A GENETIC C-MEANS CLUSTERING ALGORITHM APPLIED TO COLOR IMAGE QUANTIZATION

Transcription

1 A GENETIC C-MEANS CLUSTERING ALGORITHM APPLIED TO COLOR IMAGE QUANTIZATION P. Scheunders Vision Lab, Dept. of Physics, RUCA University of Antwerp, Groenenborgerlaan 171, 2020 Antwerpen, Belgium Abstract: This paper describes a novel data clustering algorithm, which is a hybrid approach combining a genetic algorithm with the classical c-means clustering algorithm (CMA). The proposed technique is superior to CMA in the sense that it converges to a nearby global optimum rather than a local one. As an application the problem of color image quantization is elaborated. Here, it is shown that substantial improvement of image quality is obtained by using the genetic approach. KEY WORDS : Genetic algorithm, Color image quantization, C-means clustering algorithm, Global optimization 1

2 1. Introduction The classical c-means clustering algorithm (CMA) is a well-known clustering technique in the field of pattern recognition (1). It is an iterative scheme which starts from an initial distribution of cluster centers in dataspace. Each datapoint is assigned to the cluster with closest cluster center, after which each cluster center is updated as the center of mass of all datapoints belonging to that particular cluster. This procedure is repeated until convergence. As is well known but neglected by many researchers is that CMA is very much dependent on the choice of the initial distribution of cluster centers. In this way, the algorithm ends up in a local optimum, which in real life applications can be far away from the real global optimum, especially when a large number of datapoints and clusters is involved. In previous work the influence of initial conditions on CMA was studied on the problem of gray-level image quantization (2). There the distribution of locally optimal solutions was reasonably well localized and results were visibly not distinguishable, mainly due to the low sensitivity of the human eye to gray-level variations. In this paper the problem of color image quantization is discussed. Color image quantization is of greater practical importance than gray-level quantization, because : i) Color images provide additional information in the image compared to monochrome images. ii) the human vision system is much more sensitive to small differences in color than in intensity. 2

3 iii) many low-cost color display and printing devices are restricted to a small number of colors that may be displayed or printed simultaneously. Color image quantization consists of two steps: palette design, in which a reduced number of palette colors (typically 6-256) is specified, and pixel mapping in which each color pixel is assigned to one of the colors in the palette. Several techniques exist for color image quantization. First, there is the class of splitting algorithms that divide the color space into disjoint regions. From each region a color is chosen to represent the region in the color palette. will describe two algorithms of this class which are commonly used, and which were used in this work for comparison. The median-cut algorithm (MCA) (3) divides the color space into rectangular boxes. At each step the longest box is split along the axis with largest range (where the range along an axis is defined as the difference between maximum and minimum color values). The splitting plane passes through the median of all color values in the given box along that axis. The variance-based algorithm (VBA) (4) also divides the color space into rectangular boxes. The splitting criterium is given by the mean squared error (MSE) between all the color values within the box and their center of mass. The box with largest MSE is chosen to be split. The MSE is projected along the three axisses and on all positions the reduction in projected MSE is calculated. The axis that contains the position of largest reduction in projected MSE is chosen to be split at that position. Another class of quantization techniques performs clustering of the color space. The most frequently used clustering algorithm applied to color image quantization is CMA (5,6), but also a fuzzy c-means clustering algorithm (7), and a hierarchical merging approach (8) were described. 3

