support that optimizes the DC of the POF between similar objects using the phase-difference histogram- equalization method. Another aspect to be considered ...
Real-time binary-amplitude phase-only filters Ignacio Moreno, Esmail Ahouzi, Juan Campos, and Marı´a J. Yzuel
A real-time binary-amplitude phase-only filter ~BAPOF! implemented in available phase-only modulators is presented. The BAPOF has an amplitude transmission equal to one only in a region of support, while the transmission is equal to zero in the complementary region. To implement zero transmission in a phase-only modulator we propose to add a linear phase to the region of support. In this way the correlation desired is obtained off axis. Computer simulations and experimental results obtained with this 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, phase-only filters ~POF’s! have received considerable interest for their high efficiency and their ability to be implemented optically with real-time spatial light modulators ~SLM’s!. However, although optimal in terms of light efficiency ~h!, POF’s are not optimal with respect to other important parameters like the signalto-noise ratio ~SNR!, peak-to-correlation energy ~PCE!, or discrimination capability ~DC!.2 Different approaches have been suggested in the literature to improve the performance of POF’s. One of them is the introduction of regions of support, i.e., regions that indicate which frequencies of the POF have unit magnitude, in what is called binary-amplitude phase-only filters ~BAPOF’s!. Several methods of optimization of POF’s in terms of different parameters that evaluate the correlation have been proposed in the literature. Kumar and Bahri3 proposed an algorithm that generates the region of support for POF’s that optimizes the SNR. In Ref. 4, Kumar et al. reported the use of the PCE measurement to determine the POF with maximally sharp correlation. Re´fre´gier et al.5 proposed the generation of regions of support for POF’s that yield the optimal trade-off among several parameters. And Ahouzi et 1
When this study was performed, the authors were with the Grupo de Optica, Departamento de Fisica, Universidad Auto´noma de Barcelona, 08193 Bellaterra, Spain. I. Moreno is now with the Departamento Interuniversitario de Optica, Universitat de Valencia, 46100 Burjassot, Spain. Received 14 November 1996; revised manuscript received 24 March 1997. 0003-6935y97y297428-05$10.00y0 © 1997 Optical Society of America 7428
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al.6 have proposed an algorithm to design the region of support that optimizes the DC of the POF between similar objects using the phase-difference histogramequalization method. Another aspect to be considered in the generation of a filter for pattern recognition in real applications is its capacity to be implemented in real time by SLM’s. For instance, twisted nematic liquid-crystal SLM’s can produce phase-only modulation under proper conditions of applied voltage and input polarization. This kind of modulation provides a useful device for generating POF’s. However, no currently available updatable SLM allows pure phase-only modulation while also permitting zero blocking of individual pixels. Consequently, it is not possible to implement BAPOF’s directly in real time with these SLM’s. In this paper we propose a technique to generate real-time BAPOF’s with a SLM that produces phaseonly modulation. Blocking is accomplished by the addition of a proper linear phase code to nonblocked frequencies of the filter. The addition of appropriate phase codes may be used to separate different diffraction orders.7 The linear phase code produces an offaxis shift in the correlation peak.8 With this technique we separate the optimized POF correlation region from the correlation region given by undesired frequencies. We call this kind of filter a BAPOF in phase-only mode. With the BAPOF in phase mode, the region 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! 5
H*~u, n! 5 exp@2if~u, n!#. uH~u, n!u
(1)
When a region of support R is designed to optimize some parameters, a binary mask b~u, n! is obtained, and it is defined as
H
1 b~u, n! 5 0
if ~u, n! [ R . if ~u, n! [ yR
HBAPOF~u, n! 5 b~u, n! HPOF~u, n!
