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To solve the pattern recognition problem, a method of synthesized phase objects is suggested. The essence of the suggested method is that synthesized phase objects are used instead of real amplitude objects. The former is object-dependent phase distributions calculated using the iterative Fourier-transform (IFT) algorithm. The method is experimentally studied with a Vander Lugt optical-digital 4F-correlator. We present the comparative analysis of recognition results using conventional and proposed methods, estimate the sensitivity of the latter to distortions of the structure of objects, and determine the applicability limits. It is demonstrated that the proposed method allows one: (а) to simplify the procedure of choice of recognition signs (criteria); (b) to obtain one-type δ-like recognition signals irrespective of the type of objects; (с) to improve signal-to-noise ratio (SNR) for correlation signals by 20 − 30 dB on average. The spatial separation of the Fourier-spectra of objects and optical noises of the correlator by means of the superposition of the phase grating on recognition objects at the recording of holographic filters and at the matched filtering has additionally improved SNR (>10 dB) for correlation signals. To introduce recognition objects in the correlator, we use a SLM LC-R 2500 device. Matched filters are recorded on a self-developing photopolymer.
©2012 Optical Society of America
1. Introduction
The problem of separation of a recognition signal against the background of cross-correlation noises remains actual for a long time because of the fact that the properties of the correlation signal depend directly on those of the object itself [1–6].
A variation in the recognition conditions or a change of the object usually requires optimization of the available methods for solutions or the development of new ones. Among known methods, we mention the method of digital synthesis of Fourier-filters [7–11], the method of discriminant curve [12–14], the method of stabilizing functional [15], the method of projection onto convex sets [16], and so on. It is necessary to emphasize that all mentioned and other known methods lead to a great number of highly specialized solutions, where the choice of distinctive signs of the object and the subsequent analysis of correlation signals represent a separate problem and are practically incapable to the unification.
In this paper, we propose a new approach to the recognition problem. The novelty of the suggested method that recognition comprises not the object itself, but some object-dependent synthesized phase object with a random phase distribution, which can be calculated using the known IFT algorithm [17]. In this case, the problem of recognition of amplitude objects of various types is reduced to that of phase objects of the same type.
In what follows, we present the experimental results of the recognition of amplitude objects within the conventional and proposed methods, evaluate the sensitivity of the latter to distortions of objects structure, and determine the limits of applicability.
The experiments were carried out with a hybrid Vander Lugt optical-digital correlator [18]. However, the method can be also realized with a joint Fourier-transform correlator [19]. To introduce the recognition objects in the correlator, we used a SLM LC-R 2500 device (Holoeye). As a holographic medium for the recording of matched filters, we used a self-developing photopolymer developed at the Institute of Physics of NAS of Ukraine [20, 21].
2. Method of synthesized phase objects
Let us consider the formation of synthesized phase objects and how they can help with the solution of recognition problem.
The classical scheme of synthesis of a Fourier-kinoform using the IFT-algorithm is given in Fig. 1 . It was described many times [17,22,23] and requires no further explanation. As known, the process of iteration involves simultaneously the formation of the phase structure of a kinoform ψ(υ,ν) in the spectral plane and one more phase structure in the object plane, namely, ϕ(x,y). In the calculation of a kinoform, the distribution θ(x,y) = exp(iϕ(x,y)) plays the role of a diffusion scatterer optimized with regard to the object f(x,y), that is necessary for smoothing of the field amplitude in the Fourier-plane. However, in the context of the correlation methods of recognition, random distributions of the θ(x,y) type can be of independent interest not related to the calculation of the kinoform.
The matter is in the following. Since the shape of ϕ(x,y) for the given number of iterations and the given initial diffuser ϕο(x,y) is uniquely defined by the shape of the function f(x,y), it is logical to raise two questions:
- 1) Is it possible to replace the object f(x,y) by a corresponding synthesized phase object (SP-object) θ(x,y) in the solution of the problem of its recognition?
- 2) Will the solution of the problem with such a change of objects be more efficient than other known methods?
Our numerical and optical experiments gave the positive answer to both questions. We call the method of recognition, where an SP-object θ(x,y) undergoes the procedure of recognition instead of the real amplitude object f(x,y), the method of synthesized phase objects (SPO-method).
