CHAPITRE I-5 ETUDE THEORIQUE DE LA ROBUSTESSE DU QUANTIFICATEUR UNIFORME OPTIMUM

Transcription

1 CHAPITRE I-5 ETUDE THEORIQUE DE LA ROBUSTESSE DU QUANTIFICATEUR UNIFORME OPTIMUM

2 Présentation Après avoir mis au point et validé une méthode d allocation de bits avec un découpage de la DCT en bandes circulaires, nous avons cherché à améliorer l optimisation du quantificateur sur lequel repose cette allocation. Jusqu alors, nos expériences étaient basées sur l hypothèse que les bandes circulaires suivaient une distribution Laplacienne. Nous avons voulu développer une approche valide pour toute distribution de type Gaussienne généralisée. Nous avons aussi cherché à réaliser une quantification non pas adaptée à chaque image, mais fixe pour une classe d images. Pour ce faire, il faut connaître l impact d un choix de quantificateur inapproprié à la distribution du signal afin de choisir au mieux les paramètres de distribution Gaussienne généralisée que l on doit appliquer avec toutes les images. Une étude du quantificateur uniforme optimum était donc nécessaire, ainsi qu une exploration des conséquences de la non adaptation du quantifieur. Le présent chapitre donne nos développements théoriques sur l optimisation et la robustesse du quantifieur scalaire uniforme. Ces travaux ont fait l objet de l article [BERE-97], qui va être soumi à la revue Signal Processing, et dont le texte est reproduit ci-dessous. Robustness of Optimum Uniform Quantizers to a Mismatched Statistical Model.. Introduction This paper gives a theoretical study of quantizer mismatch with scalar uniform quantizers. The current work was carried out in view of practical applications to transform image coding. The results are however applicable to the quantization of any waveform by a scalar uniform quantizer. The coding or compression of a digital signal facilitates its transmission and archive. Compression is often the only viable solution for transmitting large images over limited bandwidth channels, or for long term storage of large amount of data. A typical lossy compression system includes a signal transformation, commonly by discrete cosine transform or by filter banks, in order to decorrelate the signal and to compact its energy into a small number of coefficients. In such systems, called transform coding, the signal transformation is followed by quantization, a conversion of the transformed signal into a small number of levels. The quantization is non-invertible, and yields a lossy compression. The last operation of a transform coding system is an entropy coding that reduces the remaining redundancy of the quantized transformed coefficients. Quantization is the key operation of compression schemes because it must both preserve the features that are relevant to the end-user after signal reconstruction, and reduce the data rate (i.e. number of bits per signal sample). There are two main categories of quantizers: the scalar quantizer (SQ), and the vector quantizer (VQ). The SQ quantifies individual samples by mapping them into a limited set of values. In contrast, the VQ quantifies blocks of samples by mapping them into a limited set of blocks (called codewords). Vector quantization is an extension of scalar quantization to dimensional spaces higher than. In his fundamental work on rate-distortion theory, Shannon proved

