Optimization of Quantizer s Segment Treshold Using Spline Approximations for Optimal Compressor Function

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1 Applied Mathematics, 1, 3, doi:1436/am1331 Published Online October 1 ( Optimization of Quantizer s Segment Treshold Using Spline Approimations for Optimal Compressor Function azar Velimirović 1*, Zoran Perić, Miomir Stanković 3, Jelena ikolić 1 Mathematical Institute of the Serbian Academy of Sciences and Arts, Belgrade, Serbia epartment of Telecommunications, Faculty of Electronic Engineering, University of iš, iš, Serbia 3 Mathematical epartment, Faculty of Occupational Safety, University of iš, iš, Serbia * velimiroviclazar@gmailcom, zoranperic@elfakniacrs Received July 9, 1; revised August 9, 1; accepted August 16, 1 ABSTRACT In this paper, the optimization of quantizer s segment threshold is done The quantizer is designed on the basis of approimative spline functions Coefficients on which we form approimative spline functions are calculated by minimization mean square error (MSE) For coefficients determined in this way, spline functions by which optimal compressor function is approimated are obtained For the quantizer designed on the basis of approimative spline functions, segment threshold is numerically determined depending on imal value of the signal to quantization noise ratio (SQR) Thus, quantizer with optimized segment threshold is achieved It is shown that by quantizer model designed in this way and proposed in this paper, the SQR that is very close to SQR of nonlinear optimal companding quantizer is achieved Keywords: Optimization of Quantizer s Segment Threshold; Mean Square Error; Second-egree Spline Functions; Compressor Function 1 Introduction Quantization, in mathematics and digital signal processing, is the process of mapping a large set of input values to a smaller set such as rounding values to some unit of precision The most common type of quantization is known as scalar quantization A device or algorithmic function that performs quantization is called a quantizer [1-3] Quantizers can be uniform and nonuniform Uniform quantizers are suitable for signals that have approimately uniform distribution How most of the signals do not have a uniform distribution there is a need for using nonuniform quantizer [1-3] One of the most used methods for the realization of the nonuniform quantizer is companding technique, in which a specific compressor function is applied on an input signal The most often used compressor functions are optimal compressor function [1-3] The optimal compressor function gives the imum signal to quantization noise ratio (SQR) for the reference variance of the input signal By knowing compressor function, model quantizer is completely defined However, although easy for analysis, it is shown that this model is very difficult to realize [1-3] Therefore, in order to achieve easier practical realization, approi- * Corresponding author mation of the optimal compressor function and linearization of the optimal compressor function is performed A comprehensive analysis of SQR behavior in a wide range of variances for the piecewise uniform scalar quantizer PUSQ designed for a aplacian source according to the piecewise linear approimation to the optimal compressor law is reported in [4] The linearization of the optimal compressor function is done in [5] The demerit of method described by [5] is a high compleity of quantizer and high compleity of coding and decoding The analysis of compressor function for aplacian source is shown in [6] Quantizer designed on the basis of approimative spline functions, whose support region is divided on segments of equal size is described in [7] In this paper, in order to reduce compleity and maintenance of a reasonably good performance of quantizer, we develop a new method of construction quantizer which introduces optimization of the segment threshold and different number of cells per segments The number of cells per segments is determined depending on value of optimized segment threshold Also, depending on value of optimized segment threshold, approimate spline functions, by which the optimal compressor function is approimated, are determined Unlike with the quantizer described in [7], the support region of proposed quantizer model is not divided on segments of equal size Segments Copyright 1 SciRes

2 VEIMIROVIĆ ET A 1431 size at the proposed model depends on segment threshold that is being optimized The value of segment threshold depends on the size of the support region In papers [8-1] detailed analyses are performed on which the importance of support region choice could be well seen In the papers mentioned above, the influence of support region choice on scalar quantizer performances that are designed for aplacian source is analysed By designing the proposed quantizer based on approimate spline functions and optimized threshold segment, SQR that is close to that of the nonlinear optimal companding quantizer is obtained The rest of the paper is organized as follows: In Section the detailed description of spline functions of the second-degree and optimal compressor function is given esign of quantizer based on spline functions of the second-degree is described in Section 3 The procedure for optimization of the segment threshold is described in Section 4 Finally, Section 5 presents numerical results and discusses their implications Approimate Second-egree Spline Functions and Optimal Compressor Function The theory of approimation is the area of numerical mathematics that deals with problems of replacing one function with other It is shown that the spline of degree better approimates the optimal compressor function than the spline of degree 1 [7] Therefore, in this paper, the approimation of the optimal compressor function using spline function of the second-degree is done A function Q is called a spline of degree if [11]: 1) The domain of Q is an interval [a, b] ) Q and Q are continuous on [a, b] 3) There are points i (called knots) such that a = < 1 < < = b and Q is a polynomial of degree at most on each subinterval [ i, i+1 ] In brief, a quadratic spline is a continuously differenttiable piecewise quadratic function, where quadratic includes all linear combinations of the basic functions 1,, The second-degree spline, also called a quadratic spline, is consisted of parabola parts between two consecutive knotes, but elected to have the same tangent at knote [11] The approimate quadratic spline function g(), which approimates a nonlinear compressed function c(), has the following form [11]: r1 p1q1,, 1 r p q, 1, g (1) r p q, 1, In this paper, the coefficients of the second-degree spline, r i, p i and q i, i = 1,,, are determined by minimizing the mean-square error (MSE) as follows: 1 F c g i i i d, () i1 i i1 i1 F F F,,, i 1,, (3) r p q i i i The optimal compressor function c() by which the imum SQR is achieved for the reference variance of an input signal is defined as [1-3]: p d, p d c (4) p d, p d Without diminishing the generality, the quantizer design will be done for the reference input variance of ref 1 and 3 esign of Quantizer Based on Approimate Second-egree Spline Functions The performances of a quantizer are usually specified in terms of SQR [1-3]: SQR 1log 1log, g o measured in decibels [db], with denoting the variance of an input signal and with denoting the total distortion The total distortion is equal to the sum of the granular distortion g and the overload distortion o The overload distortion o is determined as follows [1-3]:, (5) y p d (6) o where p() is aplacian probability density function (PF) which is defined as follows [1]: p 1 ep In the rest of the paper we assume symmetry about zero in the proposed quantizer model design The aplacian PF is symmetrical about zero amely, without loss of generality, we assume that information source is aplacian source with memoryless property, the unit (7) Copyright 1 SciRes