Let’s consider the advantages and limitations of this method in more details. To determine its main characteristics in numerical and optical experiments, it is necessary to select a set of recognition objects, to calculate the SP-object for each one, and to perform the recognition itself.
- • Choice of recognition objects: Since the iteration method of synthesis of functions θ(x,y) for f(x,y) gives no possibility to obtain the analytic solution, we studied the SPO-method for a bounded set of recognition objects. In order to demonstrate the capability of the method as completely as possible, we chose objects with significantly different types of their Fourier-spectra.
- • Calculation of SP-objects: It is necessary to determine the conditions of calculation, under which the SP-objects θ(x,y) can be used instead of the input objects f(x,y). In this case, these conditions must ensure the homogeneity of the spectra of θ(x,y), which ensures, in turn, a δ-like recognition signal independent of the type of f(x,y).
- • Matched filtering in the optical-digital correlator: For the comparison of the recognition results for conventional and SPO methods, it is necessary to compare their sensitivities. As an estimation parameter, we chose the controlled changes in the structure of recognition objects, which were realized by means of the rearranging randomly taken pairs of the object points. The number of such rearrangements varies from zero to several hundreds.
3. Calculation of the SP-object and its basic properties
In the numerical experiments, we chose four different amplitude objects fn (n = 1,2,3,4) of the binary type 300 × 300 points in size (Fig. 5(a)). For them, we calculated the correlation functions fn⊗fn. The SP-objects θn were calculated by the iteration scheme given in Fig. 1 for the initial distribution ϕo = const. To study the degree of connection of θn with fn, which determines the degree of suitability of the use of θn instead of fn, we determined θn for various numbers of iterations N, by gradually increasing this number. For a fixed number N of iterations, we calculated the correlation functions | θn,N ⊗ θn,N | for the entire set of {θn,N}. In Fig. 2 on the left, one of the fn − objects (f4), the central fragment of its Fourier-spectrum, and the autocorrelation signal are shown. On the right, respectively, the SP-object for object f4, the shape of its spectrum, and the autocorrelation signal are given. Similar results were also obtained for objects f1 − f3.
Fig. 2 Distributions for the object f4 (a) and SP-object θ4,1(b): 1. − object; 2. − modulus of the amplitude of the Fourier-spectrum; 3. − autocorrelation signal.
The presented result is typical and demonstrates the main advantages of SP-objects such as uniform distribution of the amplitude in the Fourier-plane and the δ-like correlation signal, which do not depend practically on the shape and the effective frequency band of Fourier-spectra and the type of the correlation signals from real objects, for which they were calculated.
As a result of numerical experiments, we determined the criterion of the choice of θn from the set {θn,N} for each fn and various N. The obtained results are demonstrated by the example of object f4 (Fig. 3 ). In Fig. 3(a) (curve А), we show the changes of dispersion σ2 of the amplitude of the reconstructed image of object f4 at the calculation of its SP-object [22] and the maximum value of modulus of the Fourier-spectrum for θ4,N (Fig. 3(a) (curve B)), as N increases. In Fig. 3 (b,c,d), we can observe the redistribution of the phases of SP-object ϕ 4,N in the interval [0−2π] with increasing N.
Fig. 3 (a) Parameters σ 2(A) and |ℑ+1(θ)|max (B) versus the number of iterations N; histograms for (b) ϕ 4,1, (c) ϕ 4,13, (d) ϕ 4,45 calculated for the 1st, 13th, and 45th iterations, respectively.
These results, as well as the results for the other objects (Table 1 ) allow us to make the following conclusions:
- i) Phase structure of SP-objects on the 1-st iteration is close to the binary (0 or π). As the number of iterations increases, the binary structure “spreads,” and, eventually, the phases fill in the whole interval [0−2π];
- ii) Distributions of phases of SP-objects in the coordinate plane have a random character. As a result, their Fourier-spectra are practically uniform in the amplitude, that significantly simplifies registration of matched filters by the Vander Lugt correlator.
- iii) Autocorrelation functions of SP-objects have the δ-like shape and provide:
- 1) Maximum possible value of SNR, which is characteristic of binary phase masks with random distribution of elements [24];
- 2) Possibility to use a simple threshold criterion in the analysis of recognition results.
Table 1. Object and SP-object Parameters
- 1. If there is no correlation between objects fn and fm (fn⊗fm = 0), then there will be no correlation for SP-objects (| θn,N ⊗θm,N | = 0) as well.