3 that a VQ can always achieve better coding performances than a SQ [SHAN-59]. In practice, VQ is complex to implement. It requires a training phase in order to determine the dictionary of output codewords based on a number of test images. The coding phase consists of matching the encountered waveform blocks with the closest codeword. Both operations are complex. Only the decoding phase is very simple. Because of VQ s complexity and long coding times, SQ has been extensively studied [MAX-6], [GISH-68], [WOOD-69], [BERG-7], [GERS-78], [BERG-8], and utilized for image coding in the seventies and eighties. With the advance of computer technology, VQ was given more attention in the eighties and nineties [GERS-8], [GRAY-8]. However, scalar quantization recently regained attention with the contributions of [SHAP-9], [SAID-96], and remains widely used in transform coding because of its simplicity. Image coding standards extensively use scalar uniform quantizer, e.g. JPEG [PENN-9] and MPEG [LEGA-9]. With a scalar quantizer, the individual input samples are divided in threshold intervals, which boundaries are the threshold levels. All the values lying within a threshold interval are mapped into a single quantization level. The mapping of the input values into a limited number of quantization levels results in a distortion. Four types of scalar quantizers are principally considered in the literature. Their definition is given in the following. Definition : An N-level pdf-optimized quantizer is a quantizer that minimizes the average distortion for a fixed number of levels N. This is the Max-Lloyd quantizer [MAX-6]. The threshold and quantization levels are not uniformly spread over the input and the output range. Definition : An N-level minimum-distortion uniform-threshold quantizer is a quantizer that minimizes the average distortion for a fixed number of levels N, with uniform threshold levels and non-uniform quantization levels. Definition : An N-level minimum-distortion uniform quantizer is a quantizer that minimizes the average distortion for a fixed number of levels N, with both uniform threshold and uniform quantization levels. With the quantizers of definition,, and the bit rate is not controlled. Definition : An N-level entropy-constrained optimum quantizer is a quantizer that minimizes the average distortion at a given bit rate. A minimum-distortion uniform quantizer followed by entropy coding gives better performance than the Max-Lloyd quantizer (without entropy coding) in terms of ratedistortion [JAIN-89 pp5-7]. Other advantages of the uniform quantizer are both the small amount of overhead data, and the simplicity of its implementation. For these reasons, we limited the scope of this work to the uniform quantizer. Two approaches are possible: either the quantizer is adapted to the properties of each input signal, or it is fixed for a class of signals. In the first approach, the quantization is adaptive, and the computational cost is high. The second approach, non-adaptive quantization, is the one addressed in this paper. A non-adaptive quantizer is designed for a class of signals which are assumed to have similar properties, and in particular the same probability density function (pdf). A major concern in practical applications of coding is the robustness of the quantizer regarding possible variation of the input-signals pdf. Our objective is to address this robustness with the uniform quantizer by studying the effect of a possible mismatch of the input pdf compared with the pdf expected in the quantizer design. We consider both minimum-distortion uniform quantizers (definition ) and entropy-constrained optimum uniform quantizers (definition ). We assume that the input signal follows a generalized Gaussian (GG) distribution, which covers a wide range of signals found in practical applications. Signal modeling by the GG pdf includes the Laplacian and the Gaussian pdf. GG pdfs are encountered in DPCM [CUTL-5], [JAIN

4 89], cosine transform [REIN-8], [MULL-9], [JOSH-95], [MOSH-96], wavelet transform [MALL-89], [BARL-9] or subband coding [WOOD-86], [WEST-88]. Although scalar quantizers have been widely studied in the years 97-8, to our knowledge the robustness of the uniform quantizer has not yet been addressed. A detailed study of the robustness of the Max-Lloyd quantizer (definition ) was reported in [MAUE-79], and used in [JAYA-8]. We found no study about the mismatch of uniform quantizers from definition and. In this paper, we use the Mean Square Error (MSE) and the Signal to Noise Ratio (SNR) as measures of distortion. We present the deviation of the MSE and the SNR due to a quantizer designed with a pdf model that differs from the actual pdf of the input-signal. Section gives the mathematical expression of the MSE and the entropy with a generalized Gaussian pdf, then the analysis of the uniform quantizer properties, and finally the analytical formulation of the rate-distortion optimization of entropy constrained uniform quantizers. In section, mismatch of the quantizer relative to the shape parameter of the input pdf, and mismatch relative to the variance are addressed. Finally, section summarizes our findings and discuss them in comparison to related works.. Matched uniform quantizers with generalized Gaussian distributions. Notation A scalar quantizer is a staircase function that maps the input values into a smaller range of output levels. The quantizer maps a continuous random variable X into a discrete random variable X ~. The range of the input values is divided into N=L+ adjacent intervals, which boundaries are the threshold levels t, t,..., t N. The output belongs to a finite set of quantization levels {l, l,..., l L}. If the i th input value x(i) lies between the threshold levels t j and t j+, then it is mapped into an output value x ~ (i)= l j. A uniform quantizer is defined by the number of threshold intervals N and the quantization step size q. The number of quantization levels is also equal to N. The threshold and the quantization intervals are all constant and equal to q. Midtread quantizers are symmetrical with a central quantization level l L/ =, their number of quantization levels is always odd. Midrise quantizers have an even number of quantization levels; they cannot reconstruct a zero-value because zero is a threshold level. We limited this study to scalar uniform midtread quantizers, an example of which is shown in figure. Extension to midrise quantizers would follow similar derivations