4 CMA is commonly accepted as an optimal quantization approach, but also known as a very time consuming approach. In general, to test the performance of a quantization algorithm, the resulting color palette is used as an initial palette for CMA, and further improvement of the result is a measure for the performance of the particular algorithm. In this work, the statement that CMA is an optimal color image quantization algorithm is critically investigated, by examining its dependence on the initial conditions. Color image quantization is a complex data clustering problem due to its three dimensional color space. Therefore the distribution of local optima is expected to be broad. This effect is demonstrated by studying the behaviour of CMA with respect to randomly chosen initial conditions. Furthermore, because of the high sensitivity of humans to colors, visual image quality is expected to be affected by the local optimality. A way to deal with local optimality is to use stochastic optimization schemes. In this paper a quantizer, based on a genetic algorithm (GA), is defined which is more insensitive to initial conditions. Invented in the early 70 s (9), GA s became more and more popular over the last years. A GA is inspired by biological evolution, and is widely believed to be an effective global optimization algorithm. A genetic algorithm consists of a population of genetic strings, which are evaluated using a fitness function. The fittest strings are then regenerated at the expense of the others. Furthermore genetic operations such as crossover and mutation are defined. The mutation operator changes individual elements of a string, the crossover operation interchanges parts between strings. The combination of these operations is then repeated during several generations. The intrinsic parallelism of a genetic algorithm, i.e. the ability of manipulating in parallel large numbers of strings, and the crossover operation 4

5 whereby good portions of different strings are combined both make the technique a very effective optimization technique (10). The usefulness of GA s in pattern recognition and image processing has been demonstrated (11). In the field of image segmentation, genetic algorithms were already introduced (12,13,14). Our approach is a hybrid structure between GA and optimal quantization and will henceforth be referred to as genetic c-means clustering algorithm (GCMA). The performance of GCMA is compared to that of the splitting algorithms MCA and VBA, and to that of CMA where the result of the splitting algorithms is used as initial condition. It is demonstrated that the visual quality is improved by using GCMA. The outline of the paper is as follows. In the next section the technique of GCMA will be elaborated in detail. In section 3, the problem of color image quantization is explained. Several experiments are discussed in section The genetic c-means clustering algorithm The classical CMA is a clustering algorithm which optimizes an objective function based on a measure between the data points and the cluster centers in the data space. When M data points in a N-dimensional space are clustered into C clusters, the objective function can be σ 1 M 2 C = i=1 x C i ( x - vi ) 2 chosen as: 5

6 where x is a N-dimensional vector containing the coordinates of datapoint x and v i the N- dimensional vector containing the coordinates of the center of cluster i, C i is the collection of data points belonging to cluster i. This objective function describes the mean squared error (MSE) when replacing each data point by the center of the cluster to which it belongs. v i = x C i x C i x 1 Minimizing the objective function with respect to v i leads to the following conditions : i.e. the cluster centers are positioned at the center of mass of the data points belonging to the cluster. The second set of conditions that minimizes the MSE states that a data point should be associated with the closest cluster center. 2 x C j _ j = MINi [( x - vi ) ] A set of cluster centers which satisfies (2) and (3), satisfies minimal objective function conditions and is called a local optimum. One way to satisfy both sets of equations is by using CMA. Here, one starts with an initial set of cluster centers {v 1,..., v C }. The cluster centers can be chosen randomly, or as the output of another procedure, e.g. a splitting procedure. Each datapoint is assigned to the cluster with closest cluster center using (3). The position of the cluster centers is then updated using (2). This procedure is repeated until convergence. This iterative scheme is known to converge sufficiently fast. However more than one solution satisfies (2) and (3). Depending on the initial choice of cluster centers, the algorithm ends up in a local optimum that can be far away from the global one. In this paper the problem of local optima is solved by developing a hybrid algorithm combining CMA and a genetic approach. When using genetic algorithms, a population of 6

7 genetic strings (binary or other) is generated. A fitness function is defined which assigns to each string a value of fitness. Through genetic evolution, strings regenerate, mutate and interchange information. Since these genetic operations favorize strings with high fitness value, the population evolves to an optimal one. A genetic design for clustering can be constructed in the following way : - The number of clusters is fixed to a value C. - A population of P random strings is generated. Each string r represents a set of C cluster centers {v 1,..., v C } r. Notice that these strings are strings of vectors, the elements of which are the coordinates of a cluster center in dataspace. In practice, each coordinate is put into a separate string, so that each vector string is in fact a structure of several component strings: {v 1,..., v C } = ({v 1 1,..., v C 1 }, {v 1 2,..., v C 2 },..., {v 1 N.., v C N }). Henceforth we will refer to the vector string as string. All operations on a string are in fact performed on each component string separately. Notice also that the strings are not binary strings, as is the case in many genetic algorithm applications. Our strings contain integer or floating point values. Three genetic operators are defined: Regeneration: On each string of the population MSE is calculated using (1). The inverse of the MSE is used as fitness function. All strings are pairwise compared. The string with lowest MSE is copied into the other. Crossover: On each string { v 1,..., v C } one-point crossover is applied with probability P c (P c =0.8) (i.e. a uniform random number r between 0 and 1 is generated and crossover is applied if r < P c ). Out of the total population a partner string { v 1,..., v C } is randomly chosen 7

