A hybrid approach for image half-toning combining simulated annealing and Neural Networks based techniques

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1 A hybrid approach for image half-toning combining simulated annealing and Neural Networks based techniques Kurosh Madani Intelligence in Instrumentation and Systems Lab. (I 2 S) SENART Institute of Technology - University PARIS XII Avenue Pierre POINT - F LIEUSAINT France Abstract: A classes of stochastic algorithms, which are very powerful in the case of the degraded image reconstruction, are simulated annealing based algorithms. However, the reconstruction of a degraded image using iterative stochastic process require a large number of operations and is still out of real time. On the other hand, learning and generalization capability of ANN models allows a large panel of techniques improving classical techniques limitations. We are investigating in parallel implementation of image processing techniques. In this paper, we present a hybrid approach for image half-toning combining simulated annealing and neural network based techniques. Simulation and experimental results will be reported. Key-Words: Hybrid Technique, Simulated Annealing, Neural Networks, Degraded Image Reconstruction, ZISC-036, Hardware Implementation, Parallel, Image Processing, Half-Toning. 1 Introduction A classes of stochastic algorithms, which are very powerful in the case of the degraded image reconstruction, are simulated annealing based algorithms. The main advantage of such approach is related to the fact that in this kind of processing, there is not necessity to any prior hypothesis on nature of information to be restored. However, the reconstruction of a degraded image using iterative stochastic process require a large number of operations and is still out of real time. On the other hand, learning and generalization capability of ANN models allows a large panel of techniques improving classical techniques limitations. We are investigating in parallel implementation of image processing techniques. In this paper, we present an hybrid approach for image half-toning combining simulated annealing and neural network based techniques. Theoretical studies on simulated annealing show that the convergence of such process is obtained for the number of infinite iterations : so, an implementation of such algorithms become uninteresting from the point view of computation time. The goal is here to use the neural network s learning and generalization capabilities to complete an unfinished simulated annealing process. The present paper has been organized as following: The section 2 presents the principle of Simulated Annealing (SA) based image processing, especially the half-toning. A brief description of RCE based neural model and the basic properties of the IBM ZISC-036 component implementing such model are reported in section 3. The section 4 is dedicated to the hybrid approach combining simulated annealing and neural based techniques. Finally the section 5 conclude this paper and gives perspectives to the present work. 2 About Simulated Annealing approach and derived image half-toning technique Since it's introduction by Metropolis and Ulam [1] in 1949, and later by Kirkpatrick and others ([2] to [5]), the importance of Monte-Carlo algorithms for solving computation problems in high dimensional spaces is well known. In 1985, Carnevali & al. show the equivalence between the Ising model (a

