coding algorithm based on the simplified semilogarithmic

ELEKTRONIKA IR ELEKTROTECHNIKA, ISSN 1392-1215, VOL. 20, NO. 3, 2014

1Abstract—In this paper, a novel simple algorithm based onthe simplified semilogarithmic quantizer is proposed forLaplacian source coding. An analysis of the signal toquantization noise ratio (SQNR) of the simplifiedsemilogarithmic quantizer is provided in the wide variancerange for the cases of discrete input samples and continualinput samples with Laplacian distribution. For the assumeddiscrete input samples and continual input samples it is shownthat the higher SQNR is achieved by the proposedsemilogarithmic quantizer when compared to the uniformquantizer. Another major contribution of this paper is theproof that in the area of smaller variance values the simplifiedsemilogarithmic quantizer provides significantly higher SQNRvalues than the uniform quantizer, which is of greatimportance, especially in image coding, where the range ofsmaller variances is more probable than the range of highervariance values. Due to the simple realization structure andhigher achieved SQNR than the uniform quantizer, one canexpect that the proposed algorithm will be very applicable incoding of signals which, as well as speech signals and signals ofdifference between adjacent pixel values of image, followLaplacian distribution.

Index Terms—Coding algorithm, uniform quantizer,simplified semilogarithmic quantizer, discrete input samples.

I. INTRODUCTION

The A/D conversion is the process of converting acontinuous analog signal into a digital signal [1].Quantization is one of the main steps in A/D conversion andit is usually performed in two phases [2]. In the first phase,quantizers with a large number of quantization levels areusually used [3]. After that, with the aim to compress thequantized samples, which have discrete amplitudes, in thesecond phase, quantizers with much lower number of levelsare used. Quantizers in the first phase have continual inputsamples (i.e. samples with continual amplitudes) while thequantizers in the second phase have discrete input samples(i.e. samples with discrete amplitudes). The goal of thispaper is to design the quantizer for the second phase, i.e. thequantizer for discrete input samples. The uniform quantizerfor discrete input samples has already been considered in[4]. Although the semilogarithmic quantizer for continual

Manuscript received April 4, 2013; accepted November 14, 2013.This research was funded the Serbian Ministry of Science under the

project III 044006

input samples has been discussed in [5]–[11], however, tothe best of the authors’ knowledge, the semilogarithmicquantization of discrete input samples has not been reportedin the literature yet. Accordingly, in this paper, thesimplified semilogarithmic quantizer is proposed for thequantization of discrete input samples with the Laplacianprobability density function (PDF). The simplifiedsemilogarithmic quantizer we propose in this paper issimpler for practical realization compared to the classicsemilogarithmic quantizer. In general, a semilogarithmicquantizer is a kind of hybrid quantizer consisting of auniform and a nonuniform logarithmic quantizer [1]. Themain advantage of the simplified semilogarithmic quantizerover the classic semilogarithmic quantizer is based on thefact that his uniform quantizer has the unit gain. This factenables independent design of the uniform and thenonuniform logarithmic quantizer constituting the simplifiedsemilogarithmic quantizer. However, in the case of theclassic semilogarithmic quantizer, his uniform quantizergenerally has an arbitrary defined gain and its design isdependent on the design of his nonuniform logarithmicquantizer [1]. The simplified semilogarithmic compressorfunction need not be continual, which enables independentdesign of the uniform and the logarithmic quantizer. On theother side, the compressor function of the classicsemilogarithmic quantizer is continual, i.e. it has a uniquevalue at the point that separates the region where theuniform quantizer is defined from the region where thelogarithmic quantizer is defined [1]. Accordingly, the designof the uniform quantizer and the logarithmic quantizerconstituting the classic semilogarithmic quantizer aremutually dependent. Moreover, the realization of theuniform quantizer with an arbitrary defined gain, which isthe part of the classic semilogarithmic quantizer, is a morecomplex than realization of the uniform quantizer with theunit gain, which is the part of the simplified semilogarithmicquantizer, because in the first case, it is necessary toadditionally use amplifiers. These are the reasons that pointout the advantages of using the simplified semilogarithmicquantizer rather than the classic semilogarithmic quantizer.

