Color reduction using the combination of the kohonen self organized feature map and the gustafson-kessel fuzzy algorithm

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The color of the digital images is one of the most important components of the image processing research area. In many applications such as image segmentation, analysis, compression and transition, it is preferable to reduce the colors as much as possible. In this paper, a color clustering technique which is the combination of a neural network and a fuzzy algorithm is proposed. Initially, the Kohonen Self Organized Featured Map (KSOFM) is applied to the original image. Then, the KSOFM results are fed to the Gustafson-Kessel (GK) fuzzy clustering algorithm as starting values. Finally, the output classes of GK algorithm define the numbers of colors of which the image will be reduced.

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Color reduction using the combination of the kohonen self organized feature map and the gustafson-kessel fuzzy algorithm

  1. 1. DEMOCRITUS UNIVERSITY OF THRACE - GREECE 1 Color Reduction using the combination of the Kohonen Self- Organized Feature Map and the Gustafson-Kessel fuzzy algorithm Konstantinos Zagoris, Nikos Papamarkos and Ioannis Koustoudis Image Processing and Multimedia Laboratory Department of Electrical & Computer Engineering Democritus University of Thrace 67100 Xanthi, Greece Email: papamark@ee.duth.gr http://ipml.ee.duth.gr/~papamark/
  2. 2. DEMOCRITUS UNIVERSITY OF THRACE - GREECE 2 Problem definition • The main objective of this work is to propose a novel Color Clustering technique which is the combination of a neural network and a fuzzy algorithm. • Quantization of image colors is a very useful tool for segmentation, compression, presentation and transmission of images. • Reduction of the image’s colors to a small number (to its dominant colors) is important mainly for image segmentation (for example, segmentation of color documents).
  3. 3. DEMOCRITUS UNIVERSITY OF THRACE - GREECE 3 Fig. 1 Original image Fig. 2 RGB color distribution Fig. 3 Image with only 20 dominant colors Fig. 4 Distribution of 20 colors
  4. 4. DEMOCRITUS UNIVERSITY OF THRACE - GREECE 4 Color Reduction techniques • Several techniques have been proposed for color quantization which can be classified in the following main categories: – First, there is the class of splitting-merging algorithms that divide the color space into disjoint regions, by consecutive splitting up the color space. – Another class of quantization techniques consider the problem as a clustering approach. – Finally, there are general color segmentation techniques, which can be considered as color reduction algorithms.
  5. 5. DEMOCRITUS UNIVERSITY OF THRACE - GREECE 5 Splitting-merging algorithms • P. Heckbert, "Color image quantization for frame buffer display", Computer & Graphics, vol. 16, pp. 297-307, 1982. • S. Wan, S. Wong, and P. Prusinkiewicz, “An Algorithm for Multidimentional Data Clustering”, ACM Trans. Math. Softw., vol. 14 no. 2, pp. 153-162, June 1988. • I. Ashdown, "Octree color quantization", from the book: Radiosity-A Programmer's Perspective, Wiley, New York, 1994. Algorithms based on clustering • A.H. Dekker, "Kohonen neural networks for optimal color quantization", Network: Computation in Neural Systems, vol. 5, pp. 351-367, 1994. • N. Papamarkos, "Color reduction using local features and a SOFM neural network", Int. Journal of Imaging Systems and Technology, vol. 10, no 5, pp. 404-409, 1999. • N. Papamarkos, A. Atsalakis and C. Strouthopoulos, "Adaptive Color Reduction", IEEE Trans. On Systems, Man, and Cybernetics, IEEE Trans. on Systems, Man, and Cybernetics-Part B, vol. 32, no. 1, Feb. 2002. • A. Atsalakis, N. Papamarkos , N. Kroupis , D. Soudris and A. Thanailakis, "A Color Quantization Technique Based on Image Decomposition And Its Hardware Implementation", IEE Proceedings Vision, Image and Signal Processing, Vol. 151, Issue 6, pp. 511-524, 2004.
  6. 6. DEMOCRITUS UNIVERSITY OF THRACE - GREECE 6 General Color Segmentation Algorithms • Comaniciu, D., Meer, P.: Mean shift: a robust approach toward feature space analysis. IEEE Transactions on Pattern Analysis and Machine Intelligence 24 (5). (2002) 603–619.. • Nikolaou, N., Papamarkos, N.: Color segmentation of complex document images. International Conference on Computer Vision Theory and Applications. Setúbal, Portugal, (2006) 220- 227
  7. 7. DEMOCRITUS UNIVERSITY OF THRACE - GREECE 7 Overview of the Proposed Method
  8. 8. DEMOCRITUS UNIVERSITY OF THRACE - GREECE 8 Fractal Sub-Sampling Procedure Hilbert ‘s curve Benefits: •Smaller number of sampling pixels •Capture of neighborhood regions Fig. 1 Fig. 2 Fig. 3 Fig. 4