5 N=L+=7 l L=L.q l l 6=.. t =- t = -L/.q+q/ t l l = t 6=t L L/q-q/ t N=+ x l l l =-L.q/ Figure : Characteristic of a midtread quantizer.. Mean Square Error of uniform quantizers with GG pdf The mean square error (MSE) of a quantizer is defined by: D(q) = ( x x)² p( x) dx ~ where p(x) is the pdf of the input random variable X. () Without lack of generality, we assume that X is a zero-mean random variable, with variance σ. For a uniform midtread quantizer with N=L+ quantization levels, with a quantization step q, the MSE is: D(q)= D g (q) + D o (q) () where D g (q) = { L j= ( j / ) q [ x ( j ) q]² p( x) dx+ ( j ) q and L D o (q)= L ( x q )² p ( x ) dx q L j= jq ( x jq)² p( x) dx} ( j / ) q The terms D g (q) and D o (q) refer to two different kinds of errors. D o (q) becomes important when extreme values of the input are saturated by the quantizer, i.e. the range of the input values exceeds the range of the quantizer threshold levels. This is commonly referred to as the overload distortion. This distortion is high if some input values that have a rather high probability are saturated. Conversely, D g (q) becomes high when the full range of the input values is quantized, but in a coarse manner. This is called the granular noise. Granular noise and overload distortion have a different impact on the perceptual annoyance in image coding. - -

6 The input-signal may follow a great variety of distributions, and for a broad investigation of the quantization error we used the generalized Gaussian distribution p X (x) = K e ( x ) α () where K =, α and >, and where Γ (x) is the gamma distribution. αγ( ) Γ( ) The variance of this pdf is σ² = α². Γ( ) Particular cases of the generalized Gaussian distribution are the Laplacian pdf: =, and the Gaussian pdf: =. When, the distribution tends towards an impulse. When, the distribution is uniform, it can easily be demonstrated that its height is σ and its width σ. Similarly, when σ² the distribution tends towards an impulse. When σ², the distribution becomes wider and wider, its amplitude tends towards zero. σ = = (a) (b) Figure : Pdf of Generalized Gaussian distributions. Figure gives plots of generalized Gaussian distributions for various values of and σ². As shown in figure, the shape of the distribution can be modified by varying the parameter without changing the variance. The parameter is referred to as the shape parameter. In image transform coding based on Discrete Cosine Transform (DCT) or Subband Band Coding (SBC), the values of encountered in practice are often in the range of.5 to. - -

7 The quantization error D(q) can be derived by incorporating () into (). The result is D(q)= D g (q) + D o (q) () where L D g (q) =K α {α ² Γ( C)[ γ ( b, C) γ ( b, C)] j = jqαγ( D)[ γ ( b, D) γ ( b, D)] + jq ² ² Γ( E)[ γ ( b, E) γ ( b, E)] + qαγ( D)[ γ ( b, D) γ ( b, D)] ( j ) q² Γ( E)[ γ ( b, E) γ ( b, E)] } and D o (q) = K α { α ² Γ ( C )[ γ ( b, C )] LqαΓ( D)[ γ ( b, D)] Lq ²² + Γ ( E )[ γ ( b, E )] } with Γ( ) α² = σ², K = Γ( ) αγ( ), C = /, D = /, E = /, b = ( j ) q ; b = ( j / ) q ; b α α = jq ; b α = and where γ denotes the incomplete gamma function. Figure illustrates the quantization error () as a function of the quantization step q, for a fixed number of levels, with =, and σ ²=. = σ = L q α D Figure : Quantization error D as a function of the quantization step q. The quantization error reaches a minimum for a value of q denoted here q opt/d. The MSE is also a function of the number of steps N, of the shape parameter, and of the variance σ². - -

