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

DEMOCRITUS UNIVERSITY OF THRACE - GREECE

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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 LaboratoryDepartment of Electrical & Computer Engineering

Democritus University of Thrace67100 Xanthi, Greece

Email: [email protected] http://ipml.ee.duth.gr/~papamark/


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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).


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Fig. 1 Original image Fig. 2 RGB color distribution

Fig. 3 Image with only 20 dominant colors

Fig. 4 Distribution of 20 colors


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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.


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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.


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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


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Overview of the Proposed Method

Original Image

Sub Sampling

Define the Number of Final Colors

Train KSOFM

Classification through KSOFM

Initialize GK with the KSOFM classification results

Run GK Algorithm

Final Classification through GK algorithm

Final Image


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Fractal Sub-Sampling Procedure

Hilbert ‘s curveBenefits:

•Smaller number of sampling pixels

•Capture of neighborhood regions

Fig. 1 Fig. 2

Fig. 3 Fig. 4


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Kohonen Self Organized Featured Map (KSOFM)

X1

Xk

X3

X2

Yj

Y3

Y2

Y1

Competitive LayerInput Layer

......

......

Wj1

Wj2

Wj3

Wjk

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


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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


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 w jk 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:

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nm

ik kk 1

i nm

ikk 1

u xv , i 1,c and k 1,n

u

nm T

ik k i k ik 1

i nm

ikk 1

u (x v )(x v )F , i [1,c]

u


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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.

1hi i i iA det F F , i [1,c]

T2ik k i i k id x v A x v

ik 2

m 1cik

j 1 ij

1u , i 1,c and k 1,n

d

d

( ) ( 1)max U U 1


The parameters of the algorithms during the testing

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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: ε = 5∙10-5

Step of the Learning Rate: nstep = 10-5

KSOFM Termination Tolerance: ε = 5∙10-5

m = 1.2

GK termination Tolerance: ε = 5∙10-4

Iterations: λ = 100


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Original Image22410 colors

KSOFM4 colors

FCM4 colors


KSOFM – GK4 colors

Median Cut4 colors

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KSOFM7 colors

FCM7 colors



Median Cut7 colors

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KSOFM5 colors

FCM5 colors



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Median Cut5 colors


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KSOFM4 colors

FCM4 colors


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Median Cut4 colors


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KSOFM8 colors

FCM8 colors


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Median Cut8 colors


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

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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:

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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).


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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.