Complex data mapping on a binary ferroelectric liquid crystal electrically addressed spatial light modulator for target recognition

Transcription

1 Indian Journal of Pure & Applied Physics Vol. 45, August 2007, pp Complex data mapping on a binary ferroelectric liquid crystal electrically addressed spatial light modulator for target recognition Shilpi Goyal, Naveen K Nishchal, Vinod K Beri & Arun K Gupta* Photonics Division, Instruments Research & Development Establishment, Raipur Road, Dehradun * akgupta@irde.res.in Received 12 October 2006; revised 29 May 2007; accepted 1 June 2007 For displaying a complex valued function onto a binary amplitude-only or phase-only spatial light modulator, an encoding technique is required. In this paper, we implement an encoding scheme proposed by Davis et al. [Appl Opt, 42 (2003) 2003] for optical processing the complex product function in the hybrid digital-optical correlator. The output of this correlator consists of two autocorrelation peaks along with a strong dc. Out of these three terms, our interest is in capturing only one of the correlation peaks, while discarding the other correlation peak and strong dc. For capturing only one of the correlation peaks, we multiplied a chirp function with the rotation-invariant maximum average correlation height filter. The product function, that is basically a multiplied product of the input scene s Fourier spectrum and the chirp-encoded filter, is inverse Fourier transformed to obtain the correlation peaks. Due to chirp encoding, the correlation signals are focused in three different planes. Thus, placing our peak capturing CCD camera at a particular plane we can capture only one autocorrelation peak. Results with almost similar tank images having different energy values have also been shown. Keywords: Hybrid digital-optical correlator, Chirp encoding, MACH filter IPC Code: C09K 19/00 1 Introduction Exploiting the inherent Fourier transforming property of an optical lens-system, optical correlation enables the rapid identification of targets within an input signal. A lot of research work in the pattern recognition domain has been devoted to the achievement of real-time operation of optical correlators Two of the most frequently used optical architectures are VanderLugt correlator (VLC) and joint transform correlator (JTC). In JTC architecture only one spatial light modulator (SLM) is required to display both input and the reference image. The Fourier transform of the joint image is captured through an intensity-sensing device and it is displayed on the same SLM to get the correlation outputs. In the VLC architecture, independent SLMs are required for input image and filter planes. In this technique, optical Fourier transform of the image is multiplied optically with the filter. Alignment, i.e. pixel-to-pixel matching of filter with the input image s frequency spectrum, is very critical in this approach. A hybrid digital/optical scheme was proposed to overcome the alignment problem In this technique, computation of Fourier transform of the target and its multiplication with the filter are done digitally. This product function is inverse Fourier transformed optically to obtain the correlation outputs. Several advantages of this scheme over the VLC have been reported. Irrespective of the implementation geometry, a complex data-encoding scheme is essential for faithful optical processing. Usually, a filter is a complex function. The product function available for optical processing in the hybrid approach is also a complex function. Currently, no commonly available SLMs produce all complex values. SLMs are classified as either amplitude-only or phase-only. In practice, SLMs usually exhibit some degree of coupling between amplitude and phase. A complex function cannot be directly displayed onto the SLM. In literature, several techniques such as pseudorandom encoding 22, minimum Euclidean distance 23, complex data encoding 24,25 for complex data encoding have been reported. The technique proposed by Davis et al. 24,25 is easy to implement. In this technique, amplitude part and the phase parts are separated from the complex function. The amplitude part is rescaled in the range [0 1] and the phase part is rescaled in the range [-1 1]. Now these two terms are added and a threshold value +1 is chosen. After binarizing, amplitude-encoded binary phase grating is obtained. This can now be displayed onto the SLM. One major problem associated with this hybrid correlation approach is that in the correlation plane

