Application of Totalistic Cellular Automata for Noise Filtering in Image Processing

Transcription

1 Journal of Cellular Automata, Vol. 7, pp Reprints available directly from the publisher Photocopying permitted by license only 2012 Old City Publishing, Inc. Published by license under the OCP Science imprint, a member of the Old City Publishing Group. Application of Totalistic Cellular Automata for Noise Filtering in Image Processing Y. ZHAO 1, H.M. GUO 2 AND S.A. BILLINGS 1 1 Department of Automatic Control and System Engineering, University of Sheffield, UK. s.billings@shef.ac.uk 2 School of Mechanical, Aerospace and Civil Engineering, University of Manchester, UK. Received: March 5, Accepted: November 14, The selection of the neighbourhood is a very important part of the specification and training of Cellular Automata (CA) in image processing. Rather than guessing or assuming a specific neighbourhood, this paper investigates the selection of the neighbourhood and studies how the level of added noise in the image affects the selection of an optimal neighbourhood. To enhance the performance of noise removal using Cellular Automata, a basic totalistic CA (BTCA) model and a new weighted totalistic CA (WTCA) model are introduced. Both methods require much less memory storage and are feasible in practice even for very large neighbourhoods. Several experiments are presented to demonstrate that both proposed methods produce consistently better performance than the median filter and the traditional CA method for low noise levels, and for filtering at high noise levels, the WTCA model is shown an excellent performance compared to other methods. Keywords: cellular automata, image processing, neighbourhood, noise filtering, totalistic, identification 1 INTRODUCTION Cellular automata (CA) are a class of spatially and temporally discrete mathematical systems characterized by local interactions. A cellular automata is composed of three parts: a neighbourhood, a local transition rule and a discrete lattice structure. The local transition rule updates all cells synchronously 207

2 208 Y. ZHAO et al. by assigning to each cell, at a given step, a value that depends only on the neighbourhood. Because of its simple mathematical constructs and distinguishing features, CA have been widely used to model aspects of advanced computation,evolutionary computation, and for simulating a wide variety of complex systems in the real world [8, 4, 3, 17]. Based on the properties of the transition rule, CA can be classified into two types: deterministic CA (DCA) and probabilistic CA (PCA). The transition rule for DCA is deterministic while that for PCA is statistical because some state flipping of cell values occurs during the evolution. Most traditional image processing methods, based on a single iteration, can be categorised as DCA, for example edge detection, smoothing and sharpening etc, where only one deterministic rule can completely represent the evolution. Recently, many impressive results relating to DCA in image processing have been published. Chen [5] introduced a DCA model to represent the procedure of edge detection, which can then be applied in medical images. Hrebien [7] proposed a CA model to segment the medical images based on region growing. A DCA model was introduced by Zhao [15] to compress images. Some image processing methods, which are based on multi-iterations, can be categorised as PCA, such as noise filtering, thinning and convex hulls, where the representation of the evolution is not exclusive and there is always a compromise between computational time and model accuracy. Due to the presence of noise, it is much more challenging to identify the rule of a PCA than it is for that of a DCA. Billings [2, 16] have published some methods for the identification of PCA, which has been shown to be effective in many applications. Rosin [12, 13] has developed several schemes to train the best CA rule set to represent the operation of noise filtering, thinning and convex hulls for black and white and gray scale images. The results show that the detected CA rule set based on the Moore neighbourhood can significantly improve the performance of noise removal compared to the median filter. In the present paper, two approaches to represent the evolution of the filter, based on a basic totalistic CA model and a new weighted totalistic CA model to remove noise in image processing are introduced. The main contribution of the paper is to introduce a relatively simple class of CA model along with an opening neighbourhood selection, which can not only improve the performance of noise filtering but also enhance the feasibility of a CA model with a large neighbourhood, and significantly reduce the computational time compared to traditional methods. Removing salt and pepper noise is a very important topic in image processing, because this kind of noise often occurs in the real world. Many methods have been developed specifically for this kind of noise. In this paper, all experiments used salt and pepper noise. The paper is organized as follows. The selection of the lattice type and neighbourhood is discussed in Section 2. Section 3.1 proposes a basically totalistic CA model with an equally-weighted neighbourhood and corresponding identification routine. A weighted totalistic CA model is then introduced in