3 143 VEIMIROVIĆ ET A variance and zero mean value The reproduction level y is determined from the condition: y g 1 (8) The g presents the approimate spline function of the last segment (see Figure 1) The step size is equal to: (9) The optimal support region value of the proposed quantizer is as follows [5]: 3 ln 1 (1) The total number of the reproduction levels per segments in the first quadrant is: i, (11) i1 where the optimal number of reproduction levels per segments, i, is determined from the following condition: ci i ci 1( i 1) i, i 1,,, (1) c Reproduction levels are determined as the solution of the approimate spline function as follows: c ( ) c 1 ( 1 ) 1 1 y gi, i 1, 1,, i (13) y g i1 1 i k k 1 g 1 () 1, (14) 1 g () second-degree spline function nonlinear optimal compressor function = Figure 1 Second-degree spline function and nonlinear optimal compressor function for the number of segments = 4 where i,,, 1,, i For the y is taken the solution that belongs to the spline function domain Observe that indees i and indicate on the -th reproduction levels within the i-th segment Cells lengths per segments of the considered quantizer is equal to:, i 1,,, 1,, i (15) g y i enoted by the -th cells lengths within the i-th segment The granular distortion is determined by Bennett s integral [1-3]: g p d 3 (16) c The granular distortion for the proposed model, based on formula (16), is equal to: g p y, 3 g y where i 1,,, 1,, i (17) ' i 4 Optimization of Quantizer s Segment Threshold In this Section, the segment threshold optimization is described Solving the system of equations defined by Equation (3), where is the optimal value of the support region, =, defined by Equation (1), the epressions for determining coefficents r i, p i, q i, i = 1,, are obtained The values of coefficents r i, p i, q i, i = 1,, depend on the value of the segment threshold i The segment threshold i is determined in the following: Step 1 Based on the coefficents r i, p i, q i, i = 1,, epressed depending on the segment threshold i, the second-degree spline functions, g i (), defined by Equation (1) is desgined Step On the basis of the second-degree spline functions obtained in this way, g i (), the proposed quantizer as described in Section 3 is designed This way, an epression for SQR of proposed quantizer depending on the segment threshold i is achieved Step 3 The segment threshold i is numerically determined so that imal SQR value of proposed quantizer is achieved (see Figures and 3) 5 umerical Results and Conclusions umerical results presented in this section are obtained for the case when the number of segments is equal to = 4 and for number of levels = 16 and = 3 For the number of segments = 4, the approimate quadratic spline function g() defined by Equation (1) is equal to: Copyright 1 SciRes