2 648 INDIAN J PURE & APPL PHYS, VOL 45, AUGUST 2007 we get two autocorrelation peaks along with a strong dc. But our interest is in capturing only one autocorrelation peak. Lowans and Lewis 12 proposed a scheme to overcome this problem. In this technique, chirp function is used with the distortion-invariant maximum average correlation height (MACH) filter. Due to the use of chirp function, correlation signals are focused at different planes 13. By placing a detector at a particular plane, we can capture only one autocorrelation peak. In this paper, for complex data encoding, we implemented the algorithm proposed by Davis et al. 25. This encoding scheme has been used in the hybrid digital-optical correlator. The distortion-invariant filter used is the MACH filter. For discarding one of the correlation peaks and the dc, we implemented the scheme proposed by Lowans and Lewis 12. Results with almost similar tank images having different energy values have been shown. 2 Encoding of Complex Functions onto Binary SLM Davis et al 24,25. reported complex function encoding onto a binary SLM. Considering a binary phase mask with period d, a width w for the +1 region. Each region is assigned a value of +1 or 1 (indicating a π phase shift). For the SLMs, both width w and period d are formed with integer numbers of pixels. The diffraction efficiency into the first order is given by the following expression: η + 1 æ2 ö æ wö = sin π ç èπ ø çè d ø (1) The diffraction efficiency is symmetric around w = d/2 and can be modified by changing the ratio of w/d. Consider a linear phase mask with phase defined in the interval [-π, π]. To modify the ratio of w/d, we change the value for threshold and binarize the function. A general complex amplitude and phase mask to be encoded is given as ( [ ]) G = G( x) exp iφ( x) + 2πAx (2) In Eq. (2), G(x) represents amplitude of the function with values in the range [0,1]. In order to separate the diffraction orders a linear phase term is added. The spatial period of the grating is 1/A. Basically, this linear phase term, is equivalent to using an input pattern that is displaced from the origin. We apply amplitude modulation by spatially modifying the w/d ratio as a function of position to follow the desired value of G(x). Davis et al. 25 reported that it could be implemented by locally varying the threshold level applied to this function. They also mentioned that it is difficult to implement a spatially varying threshold level and hence, they proposed a new approach. In this paper, we have followed that approach. Here, we rescale the linear phase mask in the range [ 1 1]. Then the two functions (linear phase mask and amplitude function) are added together and a threshold value of +1 is selected. This new function is binarized. Here,the w/d ratio varies from 0 to ½, thus diffraction efficiency is spatially modulated. The ratio w/d is linearly dependent on the desired amplitude modulation G(x). The modified diffraction efficiency into the first order is given as 25 æ2 ö æ G( x) ö η + 1( x) = sin π èç π ø çè 2 ø (3) 3 MACH Filter In an optical correlator, correlation filter is an integral part of the system. Various types of filters have been proposed in the literature 1-3,11, Mahalanobis et al. 19 introduced the MACH filter in 1994, which has been proved to be a powerful correlation filter. It is a variant of the well-known synthetic discriminant function filter. The filter offers good performances in three major criteria simultaneously, i.e. the easy detection of correlation peak, good distortion tolerance, and the ability to suppress clutter noise. The filter algorithm is statistically optimized and it depends on a realistic and mathematically rigorous optimization procedure. The filter is designed to maximize the intensity of the average correlation output at the origin due to training images while maintaining the average similarity measure (ASM). The MACH filter optimizes the distortion tolerance by minimizing the ASM. The smaller the value of ASM, the more invariant the response of the filter will be. The MACH filter h is given by 19 h = S 1 m (4) where S is a diagonal matrix, is called the ASM, and m is the average of the training images that have undergone a two-dimensional fast Fourier transform. The ASM is defined as 19

