LPCC filters realization as binary amplitude hologram in 4-f correlator: range limitation of hologram pixels representation N.N. Evtikhiev, S.N. Starikov, R.S. Starikov, E.Yu. Zlokazov Moscow Engineering Physics Institute (State University) 115409, Kashirskoe rd., 31, Moscow, Russia ABSTRACT The results of realization of linear phase coefficient composite (LPCC) filters in 4-f correlator are presented in this paper. LPCC filters application allows achieving invariance of correlation peak in the presence of geometric distortions of contour objects. Invariant correlation filters were synthesized as amplitude holograms with continuous transparency and realized as binary amplitude holograms. The results of correlator signals modeling for the cases with different numbers of hologram counts which used to represent grayscale level gradation are presented. Keywords: invariant pattern recognition, optoelectronic correlator, invariant correlation filters, LPCC filter, computer generated hologram 1. INTRODUCTION Invariant correlation filters application is one of the major methods to achieve invariance of object recognition in the presence of identification object geometric digressions relatively to the object taken as reference object. Composite filter with linear phase coefficients (LPCC filter) is one of the perspective types of correlation filters 1, 2. Filter realization in optoelectronic correlator allows to provide object recognition in real time conditions. However this approach demands the problem decision of diffraction element synthesis in order to form a proper impulse response of an optical system. LPCC filters were realized in 4-f correlator scheme 3, 4 ; binary stochastic screening was used to represent grayscale level gradations. Possibility to realize filter as amplitude diffraction element placed on the spatial frequency plane of an optical system using media with limited dynamic range such as spatial light modulator (SLM) discussed in this paper. Such element can be realized as binary amplitude hologram synthesized on the basis of Vander Lugt widespread method of hologram synthesis. Contemporary input devices allow to realize binary images with the maximum resolution approximately 7 micrometers per pixel. This means that realization of the grayscale hologram with the resolution of 1024 1024 pixels with pixel accuracy of 8 bit in full grayscale range using stochastic screening leads to the transparency with geometrical sizes approximately 10 10 centimeters. On the basis of the widespread Fourier optics theory this is obvious that in the case of maximum input image resolution of 7 micrometers per pixel the focal length of Fourier transformation lens of 4-f correlator must be equal to 0,5 meters which consequently leads to incompact sizes of correlator. These arguments show the necessity of dependence analysis between pixels grayscale gradation value and recognition capabilities of the hologram. 2. COMPOSITE FILTER WITH LINEAR PHASE COEFFICIENTS LPCC filters possess such advantages comparing with other known types of correlation filters as good discrimination capabilities for binary contoured images due to high signal-to-noise ratio and correlation function obtained with LPCC filter is pretty similar to autocorrelation function. LPCC filter is also attractive for application as it saves calculation time contrary to the competing filter types which require matrix inversion during its synthesis. LPCC filter is 1, 2 : h * k ( N 1) 2π ( x, y) = exp( j nk) sn ( x, y), k = 0,1,... ( N 1) (1) N n= 0 Where s 0, s 1, s N-1 are images from the training set. Early works on LPCC filters modeling show possibilities to obtain high and sharp correlation peak in the cases when more than 20 images of reference object from the training set were used. Optical Pattern Recognition XX, edited by David P. Casasent, Tien-Hsin Chao, Proc. of SPIE Vol. 7340, 73400C 2009 SPIE CCC code: 0277-786X/09/$18 doi: 10.1117/12.819105 Proc. of SPIE Vol. 7340 73400C-1
MatLab v7 development environment features were used to synthesize LPCC filter invariant to rotation of input object in the range from 0 to 170 degrees. Filter was synthesized on the basis of 18 images of the contoured reference object presented on figure 1 rotated in observable range with the step of 10 degrees. Resolution of all images was 256 256 pixels. Amplitude distribution of the filter presented on figure 2. Fig. 1: Reference object in the basic state (0 degrees of rotation) 0.9 50 0.8 0.7 100 0.6 0.5 150 0.4 0.3 200 0.2 0.1 250 50 100 150 200 250 Fig. 2: amplitude distribution of synthesized filter; 3. HOLOGRAPHIC FILTERS SYNTHESIS LPCC filters realization in 4-f correlator demands the decision of the problem of diffraction element synthesis which allows to form a proper impulse response of the optical system. Most of all widespread devices such as SLM allow inputting only amplitude or phase information of the filter into the spatial frequencies plane of the optical system cutting out the significant rest of the information incorporated into the filter. This approach leads to loss of recognition and discrimination capabilities of the filter. Vander Lugt method of holographic synthesis allows to keep amplitude and phase information of the filter as amplitude interference distribution. Thus LPCC filter can be realized as amplitude mask located in spatial frequencies plane of the optical system. Specially developed software in MatLab v7 development environment was used to provide calculation of the hologram. Grayscale holograms were synthesized with the resolution of 1024 1024 pixels and accuracy of grayscale representation of 8 bits or 256 levels of grayscale gradation. Figure 3 contains zoomed central part of the hologram Figure 4 presents numerically recovered amplitude of impulse response of the hologram. Impulse response of the hologram contains delta function located in zero diffraction order and two conjugated to each other images corresponding to filter by amplitude and phase located in 1 and -1 orders. Figure 5 represents a side view of correlation field obtained during calculation of correlation between reference object and hologram. Proc. of SPIE Vol. 7340 73400C-2