8 . Minimum-MSE uniform quantizer The quantization step that minimizes the MSE is obtained by differentiating D(q) with respect to q and equating the result to zero. The differentiate of D(q) is: dd (q) dq where dd g (q) dq and ddo( q ) dq = dd g (q) dq = K α j L = = KL α { L + dd o( q) dq { -j αγ( D)[ γ ( b, D) γ ( b, D)] + j q Γ( E)[ γ ( b, E) γ ( b, E)] + α Γ( D)[ γ ( b, D) γ ( b, D)] - (j-) q Γ( E)[ γ ( b, E) γ ( b, E)] q Γ( E )[ γ ( b, E )] - αγ( D)[ γ ( b, D)] } (5) Our equation allows the calculation of q opt/d for GG pdf with any value of the parameters. Usually books give tables of q opt/d for a set of values of N, and only for a few distributions like the Laplacian and the Gaussian ones [JAIN-89]. Formula (5) allows to determine the quantization step of the minimum-mse uniform quantizer by solving the non-linear equation dd ( q ) =. We used toolboxes of the MatLab package from MatWorks, which dq resolves non-linear equations by the Gauss-Newton method. Figure shows that the penalty for choosing q too small compared with q opt/d is much more important than the penalty for choosing q too high. The overload and granularity error curves provide an insight to the penalties observed when q departs from q opt/d. The optimum quantization step is reached when the sum of the overload and granularity noise is minimal. Below q opt/d the overload error is dominant, and above q opt/d, the granularity error is dominant. The MSE increases more rapidely with increasing overload than with increasing granularity. If the quantization step q is not the minimum-distorsion value, overload distortion should be carefully avoided because it is more penalizing than granularity. In practical situations, it is preferable to over-estimate q as compared with q opt/d rather than under-estimate it. - -

9 = σ = = σ = D (a) (b) σ = σ = D = (c) (d) = D (e) (f) Figure : MSE and SNR as a function of the quantization steps, for different values of N,, and σ². Figure shows the MSE as a function of q, given various numbers of quantization levels N (figure -a), shape parameters (figure -c), and variances σ (figure -e). A plot of the Signal to Noise Ratio (SNR), with SNR = log (σ ²/MSE), is added as a companion of the MSE (figure -b, -d, and -f). Figure -a & b show, as expected, that large values of N result in a low distortion. In addition, as q, all the MSE curves for various values of N converge towards the same asymptotic curve. In spite of what a prime interpretation of the influence of on q opt/d could suggest, figure - c & d show that the values of the optimal quantization step decrease as the shape parameter increases. One could have expected q opt/d to increase with, in order to limit - -

10 the overload. However, when the shape parameter increases, the pdf tends toward the uniform distribution, as seen on figure -a. For uniform quantizers with a uniform pdf, the whole input range is used, the saturation distortion tends toward zero. Thus, when increases, the pdf-optimization procedure yields small values of q opt/d in order to limit the granularity distortion. When properly optimized, quantizers with large pdf will perform better than quantizers with small pdf. However, a quantization step different from q opt/d is more penalizing for large than for small. In practical situations, minimum-mse uniform quantizers with input pdf of large shape parameters will perform better than with input pdf of small shape parameters. But the penalty for having a poor optimization of the quantization step is more important with large shape parameter pdfs. Concerning the influence of σ² on q opt/d, figure -e & f show that the values of the optimal quantization step increase as the variance increases. Pdfs with a large variance are widely spread but they do not tend towards a uniform pdf, their shape remains that of a generalized Gaussian distribution. The saturation has to be limited during the quantizer optimization process by having a large quantization step. Figure -f shows that uniform quantizers with various input pdf variance have all the same maximum SNR value. However, the penalty for poorly optimized quantizers is higher with small variances. In practical situations, minimum-mse uniform quantizers having different variances perform identically in terms of SNR. But the penalty for having a poor optimization of the quantization step is more important with small variance pdfs.. Entropy of uniform quantizers with GG pdf The entropy of the output of a quantizer is the minimum amount of information to be transmitted in order to be able to reconstruct the quantizer output with an arbitrarily small error. It is also referred to as the lower bound data rate, or bit rate for a given distortion. It is expressed in bits per sample (bps). It is given by: p p H Q = - log ( ) bits/sample (6) i i i Application of formula (6) to uniform midtread quantizers yields: H Q = - { where p j = p = p = ( j+ / ) q ( j / ) q q / L q L p p j j j = pxdx ( ) pxdx ( ) pxdx ( ) log ( ) + p log (p ) + p log (p ) } (7) - 5 -