3 APPLICATION OF TOTALISTIC CA FOR NOISE FILTERING IN IMAGE PROCESSING 209 Section 3.2 to further improve the performance. Several examples are demonstrated to evaluate the effectiveness of both methods in Section 4. Finally, conclusions are given in Section 5. 2 LATTICE TYPE AND NEIGHBOURHOOD A CA consists of three components: a discrete lattice, a finite neighbourhood R, and a transition rule. The commonly used lattice types in CA are square, triangular and hexagonal lattices, as illustrated by Figure 1. In terms of the alignment of pixels in a 2-D image, the square lattice is the natural selection for a CA model applied in image processing. However, for filtering large blocks of noise in a high resolution image, it is promising to use other types of lattice, which may require further study. Many investigators have studied CA systems based on a triangular lattice or a hexagonal lattice [10, 11] and some impressive results have been achieved. In this paper, only the square lattice is considered. The neighbourhood of a cell is the set of cells in both the spatial and temporal dimensions that are directly involved in the evolution of the cell. In previous research on system modelling using CA, the neighbourhood is always guessed or assumed as one of a small set before identification of the rule. However, many studies have revealed that different kinds of neighbourhood may have significant influence over the results. Hence, in this paper different types of neighbourhood are considered and the effects of the selection of the neighbourhood on the performance of the CA filter are studied. We start with the definition of a cellular space in which the automata is defined. A regular lattice L R d consists of a set of cells (pixels in image), which homogenously cover a d-dimensional Euclidean space. Each cell is labelled by its position p L. The spatial arrangement of the cells is specified by the nearest neighbour connections, which are usually in a regular arrangement. For any cell p L, its neighbourhood R b (p) is a finite set of neighbouring cells and is defined as R b (p) = {p + c i : c i R d, i = 1,...,b} (1) (a) (b) (c) FIGURE 1 Typical lattice types of CA. (a) square lattice; (b) triangular lattice; (c) hexagonal lattice

4 210 Y. ZHAO et al. where b is the number of nearest neighbours on the lattice. Commonly used neighbourhoods in a 2-dimensional CA on a square lattice are the von Neumann neighbourhood (illustrated in Figure 2.(a)), expressed as R 4 (p) = {(p x, p y ) + (c x, c y ):(c x, c y ) {(1, 0), (0, 1), ( 1, 0), (0, 1)}} where p x, p y denote the coordinate position of p, and the Moore neighbourhood (illustrated in Figure 2.(b)), expressed as R 8 (p) = {(p x, p y ) + (c x, c y ):(c x, c y ) {(1, 0), (1, 1), (0, 1), ( 1, 1), ( 1, 0), ( 1, 1), (0, 1), (1, 1)}} where the cell p itself is not normally included in the neighbourhood, although in some applications it is included [6]. In most applications, the neighbourhood is chosen to be symmetrical as it is easy to be understood, but asymmetrical neighbourhoods have also been studied in some applications, for example, in modelling crystal growth using CA [9]. Larger symmetrical neighbourhoods are also considered in this paper and they are defined as the r-radial: R b (p) = {(p x, p y ) + (c x, c y ):c x, c y [ r, r] c x2 + c y2 r} and the r axial neighbourhood R b (p) = {(p x, p y ) + (c x, c y ):c x, c y [ r, r]} The von Neumann and Moore neighbourhoods are special cases of these when r = 1. Figure 2.(c) and (d) illustrate the 2-radial and 2-axial neighbourhoods respectively, where the gray cells denote the neighbourhood of the central white cell. (a) (b) (c) (d) FIGURE 2 Four types of neighbourhood studied in this paper. (a) von Neumann; (b) Moore; (c) 2-radial; (d) 2-axial