8 (i.e. a uniform integer random number i between 1 and P is generated, and the string i is chosen as a partner string). Then an integer random number j between 1 and C is generated. Both strings are cut in two portions at position j and the portions {v j+1,..., v C } are mutually interchanged, i.e.: {v 1,..., v C } Ð {v 1,..., v j, v j+1,..., v C } {v 1,..., v C } Ð {v 1,..., v j, v j+1,...,v C } Mutation: Mutation is performed on each element v j of each string with a small probability P m (P m =0.05). From v j one of the N component is chosen at random. Then a random number, taking the binary values -1 or 1, is generated, and is added to the chosen component. The total of these three operations is called a generation. After a generation, the string with lowest MSE is stored. Then the next generation is performed. If after the next generation a string with lower MSE is created, it replaces the stored one. The string which is stored after G generations is chosen as the optimal result. After applying several generations, the stored string approaches an optimum. For problems with high dimensionality, large populations may have to be defined, and a large number of generations may be necessary before the system converges. To reduce the search space drastically, the genetic approach is combined with CMA. Hereby CMA is applied to all strings of the population during each generation, before the regeneration step. By doing this, all strings are forced into local optima. This procedure will be referred to as the genetic c-means algorithm (GCMA). 8

9 It is clear that the described algorithm is more computer power demanding than CMA. CMA is applied PxG times. A faster alternative approach applies CMA during each generation only on the string with lowest MSE, hereby reducing the complexity to O(G). On the other strings of the population only a rude approximation of CMA is applied. This approximation can be e.g. one iteration of CMA. We propose to use the following fast randomized version of CMA. A small subset of datapoints is chosen randomly, after which one iteration step of CMA is applied on this subset, i.e. each datapoint of the subset is assigned to its closest cluster center after which the cluster centers are updated as the centers of mass of the datapoints of the subset which are assigned to the cluster. This procedure is repeated for several times using different subsets. Typical values are 10 subsets of a few hundred datapoints. The result of this randomized approximation of CMA leads to suboptimal results which are sufficient for the population to produce useful information to interchange and to allow for the production of a more optimal solution. The complexity of the genetic operators remains of O(PxG), but they are extremely suitable for parallel processing. Moreover the number of iterations, required for CMA to converge, decreases gradually with the number of generations, because strings become fitter. All this limits the execution time of GCMA. On sequential computers, a run of GCMA on P strings during G generations is about 10 times faster than PxG independent runs of CMA. 3. Color image quantization CMA and GCMA are applied to the problem of color image quantization. The aim is to 9

10 reduce the total amount of colors of an RGB-image ((256) 3 ) to an appropriate amount for display or printing devices (typically between 16 and 256). Other color coordinate systems are known, which are more suited for interpretation by the human visual system, but this falls outside the scope of the proposed work. Color space is filled by the occurence frequencies of colors which define a three dimensional histogram. The complete histogram is too large to fit in main memory of most computers. A frequently used preprocessing step is reducing the color resolution of the image to 5 bits for each color component, hereby reducing the image histogram to 32x32x32 elements. The use of the histogram is necessary for splitting algorithms as MCA and VBA. Altough this reduction is commonly known as sufficient for most color image quantization problems, it is a restriction which should not necessarily be taken using CMA and GCMA. Instead of scanning the histogram, the color image itself can be scanned during the clustering procedure. As most colors do appear not at all or only once in a typical color image, this procedure is even less time consuming. CMA is applied as follows: i) An initial set of C quantized colors is generated. The set can be randomly chosen over the color space, or it can be a result from other quantization procedures, as MCA and VBA. Each quantized color consists of it red, green and blue component, and has the meaning of a cluster center in color space. The quantized colors are stored into a color. ii) Equation (3) is applied, i.e. each pixel color is assigned to the closest palette color. A palette image is constructed, with the size of the color image. For all pixels, the mean squared 10

