial de Ciencia y Tecnologıa through projectTAP96-1015-C03-01. I. Moreno and E. Ahouzithank the Spanish Ministry of Education for theirrespective grants.

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