3 GOYAL et al.: BINARY FERROELECTRIC LIQUID CRYSTAL s = X - M X - M N N å ( i )( i + ) (5) i=1 where X i are the individual training images in the Fourier domain. The symbol + is used to indicate conjugate transpose. The training image is lexicographically ordered and its elements are placed on the diagonal of X i, and M is the mean of training images, arranged similarly to X i. 4 Chirp-encoded MACH Filter In the output plane of a hybrid digital-optical correlator, two-autocorrelation peaks along with a strong dc are obtained. Out of these three output terms, our interest is in capturing only one autocorrelation peak. Hence one autocorrelation peak and the dc are undesired terms. Lowans and Lewis 12 proposed a technique to remove the problematic dc and symmetric correlation peak by using a chirp encoded binary phase-only filter. The chirp can be given by the following expression 12,13 : 5 Computer Simulation A computer simulation has been done on MATLAB platform to verify the idea of complex function mapping onto a binary amplitude or phase function and chirp encoding. Fig. 1(a) shows the original image IRDE, (b) shows the binary-phase only filter of (a), and (c) shows the reconstructed image. As is clear from Fig. 1(c) that this is the edgeenhanced version of the original image [Fig. 1(a)]. The complex function as the amplitude encoded binary phase grating has been shown in Fig. 1(d). When its Fourier transformation is obtained,we observe the complete original intensity image as shown in Fig. 1(e). To verify the idea of chirp encoding with the developed complex functionencoding algorithm, we carried out simulation study. Figs 2(a and b) show the images of size pixels, used for synthesizing the MACH filter. These é êë 2 2 C( x, y) = exp ik( x + y ) / 2z ù úû (6) In Eq.(6), z is the focal length of the chirp and k = 2π/λ, and λ is the wavelength of light. This chirp function is multiplied with the pre-synthesized MACH filter. The input scene or target s Fourier spectra is multiplied with this chirp-encoded MACH filter. Let t(x,y) represent a target image and T(u,v) its Fourier transform. The chirp function can be either multiplied with the filter or with frequency spectrum of the target. The product function P(u,v) is given as P(u,v) = [C(x,y) {S 1 m}] T(u,v) = [T(u,v) C(x,y)] {S 1 m} (7) The finally obtained product function P(u,v) is mapped into a binary amplitude function and then displayed onto the SLM for obtaining the correlation output. Due to chirp encoding, the correlation terms contain chirp phase functions of opposite sign and hence, different focal powers 12. One term acts as a convex lens while the other as a concave lens. In combination with the lens, the correlation terms are focused in different planes. Therefore, an output camera will only detect one correlation peak in focus. Thus, placing our peak capturing CCD camera at a particular plane, we can capture only one autocorrelation peak. Fig. 1 Simulation results: (a) original image IRDE, (b) binary-phase only filter, (c) reconstructed edge enhanced image, (d) amplitude-encoded binary phase grating, and (e) reconstructed intensity image. Fig. 2 (a and b)- images used for synthesizing the MACH filter

4 650 INDIAN J PURE & APPL PHYS, VOL 45, AUGUST 2007 were zero padded to make pixels size. Fig. 3(a) shows the MACH filter and (b) shows the correlation outputs. We observe that a strong dc and two autocorrelation peaks are obtained in the output plane. Fig. 4(a) shows the chirp-encoded MACH filter and (b) shows the first autocorrelation peak. Figs 4(c) and (d) show the strong dc and second autocorrelation peak, respectively. Thus, we observe that after chirp encoding with MACH filter the desired output term can be recorded. Here, the desired output may be either the first or second autocorrelation peak. Fig. 5 shows the simulation results with almost similar tank images having different energy values. The tank images have been shown in Figs 5 (a,d,g,j,m,p) and their corresponding correlation outputs with MACH filter in Figs 5(b,e,h,k,n,q). Figs. 