11 Incorporating the pdf definition () into (7) results in: p j = K α Γ( )[ (, ) (, )] E γ b E γ b E 5 p = K α Γ( E) γ ( b, E) 6 p = K α Γ( )[ (, )] E γ b E 7 (8) with ( j / ) q b = α ( j + / ) q b 5 = α b 6 = q / α L q b 7 = α = σ = (a) σ = Ν=5 = (b) Figure 5: Entropy as a function of the quantization steps for different values of N,, and σ². Figure 5 plots the entropy of formula (7) as a function of q, for different values of N,, and σ². There is a value of q that maximizes the entropy, denoted q opt/r. As expected, the entropy of quantizers with a large number of levels N is greater than the entropy of quantizers with small N, and q opt/r decreases with increasing N. q opt/r increases with increasing (on the contrary, q opt/d decreases with increasing ). The entropy of minimum-mse quantizers increases with increasing. q opt/r increases with increasing σ² (similarly, q opt/d increases with increasing σ²). The entropy of minimum-mse quantizers is independant of σ² (c)

12 .5 Entropy-constrained uniform quantizer with GG pdf The quantization step of an entropy-constrained quantizer should minimize the distortion subject to a fixed entropy constraint H. Using to the Lagrangian multiplier method, the solution of this problem minimizes the following functional: J(q) = D(q) + λ [H Q (q)- H ] (9) By differentiating J(q) with respect to q and λ, equating the result to zero, and choosing λ so that H(q) = H, the problem is to solve the system of non-linear equations: dj dd( q) dh( q) = + λ = dq dq dq dj = Hq ( ) H = dλ () After incorporating formulas (5) and (A), from the appendix, into (), we are able to resolve () for any a priori number of levels N, any fixed entropy H, and any GG input pdf. We used MatLab toolboxes that resolve sets of non-linear equations with the Gauss- Newton method. Our approach results in a practical method for designing N-level entropy-constrained uniform quantizers (definition ). Note that minimum-mse uniform quantizers (definition ) are designed simply by taking λ= and relaxing the constraint H(q)- H.=. The performance of an entropy-constrained optimum quantizer is assessed by its ratedistortion curve R(D). It is well known that for each pdf there exists a bound, called the rate-distortion bound R B (D), such that: R(D) R B (D) () The minimum bit rate needed to transmit a quantized signal is determined by the entropy of the quantizer output. This entropy H Q is given by: H Q H s - log (q) () Where H s is the differential entropy of the source: + H s = - px ( )log pxdx ( ) () For minimum-mse uniform quantizers, the uniform pdf yields the lowest possible distortion. According to the well-known formula of the distortion for uniform quantizers with a uniform pdf, we have: D = q = (5) The entropy of the quantizer output can be bounded by H Q for q =. Incorporating () into () results in the Gish-Pierce asymptote of the rate-distortion performance. Figure 6 shows an example of R(D) curve for a 5-levels uniform quantizer with a Laplacian unity variance pdf (= and σ²=). The relation between the entropy R, the quantization step q, and the distortion D is shown on the figure. The R(D) curve (figure 6- b) has an optimum that is reached when the best compromise between the highest rate and the lowest distortion is achieved. This point corresponds to the optimum entropy

13 constrained uniform quantizer. Its quantization step is denoted q opt/r-d. Below or above this optimum quantization step, the distortion is higher (figure 6-b and c). For q> q opt/r-d the entropy is lower, and for q< q opt/r-d the entropy is higher which is less favorable to compression (figure 6-a). Similarly to minimum-mse quantizers, for entropy-constrained quantizers it is preferable to over-estimate q as compared to q opt/r-d rather than to under-estimate it Gish-Pierce lower bound (a) = σ²= (b) Figure 6: Rate-distortion performance R(D) of matched uniform quantizers (6-b), correspondence with the curves R(q) (6-a), and q(d) ( 6-c). (c) - 8 -

