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Linearization of the Optimal Compressor Function for Gaussian Mixture Model Zoran H. Perić, Jelena R. Nikolić, Vladica N. Đor đ ević

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In this paper, the linearization of the optimal compressor functions were done by using a method with a variable number of representational levels within segments. Design of linearized compandor for the proposed probability density function (GMM) was done, followed by analysis of the results. The paper is structured as following: 1. Introduction 2. GMM (Gaussian mixture model) 3. Linearization of the Optimal Compressor Function 4. Numerical Results 5. Conclusion

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1. Introduction Scalar quantization. Uniform and non-uniform quantizers. Compression as the equivalent to a non-uniform quantization (Bennet). Which is the most commonly used compressor functions? What is the advantage of optimal compressor function?

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2. GMM (Gaussian mixture model) In order to achieve a better approximation of the histogram of probability density function of real signals, than in the case of Laplacian and Gaussian sources, the Gaussian mixture model, was used. In this case, the Gaussian mixture model, which consists of two Gaussian components, was used:

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Fig. 1. GMM in the case of different mean values of Gaussian components

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3. Linearization of the Optimal Compressor Function Because of the very complex practical realization of compressor function, its linearization must be performed. Optimal compressor function: Optimal compressor function for the proposed Gaussian mixture model:

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Fig. 2. Illustration of the linearized optimal compressor function

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Segments are determined using the following expression: where L-is the number of segments. Number of cells within the segments: where c i and c i-1 - are the output values of the compressor for the corresponding values t i and t i-1, respectively.

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Size of the amplitude quantum: Granular distortion in the case of linearized models: Probability that the current value of the input signal belongs to the i-th segment:

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Overload distortion: where y ov - is the representational level with the highest value, Total distortion:

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SQNR (Signal-to-quantization-noise ratio): where σ²-is the variance of the input signal,

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4. Numerical Results Fig. 3. Optimal compressor functions for three different mean values of the Gaussian component, m 2.

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Piecewise linear compressor fuction for m2=1

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Piecewise linear compressor fuction for m2=4

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Piecewise linear compressor fuction for m2=8

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Table I Values of SQNR for different number of representational levels of quantizer and for different mean values of Gaussian components 5. Conclusion The significance of this work lies in the fact that in order to achieve a better approximation of the histogram of probability density function of real signals, than in the case of Laplacian and Gaussian sources [7,8], the Gaussian mixture model, which consists of two Gaussian components, was used [4]. This is reflected in the significantly higher value of SQNR than in the case of linearization of the optimal compressor functions reported in [7,8].

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References [1] A. Gersho and R. M. Gray, Vector Quantization and Signal Compression. Boston, Dordrecht, London: Kluwer Academic Publishers, 1992, ch. 5, pp. 133-172. [2] J. D. Gibson, The Communications Handbook. New York: Crc Press Inc, 2002, 2nd ed., ch. 3, pp. 31-39. [3] L. R. Rabiner and R. W. Schafer, "Introduction to Digital Speech Processing", Foundations and Trends in Signal Processing, vol. 1, pp. 1-194, Jan. 2007. [4] N. S. Jayant and P. Noll, Digital Coding Of Waveforms, Principles and Applications to Speech and Video. New Jersey: Prentice Hall, 1984, 2nd ed., chs. 4-5, pp. 115-251. [5] W. C. Chu, Speech Coding Algorithms, Foundation and Evolution of Standardized Coders. New Jersey: John Wiley & Sons, 2003, ch 5-6, pp. 143-183. [6] Y. Q. Shi and H. Sun, Image and Video Compression for Multimedia Engineering: Fundamentals, Algorithms, and Standards. New York: Crc Press Inc, 2008, 2nd ed., ch. 2, pp. 31- 44. [7] Z. Perić, J. Nikolić, "Linearizacija optimalne kompresorske funkcije", Zbornik radova XVII telekomunikacionog foruma TELFOR 2009, CD izdanje, str. 572-575, Beograd, 24- 26. novembar, 2009. [8] Z. Perić, J. Nikolić, Z. Eskić, "Projektovanje linearizovanog hibridnog skalarnog kvantizera za Gausov izvor", Zbornik radova INFOTEH JAHORINA 2009, Vol.8, Ref. B- I-10, str. 122-125, Jahorina, 18-20. mart, 2009.

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Thank you for your attention!

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