11 distance between the pixel color and each palette color is calculated. The palette color with lowest distance is then stored into the palette image, at the current pixel position. iii) Equation (2) is applied. Each palette color is updated as the center of mass of all pixel colors which were assigned to it, i.e which have a palette image value equal to the palette color. ii) and iii) are applied alternatively until convergence. The resulting color palette is fed into the lookup table devices of the printing or display device. To display the quantized image, step ii) of the procedure is repeated which generates the palette image which can be used to find the corresponding palette colors, and to calculate the MSE. GCMA is applied as follow: A population of strings of color palettes is randomly generated. In the most time consuming algorithm, each string is fed into CMA separately, after which the genetic operators are applied on the resulting color palettes. This is repeated during G generations. In the less time consuming algorithm, for each string the randomized CMA is performed and an approximated MSE is calculated. The palette with lowest MSE is used as input for CMA. The genetic operators are applied on all strings of the population. This procedure is repeated during G generations. 4. Experiments and discussion 11

12 In this section several experiments are performed to demonstrate the performance of GCMA compared to that of the splitting algorithms and CMA. The images used are RGB color images of 512x512 pixels. A set of 5 commonly used images namely Lena, peppers, airplane, mandrill and sailboat are taken from a standard image library (e.g. WWW at site Images are used with full 8 bit/color or are reduced to 5bit/color. All images are quantized to 16, 32 and 64 colors. In the first experiment the dependence of CMA on the initial conditions is elaborated. Several strategies are possible to obtain an initial set {v 1,..., v C } for starting CMA. An obvious choice is a random initial set. Applying the quantizer on different initial sets independently, allows to study in a statistical way the influence of the initial conditions on the behaviour of CMA. A statistically representative number of initial sets is constructed and CMA is applied on each set independently. The distribution of obtained MSE s is a statistical representation of the distribution of local optima obtained by the quantizer. In figure 1a the distribution of results is shown for the middle 128x128 part of test image Lena with 8 bit/color using CMA with a reduction downto 16 colors. The result is shown after 5000 independent runs. On the x-axis tthe obtained MSE s and on the y-axis the frequency distribution of obtained results are shown. Adding results of more runs did not change the distribution anymore. In figure 1b the same is shown using GCMA on a population of 500 strings during 10 generations. The most time consuming approach was used, i.e. the approach where the total population undergoes CMA during each generation. After each generation the obtained values of MSE of all strings were accumulated in the histogram, such that in total a distribution of 5000 MSE values was obtained. The evolution 12

13 of the population after more than 10 generations did not change the distribution anymore. Using CMA, a broad distribution of local optima is observed. Results between best and worst case differed by more than 100%. The best result obtained using CMA was an MSE of 355 while the optimal result after GCMA lead to an MSE of 310. Using GCMA, more than 10% of the population ended up in the optimal result and more than 90% ended up in results which were better than the best result obtained by CMA. By repeating the experiment using GCMA, the optimal MSE of 310 was reproduced with numerical precision, which indicates that this was indeed the global optimum. In the second experiment, the performance of GCMA was quantitatively compared to that of the splitting algorithms MCA and VBA, and to that of CMA. Resulting color palettes from the splitting algorithms were used as initial conditions for CMA. Using GCMA, the less time consuming approach was chosen, i.e. CMA was applied only on the fittest string out of the population during each generation. On the other strings the randomized CMA was applied. A population of 20 strings was processed during 50 generations. The best result out of the population is shown. All results are shown in table 1, MSE s are displayed for the 5 test images, reduced to 16, 32 and 64 colors. All images were reduced on beforehand to 5 bit/color to allow to use the color histogram for the splitting algorithms. For comparison, in the last column GCMA was applied on the full 8bit/color images. \The first conclusion that can be drawn from the table is that VBA is superior to MCA. Secondly, substantial improvement is obtained by applying CMA on the results of the splitting algorithms. Furthermore, altough the final result is commonly accepted to be 13