14 6 5 N=6 GP: Gish-Pierce lower bound N= N=7 = σ²= N= (a) =. GP =. σ²= =. =. =.5 5 σ =.5,.,., &. σ²= GP =. GP =. GP σ=.5,.,., &. GP = (b) (c) Figure 7: Rate-distortion performance R(D) of matched uniform quantizers. Figure 7 shows the rate-distortion performance of entropy-constrained uniform quantizers, given various values of N (figure 7-a), (figure 7-b), and σ² (figure 7-c). As expected, large values of N result in entropy-constrained quantizers with a high entropy and a low distortion (figure 7-a). The R(D) performance is closer to the Gish-Pierce lower bound for large N. The difference between the lower bound is less than.5 bit with 6 levels and more than bit with 7 levels and less (= and σ²=). The performance of optimum entropy-constrained uniform quantizers increases with increasing (figure 7-b). But the distortion increase, when one departs from the optimum, also increases with. The performance of entropy-constrained uniform quantizers is independant of the variance, whether they are optimum or not. These observations are similar to the findings of section. for minimum-mse quantizers.. Mismatched uniform quantizers. Mismatch relative to the shape Quantizer mismatch refers to practical situation of non-adaptive quantization when the input pdf is different from the pdf expected for the design of the quantizer. Shape mismatch occurs when the shape parameter X of the input signal pdf differs from the - 9 -

15 shape parameter Q used for the quantizer design (i.e. for determining the optimum opt quantization step ). Various generalized Gaussian pdf shapes were given in figure -a q Q with a fixed variance σ ²=. Without lack of generality, unit variance pdfs will be considered throughout of the current section... Minimum-MSE uniform quantizers As a first insight regarding the effect of the shape parameter on the quantizer, figure 8 gives the distortion as a function of for MSE-optimized quantizers, i.e. when q opt/d is evaluated and used for each point of the curve. Figure 8 shows that the distortion of minimum-mse uniform quantizers decreases with increasing. D N= N=7 N= N= N=6 N= N=7 N= Figure 8: Distortion of matched minimum-mse uniform quantizers as a function of the shape parameter. Figure 9 illustrates the relative performance of uniform quantizers when the quantization step departs from the optimum, i.e. in case of mismatch relative to the shape parameter. It shows the distortion as a function of, each curve being computed with only one value of q opt/d. Figure 9-b shows that when X < Q, the SNR is much lower than at the expected optimal point where X = Q. If the input pdf has a shape parameter lower than the quantizer shape parameter, the performance of the quantizer for this input pdf will be poor. Conversely, when X > Q, the SNR is slightly higher than expected. For X much greater than Q, the SNR reaches an asymptote. If the input pdf has a shape parameter higher than the quantizer shape parameter, the quantizer performance for this input pdf will be slightly better than expected because the input pdf is closer to the uniform pdf than the quantizer pdf itself. - -

16 q opt/d, = σ = 8 6 q opt/d, = q opt/d, = q opt/d, = D q opt/d, = q opt/d, = q opt/d, =.5 SNR 8 q opt/d, = beta beta Figure 9: MSE and SNR as a function of the shape parameter for different values of q opt/d : shape mismatch. As an example using data from figure 9-b, if the quantizer was optimized for Q =, and the input is X =.5, then the expected SNR is 8.5 db, but the observed SNR is only 8 db. Choosing a model pdf with a too high shape parameter compared with the real input results in a poor quantizer performance compared with expection. If Q =.5, and X =, then the expected SNR is.5 db, but the observed SNR is slightly higher: db. Choosing a too small shape parameter for the quantizer compared with observed input pdf parameters does not degrade the quantizer performance. It slightly increases it compared with expectation, but the global quantizer performance remains relatively poor. The study of the quantizer robustness in terms of SNR when the input shape parameter deviates from its expected is of major interest. Figure shows the deviation / about the MSE-optimum Q, assuming that a SNR deviation of ±.5 db is acceptable. A deviation of -.5 db is observed when X < Q, and a deviation of +.5dB is observed when X > Q. A small deviation of X is enough to yield a loss of.5db, especially for small input pdf shape parameters X. Clearly, the robustness increases with. This finding is in accordance with the discussion of figure -c&d, and σ²= +.5dB -.5dB Figure : Range of the shape parameter for a deviation of +/-.5dB.. Entropy-constrained uniform quantizers This section addresses the relative performance of matched and mismatched entropyconstrained quantizers with respect to the shape parameter. - -