H
exp@2if~u, n!# 0
if ~u, n! [ R . if ~u, n! [ yR
(3)
SLM’s that produce phase-only modulation can be suitable devices for implementing POF’s. However, if we want to implement binary phase-only filters, zero transmission is needed simultaneously with phase modulation. So it is necessary to have a modulator that controls both phase and amplitude modulations. Available SLM’s do not produce this kind of modulation, and the blocking technique cannot be applied 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 shift in the direction determined by x0 and y0 is added to the POF in those pixels that should not be blocked in the BAPOF. On the other hand, those pixels that should be blocked keep the phase of the POF. As a result the correlation peak corresponding to the BAPOF is obtained off axis in the final correlation plane. Let cPOF~x, y! denote the correlation plane obtained when the filter HPOF~u, n! is used. If s~x, y! is the scene to be analyzed and hPOF~x, y! is the impulse response of the POF, then cPOF~x, y! 5 s~x, y! p hPOF~x, y!, where the asterisk denotes the convolution operation. Let b~u, n! be a binary mask applied to the POF and let b~x, y! be its FT. The correlation distribution, cBAPOF~x, y!, obtained with the BAPOF given by Eq. ~3! is then cBAPOF~x, y! 5 cPOF~x, y! p b~x, y!.
(5)
Equation ~5! represents the distribution we want to reproduce using the BAPOF in phase mode given by the Eq. ~4!. Taking into account that b~u, n! is a binary function that takes values of 0 or 1, we can rewrite H9BAPOF as H9BAPOF~u, n! 5 $b~u, n!exp@i~ux0 1 ny0!# 1 @1 2 b~u, n!#%HPOF~u, n!.
(6)
Let c9BAPOF~x, y! be the correlation plane when H9BAPOF is used. It is given by c9BAPOF~x, y! 5 cPOF~x, y!pFT$b~u, n!exp@i~ux0 1 ny0!# 1 @1 2 b~u, n!#%,
c9BAPOF~x, y! 5 cBAPOF~x, y!pd~x 2 x0, y 2 y0! 1 cPOF~x, y!pr~ x, y!,
(2)
The BAPOF HBAPOF is defined as
5
where FT$. . .% denotes the FT operation. By using Eq. ~5! we obtain
(7)
(8)
where r~x, y! is the FT of the complementary mask 1 2 b~u, n!. The first term on the right-hand side of in 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 phase code give rise to an optimized correlation peak shifted from the center. On the other hand, the second term is the convolution of cPOF~x, y! with the FT of the complementary mask centered at the coordinates ~0, 0!. 3. Results
We test the performance of the implementation method by using optimized BAPOF’s for different criteria. In the experiments we use a twisted nematic liquid-crystal TV from an Epson video projector ~Model VP-100PS!. Previously we characterized this SLM and determined the configuration of the input polarization and the positions of the potentiometers of control of the video projector that yield phasemostly modulation.8 The scene we studied is shown in Fig. 1~a! and consists of two similarly shaped letters, E and F. The target is the letter E. The phase-only distribution of the POF matched to the letter E is shown in Fig. 1~b!. Black areas correspond to a zero phase and white areas to a 2p phase. The proposed technique is applied to a BAPOF optimized for different criteria ~the SNR,3 a trade-off between the SNR and the PCE,5 and the DC6!. As an example of a binary mask for a BAPOF, Fig. 1~c! shows one corresponding to the region of support that optimizes a trade-off between the SNR and the PCE.5 Figure 1~d! shows the phase distribution corresponding to the BAPOF of Fig. 1~c!. A linear phase is added to those pixels that are not blocked in Fig. 1~c!, and we can see that the region to be blocked keeps the phase distribution of the POF. Figure 2~a! shows the correlation plane obtained by computer simulation when the optimized POF in terms of the SNR3 is used with the blocking technique. We can see that the peaks are wide, which is a characteristic of this kind of optimization. Figure 2~b! shows the computer-simulated correlation plane when the phase-only version of the BAPOF is used. We can see that the distribution of Fig. 2~a! is reproduced in Fig. 2~b! but is shifted from the center. In the center, other peaks appear as a result of the nonblocked pixels. Figure 2~c! shows the intensity distribution of the correlation plane obtained with the liquid-crystal TV. Figure 2~d! shows a threedimensional plot of the distribution of Fig. 2~c!. We can see that there is very good agreement with the simulated results @Fig. 2~b!#. The main difference between Figs. 2~b! and 2~d! is in the peaks that appear centered. In the optical experiment the scene 10 October 1997 y Vol. 36, No. 29 y APPLIED OPTICS
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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 corresponding to the BAPOF that optimizes a trade-off between the SNR and the PCE. ~d! Phase-only distribution corresponding to the BAPOF in phase mode.