- 2. If the signal of cross-correlation between objects f n and f m exists (fn⊗fm ≠ 0), then it exists also for SP-objects (| θn,N ⊗θm,N | ≠ 0).
The first condition indicates that the SP-objects for uncorrelated objects are statistically independent, whereas the second condition makes it possible to obtain a bijective correspondence between cross-correlation curves for real objects and SP-objects.
Thus, for SP-objects, the highest degree of uniformity of their Fourier-spectra is ensured already in the 1-st iteration, and conditions 1 and 2 are satisfied. For real objects f(x,y), their properties (significant signs) are integrally reflected in the distribution of the binary phase elements of SP-objects in the coordinate plane. Any changes in the structure of f(x,y) cause changes in the distribution of θ(x,y), that can be quantitatively determined, in turn, by the level of the mutual correlation signal for SP-objects.
This allows us to assert that such SP-objects can be applied to the recognition instead of real amplitude objects. The next step will be the estimation of practical significance of the method. For this purpose, we perform the comparative analysis of the recognition results for the conventional and SPO methods with the Vander Lugt correlator.
4. Optical experiment
4.1. Experimental procedure
In Fig. 4 , we present the setup (а) and the photo (b) of the optical-digital Vander Lugt correlator. To introduce the images in the object plane of the correlator, we use a SLM LC-R 2500 device based on a reflective LCOS microdisplay (“HoloEye”). A SLM device operates in the phase modulation mode of the wavefront. Images of all objects were phase-only encoded [25] and converted into a conventional graphic files according to the characteristic curve of a SLM device. Let us consider the operation of the correlator in the recording mode of matched filters and the mode of matched filtering.
Fig. 4 (a) Scheme and (b) photo of experimental setup of Vander Lugt optical-digital correlator: CCD1, PC1, k, Fr, P, Bs, SLM, Mr, Sh, MF, L1, A, L2, CCD2 – camera with a computer for the object domain, collimator, Fresnel rhomb, polarizer, beam splitter, spatial light modulator LC-R 2500, mirror, shutter, matched filter, Fourier lens, analyzer, lens, a camera and a computer for the Fourier domain and correlation plane, respectively.
- • Recording of matched filters: The beam of a He-Cd laser passes through collimator k and splitter Bs, and is divided into the reference and object beams. Fresnel rhomb Fr and analyzer А set the necessary polarization of the object beam, by ensuring the phase mode of operation of a SLM device. Polarizer P1 and shutter Sh are not used, whereas polarizer P2 plays the role of a variable attenuator for the reference beam. To the SLM device using CCD1 and computer PC1, we supply the graphic file containing the image of the reference object in the gray-scale format with regard for the characteristic curve of the SLM device. The object beam and the collimated reference beam form a matched filter on self-developing photopolymer PPC-488 [20] in the Fourier-plane Pmf of the correlator. The conditions of the recording of matched filters were optimized to attain the maximum diffraction efficiency (η) at minimum level of intrinsic noises.
- • Matched filtering:The operation of the correlator in the mode of matched filtering includes the following steps. The collimated laser beam of required polarization direction reflects from the mirror of the SLM device, where the image of the object is supplied, passes lens L1, and falls on the plane Pmf, where the matched filter MF for the reference object is positioned. Then, camera CCD2 in the correlation plane fixes mutual correlation signal obtained as a result of the inverse Fourier-transformation of the product of Fourier-images of the input and reference images of the objects, which is realized by lens L2.
To register the correlation signal, we use Sony4800 camera with the size of a pixel of 30μm × 30μm and the size of a sensor of 580 × 470 pixels with ADC panel (8-bit) by the “Spiricon” firm with specialized software for the analysis of laser beams. The relative calculation error of SNR is 5% at most.
We now define the procedure of recognition within the SPO-method:
Stage 1:
- i) For the reference object fref, the SP-object θref is calculated using the IFT-algorithm;
- ii) θref is placed on the object plane of the correlator instead of fref, and the recording of a matched filter is produced;
Stage 2:
- i) For the comparison object fin, the SP-object θin is calculated using the IFT-algorithm;
- ii) θin is placed on the object plane of the correlator instead of fin, and the matched filtering is realized;
- iii) The mutual correlation signal Icorr = | θref ⊗θin |2 is registered in the correlation plane.