This paper is organized as follows. Section II describesperformances of the simplified semilogarithmic quantizerand the uniform quantizer, both designed for the continualinput and Laplacian PDF. Section III derives performances

Coding Algorithm Based on the SimplifiedSemilogarithmic Compression Law for Discrete

Input Samples with Laplacian DistributionZ. Peric1, M. Savic2, J. Nikolic1

1Faculty of Electronic Engineering, University of Nis, Serbia2Mathematical Institute of Serbian Academy of Science and Arts, Belgrade Serbia

[email protected]

http://dx.doi.org/10.5755/j01.eee.20.3.6679

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of the simplified semilogarithmic quantizer and the uniformquantizer designed for the discrete input and Laplacian PDF.In addition, the design procedure of the simplifiedsemilogarithmic quantizer and novel coding algorithmdescription is given. Numerical results are given in SectionIV. In Section V, experimental results are presented. Finally,Section VI is devoted to the conclusions which summarizethe contribution achieved in the paper.

II. PERFORMANCES OF THE UNIFORM QUANTIZER AND THESIMPLIFIED SEMILOGARITHMIC QUANTIZER DESIGNED FOR

CONTINUAL INPUT SIGNAL WITH LAPLACIAN PDFSamples of a signal, which come onto the input of

quantizers, are usually continual in amplitude, i.e. they cantake any real value from the interval (-∞, +∞). In this paperwe consider Laplacian PDF of the input samples, which isone of the most commonly used distributions [1]. Withoutany loss of generality, we assume that the informationsource is Laplacian source with memoryless property andzero mean value. The probability density function for arandom variable with a Laplacian distribution, zero meanand variance σ2 is given by( ) = √ exp − √ | | . (1)

During quantization, an error is made, which can bemeasured by distortion. In general, the total distortion D isequal to the sum of the granular Dg and the overload Do

distortion, i.e. D = Dg + Do. The quality of the quantizedsignal along with distortion is usually measured by signal toquantization noise ratio (SQNR) [1]SQNR = 10 log , (2)

where σ2 is a variance of an input signal, and D is the totaldistortion. In this paper, the analysis of numerical results isconducted using SQNR rather than distortion.

A. Performances of the Uniform Quantizer Designed forContinual Laplacian Source

A uniform quantizer Qu is defined by the followingparameters: N - the number of quantization levels, xmax - thesupport region threshold and Δ = 2xmax / N - the quantizerstep size [4]. The granular Dg and the overload Do distortionof a uniform quantizer for the assumed Laplacian PDF ofvariance σ2 are respectively given by [4]:( ) = 1 − exp −√2 , (3)( ) = exp −√2 (1 + √2 + ), (4)

where t = σ0 / σ and k = xmax / σ0. denotes a referentvariance for which the uniform quantizer is designed.

B. Performances of the Simplified SemilogarithmicQuantizer Designed for Continual Laplacian Source

The simplified semilogarithmic quantizer Qs is a kind of ahybrid quantizer consisting of a uniform and a nonuniformlogarithmic quantizer [9]. Its support region [-xmax, xmax] iscomposed of the uniform part [-xmin, xmin] and thelogarithmic part [-xmax, -xmin) ∪ (xmin, xmax]. xmin is the

threshold between these two parts and xmax is the supportregion threshold of the simplified semilogarithmic quantizer.The simplified semilogarithmic quantizer is definedaccording to the compressor function proposed in [9]( ) == , | | ≤ ,(1 + log | | sgn( ), < | | ≤ . (5)