5 APPLICATION OF TOTALISTIC CA FOR NOISE FILTERING IN IMAGE PROCESSING METHODS 3.1 Basic Totalistic CA Rule and Identification Each cell p L on the lattice is assigned a state value s(p) ε chosen from the finite set of elementary states ε. For the binary black and white image discussed in this paper, only two states are available in the set, expressed as ε ={0, 1}. A totalistic cellular automata, introduced by Wolfram[14], is a cellular automata in which the rules depend only on the total (or equivalently, the average) of the values of the cells in a neighbourhood. Considering the application of filtering salt and pepper noise in a black and white image, this paper proposes a basic totalistic cellular automata (BTCA) that can be defined as: Consider a pixel labelled as p ={(p x, p y )} in a black and white image at time t. The state value of p at the next time t + 1, s(p t+1 ), is inverted, only if the number of cells in its neighbourhood whose state value is the same as s(p t ) is not larger than a thresholding β. This also can be mathematically expressed as: { invert(s(pt )) F(R s(p t+1 ) = b (p t )) β s(p t ) else (2) where two components are required to be determined: the neighbourhood R b (p t ) and the thresholding β. F( ) denotes the operation to count the number of the cells in the neighbourhood whose values are the same as s(p t ). The structure of this model is relatively simple as there is only one parameter which has to be stored to represent a rule if the neighbourhood is fixed. This in turn can considerably reduce the memory requirements and makes it feasible to run this algorithm in practice for a very large neighbourhood. Furthermore, the number of possible values of β is limited at b + 1, where b is the number of cells in neighbourhood, which indicates that the computational complexity for the best rule based on this model, is reduced and the search strategy for the best rule can be relatively simple. Consider the Moore neighbourhood for example, there are only 9 possible values of β : β {0,1,2,...,8} available to be chosen. For the traditional CA model, there are 2 8 = 256 possible rules. Consider a large neighbourhood, for example, 2-axial neighbourhood, there are 2 24 = possible rules, but for the presented model, the number of possible rules is reduced to only 25. Hence, this model can simplify the search strategy by significantly reducing the number of possible rules. The following part discusses the performance of this model in noise removal. The root mean-square (RMS) error is employed in this paper as the objective function. As discussed in Rosin [12], more than one iteration of the CA rule may be required to achieve the optimal RMS error (reaching convergence) for a specific neighbourhood and β. The number of required iterations

6 212 Y. ZHAO et al. depends on the percentage of the noise, the characteristics of the original image and the class of CA model used. Let J denote the calculated RMS error for a selected neighbourhood R b and thresholding β. The identification of the BTCA model can be described as follows and illustrated by Figure Select a neighbourhood from the pool of the candidate neighbourhoods. 2. Assign the β with a value chosen from 0, 1, 2...,b. 3. Iteratively implement the equal-weighted totalistic cellular automata until the optimal J is found. 4. Repeat step 2 until all possible values of β are scanned and store the value of β with the optimal J. 5. Repeat step 1 until all possible neighbourhoods are scanned and the best CA rule is defined as the combination of neighbourhood and β with the optimal J. The complexity of the scheme depends on the number of candidate neighbourhoods, the size of each neighbourhood and the size of the considered image. From the system identification point of view, the size of neighbourhood determines the rule complexity. Theoretically, a BTCA model with a larger neighbourhood can always produce less RMS for the sampled image because it has more number of input patterns to represent a more complicate map of inputs and outputs. However, if the identified model is used in other images with the same level of noise, the generated RMS is often not optimal because the model can be overestimated, which will be demonstrated in the examples of this paper. That is the reason why a parsimonious model is always pursued in system identification. FIGURE 3 The block diagram for the identification of the BTCA model.