2 physical system) and the image reconstruction dilemma proposing a powerful iterative stochastic relaxation based algorithm for picture half-toning and picture smoothing [6] : an image is considered as some global state of a 2-D physical system and it's processing is based on global energy minimization of such physical system. The image half-toning consists on encoding a multilevel (gray level) image to a binary image in which the gray levels are represented by some spatial (2-D) density of binary states (for example, a spatial density of "1" in a given region of the halftoned image). Let us to consider this case. We have chosen the following notation : ai,j : pixel of the multilevel image, with ai,j [ -1, +1 ], bi,j : corresponding pixel of the half-toned image and Vi,j,k,l : some kernel with the following properties : V i, j, k, l 0 and V i, j, k, l = 1 The pixel's energy then could be defined by relation (1), where ri,j is some representation relative to the corresponding region in the half-toned image. E = i j r i, j = k 2 ( a i, j r i, j ) k l l V i, j, k, l (1) with Developing relation (1), one can show (Carnevali & al.) that the energy function could be written as relation (2). 1 E = λ âi, j bi, j + i j 1 λ with λ = 2, i, j = I i, j, k, l = n i1 j1 i2 j2 m k l I V i, j, n, m b k, l b b i2, j1, i2, j2 i1, j1 i2, j2 V i, j, k, l V n, m, k, l a k, l and To update the pixel's value, two dynamics are generally used. In both of them, the updating is performed randomly. The difference is related to the probabilistic decision function that will be used to valid the new value of the pixel (in the case of the half-toning process, the decision concerns the validation of pixel's value to be "1" or "0"). (2) The original Carnevali's algorithm uses pixels values belonging to the continuous interval [-1, +1]. Moreover, the energy is also encoded supposing an infinite precision (all possible values belonging to the continuos interval [-1, +1]). Such hypothesis could not be implemented simply because of the computer's precision limitation. We have considered the following modifications : all used values in our version of this algorithm are integers with finite number of bits (number which should be determined). Moreover : all Vi,j,k,l are supposed to be constants and identical, the value of λ parameter in relation (2) is supposed to be 1, and the simulated annealing control parameter T is also supposed to be an integer. The first consequence of such hypothesis is related to the energy value that will be also integer and not normalized. According to the above mentioned hypothesis (relative to the algorithm's modifications), and considering the fact that in the Carnevali's process the neighborhood doesn't change, the energy variation will be given by the relation (3) with : b i, j { 0, 1 } and b i, j { 1, 0, 1 }. E = E E = i 0, j 0 b i, j + I b i, j b i, j k i 0 l j 0 The decision dynamics is based on the Glauber dynamics (relation (4)). ( ) = 1 n + 1 P X i " 0 " P X n + 1 " 1 " i E 1 + exp i T n ( ) = 1 P ( X i ) 3 About RCE Model and the ZISC-036 Neuroprocessor The RCE (Restricted Coulomb Energy) like ANNs include three layers: an input layer, a hidden layer and an output layer. Each node of the hidden layer is a processing unit which computes the distance between the input layer and the prototype stored within each node (thanks to a connection between the input and the hidden layer). The output layer is used to give the categories which correspond to the input data. Connections between hidden and output layers are dynamically established during the learning phase. These models consist of mapping an

n-dimensional space by prototypes where each prototype is associated with a category and a threshold, influence field (for RCE), a part of the n- dimensional space around the prototype where

3 n-dimensional space by prototypes where each prototype is associated with a category and a threshold, influence field (for RCE), a part of the n- dimensional space around the prototype where generalization is possible (see figure 1). A prototype is a vector defining the coordinates of the prototype within the n-dimensional space. Within the network, several prototypes may be associated with one category, and influence fields can partially overlap one another. The ZISC-036 hardware can implement the RCE and the KNN algorithms. Based on this model, the ZISC operates in two phases, the learning phase and the recognition phase [11][12] [12]. were P represents the memorized prototype and V is the input pattern. compare the distance to its threshold, communicate with other neurons (in order to find the minimum distance, category, etc.), adjust its threshold (during learning phase). V 1 Fig.2 : Example of an input mapping in a 2-D space: ROI (a) and 1-NN (b) using norm L1 in the case of the ZISC-036 RCE I/O bus HOST Neuron 1 Neuron n weight weight Fig.1 : Example of RCE-like map of a two dimensional space. V 0 Norm distance evaluator... Norm distance evaluator The IBM ZISC-036 is a parallel neural processor based on the RCE and KNN algorithms. Each chip is capable of performing up to recognitions per second. Thanks to the integration of an incremental learning algorithm, this circuit is very easy to program in order to develop applications; a very few number of functions (about ten functions) are necessary to control it. ZISC-036 is composed of 36 neurons. This chip is fully cascadable which allows the use of as many neurons as the user needs (a PCI card has been developed with a capacity of 684 neurons). A neuron is an element which is able to: memorize a prototype (64 components coded on 8 bits), the associated category (14 bits), a threshold (14 bits), a context (7 bits), compute the distance, based on the selected norm (norm L1 given by relation 2 or LSUP given by relation 3) between its memorized prototype and the input vector (the distance is coded on fourteen bits), n L1 : dist = V i P i (2) and i= 0 LSUP : dist = max i = 0... n V i P i (3) category context (sub-network index) thresholding Inter-neuron communication bus category context (sub-network index) Fig.3 : IBM ZISC-036 bloc diagram. thresholding Figure 2 shows an example of the input mapping of a 2-dimensional space with the ZISC-036. Figure 2(a) using ROI shows that due to the close proximity of neighboring neurons, the ROI are adjusted while figure 2(b) is an example of the KNN (in the case of the figure a 1-NN has been considered). Figures 3 gives the ZISC-036 structure s bloc diagram. 4 The hybrid approach combining RCE and SA techniques As it has been mentioned previously, the convergence of a simulated annealing based process is obtained for infinite iterations and so, an implementation of such algorithms become uninteresting from the point view of computation time. The goal is here to use the neural network s