17 In figure -a, the quantizer is matched for a Laplacian pdf (=). When the targeted entropy is above the optimum point of the matched quantizer, it is favorable to have an input pdf with X > Q because the real entropy is comparable to expectations, and the distortion is much lower. Here, X < Q yields a slightly better entropy than expected but a worse distortion. When the targeted entropy is below the optimal point, having X > Q does not make much difference. Having X < Q increases the distortion. As the entropy gets even lower, there is an entropy below which the real entropy and distortion are a little better than expected. Such a reversed situation where X < Q is more favorable is not observed in figure -b where Q =. Globally, with entropy constrained quantizers, under-estimation of the quantizer shape parameter Q compared with the input pdf X is more favorable because the real distortion is better than expected. At low bit rates, the robustness of the quantizer relative to shape mismatch is higher..5 mismatched, = matched, = mismatched, =.5 mismatched, =.5 matched, = mismatched, = σ = σ = Figure : Rate-distorsion performance of shape mismatched entropy-constrained uniform quantizers.. Mismatch relative to the variance Variance mismatch occurs when the variance σ ² X of the input signal pdf differs from the variance σ ² Q used for the quantizer design (i.e. for determining the optimal quantization opt step ). Various generalized Gaussian pdf variances were illustrated in figure -b with q σ ² Q a fixed shape parameter =... Minimum-MSE uniform quantizers We first study both the quantization distortion and SNR as a function of σ ² for the minimum-mse quantizer, i.e. when q opt/d is evaluated and used for each value of σ ². Figure shows such plots for various values of quantization steps, each point was drawn with the exact optimum q corresponding to the value of σ ². The MSE of minimum-mse quantizers is a linear function of the variance, and as a result the SNR is constant. This can obviously be also deducted from section. and figure -e & f. - -

18 .9 = N= 6 55 N= N= D N=7 5 N= N= N= N= Figure : Distortion of minimum-mse quantizers as a function of the variance σ ². The value of the input pdf variance does not influence the SNR performance of minimum-mse uniform quantizers. This can obviously be also deducted from section. and figure -e & f. This results is known [JAIN-89], but the equation () does not shoew an obvious linear relationship between the distostion and the variance. Figure illustrates the relative performance of uniform quantizers when the quantization step departs from the optimum. It shows the distortion as a function of σ ², each curve being computed with only one value of q. Figure -b shows that whenσ ² X σ ² Q,, the SNR is lower than at the optimal point σ ² X =σ ² Q. The penalty for under-estimating the variance is slightly higher than the penalty for over-estimating the variance because the overload distortion rises very rapidely when the variance is greater than expected (see figure and ) q opt/d, σ = q opt/d, σ = q opt/d, σ = D s q opt/d, σ = q opt/d, σ = q opt/d, σ = SNR s q opt/d, σ = q opt/d, σ =.5 Figure : Distortion as a function of the variance σ² for different values of q opt/d. As an example using data from figure -b, if σ ² Q =, and σ ² X =.5, then the expected SNR is 5.5 db, but the observed SNR is only. db. If σ ² Q =.5, and σ ² X =, then the expected SNR is also 5.5 db (because the minimum-mse quantizer performance is independent of the variance), but the observed SNR is only. db. - -