7 APPLICATION OF TOTALISTIC CA FOR NOISE FILTERING IN IMAGE PROCESSING Weighted Totalistic CA Rule and Identification The members of the neighbourhood in a BTCA model make the same contributions to calculate F( ). The advantages of such a method are the relatively simple model produced and the less computational complexity, which is often vital for a very large neighbourhood. However, in the field of spatio-temporal systems, a few publications have recently suggested that the contributions of each cell in a neighbourhood to the evolution of the center cell can be inconsistent [1, 18], which inspires this paper to introduce a weighted totalistic CA (WTCA) model by considering the spatial inconsistency of cells in the neighbourhood. The weight coefficient of each cell in the neighbourhood is inversely proportional to the distance from this cell to the center cell and symmetrical to the central cell. Figure 4 illustrates the weight coefficients for four types of neighbourhood discussed in this paper. A description of the rule can still be represented by Equation (2), but F( ) has to be replaced by F( ) = b i=1 w id i { 1 s(ci ) = s(p) d i = 0 else (3) where c i R b (p) denotes the state value of the cell in the neighbourhood and w i denotes the corresponding weight coefficient. The number of possible choices of β, denoted by n, is significantly increased compared to the BTCA model, but this is still much less than the number of possible rules of the traditional CA model, as shown in Table 1. Note the value of β does not have to be integer any more. The search strategy of the WTCA model is the same as that of BTCA model. However, an effective way to reduce the computational time associated with a large neighbourhood is to fix the increment step of β, to produce an approximate value of the best β. Because the accuracy of the approximated β depends on the increment step, there is a clear tradeoff between computation complexity and model accuracy. (a) von Neumann (b) Moore (c) 2-radial (d) 2-axial FIGURE 4 Weight coefficients of four types of neighbourhood for the weighted totalistic CA model

8 214 Y. ZHAO et al. n Neighbourhood CA BTCA WTCA von Neumann Moore radial axial TABLE 1 Comparison of the numbers of possible rules for different neighbourhoods for the traditional legal CA model, BTCA model and WTCA model 4 EXPERIMENTAL RESULTS 4.1 Experiment 1: Identification of a BTCA model The purpose of the first experiment is to demonstrate the identification of a BTCA model and then inspect if the identified BTCA models from several corrupted images that have the same noise level and the same original image are consistent or not. The size of the tested images is pixels, and each of them was corrupted by salt and pepper noise with 0. 3 probability. This experiment tested three groups of data that have different original images, which are shown in Figure 5. The candidate neighbourhoods consist of the: von Neumann, Moore, 2-radial and 2-axial neighbourhoods. As shown in Table 2, for a specific tested group, the combination of 2-radial neighbourhood and β = 3 consistently forms the best BTCA model in term of the RMS errors. Although it has slightly different selections of β for 2-axial neighbourhood, the final model is not affected. Moreover, for the three tested groups, the detected rules are also consistent, which indicates that under noise level p = 0. 3 of single salt and pepper noise, the tested three original images have the identical best BTCA models to represent the noise filter, and which (a) Lena (b) Camera (c) man FIGURE 5 Three original black and white images for Experiment 1, where each pixel takes only two possible values :0or1.

9 APPLICATION OF TOTALISTIC CA FOR NOISE FILTERING IN IMAGE PROCESSING 215 test1 test2 test3 test4 Group R b J β J β J β J β von Neu Lena Moore radial axial von Neu Camera Moore radial axial von Neu man Moore radial axial TABLE 2 RMS Errors after filtering of the three groups of test image corrupted by salt and pepper noise with noise level The parameter β was scanned from 0 to b and R b was chosen from the pool of candidate neighbourhoods. can be expressed as { invert(s(pt )) F(R s(p t+1 ) = b (p t )) 3 s(p t ) else (4) where R b is chosen as 2-radial neighbourhood. Lots of experiments have been implemented for different noise levels and the results are exactly the same, which shows that the detected local BTCA model from a training image can optimally represent the evolution rule for the other two groups of image. This paper can not guarantee the detected model is globally optimal for all images with the same noise level, but it is definitely optimal for the tested groups in this experiment. There is always a compromise between performance and commonality of the model for this application. Figure 6 illustrates the trends of RMS error by β for four types of neighbourhood. The data were averaged after evaluating 4 cases of the same noise level. It shows that the result of a large neighbourhood is a high resolution version of that of a small neighbourhood with the same main trend, but provides more chance to find the better rule. The performance of this model compared to previous methods will be discussed in the next section. It can be noticed in Table 2 that the larger neighbourhood does not always have a better performance than that of the smaller one. For example, the 2-axial neighbourhood has worse results compared to a 2-radial neighbourhood in this example. One of the reasons is that the members of the neighbourhood have the same weight in this algorithm, which neglects spatial information of each member. This is