5(c,f,I,l,o,r) show corresponding single autocorrelation peak obtained with chirp-encoded MACH filter. The detector is placed in this case at 190 mm. 6 Performance Study Four parameters signal-to-noise ratio (SNR), peakto-sidelobe ratio (PSR), discrimination ratio (DR), and peak-to-correlation energy (PCE) have been studied to check the performance. To calculate the values of SNR, we used the formula given by Vijaya Kumar [Chap. 2, Ref. 3]. Fig. 3 Simulation results: (a) MACH filter, and (b) correlation outputs Fig. 4 Simulation results: (a) chirp-encoded MACH filter, (b) first autocorrelation peak, (c) strong dc, and (d) second autocorrelation peak Fig. 5 Simulation results: (a), (d), (g), (j), (m) and (p) are the images used for synthesizing MACH filter, (b), (e), (h), (k), n) and (q) are the correlation outputs obtained with MACH filter, (c), (f), (i), (o) and (r) are the single autocorrelation peaks obtained with chirp-encoded MACH filer

5 GOYAL et al.: BINARY FERROELECTRIC LIQUID CRYSTAL E{η/ H} SNR = (8) var{η} Here, η is the peak of the filter output and E{.} and var{.} denote the expected value and variance, respectively. Considering the Gaussian noise n(x) we calculated SNR. It is said that for valid targets, the MACH filter maximizes the PSR, defined as [Chap. 6, Ref. 3] peak - µ PSR = (9) σ where µ and σ are the mean and standard deviation of the correlation values in some neighbourhood of the peak. We calculated the values of PSR after employing the MACH and chirp-encoded MACH filter. Discrimination ratio is defined as: highest peak due to crosscorrelation DR = 1- (10) autocorrelation peak height PCE is defined as the ratio of correlation peak intensity value to the total energy of correlation plane. We used the computer simulation results for calculating these performance measures. All the calculated values for SNR, PSR, DR, and PCE have been shown in Table 1. It was observed that there are not much difference in the values of the performance metric between MACH and chirp-encoded MACH filter. Hence, chirp encoding serves the desired purpose without any additional cost. reflection type and binary in nature, has been used to display the product function. A CCD camera of size pixels (Sony, Japan; with pixel size µm) has been used for capturing the correlation peaks. Focal length of the Fourier transforming lens used was 105 mm. Experimental results of complex function encoding onto the binary SLM are shown in Fig. 7. Fig. 7(a) shows the reconstructed edge-enhanced image and (b) the reconstructed intensity image. The pre-synthesized MACH filter is shown in Fig. 8(a) and the encoded MACH filter has been shown in Fig. 8(b). The correlation outputs are shown in Fig. 8(c). Fig. 6 The experimental set-up used. SF: spatial filter, CL: collimating lens, PBS: polarizing beam splitter, SLM: spatial light modulator, CCD: charge-coupled device Fig. 7 Experimental results: (a) reconstructed edge-enhanced image, and (b) reconstructed intensity image 7 Experimental Results The experimental set-up 15 used to implement the hybrid approach is shown in Fig. 6. A diode laser (λ = 670 nm) has been used to generate an expanded collimated beam, which illuminates the input SLM. A ferroelectric liquid crystal SLM (Displaytech, USA) of size pixels (with pixel size µm), Table 1 Calculated values of performance measure parameters for MACH and chirp-encoded MACH filter Performance metrics MACH filter Chirp-encoded MACH filter SNR PSR DR PCE Fig. 8 Experimental results: (a) MACH Filter, (b) chirpencoded MACH filter, and (c) correlation output

6 652 INDIAN J PURE & APPL PHYS, VOL 45, AUGUST 2007 Experimental results for chirp encoded MACH filter are shown in Fig. 9. Fig. 9(a) shows the correlation outputs for true class image with MACH filter. Fig. 9(b) shows the correlation output with chirp-encoded MACH filter. In this case, a single peak that is one autocorrelation peak is obtained. Fig. 9(c) shows the output of chirp-encoded MACH filter; here only strong dc is available in the output plane. We have also carried out the study for false class images and the results are shown in Fig. 10. Fig. 10(a) shows the correlation output with MACH filter. Fig. 10(b) shows the output with chirp-encoded MACH filter; here no correlation peak is seen. Fig. 10(c) shows