19 All matched minimum-mse quantizers perform the same regarding the variance. Mismatch of the quantizer relatively to the input variance is always penalizing especially if the input variance is smaller than the variance of the quantizer... Entropy-constrained uniform quantizer This section addresses the relative performance of matched and mismatched entropyconstrained quantizers with respect to the variance. Figure shows the rate-distortion of the entropy-constrained quantizer with a variance mismatch, when the quantizer design is matched σ²=. Variance mismatch has no effect on the performance of entropy-constrained quantizers. The only difference lies on the existence of points on the R(D) curve. When σ ² X <σ ², more points exist with entropy below the optimum, and vice-versa..5.5 matched σ = and mismatched, σ =.5 &.5.5 = mismatched, σ = matched, σ = mismatched, σ = Figure : Rate-distorsion performance of variance mismatched entropyconstrained uniform quantizers.. Discussion and conclusion We have focused this paper on scalar quantizers with Generalized Gaussian distributions. GG distributions cover a wide range of possible distributions and relate well with distributions encountered in coding applications. We limited this work to the uniform quantizers because they have interesting theoritical and practical properties. They give nearly optimum solutions to entropy constrained quantization ([WOOD-69], [NOLL-78], [BERG-8]). Their simple implementation makes them attractive to waveform and image coding. A mathematical formulation of the quantizer distortion with a midtread quantizer and of its derivate, as well as a formulation of the entropy and its derivate give a practical method - -

20 for determining minimum-mse and entropy-constrained quantizers. With our quantizer design method, it is possible to study in detail the properties of minimum-mse and entropy-constrained quantizers. Particularly, quantizer mismatch, when the input pdf differs from the pdf used for the quantizer design has not been extensively studied dispite its practical interest for memoryless source coding or non-adaptive quantization. [MAUE- 79] reported results for mismatched Max-Lloyd quantizers. He found that the quantizer shape parameter should be chosen as a lower bound to the input shapes ( Q X ) and that variance mismatch is not very critical. [JAYA-8] gives results of mismatch for nonuniform and uniform minimum-mse quantizers, only with levels. He suggests that in these conditions, the performance of uniform and non-uniform quantizers are very similar, and that the difference would be more significant at higher bit rates. Our results are in agreement with the previous findings, and extend them to more pdfs, more bite-rates, and to entropy-constrained quantizers. They lead to practical conclusions for the design of uniform midtread quantizers: Influence of the shape parameter Let us assume that the input pdf shape parameters X lies in an interval [ Xmin, Xmax ]. If the quantizer is designed with Q = Xmin, i.e. the input shape parameter is always larger than the quantizer shape parameter, then the quantizer output distortion is lower than expected. The approach yields better performance than expected, but is conservative (better performance could be achieved for the highest values of X ). In this situation, the quantizer mismatch corresponds to quantization steps always higher than the minimum- MSE optimum, and granularity error is important. The R(D) performance of an entropyconstrained quantizer with Q = Xmin is globally robust to shape mismatch, or gives better performance than expected. Conversly, if Q = Xmax the distortion will be higher than expected for small shape parameters. Here, the poor optimization results in a too small quantization step compared with the minimum-mse, and the overload distortion is important. The R(D) performance of an entropy-constrained quantizer with Q = Xmin is not very robust to shape mismatch, yielding higher distortions than expected especially at higher rates. Under-estimating the shape parameter of the quantizer is penalizing, over-estimated it is slightly advantageous. The robustness of entropy-constrained quantizers increases with decreasing bit-rates. Influence of the variance Let us assume that the input pdf variance σ X lies in an interval [σ Xmin, σ Xmax ]. If the quantizer is designed with σ Q = σ Xmin, i.e. the input variance is always larger than the quantizer variance, then the quantizer output distortion is higher than expected. The approach yields worse results than expected, and worse than if σ Q = σ Xmax. In this situation, the quantizer mismatch corresponds to a quantization step always smaller than the minimum-mse optimum, and overload error is important. The R(D) performance of an entropy-constrained quantizer with σ Q = σ Xmin or σ Q = σ Xmax is robust to variance mismatch. If σ Q = σ Xmax the distortion will be higher than expected. Here, the poor optimization results in a too large qiantization step compared with the minimum-mse, and granularity distortion is happening. Mismatch of the quantizer relatively to the input variance is always penalizing. The entropy-constrained quantizers are robust to variance mismatch