Parameter B denotes the ratio B = xmax / xmin. N = N1 + N2

and N is the total number of quantization levels, whereas N1

and N2 are the number of quantization levels in the uniformand in the logarithmic part, respectively. It is obvious thatthe uniform part has a unit gain. Accordingly, there arecertain advantages in the realization of the uniformquantizer, which is an integral part of the simplifiedsemilogarithmic quantizer, and, accordingly, in therealization of the whole simplified semilogarithmicquantizer. One of the simplified semilogarithmic quantizer’sadvantages is the fact that his uniform quantizer can bedesigned apart from his nonuniform logarithmic quantizer.That is the main reason we have named this semilogarithmicquantizer the simplified semilogarithmic quantizer [9]. Inthe case of the simplified semilogarithmic quantizer Qs, thegranular distortion consists of the distortions from theuniform Dg1(Qs) and logarithmic part Dg2(Qs), i.e.Dg = Dg1(Qs) + Dg2(Qs). Accordingly, the total distortion ofthe simplified semilogarithmic quantizer D(Qs) consists ofthree components, two granular distortions Dg1(Qs) andDg2(Qs), and one overload distortion Do(Qs). For theassumed Laplacian source of variance σ2 the followingexpressions for these distortions are obtained in [9]:( ) = 1 − exp − √ , (6)( ) = ( ) × exp − √ + √2 ++ − exp − √ + √2 + , (7)( ) = exp(− √ ). (8)

III. PERFORMANCES OF THE UNIFORM QUANTIZER AND THESIMPLIFIED SEMILOGARITHMIC QUANTIZER DESIGNED FOR

THE DISCRETE INPUT SAMPLES WITH LAPLACIAN PDFAssume that continual amplitude samples having

Laplacian PDF are quantized by the uniform quantizer Qu

having Nu output levels = ,… , . Let us denote themaximal amplitude of the uniform quantizer Qu with xmaxcont.This amplitude depends on the input signal amplitude range.The design of the quantizer Qu can be described in anotherway: firstly, the uniform quantizer for the unit standarddeviation (σ = 1) is designed with the maximal amplitude

[1] and after that, the denormalization is done, bydividing all thresholds and representation levels with =

(xmax is maximal amplitude of discrete inputsamples). The denormalization is performed with the aim toadjust the support region [− , ] to thesupport region [−xmax, xmax]. In the second step, sampleshaving discrete amplitudes that are the output of thequantizer Qu are further quantized by the second quantizer Q

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having N levels, N < Nu. The input samples of the quantizerQ can take Nu discrete values from the set X of the outputlevels of the quantizer Qu. The probabilities of these discretelevels for Laplacian PDF are ( ) = ( )∆ =√ exp − √ | | ∆ , i = 2, ... , Nu-1, ∆ = , and( ) = = exp(− √ ) [10]. Since discrete

samples are limited in amplitude by xmax quantizer Q, havingthe support region [-xmax, xmax], produces only the granulardistortion Dg, i.e. the total distortion is equal to the granulardistortion D = Dg. The aim of this paper is to design thequantizer Q.

A. Design of the Uniform Quantizer for Discrete InputSamples

Let us recall shortly explanation of the uniformquantizer for discrete input samples that is reported in [4].Output levels of the N-level quantizer Q are denoted byyj, j = 1, ... , N and it is valid that Nu = N∙L, where L is aninteger. This means that L discrete input levelsXj = {xj1, ... , xjL}∈X are mapped to one output level yj,j = 1, ... , N. Since the quantizer Q is symmetric duringthe design we take into account only the positive range.The quantization step size of the considered quantizerdesigned for σ = 1 is ∆= 2 ⁄ and, accordingly, thethresholds and representation levels are given by [4]:= × ∆, = 0,… , 2⁄ , (9)= ( − 1 2⁄ ) × ∆, = 1,… , 2⁄ . (10)

The support region of the uniform quantizer obtainedin this way [- , ] is different from the supportregion [-xmax, xmax]. In order to adjust the support regionof the uniform quantizer, the thresholds andrepresentation levels should be divided by= ⁄ := , = 0,… , 2⁄⁄ , (11)= ⁄ , = 1,… , 2.⁄ (12)

The granular distortion of the uniform quantizer fordiscrete input samples is given by [4]= ∑ ∑ − ( ). (13)

B. Design of the Simplified Semilogarithmic Quantizer forDiscrete Input Samples

The design of the simplified semilogarithmic quantizer isbased on the companding technique application, whosescheme is shown on Fig. 1.