is reproduced in the correlation plane centered at the origin because of the nonlinearities of the SLM.8 This results in some noise superimposed on the central peaks, which are those that do not give the desired correlation. Figures 3~a! and 3~b! show computer-simulation and experimental results of the regions of interest in the correlation plane corresponding to optimization of the trade-off between the SNR and the PCE.5 We can see that the peaks are much narrower than those in Figs. 2~b! and 2~d!. Figures 4~a! and 4~b! correspond to the case of optimization of the DC by use of the phase-difference histogram method.6 For the case of the POF, the cross-correlation peak corre7430
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sponding to the letter F has an intensity equal to 64% of the autocorrelation peak. With the application of the binary mask designed with the phase-difference histogram method the cross-correlation peak decreases to 43% of the autocorrelation peak. Good agreement between the simulated and experimental results is also obtained in this case.
4. Conclusions
In this study we have proposed a method to implement BAPOF’s in devices that produce phase-only modulation. 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 the BAPOF. ~b! Computer simulation for the BAPOF in phase mode. ~c! Experimental results for the BAPOF in phase mode by use of a liquid-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 those pixels that should not be blocked. The result is a new phase-only distribution that can be implemented in a phase-mostly SLM and reproduces, in an off-axis
direction, the intensity distribution of the correlation plane obtained with the BAPOF. We have demonstrated by computer simulations and experiments the 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-crystal SLM. 10 October 1997 y Vol. 36, No. 29 y APPLIED OPTICS
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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 the BAPOF 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 Comisio´n Interministerial de Ciencia y Tecnologı´a through project TAP96-1015-C03-01. I. Moreno and E. Ahouzi thank the Spanish Ministry of Education for their respective grants.
5.
6.
References 1. J. L. Horner and P. D. Gianino, “Phase-only matched filtering,” Appl. Opt. 23, 812– 816 ~1984!. 2. B. V. K. Vijaya Kumar and L. Hassebrook, “Performance measures for correlation filters,” Appl. Opt. 29, 2997–3006 ~1990!. 3. B. V. K. Vijaya Kumar and Z. Bahri, “Phase-only filters with improved signal-to-noise ratio,” Appl. Opt. 28, 250 –257 ~1989!. 4. B. V. K. Vijaya Kumar, W. Shi, and C. Hendrix, “Phase-only
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filters with maximally sharp correlation peaks,” Opt. Lett. 15, 807– 809 ~1990!. Ph. Re´fre´gier, B. V. K. Vijaya Kumar, and C. Hendrix, “Multicriteria optimal binary amplitude phase-only filters,” J. Opt. Soc. Am. A 9, 2118 –2125 ~1992!. E. Ahouzi, J. Campos, and M. J. Yzuel, “Phase-only filter with improved discrimination,” Opt. Lett. 19, 1340 –1342 ~1994!. J. A. Davis, D. M. Cottrell, J. E. Davis, and R. A. Lilly, “Fresnel lens-encoded binary phase-only filters for optical pattern recognition,” Opt. Lett. 14, 659 – 661 ~1989!. I. Moreno, J. Campos, C. Gorecki, and M. J. Yzuel, “Effects of amplitude and phase mismatching errors in the generation of a kinoform for pattern recognition,” Jpn. J. Appl. Phys. 34, 6423– 6432 ~1995!.