learning and generalization capabilities to complete an unfinished simulated annealing process.

input image (issued from an unfinished simulated annealing) to a pixel of the output image (in this case the category is a pixel.

The second association strategy consists on association a region of the input image to a category which is also a region of the output image.

4 learning and generalization capabilities to complete an unfinished simulated annealing process. The approach we propose is a hybrid solution combining a simulated annealing based module and an RCE (RBF) based neural unit. Figure 4 shows the bloc diagram of such hybrid processing unit. input image (issued from an unfinished simulated annealing) to a pixel of the output image (in this case the category is a pixel. This association dose not lead to good results because the neighborhood correlation is not taken into account. The second association strategy consists on association a region of the input image to a category which is also a region of the output image. SIMULATED ANNEALING BASE MODULE NERAL NETWORK BASE UNIT Fig.4 : Bloc diagram of the proposed hybrid processing unit. The simulated annealing module stops the process after a finite number of iterations, leading to an unfinished result. Then the neural network based unit improves the result basing on the learning and generalization capabilities of such process unit. The interest here is to take advantage from the fact that the simulated annealing based process will process the input information (images in our case) without any prior hypothesis on nature of that input information : minimizing some global energy associated to the system. However, as the simulated annealing based process has been stopped after a finite number of iterations the results quality will not be acceptable. Then, the neural network based unit will act as a function approximation unit approximating the unfinished process by the most appropriated function. The approximation is performed locally (considering some neighborhood for each pixel of the image). Fig.6 : Original multi-leveled image (Left) and an unfinished simulated annealing based result with a insufficient number of iterations. The figure 5 compares results obtained for each association relative to a 3 by 3 neighborhood. The figure 6 shows an unfinished simulated annealing based image half-toning obtained from the original multi-leveled image of Lena. The figure 7 represents the output image of the ZISC-036 based neural unit considering a 3 by 3 neighborhood around each pixel. For the figure 7, a randomly pixel to pixel learning process has been performed. The first image of the figure 8 (left side) corresponds to the half-toned image obtained after a large number of iterations using the presented simulated annealing technique. The second image of the same figure (right side) reproduces the result obtained from an unfinished simulated annealing half-toning using the presented hybrid approach only after one single iteration (reconstruction). Fig.5.: Results obtained for a region to pixel association (left) and region to region association (right) relative to a 3 by 3 neighborhood. Two kind of associations (between input and output images) could be considered : the most simplest is to associate a region (some neighborhood) of the Fig.7 : Hybrid technique based result with a pixel to pixel learning process after only one iteration.

Concerning future perspective of the present work, we are working on a significant learning data base construction including a panel of various half-toned images.

8 : Left : Half-toned Lena obtained from simulated annealing based process after a large number of iterations.