Fig. 1. Companding quantizer.

From Fig. 1 one can observe that nonuniform quantizationcan be achieved in the following way: first, by compressingthe input signal x using a nonuniform compressorcharacteristic c(x); next by quantizing the compressed signalc(x) by employing a uniform quantizer Q(c(x)); and finallyby expanding the quantized version of the compressed

signal using a nonuniform transfer characteristic c-

1(Q(c(x))), which is inverse to the characteristic of thecompressor. The overall structure, which consists of acompressor, a uniform quantizer and an expandor incascade, is called a compandor [1], or a compandingquantizer.

Figure 2 shows the positive range of the simplifiedsemilogarithmic compressor function c(x). Therepresentation levels xi of the simplified semilogarithmicquantizer are denoted on the x axis. Since the consideredsimplified semilogarithmic quantizer is symmetric whiledesigning we take into account only the positive range.

Fig. 2. The simplified semilogarithmic compressor function.

The thresholds and representation levels in the uniformpart are defined by the following expressions:= × ∆ , = 0,… , 2⁄ , (14)= ( − 1 2⁄ ) × ∆ , = 1,… , 2⁄ , (15)

where ∆ = 2 ⁄ and ∆ = 2 log ⁄ , N = N1 +N2. The thresholds and representation levels in thelogarithmic part of the simplified semilogarithmic quantizerare defined by:( ) = 1 + log = + ∆ ,= 0,… , 2⁄ , (16)( ) = 1 + log = + ( − 1 2⁄ )∆ ,= 1,… , 2⁄ . (17)

By solving (16) and (17), the following expressions forthe thresholds and representation levels in the logarithmicpart are obtained:= ⁄ , = 0, … , 2,⁄ (18)= ( )⁄ , = 1, … , 2⁄ , (19)

where B denotes ratio = ⁄ , is the supportregion threshold of the simplified semilogarithmic quantizerand is the threshold between the uniform and thelogarithmic part of the simplified semilogarithmicquantizer. The support region of the simplifiedsemilogarithmic quantizer obtained in this wayis [− , ] and it has to be adjusted to the range[−xmax, xmax]. This means that the thresholds and

xiux(i-1)uxmaxxmin

c(xmax)

xmin

x(i-1)lxi l

u

l

x

c(x)

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representation levels should be divided by = ⁄ := , = 0,… , 2⁄⁄ , (20)= ⁄ , = 1,… , 2⁄ , (21)= , = 0,… , 2⁄⁄ , (22)= ⁄ , = 1,… , 2⁄ . (23)

Granular distortion of the simplified semilogarithmicquantizer is given by

= − +⁄+2∑ ∑ ( ⁄ ) − ( ⁄ )⁄⁄ , (24)

where μi, i = 1, ... , N is the parameter which stands for thenumber of the input levels that are maped to one outputlevel. Also, it is valid that = ∑ . Finally, xij∈X areinput samples of the simplified semilogarithmic quantizer.

C. Novel Coding AlgorithmEvery quantizer can be viewed as the combination of an

encoder and a decoder. In the case of a compandor, theencoder consists of a compressor and a uniform quantizerwhile the decoder is composed of an expandor [1]. Assumethat onto the input of an encoder come M samples xi,i = 1, ... , M, which belong to the set of N0 different samplesfrom the set X (xi∈X). Therefore, at the output of theuniform quantizer there are M quantization levels qi,i = 1, ... , М which can take the values from the set of Ndifferent values that are defined by the representation levelsof the uniform quantizer.