the results for chirp-encoded MACH filter; here only strong dc is available for recording. Fig. 11 shows correlation outputs recorded at different planes. Figs 11(a,d,g,j,m,p) show the correlation outputs with MACH filter at Fourier transform plane. Figs 11(b,e,h,k,n,q) show the outputs with chirpencoded MACH filter, recorded at the plane where only one autocorrelation peak is seen. Figs 11(c,f,i,l,o,r) are the corresponding dc term recorded at the Fourier plane. Note that correlation peaks are seen only in Figs 11(a,b), which correspond to the true class image. For false class images no peaks are obtained. Fig. 9 Experimental results: correlation output for true class image with (a) MACH filter, (b) chirp-encoded MACH filter, one autocorrelation peak, and (c) chirp-encoded MACH filter, strong dc only Fig. 10 Experimental results: correlation output for false class images with (a) MACH filter, (b) chirp-encoded MACH filter, no correlation peak, and (c) chirp-encoded MACH filter, strong dc only Fig. 11 Experimental results: correlation output with (a), (d), (g), (j), (m) and (p) MACH filter, correlation peaks for true class image and no peaks for false class images, (b), (e), (h), (k), (n) and (q) chirp-encoded MACH filter, one autocorrelation peak with true class image and no peak with false class images, (c), (f), (i), (o) and (r) chirp-encoded MACH filter, strong dc only

7 GOYAL et al.: BINARY FERROELECTRIC LIQUID CRYSTAL Conclusion In the hybrid digital-optical correlation approach the product function, which needs to be written onto the SLM is a complex function. To display the product function on the SLM we have implemented the complex data encoding technique proposed by Davis et al 24,25. Encoding provides acceptable performance and numerically efficient and direct method of representing fully complex functions with SLMs that are not fully complex. In this paper, we have also implemented the chirp encoding with the distortion-invariant MACH filter to avoid recording of the dc and one of the autocorrelation peaks. Due to chirp encoding, the correlation signals are focused in three different planes. Thus, placing our peakcapturing detector at a particular plane, we captured only one autocorrelation peak. Results with almost similar tank images having different energy values have also been shown. Acknowledgement The authors wish to acknowledge Shri J A R Krishna Moorty, Director, IRDE Dehradun, for encouragement and permission to publish this paper. References 1 Yu F T S & Jutamulia S, Eds, Optical Pattern Recognition (Cambridge Univ Press, Cambridge, 1998). 2 Javidi B, Ed, Image Recognition and Classification; Algorithms, Systems, and Applications (Marcel Dekker, New York, 2002). 3 Vijaya Kumar B V K, Mahalanobis A & Juday R D, Correlation Pattern Recognition (Cambridge Univ Press, Cambridge, 2005). 4 Pati G S & Singh K, Opt Eng, 37 (1998) Pati G S, Tripathi R & Singh K, Opt Commun, 151 (1998) Nishchal N K, Goyal S, Aran A, Beri V K & Gupta A K, Opt Eng, 44 (2005) Young R, Chatwin C & Scott B, Opt Eng, 32 (1993) Birch P, Young R, Claret-Tournier F, Budgett D & Chatwin C, Opt Eng, 41 (2002) Birch P M, Li G, Claret-Tournier F, Young R, Budgett D & Chatwin C, Opt Eng, 41 (2002) Birch P M, Claret-Tournier F, Budgett D, Young R & Chatwin C, Opt Eng, 41 (2002) Bone P, Young R & Chatwin C, Opt Eng, 45 (2006) Lowans B S & Lewis M F, Opt Lett, 25 (2000) Tang Q & Javidi B, Appl Opt, 32 (1993) Bhagatji A, Nishchal N K, Gupta A K & Tyagi B P, Proc SPIE, 6405 (2006) Beri V K, Aran A, Goyal S, Bhagatji A & Gupta A K, Jour Opt (India), 34 (2005) Gupta A K, Nishchal N K & Beri V K, Proc SPIE, 6234 (2006) Bhagatji A, Nishchal N K, Beri V K & Gupta A K, Opt Lasers Eng, 45 (2007) Goyal S, Nishchal N K, Beri V K & Gupta A K, Wavelet modified maximum average correlation height filter for outof-plane rotation invariance, (Communicated). 19 Mahalanobis A, Vijaya Kumar B V K, Song S, Sims S R F & Epperson J F, Appl Opt, 33 (1994) Mahalanobis A & Vijaya Kumar B V K, Opt Eng, 36 (1997) Bhuiyan S M A, Alam M S & Sims S R F, Proc SPIE, 5816 (2005) Cohn R W & Liang M, Appl Opt, 35 (1996) Juday R D, J Opt Soc Am A, 18 (2001) Davis J A, Cottrell D M, Campos J, Yzuel M J & Moreno I, Appl Opt, 38 (1999) Davis J A, Valadez K O & Cottrell D M, Appl Opt, 42 (2003) 2003.