4 VEIMIROVIĆ ET A , , , , ,11 SQR [db] , , , , , ,5 43 4, , , , , , , , ,7 1 Figure umerical determination of the segment threshold 1 for the number of levels = 16 4, , ,9 3,85 SQR385 [db] 38 3, , , , ,6 55 5,5 56 5,6 57 5,7 58 5,8 59 5,9 6 6, 61 6,1 6 6, 63 6,3 64 6,4 65 6,5 1 Figure 3 umerical determination of the segment threshold 1 for the number of levels = 3 r p q,, g r p q , 1, (18) The number of conditions set is equal to the number of coefficients that should be determined In fact, as we have three knotes and two subintervals (see Figure 1), and each second-degree polynomial has three coefficients, means to determinate the si coefficients in total [11] Solving the system of equations defined by Equation (3), the coefficents r 1, p 1, q 1, r, p, q are determined The values of coefficents r 1, p 1, q 1, r, p, q depend on the value of the segment threshold 1 which is numerically determined as described in Section 4 In Figures and 3 the dependence of SQR of the proposed quantizer model on the segment threshold 1 for the number of levels = 16 and = 3 is shown Based on Figure it can be concluded that the imal value of SQR of the proposed quantizer is achieved for the value of the segment threshold 1 = 451 Also, based on Figure 3 it can be concluded that the imal value of SQR of the proposed quantizer is achieved for the value of the segment threshold 1 = 594 Table 1 shows the values of SQR of the proposed quantizer which segment threshold 1 numerically determined, (SQR ), the values of SQR of the proposed quantizer having equidistant segment thresholds (the segment threshold 1 is at the middle of support region) [7], (SQR S ), and the values of SQR of nonlinear optimal companding quantizer, c():,,, (SQR O ), for the number of levels = 16 and = 3 The granular distortion g and the overload distortion o of nonlinear optimal companding quantizer defined as in [1-3]: O o O g 9 1e 3, (19) e () In Figure 4 dependency of SQR, SQR S and SQR O on the number bits per sample (R = 4 bit/sample and R = 5 bit/sample) is shown Analyzing the results shown in Table 1 and Figure 4, one can notice that design of quantizer based on approimate spline of degree, with the optimized segment threshold 1, achieved higher SQR than design of quantizer based on approimate spline of degree with the segment threshold 1 that is it the middle of the support region Also, analyzing the results shown in Table 1 and Figure 4, one can conclude that design of quantizer based on approimate spline of degree, with the optimized segment threshold 1, achieved SQR very close to that of nonlinear optimal companding quantizer Comparing the performance of the proposed quantizer model with the optimized segment threshold 1, with the quantizer model having equidistant segment thresholds [7], and with the nonlinear optimal companding quantizer [1-3], it can be concluded that the proposed model is a very effective solution because a simple quantizer model achieves a high quality signal For the case when the number of levels is lower than 16, the compleity of the Table 1 The values of SQR, SQR S and SQR O for the number of levels = 16 and = 3 SQR [db] SQR S [db] SQR O [db] Copyright 1 SciRes

5 1434 VEIMIROVIĆ ET A SQR [db] SQR SQR S SQR O , 41 4,1 4 4, 43 4,3 4,4 44 4,5 45 4,6 46 4,7 47 4,8 48 4,9 49 5, 5 R [bit/sample] Figure 4 ependency of SQR, SQR S and SQR O on the number bits per sample proposed quantizer model may be larger than the optimal loyd-ma s scalar quantizer model [1-3] For the case when the number of levels is more than 3, the compleity of the proposed quantizer model becomes high ue to these limitations, this paper analyzes the quantizer model for the number of levels = 16 and = 3, ie for the number bits per sample R = 4 and R = 5 The proposed quantizer model besides aplacian PF, is convenient for other probability density functions 6 Acknowledgements This work is partially supported by Serbian Ministry of Education and Science through Mathematical Institute of Serbian Academy of Sciences and Arts (Proect III446) and by Serbian Ministry of Education and Science (Proect TR335) REFERECES [1] S Jayant and P oll, igital Coding of Waveforms, Principles and Applications to Speech and Video, Prentice Hall Secondary Education ivision, ew Jersey, 1984 [] A Gersho and R M Gray, Vector Quantization and Signal Compression, Kluwer Academic Publishers, Boston, ordrecht, ondon, 199 doi:117/ [3] R Rabiner and R W Schafer, Introduction to igital Speech Processing, Foundations and Trends in Signal Processing, Hanover, 7 [4] J ikolić, Z Perić, Antić, A Jovanović and enić, ow Comple Forward Adaptive oss Compression Algorithm and ITS Aplication in Speech Coding, Journal of Electrical Engineering, Vol 6, o 1, 11, pp 19-4 doi:1478/v [5] Z Perić, M Petković and M inčić, Simple Compression Algorithm for Memoryless aplacian Source Based on the Optimal Companding Technique, Informatica, Vol, o1, 9, pp [6] Z Perić and J ikolić, Analysis of Compressor Function for aplacian Source s Scalar Compandor Construction, ata Recording, Storage and Processing, Vol 8, o, 6, pp 15-4 [7] Velimirović, Z Perić, J ikolić and M Stanković, esign of Compandor Quantizer for aplacian Source for Medium Bit Rate Using Spline Approimations, Facta Universitatis, Vol 5, o 1, 1, pp 9-1 [8] S a, On the Support of Fied-Rate Minimum Mean- Squared Error Scalar Quantizers for a aplacian Source, IEEE Transactions on Information Theory, Vol 5, o 5, 4, pp doi:1119/tit [9] Z Perić, J ikolić and Pokraac, Estimation of the Support Region for aplacian Source Scalar Quantizers, Journal of Electrical Engineering, Vol 58, o 1, 7, pp [1] Z Perić, J ikolić and Pokraac, Analysis of Support Region for aplacian Source s Scalar Quantizers, Proceedings of 7th IEEE Conference on Telecommunications in Modern Satelite, Cable and Broadcasting Services TESIKS 5, iš, 8-3 September 5, Vol, pp [11] W Cheney and Kincaid, umerical Mathematics and Computing, 6th Edition, Thomson Higher Education, Belmont, 8 Copyright 1 SciRes