The novel coding algorithm for the N-level quantizerdefined by the considered simplified semilogarithmiccompressor function c(x) (5) consists of the following steps:

1. The compression of samples xi by applying thesimplified semilogarithmic compressor function c(x) inorder to obtain the compressed samples c(xi), i = 1, ... , М;2. The uniform quantization of the compressed samplesc(xi), i = 1, ... , М, where quantization levels qi,i = 1, ... , М are obtained;

3. The encoding of the obtained quantization levels qi,i = 1, ... , М with binary words Ii, i = 1, ... , М, (qi→Ii,i = 1, ... , М) and further transmission to the decoder, viathe communication channel;4. The decoding of the transmitted binary words Ii,i = 1, ... , М, whereby corresponding quantization levelsqi, i = 1, ... , М (Ii→qi, i = 1, ... , М) are obtained;5. The expanding of the obtained quantization levels, byapplying the inverse simplified semilogarithmiccompressor function c-1(x) whereby representation levelsyi, i = 1, ... , М are obtained.

IV. NUMERICAL RESULTS

This section discusses the performances that we haveascertained by applying the simplified semilogarithmicquantizer and the uniform quantizer in the quantization ofsignals having Laplacian PDF and a wide variance range. Avariance σ2 can be expressed in the logarithmic domain asσ [dB] = 20log10(σ/σ0), where σ0

2 is some referent variance.By assuming σ0 = 255 as in [10], σ[dB] in our analysisranges within [-48.13 dB, 0 dB], (20log10(1/255) = -48.13 dB, 20log10(255/255) = 0 dB). The comparison of theSQNR characteristics for the uniform and the simplifiedsemilogarithmic quantizer with N = 32 and 64 quantizationlevels for the case of the continual and discrete input withLaplacian PDF are shown on Fig. 3 and Fig. 4. From theFig. 3 and Fig. 4 one can notice that in the case of thecontinual input samples SQNR characteristic of the uniformquantizer, at first increases to its maximum and then rapidlydeclines. However, the simplified semilogarithmic quantizerprovides a more constant and, in almost all of the observedvariance range, higher level of SQNR, which can beconsidered via the gain in the average SQNR of about 8 dB,10.6 dB, for the case of N = 32 and 64 quantization levels,respectively. Accordingly, in the case of the continual inputsamples, a better robustness of SQNR is obtained by thesimplified semilogarithmic quantizer compared to theuniform quantizer. In the Fig. 3 and Fig. 4, it is interesting tonotice that SQNR characteristics differ depending onwhether the input signal is discrete or continuous, regardlessof the same assumed variance range of the input signal.

a) b)Fig. 3. SQNR characteristics of the uniform and the simplified semilogarithmic quantizer with N = 32 quantization levels for the case of: a) continual inputand b) discrete input.

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5

10

15

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N=32; N1=8; N2=24

1 uniform; k=0.18;SQNRav=9.30 [dB]2 semilogarithmic; xmin=0.014; xmax=2.01; SQNRav=17.36 [dB]

SQN

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20log(/0) [dB]

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1 semilogarithmic; xmin=0.13; xmax=8.9; SQNRav=25.44 [dB]2 uniform; SQNRav= -11.04 [dB]

SQN

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B]

20log(/0)[dB]

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2

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a) b)Fig. 4. SQNR characteristics of the uniform and the simplified semilogarithmic quantizer with N = 64 quantization levels for the case of: a) continual inputand b) discrete input.