14 optimal, a substantial improvement can be noticed after applying GCMA. Finally, when applying GCMA on the full 8bit/color images, a further improvement is noticed, which indicates that a prereduction to 5bit/color is not always sufficient for color image quantization. Finally a comparison was made between the different color image quantization algorithms with respect to image quality. Therefore the image Lena was quantized to 32 colors using the 5 described techniques on the reduced 5bit/color image: MCA before (MSE=1755) and after CMA (MSE=240), VBA before (MSE=304) and after CMA (MSE=217) and GCMA (MSE=201). The resulting images are shown in figure 2. As can be noticed, MCA leads to bad image quality while all others have reasonable quality. The image quality of the splitting algorithms improves after applying CMA. A further improvement is clearly visible when using GCMA. Differences are noticed especially in slow varying regions. The better the algorithm, the smoother transitions between adjacent colors become. For completeness the quantization results of the other four test images are shown in figure 3. The gray-level figures show the luminance (Y = R G B) after transformation to UCS scale. On the left the original images are displayed; on the right the images are quantized to 32 colors, using GCMA on the full 8 bit/color images. Only small visual differences are noticeable, which indicates that the use of such small color palettes can be sufficient for display or print purposes. 14

15 5. Conclusions In this paper a genetic c-means clustering algorithm (GCMA) is proposed which is a hybrid technique combining the c-means clustering algorithm (CMA) with a genetic algorithm. We applied the algorithm to the problem of color image quantization. It is shown that this algorithm is less sensitive to the initial conditions than CMA. Furthermore GCMA is compared to several classical color image quantization algorithms, and its performance is shown to outperform the others, an effect which affects the visual image quality. The measured effects of local optimality on optimal clustering techniques is not only of importance for color image quantization. The genetic approach can also lead to important improvements for color image segmentation techniques, where local constraints between neighbouring pixels are taken into account, or for data clustering techniques in general in the case where large amounts of multidimensional data are to be clustered. REFERENCES 1. J.T. Tou and R.C. Gonzalez, Pattern Recognition Principles, A. Wesley (1974). 2. P. Scheunders, A genetic Lloyd-Max image quantization algorithm; Pattern Recognition Letters, Vol. 17, 5, (1995). 3. P. Heckbert, Color image quantization for frame buffer display, Comput. Graphics 16 (3), (1982). 15

16 4. S.J. Wan, P. Prusinkiewicz and S.K.M. Wong, Variance-based color image quantization for frame buffer display, Color. Res. Appl. 15, (1990). 5. S.A. Shafer and T. Kanade, Color vision, Encyclopedia of artificial intelligence; S.C. Shapiro, D. Eckroth E DS., , Wiley, N.Y. (1987). 6. M. Celenk, A color clustering technique for image segmentation, Computer Vision, Graphics and Image Processing 52, (1990). 7. R. Balasubramanian and J.P. Allebach, A new approach to palette selection for color images, J. Imag. Technol. 17, (1990). 8. Y.W. Lim and S.U. Lee, On the color image segmentation algorithm based on the thresholding and the fuzzy c-means techniques, Pattern Recognition Vol. 23, nr. 9 (1990). 9. J.H. Holland, Adaption in natural and artificial systems, Univ. Of Michigan Press (1975). 10. L. Davis, Handbook of genetic algorithms, Van Nostrand Reinhold, N.Y. (1991). 11. J.T. Alander, An indexed bibliography of genetic algorithms, Year , Techn. Report 94-1 Univ. of Vaasa, Dept of Information Technology and Production Economics, Vaasa Finland (1994). 16