  9. 9. DEMOCRITUS UNIVERSITY OF THRACE - GREECE 9 Kohonen Self Organized Featured Map (KSOFM) j k jky arg min x w= − • The training algorithm of the KSOFM is based on competitive learning: • The winner output neuron changes its connections weights as follows: ( )jk k jkw n x w∆ = −
  10. 10. DEMOCRITUS UNIVERSITY OF THRACE - GREECE 10 Gustafson – Kessel Fuzzy Algorithm The Gustafson – Kessel fuzzy algorithm is an extension of the fuzzy c-mean algorithm that produces ellipsoidal classes by using a covariance matrix: Figure 1 Figure 2 Figure 3
  11. 11. DEMOCRITUS UNIVERSITY OF THRACE - GREECE Gustafson – Kessel Fuzzy Algorithm 1. Define the number of the classes c, the weighting parameter m and the cluster volumes ρi. 2. Define the termination tolerance ε>0 and the number of iterations λ. Set a counter α equal to one ( α = 1). 3. Initialize randomly the partition matrix U=[uik]. In this work, the partition matrix is initialized not randomly but from the connections weights wjk of the KSOFM for each output class. 4. Compute the centers of the classes according to the following equation: 5. Compute the covariance matrix Fi for each class according to the following equation: 11 ( ) ( ) [ ] [ ] n m ik k k 1 i n m ik k 1 u x v , i 1,c and k 1,n u = = = ∈ ∈ ∑ ∑ ( ) ( ) n m T ik k i k i k 1 i n m ik k 1 u (x v )(x v ) F , i [1,c] u = = − − = ∈ ∑ ∑
  12. 12. DEMOCRITUS UNIVERSITY OF THRACE - GREECE 12 6. Compute the matrix Ai for each class according to the following equation: 7. Compute the distance dik of every sample xk from the center of each class vi according to the following equation: 8. Update the partition matrix U=[uik] for each sample xk according to the following equation: 9. if or stop, else set and go to step 4. ( ) 1h i i i iA det F F , i [1,c]− = ρ ∈ ( ) ( )T2 ik k i i k id x v A x v= − − [ ] [ ]ik 2 m 1c ik j 1 ij 1 u , i 1,c and k 1,n d d − = = ∈ ∈    ÷ ÷   ∑ ( ) ( 1) max U Uα α− − < ε α ≥ λ 1α = α +
  13. 13. DEMOCRITUS UNIVERSITY OF THRACE - GREECE The parameters of the algorithms during the testing 13 KSOFM Fuzzy C-Mean KSOFM - GK Initially Learning Rate: ninitially = 10-2 m = 1.2 Initially Learning Rate: ninitially = 10-2 Final Learning Rate: nfinal = 10-4 Epochs = 2000 Final Learning Rate: nfinal = 10-4 Step of the Learning Rate: nstep = 10-5 Termination Tolerance: = 5ε ∙10-5 Step of the Learning Rate: nstep = 10-5 KSOFM Termination Tolerance: = 5ε ∙10-5 m = 1.2 GK termination Tolerance: = 5ε ∙10-4 Iterations: = 100λ
  14. 14. DEMOCRITUS UNIVERSITY OF THRACE - GREECE 14 Original Image 22410 colors KSOFM 4 colors FCM 4 colors
  15. 15. DEMOCRITUS UNIVERSITY OF THRACE - GREECE KSOFM – GK 4 colors Median Cut 4 colors 15
  16. 16. DEMOCRITUS UNIVERSITY OF THRACE - GREECE 16 Original Image 99760 colors KSOFM 7 colors FCM 7 colors
  17. 17. DEMOCRITUS UNIVERSITY OF THRACE - GREECE KSOFM – GK 7 colors Median Cut 7 colors 17
  18. 18. DEMOCRITUS UNIVERSITY OF THRACE - GREECE 18 Original Image 33784 colors KSOFM 5 colors FCM 5 colors
  19. 19. DEMOCRITUS UNIVERSITY OF THRACE - GREECE KSOFM – GK 5 colors 19 Median Cut 5 colors
  20. 20. DEMOCRITUS UNIVERSITY OF THRACE - GREECE 20 Original Image 31655 colors KSOFM 4 colors FCM 4 colors
  21. 21. DEMOCRITUS UNIVERSITY OF THRACE - GREECE 21 KSOFM – GK 4 colors Median Cut 4 colors
  22. 22. DEMOCRITUS UNIVERSITY OF THRACE - GREECE 22 Original Image 69656 colors KSOFM 8 colors FCM 8 colors
  23. 23. DEMOCRITUS UNIVERSITY OF THRACE - GREECE 23 KSOFM – GK 8 colors Median Cut 8 colors
  24. 24. DEMOCRITUS UNIVERSITY OF THRACE - GREECE Advatages • The experimental results have shown that the proposed technique has the ability to retain the dominant colors even if the final image consists of a very small number of unique colors. • It can merge areas of the image having similar colors. In this point of view, it can be considered as a powerful color image segmentation procedure 24
  25. 25. DEMOCRITUS UNIVERSITY OF THRACE - GREECE Disadvatage High Computation Cost which comes from the determination of the Mahalanobis distance. For an AMD Athlon 64 3000+ (2GHz) based PC with 1GByte RAM, the processing time for a 512x384 image with 119143 colors for all the algorithms are: 25 KSOFM Fuzzy C-Mean KSOFM - GK 2.43 seconds 8.32 seconds 43.27 seconds The number of colors of which the image is reduced is six (6).
  26. 26. DEMOCRITUS UNIVERSITY OF THRACE - GREECE 26 Conclusions • A new Color Clustering technique is proposed which is based on a combination of a KSOFM neural network and the Gustafson-Kessel fuzzy algorithm. • The main advantages is that it can merge areas of the image with similar colors and has the ability to retain the image’s dominant colors. • Future directions should include the ability to detect the optimal number of final colors and reduce the high computational cost.

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