Also, an interesting observation is that in the consideredvariance range the simplified semilogarithmic quantizer,designed for the discrete input samples, reaches a higherSQNR compared to the uniform quantizer [4] designed forthe discrete input samples and the equal number ofquantization levels. In addition, it is important to perceivethat SQNR characteristics of both quantizers, designed forthe discrete input samples, vary a lot in the observedvariance range. As in the case of the continual inputsamples, in the case of discrete input samples we havecalculated the gain in the average SQNR that amounts toabout 36.5 dB, 75.4 dB, for the case of N = 32 and 64quantization levels, respectively.

Finally, it is important to highlight that in the area ofsmaller variance values the simplified semilogarithmicquantizer provides a significantly higher SQNR values thanthe uniform quantizer. The range of smaller variances ismore often present in the practical applications (grayscaleimage coding) than the range of higher variance values [10].Therefore, the application of the simplified semilogarithmicquantizer yields results that are significantly better thanwhat can be concluded from the theoretical point of view,i.e. from the comparison of the average SQNR values.

V. EXPERIMENTAL RESULTS

In Table I, experimental results for the simplified

semilogarithmic quantizer for discrete input samples aregiven, for different values of N1 and N2. Values in Table Iare averaged values for three standard test images (Lena,Street and Boat) shown on Fig. 5.

PSQNR values in Table I are very high, i.e. near losslesscompression is achieved using simplified semilogarithmicquantizer designed for discrete input samples [10].

TABLE I. EXPERIMENTAL RESULTS FOR THE SIMPLIFIEDSEMILOGARITHMIC QUANTIZER DESIGNED FOR DISCRETE

INPUT SAMPLES.PSQNR[dB]

Lena Street Boat

N = 64; N1 = 32;N2 = 32

54.40 56.61 51.94

N = 32; N1 = 16;N2 = 16

47.60 48.24 45.28

In Fig. 6, three images from Fig. 5, after compressionwith the simplified semilogarithmic quantizer with N = 32levels are shown. We can observe that the reconstructedimages are almost visually identical to the original images.Based on these facts, conclusion arises, that proposed modelof the simplified semilogarithmic quantizer designed fordiscrete input samples, can be used with great success, forimage compression.

a) b) c)Fig. 5. The grayscale images, size 512 x 512 pixels a) Lena b) Street c) Boat.

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5

10

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25

SQN

R[d

B]

20log(/0) [dB]

1 uniform; k=0.3; SQNRav=12.18 [dB]2 semilogarithmic; xmin=0.02; xmax=3.1; SQNRav=22.8 [dB]

N=64; N1=16; N2=48

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-50

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1 semilogarithmic; xmin=0.11; xmax=7.0; SQNRav=71.86 [dB]2 uniform; SQNRav= -3.53 [dB]

1

2

N=64; N1=16; N2=48

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a) b) c)Fig. 6. The Grayscale images from Fig. 5 after compression with the simplified semilogarithmic quantizer with N = 32 levels a) Lena b) Street c) Boat.

VI. CONCLUSION

This paper has demonstrated the significance of the novelcoding algorithm development via the gain in theperformances that have been ascertained by applying theproposed simplified semilogarithmic quantizer in thequantization of signals having Laplacian PDF and a widevariance range over the uniform quantizer designed for thesame PDF. Particularly, by comparing the SQNRcharacteristics of the proposed simplified semilogarithmicquantizer and the uniform quantizer, both designed for thediscrete input samples with Laplacian PDF, it has beenrevealed that the proposed simplified semilogarithmicquantizer, along with the gain in the average SQNR thatranges up to 75.4 dB, in the whole of the consideredvariance range provides a higher level of SQNR. Moreover,it has been shown that in the area of smaller variance valuesthe simplified semilogarithmic quantizer provides asignificantly higher SQNR values than the uniformquantizer. Experimental results are very well matched withtheoretical expectations i.e. obtained PSQNR values indicatea high quality level of reconstructed images. Accordingly,one can believe that the proposed simplifiedsemilogarithmic quantizer will be of great importance formany practical image coder realizations.

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coding algorithm based on the simplified semilogarithmic

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