17 12. P. Andrey and P. Tarroux, Unsupervised image segmentation using a distributed genetic algorithm, Pattern Recognition Vol 27, No. 5, (1994). 13. B. Bhanu, S. Lee and J. Ming, Self-optimizing image segmentation system using a genetic algorithm, Proc. Fourth Internat. Conf. on Genetic Algorithms, (1991). 14. S. Cagnoni, A.B. Dobrzeniecki, J.C. Yanch and R. Poli, Interactive segmentation of multi-dimensional medical data with contour-based application of genetic algorithms, Proc. First IEEE Internat. Conf. on Image Processing, Austin, Texas Vol.III, (1994). 17

18 Figure Captions Fig. 1a: Distribution of local optima quantizing part of Lena to 16 colors using CMA. Here, results of 5000 independent runs are displayed in a histogram. Fig. 1b: The same using GCMA with a population of 500 strings after 10 generations. All 5000 intermediate results are shown in the histogram. Table 1: Comparison of the performance of GCMA compared to other techniques. Results of the first 5 columns are from prequantized images to 5 bit/color. Column 1: MCA; 2: CMA using the result of column 1 as initial conditions; 3: VBA; 4: CMA using the result of column 3 as initial conditions; 5: GCMA; 6: GCMA using full 8 bit/color images. Fig. 2: Comparison of the visual quality of Lena after quantization to 32 colors; 2a: original; 2b: using MCA; 2c: using CMA with the result of 2b as initial conditions; 2d: using VBA; 2e: using CMA with the result of 2d as initial conditions; 2f: using GCMA. Fig. 3: Quantization results for the other four test images. The image luminance is displayed. Left: originals; right: quantized to 32 colors, using GCMA on the full 8bit/color images.

19 Figure 1a:

20 Figure 1b:

21 Table 1: Image C Median-cut Variance-based GCMA GCMA before after before after (8bit/col) Lena peppers airplane mandrill sailboat

22

23 Figure 2 a) b) c) d)

24 e) f)

25 Figure 3

26 Biographical sketch PAUL SCHEUNDERS graduated in physics at the University of Antwerp (UIA) in 1986, and received a Ph.D. in physics at the same university in 1990, with work in the field of statistical mechanics. Since 1991 he is a research associate at the University of Antwerp (RUCA) at the Vision Lab of the department of physics. He is currently working in the field of image processing and pattern recognition.

27 Summary: In this paper the problem of color image quantization is discussed. Color image quantization consists of two steps: palette design, in which a reduced number of palette colors (typically 6-256) is specified, and pixel mapping in which each color pixel is assigned to one of the colors in the palette. Several techniques exist for color image quantization. First, there is the class of splitting algorithms that divide the color space into disjoint regions. From each region a color is chosen to represent the region in the color palette. Two algorithms of this class are described which are commonly used, and which are used in this work for comparison: the median-cut algorithm (MCA) and the variance-based algorithm (VBA). Another class of quantization techniques performs clustering of the color space. The most commonly used clustering algorithm applied to color image quantization is the classical c- means clustering algorithm (CMA). CMA is a well-known clustering technique in the field of pattern recognition. It is an iterative scheme which starts from an initial distribution of cluster centers in dataspace. Each datapoint is assigned to the cluster with closest cluster center, after which each cluster center is updated as the center of mass of all datapoints belonging to that particular cluster. This procedure is repeated until convergence. As is well known but neglected by many researchers is that CMA is very much dependent on the choice of the initial distribution of cluster centers. In this way, the algorithm ends up in a local optimum, which in real life applications can be far from the real global optimum, especially when a large number of datapoints and clusters is involved. CMA is commonly accepted as the optimal quantization approach. In this work, this statement is critically investigated, by examining its dependence on the initial conditions. A way to deal with local optimality is to use stochastic optimization schemes. In this paper a quantizer, based on a genetic algorithm (GA), is defined which is less sensitive to initial conditions. Our approach is a hybrid structure between GA and clustering and will henceforth be referred to as genetic C-means clustering algorithm (GCMA). The performance of GCMA is compared to that of splitting algorithms MCA and VBA, and to that of CMA where the result of the splitting algorithms is used as initial condition. It is demonstrated that the visual quality is improved by using GCMA.