10 216 Y. ZHAO et al. (a) von Neumann (b) Moore (c) 2-radial (d) 2-axial FIGURE 6 Trends of RMS error against β after filtering of the three groups of test image corrupted by single salt and pepper noise with noise level 0. 3 a byproduct of the BTCA model because the spatial information plays a very important rule in noise removal, especially for a high noise level. Another reason, as discussed above, using a too large neighbourhood can produce a non-parsimonious model, which has a better performance for the sampled data, but always worse performance for other test images. 4.2 Experiment 2: Identification of a WTCA model The purpose of the second experiment is to demonstrate the utilization of the WTCA model in image noise filtering. The original image lena with size pixels was employed and was corrupted by noise at the levels 0. 02, 0. 2 and 0. 6 respectively. Figure 7 illustrates the results of the RMS error over β using a WTCA model. The data were averaged after evaluating 4 cases of the same noise level. The valleys of the trend indicate the position of optimal β for a specific neighbourhood. These results also show that the WTCA model can generate a smoother trend in the RMS error, compared to the BTCA model as more values of β are sampled, which indicates it has more chance to capture a better value of β.

11 APPLICATION OF TOTALISTIC CA FOR NOISE FILTERING IN IMAGE PROCESSING 217 (a) von Neumann (b) Moore (c) 2-radial (d) 2-axial FIGURE 7 Trends of RMS error against β based on the WTCA model with different neighbourhoods and noise levels 4.3 Experiment 3: Performance comparison To evaluate the performance of the WCTA and the BTCA model, in the second experiment, two large black and white images ( pixels) were constructed, one for rule identification and one for testing. Both images consisted of a composite of several 256 sub-images that have different characteristics. This experiment compared the results of three versions of CA: the WTCA model, the BTCA model, the traditional CA model [12] and 3 3 median filter (MF) at different noise levels. Three groups of experiments were repeated for each noise level. The optimal parameters (neighbourhood, β and number of iterations to reach convergence) were determined from the training image and the RMS errors were calculated by applying the detected models in the testing image. The results are shown in Table 3. At low noise levels (p=0.01), both the BTCA model and the WTCA model determined the 2-axial neighbourhood as the optimal neighbourhood but with different values of optimal parameter β. The three CA methods generated considerably better RMS error than that of the median filter, which proves again the importance of utilization of CA in image processing. The BTCA model produces even better results