Fig. 3: Zoomed central part of the hologram Fig. 4: Numerically recovered amplitude of the hologram impulse response Fig. 5: Example of side view of correlation signal Proc. of SPIE Vol. 7340 73400C-3
4. MODELING OF HOLOGRAM PIXELS REPRESENTATION RANGE LIMITATION Analysis of realized filters features variations for the cases of limited value of grayscale levels gradations was provided. Variations of grayscale level gradation value of hologram steps and numeric recovery of proper impulse response and also numeric modeling of images and holographic filters correlation in the scheme of 4-f correlator were provided. Modeling was realized for the cases of g=2, 4, 8, 16, 32, 64, 128 and 256. The examples of the impulse response amplitude numeric recovery results for different cases of the value g are presented in figure 6. This is obvious by sight that all geometrical features of recovered amplitude distribution are pretty similar to realized filter in the limits of single count in the cases of g not less than 16. Examples of image and holographic filters correlation modeling results are presented in figure 6. Represented results conform to spatial amplitude distribution of complete light signal of 4-f correlator. These signals contained by spatially separated convolution of input image and correlation filter, input image and correlation of the filter and input image. The last area is the most interesting thing for the decision of the problem defined by researches. Correlation peaks for all images and for the cases of all values of g are sufficiently localized and their magnitudes are sufficiently high even in the case of g=2. Result represented in figure 7 shows the dependence between angle of input object rotation and correlation peak value for the cases of g=8, 16, 256 when true object were used as input object. Fig. 6: The result of amplitude of the hologram impulse response numeric recovery for the cases of different grayscale levels gradation value g (left); the results of optical correlator signal modeling for the different grayscale levels gradation value g (right) Proc. of SPIE Vol. 7340 73400C-4
6000 5500 5000 4500 g=8 g=16 g=256 4000 3500 3000 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 Fig. 7: Value of correlation peak amplitude in dependence from rotation angle for g = 8, 16, 256. 5. TWO CLASS PROBLEM In order to provide multiclass pattern recognition estimation images of false objects were used. These objects were pretty equal to the true object by intensity. Linear sizes of false objects equal to the linear size of true object. Forms of false objects were similar to the true object. These objects also got the same contour line width as the objects of class true. All examples of false objects are represented in figure 8. Multiclass pattern recognition of the holographic filters for g>8 can be estimated as wholly satisfactory although it is lower than in the case of full-scale hologram representation value of 256 levels of gray. Examples of modeling results illustrating multiclass pattern recognition of the holographic filter in the case of g=16, 256 are shown in figures 9 and 10. All differences between the correlation results for the case of full-scale hologram representation and for the cases of g>32 become neglible. In conclusion it must be noted that it is necessary to use not less than 16 levels of grayscale gradation of digitally synthesized hologram on the basis of LPCC filter to keep its selectivity completely. 1 2 3 Fig. 8: False objects Proc. of SPIE Vol. 7340 73400C-5
6500 6000 5500 5000 4500 4000 3500 true false 1 false 2 false 3 3000 2500 2000 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 Fig. 9: Value of correlation peak amplitude in dependence from rotation angle for true and false objects g = 256. 5500 5000 4500 4000 3500 true false 1 false 2 false 3 3000 2500 2000 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 Fig. 10: Value of correlation peak amplitude in dependence from rotation angle for true and false objects g = 16. 6. CONCLUSIONS As a result of the work it must be noticed that invariant correlation filter with linear phase coefficient can be realized in 4-f correlator scheme as amplitude hologram using holographic media with limited dynamic range of amplitude modulation. Correlation peaks keep its satisfactory features even in the cases of grayscale gradation less than 8 levels. In order to keep multiclass pattern recognition feature of the correlation system it must be used not less than 16 levels of grayscale gradation. 7. ACKNOWLEDGEMENTS This research is supported by Ministry of Education and Science of Russia and Russian Foundation for Basic Research. REFERENCES 1. Hassebrook L., Vijaya Kumar B. V. K., Hostetler L, Opt. Eng., 29, 1033, (1990). 2. Vijaya Kumar B. V. K. Appl. Opt., 31, 4773, (1992). 3. N. N. Evtikhiev; S. N. Starikov; S. A. Sirotkin; R. S. Starikov; E. Yu. Zlokazov, proc. SPIE, 6977, 2008, 69770C. 4. N N Evtikhiev, S N Starikov, E Yu Zlokazov, S A Sirotkin, R S Starikov, Quantum Electron, 2008, 38 (2), 191-193. Proc. of SPIE Vol. 7340 73400C-6