5 Concerning future perspective of the present work, we are working on a significant learning data base construction including a panel of various half-toned images. We also working on improving the hybrid technique by associating another kind of neural models (competitive and general). Acknowledgments: Fig.8 : Left : Half-toned Lena obtained from simulated annealing based process after a large number of iterations. Right : Hybrid technique based result with a region to region association learning process after only one iteration. 5 Conclusion The image half-toning consists on encoding a multilevel (gray level) image to a binary image in which the gray levels are represented by some spatial (2-D) density of binary states. Simulated annealing based algorithms, very powerful in the case of the degraded image reconstruction, are used to obtain a half-toned image from an originally gray level image. The main advantage of such approach is related to the fact that in this kind of processing, there is not necessity to any prior hypothesis on nature of information to be restored. However, the reconstruction of a degraded image using iterative stochastic process require a large number of operations and is still out of real time. In this paper, we have presented an hybrid approach for image half-toning, combining simulated annealing and neural network based techniques. The neural part of the system we proposed is based on RCE-RBF like neural network. The goal is to use the neural network s learning and generalization capabilities to complete an unfinished simulated annealing process, and so, to accelerate the processing duration saving the above mentioned advantages related to the simulated annealing based processing. We have implemented our neural module on a IBM ZISC-036 based board including 16 neuro-processors with 36 neurons per processor. Experimental results, obtained from unfinished simulated annealing based processing, showing half-toned image quality improvement, have been presented validating our approach. The author wish thank Dr. Nabil Mesbah for his participation during his Ph.D. period to simulated annealing and neural networks aspects of the present work and for useful discussions. References: [1] N. Metropolis, S. Ulam, J. of Am. Statistical Assn. 44, [2] S. Kirkpatrick, S. D. Gelatt, M.P. Vecchi, Optimization by simulated annealing, Science 8-67, [3] S. Kirkpatrick, Optimization by simulated annealing: Quantitative studies, J. of Statistical Phys. 34, 975, [4] S. Geman, D. Geman, Stochastic relaxation, Gibbs distribution and Bayesian restoration of images, IEEE Trans. PAMI, Vol. 3, 1984, pp [5] D. W. Murray, A. Kashko, H. Buxton, A parallel approach to the picture restoration algorithm of Geman and Geman, Image Vision Computing Vol. 4, N 6, 1986, pp [6] P. Carnevalli, L. Coletti, S. Patarnello, Image processing by simulated annealing, IBM J. of Res. and Dev. Vol 29 N 6, Nov. 1985, pp [7] P. Garda, K. Madani, F. Devos, P. Chavel, P. Lallane, J. Taboury, A massively parallel image processor for stochastic relaxation using optical random number generation, Optical Computing, Ed. J.W. Goodman, P. Chavel, G. Roblin, SPIE Volume 963. [8] P. GARDA, K. MADANI, F. DEVOS, P. CHAVEL, P. LALANNE, J. TABOURY, A monolithic processor array for stochastic relaxation using optical random number generation, NATO Series, Vol. F 68, Neurocomputing, Edited by F. Fogelman Soulié and J. Hérolt, Springer - Verlag Berlin Heidelberg [9] K. MADANI, N. MESBAH, Discussion on a massively parallel implementation of simulated annealing algorithms for image processing, International Symposium on Intelligent Systems and advanced manufactoring, Unconventional Imaging for Industrial Inspection, Philadelphia, Pennsylvania, USA, 23-26october 1995.

6 [10] K. MADANI, N. MESBAH, Compromise Discussion around Two Dynamics for Implementation of Simulated Annealing based Image restoration, Aerospace remote Sensing, EUROPTO'97, IEE, London, United Kingdom, September 1997, Image and Signal Processing for Remote sensing SPIE Vol [11] ZISC036 data book, IBM Essonnes Component Development Laboratory, IBM Microelectronics, Corbeil-Essonnes, France. [12] Eide A., Lindblad Th., Lindsey C.S., Minerskjöld M., Sekhviaidze G. and Székely G.: An Implementation of the Zero Instruction Set Computer (ZISC-036) on a PC/ISA-bus Card, 1994 WNN/FNN Proc., Washington D.C., December [13] Robert David, Erin Williams, Ghislain de Trémiolles, Pascal Tannhof, Description and Practical Uses of IBM ZISC-036, VI-DYNN'98 - Virtual Intelligence - Dynamic Neural Networks Stockholm - Sweden - June 22-26, 1998.

Image Restoration using Markov Random Fields

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