12 218 Y. ZHAO et al. noise orgi. MF CA BTCA WTCA J J J J rule J rule (1) 14.16(1) 13.06(1) 12.65(2) (1) 14.40(1) 13.35(1) 2-axial(0) 12.87(2) 2-axial(0.45) (1) 14.26(1) 13.16(1) 12.68(1) (1) 31.74(2) 31.15(1) 30.91(3) (1) 31.70(2) 31.11(1) 2-radial(2) 30.97(3) 2-radial(1.25) (1) 31.92(2) 31.32(1) 31.09(3) (2) 47.40(5) 48.31(2) 47.17(2) (2) 47.40(5) 48.25(3) 2-radial(3) 47.25(2) 2-radial(2.95) (2) 47.46(5) 48.25(3) 47.26(2) TABLE 3 Comparison of RMS errors after filtering of the test images corrupted by salt and pepper noise using the different methods. The criteria for all models is expressed as Optimal RMS error(the number of iterations to reach convergence) and the detected totalistic CA rule is described as neighbourhood(β). than the traditional CA model, mainly due to the larger neighbourhood. The WTCA model produces the best results as it detected a more accurate value of β = As shown in the table, the results of three experiments repeated at this noise level are consistent. At greater noise levels, the three versions of CA continue to perform better than the median filter. At p = 0. 1, both the BTCA model and the WTCA model determined the 2-radial neighbourhood as the optimal neighbourhood but with different values of optimal β. The BTCA model has the better RMS error than that of the traditional CA model. Moreover, the BTCA model only requires one iteration to reach convergence compared to 2 iterations of the traditional CA model. The WTCA model continuously has the best RMS errors, but it needs 3 iterations to reach convergence. At p = 0. 3, the BTCA model has worse RMS errors than the traditional CA model. Due to the high noise level, the ideal CA rule to represent the filter is relatively complicated. The disadvantage of the simplicity of the BTCA model was prominent because the accuracy to approximate the ideal CA rule is out of the acceptable range. The WTCA model consistently has the the best performance with optimal RMS errors and the least number of iterations. For the WTCA model, the computational time has to be scarified compared to with the BTCA model. It takes 59. 6s to train the rule for a black and white image with resolution pixels with a noise level 0. 2 on a 2.27GHz Intel Xeon PC with programs coded in C++ for an image with such high resolution. 4.4 Experiment 4: model vs noise level The third experiment aims to learn the interaction between the noise level and the corresponding optimal neighbourhood and β. The sampled image

13 APPLICATION OF TOTALISTIC CA FOR NOISE FILTERING IN IMAGE PROCESSING 219 Noise level detected R Noise level detected R axial radial axial radial axial radial radial radial radial radial radial axial radial axial TABLE 4 List of detected optimal neighbourhoods for different levels of noise based on the WTCA model for Experiment 4. was Camera with a resolution pixels. Starting from , the noise level was increased to 0. 9, and the detected neighbourhoods based on the WTCA model are shown in Table 4, where data were averaged for 4 cases for the same noise level. It has been found that a WTCA model with a large neighbourhood has better performance in filtering very low level noise. Following the increment of noise level, the size of the detected optimal neighbourhood becomes smaller and smaller. At noise levels between to 0. 5, the 2-radial neighbourhood was reached. This is because the error produced by overestimation of a large neighbourhood becomes greater and greater than the contribution to noise filtering following the increment of the noise level. When the noise level is greater than 0. 5, a larger and larger neighbourhood is detected following the increment of noise level, which is because the contribution of noise filtering using a large neighbourhood is greater than the error introduced by overestimation. That is the reason why the size of the detected neighbourhood shows a U shape over the increment of noise level. Figure 8 shows the detected values of β against the noise level from to 0. 5, where all detected neighbourhoods are the 2-radial neighbourhood. It is expected that the value of β increases following the increment of the noise level, but the relationship is not linear. 5 CONCLUSIONS From a system identification point of view, this paper has discussed the selection of lattice types, neighbourhood and approaches of identification for two kinds of introduced totalistic CA models for noise filtering in image processing. Instead of focusing on one specific neighbourhood, this paper has investigated the selection of the neighbourhood by learning how the noise level affects the selection of the optimal neighbourhood. From the experimental results it has been found that for filtering salt and pepper noise, the