4.2. Results and discussion
4.2.1. Matched filtering
To obtain the cross-correlation dependences, we use the same set of objects f1− f4 as for numerical experiments. For each initial object, we calculate the series {fn(k)}, k∈[1-800] of derivative objects obtained by introducing distortions to the initial objects structure. The distortions, as mentioned above, are rearrangements of pairs of points (pixels) of the object taken randomly, k being the number of such rearrangements. In Fig. 5(b) , we show a fragment of object f1 for various numbers of rearrangements.
Fig. 5 Objects f1− f4 (a) and fragments of object f1 (b) for various numbers of rearrangements: 1. – k = 0; 2. – k = 150; 3. – k = 300; 4. – k = 600.
For all objects fn and series {fn(k)} under the optimum conditions (see section 3), we calculated the corresponding θn,1 and series {θn,1(k)}. Then we recorded the matched filters and carried out recognition using conventional and SPO methods. Cross-correlation signals were registered by CCD2 camera, and their SNRs were calculated. We obtained the intensity of correlation signals Icorr as a function of the degree of distortions of the structure of the compared objects. We also estimated the degree of homogeneity of the intensity of the Fourier-spectra of objects and SP-objects. The spectra were registered by CCD2 camera in plane Pmf of the correlator (Fig. 4(a)).
Figure 6 shows typical results by the example of object f1. In Fig. 6(a), curves A and B show the behavior of the signal Icorr as a function of the parameter k for f1 and θ11, respectively. The shape of correlation signals (for values of k marked by circles on the curves) is presented in Fig. 6(b,c). We present the Fourier-spectra of the object (Fig. 6(d)) and the SP-object (Fig. 6(e)) in the main order of SLM, as well as zero and ± 1 orders of SLM. In the Fourier-spectrum of the object, the zero order of SLM cannot be explicitly separated because of the presence of eigenfrequencies of the object in this region.The values of SNR for the autocorrelation signal of object f1 and SP-object θ1,1 were 2.1 dB and 24.8 dB, respectively, for the diffraction efficiencies of the corresponding matched filters η = 7% and η = 60%. The course of curves A and B in Fig. 6(a) indicates that the SPO-method possesses a high sensitivity to object structure distortions, that can, however, play a positive or a negative role depending on the character of the problem in hand.
Fig. 6 Experimental results for object f1 (left) and SP-object θ1,1(right): (a) – dependence of the intensity of the cross-correlation signals on k; (b,c) – shape of correlation signals; (d,e) – shape of Fourier-spectra.
On the basis of the results of recognition obtained for objects f1 − f4, we may conclude that the characteristic distinctions of the methods under comparison, which were indicated in numerical experiments, are also confirmed by optical experiments.
We have established that, in the applied scheme of a correlator at the recording of matched filters, a part of light non-diffracted on SLM, falls in the region of zero frequencies of the Fourier-spectrum of the object. In Fig. 6(e), this intense peak is well noticed. This peak induces the appearance of the false component of recognition signals, which masks the actual course of curves in the region with strong distortions of the object.
So, it is seen in Fig. 6(a) (curve A) that the intensity of the correlation signal does not decay to zero in the region with k > 400 where the structure of the object is sufficiently strongly distorted, as k increases, and it remains practically constant. This effect is observed for both conventional and SPO methods. Below, we present a means to remove this effect using the off-axis matched filtering. The method of separation of the recognition signal and the optical noise in the correlation plane is known [26], but we have first demonstrated the improvement of parameters of the recognition signal by means of the spatial separation of the Fourier-spectra of objects and optical noises in the Fourier-plane for the optical-digital correlator.
4.2.2. Off-axis matched filtering
In order to exclude the influence of optical noises of the correlator on the recognition result, we have realized the spatial separation of the Fourier-spectra of objects and a zero-order SLM at the recording of the filters and at the matched filtering by means of the superposition of the deflective phase grating on the input and reference objects.