14 220 Y. ZHAO et al. FIGURE 8 Detected values of β against the noise level based on the WTCA model for Experiment 4. optimal neighbourhood varies for different levels of noise. At low noise levels, a large neighbourhood always produces better performance than a small neighbourhood, but at high noise levels, the results are the converse of this. The paper introduced a basic totalistic CA to simplify the traditional CA model. Lots of experiments have shown that this model requires much less computational time, for both training and testing, and also has consistently better performance than a median filter. For low noise levels, this method also produces better RMS errors and less number of iterations to reach convergence than the traditional CA model. For high noise levels, the disadvantage of simplicity may be prominent and this leads to a worse performance, but it is still an effective way to remove noise at low noise levels in image processing. To further improve the performance, this paper also presented a weighted totalistic CA which considers the spatial inconsistency of the contributions made by the members of the neighbourhood. The results showed that this model can not only produce consistently better RMS errors, but also consistently reduce the number of iterations to reach convergence, compared to the other two versions of CA. Although, the computational time has to be sacrificed in an acceptable range compared with the BTCA model this can be a worth effort because of the improvement in the RMS error. This indicates that the WTCA model is a high quality CA model for filtering salt and pepper noise at different levels. Some interesting observations have also been found by studying the interaction between the noise level and the size of the corresponding optimal neighbourhood. Automatic detection of the optimal neighbourhood before identifying the β can significantly reduce the computational complexity and this topic will be the subject of the further studies.

15 APPLICATION OF TOTALISTIC CA FOR NOISE FILTERING IN IMAGE PROCESSING 221 ACKNOWLEDGMENT The authors gratefully acknowledge that part of this work was financed by Engineering and Physical Sciences Research Council(EPSRC), UK, and by the European Research Council(ERC). REFERENCES [1] R Alonso-Sanz. (2004). One-dimensional r=2 cellular automata with memory. International Journal of Bifurcation and Chaos, 14(9): [2] S. A. Billings and Y. X. Yang. (2003). Identification of probabilistic cellular automata. IEEE Transactions on Systems, Man, and CyberneticsłPart B, 33(2): [3] L Bull and A Adamatzky. (2007). A learning classifier system approach to the identification of cellular automata. Journal Of Cellular Automata, 2(1): [4] C.A.Reiter. (2005). A local cellular model for snow crystal growth. Chaos Solutions & Fractals, 23: [5] Z. X. Chen. (2009). Cellular automata for edge detection: Exploring the nature of cellular automata through applications in medical images. Przeglad Elektrotechniczny, 85(4): [6] A. Deutsch and S. Dormann. (2005). Cellular Automaton Modeling of Biological Pattern Formation. Birkhauser. [7] M. Hrebien, P. Stec, and T. Nieczkowski. (2008). Segmentation of breast cancer fine needle biopsy cytological images. International Journal of Applied Mathematics and Computer Science, 18(2): [8] A. Ilachinski. (2001). Cellular Automata: A Discrete University. World Scientific, London. [9] P. Meankin. (2000). Fractal, Scaling and Growth Far From Equilibrium. Cambridge University Press. [10] K. Morita, N. Margenstern, and K. Imai. (1999). Universality of reversible hexagonal cellular automata. Rairo-Informatique Theorique et Applications-Theoretical Informatics and Applications, 33(6): [11] C. Pagnutti, M. Anand, and M. Azzouz. (2005). Lattice geometry, gap formation and scale invariance in forests. Journal of Theoretical Biology, 236(1): [12] P. L. Rosin. (2006). Training cellular automata for image processing. IEEE Transaction on Image Processing, 15(7): [13] P. L. Rosin. (2010). Image processing using 3-state cellular automata. Computer Vision and Image Understanding, 114: [14] S. Wolfram. (2002). A New Kind of Science. Wolfram Media, Inc. [15] C. F. Zhao, C. C. Shi, and P.K He. (2008). A cellular automaton for image compression. 4th International Conference on Natural Computation (ICNC 2008), 7: [16] Y. Zhao and S. A. Billings. (2006). Neighborhood detection using mutual information for the identification of cellular automata. IEEE Transactions on Systems, Man, and CyberneticsłPart B: Cybernetics, 36(2): [17] Y. Zhao and S. A. Billings. (2007). The identification of cellular automata. Journal Of Cellular Automata, 2(1): [18] Q. Zhuang, B. Jia, and X. G. Li. (2009). A modified weighted probabilistic cellular automaton traffic flow model. Chinese Physics B, 18(8):