For phase functions of the type θ(x,y) = exp(iϕ(x,y)), which are introduced to the object plane of the correlator using the SLM, such grating is created by means of superposition of the linear phase 2π(xυo + yνo) on the phase ϕ(x,y) of the input function [27]. As a result, the spectrum can be described by the formula:
where ℑ+1 is the operator of direct Fourier-transformation, Θoff(υ,ν) is the shifted spectrum, Θ(υ,ν) is the axial spectrum, δ is the delta-function, the symbol ⊗ stands for the operation of convolution, and υo,νo are the shifts of the spectra along the axes υ,ν, respectively. The axis, relative to which the shift is made, passes through the center of the object plane (SLM), Fourier lens, and center of the Fourier-plane Pmf (Fig. 4(a)). The recording of the filter for the reference object with added phase grating and the subsequent matched filtering of input objects with the added phase grating can be considered as the off-axis matched filtering relative to the indicated axis.To enhance recognition efficiency, we have realized a shift of the Fourier-spectra by a value that is close to the maximally admissible one which is equal to half a size of the main diffraction order of SLM [27]. The efficiency of the proposed off-axis matched filtering is shown below by the example of objects f2 and f4. In Fig. 7 , we present the fragments of objects f2 and f4 after phase-only encoding with added phase grating (a,b), as well as their on-axis (c,d) and off-axis (e,f) Fourier-spectra.
Fig. 7 (a,b) – fragments of the objects f2 and f4 after phase-only encoding with superposed grating; (c,d) – on-axis Fourier-spectra; (e,f) – off-axis Fourier-spectra.
As a result of the performed off-axis matched filtering within the conventional and SPO methods for all objects fn and series {fn(k)}, as well as for θn,1 and series {θn,1(k)}, we have observed the proper course of cross-correlation curves in the whole range of variations of k- rearrangements, including large k that correspond to strong distortions of the objects structure, in contrast to the on-axis matched filtering. In Fig. 8(a,b) , we show the correlation curves for the on-axis (1) and off-axis (2) matched filtering for object f1 and its SP-object, respectively. As is seen, the curves corresponding to the off-axis matched filtering reveal a decrease for all k, which gives the possibility to perform a proper comparison of the sensitivity of the methods.
Fig. 8 Dependence of the intensity of the correlation signals on the parameter k for object f1 (a) and SP-object θ11 (b) for (1) on-axis and (2) off-axis matched filtering.
In addition, the off-axis matched filtering leads to a growth of SNR for correlation signals and the diffraction efficiency (η) of matched filters (Table 2 ).
Table 2. Pattern Recognition Results
The results shown in Table 2 indicate the significant advantages of the SPO-method as compared with the conventional one. It should be noted that the random distribution of samples of the phase in the plane of an SP-object and its binarity allow us to identify statistical properties of binary phase masks with a random distribution [24] and our SP-objects with sufficient degree of accuracy and, thus, to determine the upper bound of SNR for correlation signals by the formula:
where K2 is the total number of samples, i,j = 1,2, …, K are the coordinates of samples in the correlation plane (X2Y2). Formula (2) allows one to estimate the ratio of the intensity of a correlation signal at the center coordinate Icorr (K/2,K/2) to the noise intensity at any point of the correlation plane.Thus, SNRs obtained experimentally and calculated by Eq. (2) agree with each other. We note that, for the initial objects after the phase-only encoding, the estimation of SNR by Eq. (2) is possible only for test object f2, because it is a binary phase mask with a random distribution of rectangle elements (2 × 2 pixels).
The main conclusions that can be made by the results of optical experiments with the hybrid optical-digital Vander Lugt correlator are the followings: 1) the SPO-method can be practically implemented; 2) the results of numerical and optical experiments for autocorrelation signals are in good agreement with each other. In particular, the cross-correlation curves for the conventional and SPO methods demonstrate the one-to-one correspondence between themselves for weak and strong distortions of the structure of recognition objects.
5. Conclusion
We have proposed numerically and experimentally investigated the method of synthesized phase objects to solve the recognition problem. It is shown that the solution of the problem for real objects belonging to various classes can be reduced under certain conditions to that of the problem of recognition of object-dependent SP-objects that belong to the same class of binary phase masks with random distribution of elements. This method allows to unify the shape of recognition signals by reducing it to the δ-like one and to improve SNR for correlation signals by 10 − 103 times. It was also shown that the sensitivity of the proposed method to the distortions of the identified object structure is higher than that of the conventional method.
Acknowledgments
This research was partially supported by Specialized Enterprise Holography Ltd. Project No. 01/03-04, 2004, Ukraine. This research was also partially supported by the research fund from Chosun University, 2010, South Korea.
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