Dither and data compression

Citation for published version (APA): Schobben, D. W. E., Beuker, R. A., & Oomen, A. W. J. (1997). Dither and data compression. IEEE Transactions on Signal Processing, 45, 2097-2101. https://doi.org/10.1109/78.611218

DOI: 10.1109/78.611218

Document status and date: Published: 01/01/1997

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Dither and Data Compression

Daniel¨ W. E. Schobben, Rob A. Beuker, and Werner Oomen

Abstract—This correspondence presents entropy analyses for dithered and undithered quantized sources. Two methods are discussed that reduce the increase in entropy caused by the dither. The first method supplies Fig. 1. Subtractively dithered quantizer. the dither to the lossless encoding-decoding scheme. It is argued that this increases the complexity of the encoding-decoding scheme. A method to reduce this complexity is proposed. The second method is the usage of a distributed in this interval and is independent of the quantizer input dead-zone quantizer. A procedure for determining the optimal dead-zone [2]. Note that typically the dither is not transmitted to the decoder width in the mean-square sense is given. but is regenerated there. If a nonsubtractively dithered quantizer were Index Terms—Distortion, dither, entropy, quantization. used, in which the dither is not subtracted from the quantizer output, this would result in the same entropy but in an increased distortion. Derivation of the distortion for a nonsubtractively dithered quantizer I. INTRODUCTION is straightforward and is not discussed here. When a signal is quantized coarsely, signal dependent quantization errors are introduced that can be perceptually annoying. Roberts [1] III. DISTORTION AND ENTROPY first used dither in a simple PCM video system to remove false contouring. A survey of the dithering technique is provided in [2]. The stochastic variables associated with the signals xY xY~ dY rY and ~ The objective of this correspondence is to investigate the effect of z will be denoted ˆY ˆYhY‚Y and  throughout. In the sequel, the dithering on the entropy of the quantized source. Very little has been MSE will be used as the distortion measure and is calculated from P published in this area, but we wish to point out the work in [3]. We MSE a i‘@ˆ~ ˆA “ (2) will show that the use of dither can cause an increase in both entropy and mean square error (MSE). Therefore, we will introduce methods where i‘
“ denotes the expectation operator. In the undithered case, to reduce both entropy and MSE. the MSE is smaller than
PaIP for input signals that are small This correspondence is organized as follows. First, in Section II, compared with the quantizer step size
X For larger input signals, an introduction to the dithering technique is given. In Section III, the MSE is approximately equal to
PaIPX When uniform dither is the distortion and entropy are defined, and the source distribution is applied, however, the MSE equals
PaIP independent of the input modeled. It is argued that both entropy and distortion can increase signal’s distribution. Note that the MSE is an objective measure of when dither is used. Two ways of reducing the entropy in a dithered distortion and does not fully reflect the perceived quality. scheme will be considered. In Section IV, it is shown that the entropy The entropy represents the lower bound of the bit rate that can be will increase by applying dither prior to quantization if the entropy achieved by losslessly coding the quantized data. The entropy, in bit coder has no knowledge of the dither values. The effect on the entropy per sample (bps), is defined by when the dither values are known to the lossless encoding-decoding I scheme is also discussed. Another way of reducing the increase r@Aa pz @iAlogP@pz@iAA (3) in entropy using a dead-zone quantizer is discussed in Section V. iaI Furthermore, a method for designing an optimal dead-zone quantizer where pz@iA is the probability that the quantizer output equals i
X is shown. Results of experiments with dither in subband coding and For a uniform quantizer, this probability is given by transform coding schemes are discussed in Section VI. ‘iC@IaP“A
pz@iAa pr@rA dr (4) II. DITHERED QUANTIZERS ‘i@IaP“

A subtractively dithered quantizer, as shown in Fig. 1, is used with r a x C dX The probability density function (pdf) of ‚Y pr@rA is throughout this correspondence. Its transfer function is given by equal to the convolution of the pdf’s of ˆ and h, which are given p @xA p @dA x~ a @x C dA d by x and d , respectively. In this case, the entropy of the continuous random variable ‚ a ˆ C h is greater than the entropy x I @xAa
C of ˆX Typically, the same holds for the entropy of quantizer output
P (1) [4], i.e.,1 ˜
™
where indicates the floor operator, and represents the quan- r@@ˆ C hAA ! r@@ˆAAX (5) tizer step size. Note that the quantizer  is uniform and infinite. If the dither d is a white noise signal uniformly distributed on The calculation of both entropy and distortion require a model for ‘@
aPAY
aPA, the quantization error is also white and uniformly the source distribution. We use two pdf’s—the uniform pdf and the generalized Gaussian pdf (GG-pdf) [5]—and model the source as a Manuscript received February 23, 1996; revised March 27, 1997. The associate editor coordinating the review of this paper and approving it for random variable. The pdf of a video signal in a simple PCM system publication was Dr. Sawasd Tantaratana. roughly resembles the uniform pdf. The generalized Gaussian pdf D. W. E. Schobben is with the Eindhoven University of Technology, is used to model the AC coefficients and differentially coded DC Eindhoven, The Netherlands. coefficients in a transform coding system, such as, for example, R. A. Beuker and W. Oomen are with the Philips Research Laboratories, Eindhoven, The Netherlands. 1 Use h@x C #A ! h@xAY with x independent of #Y combined with [4, th. Publisher Item Identifier S 1053-587X(97)05794-2. 16.6.3, p. 496] to show that r@@x C #AA ! r@@xAAX

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Fig. 2. Increase in entropy due to dithering a uniform source. Fig. 3. Increase in entropy due to dithering a uniform source if the dither is supplied to the encoder and decoder.

DCT and subband transforms [6]. The uniform pfd is given by px@xAaw@xAwith I w w
Y `x w@xA a w P P (6) HY otherwise where w defines the width of the pdf. Given the positive constants — and ˜, which are defined by ˜ I @QaA — a Y˜a P@IaA '@IaA (7) Fig. 4. Dithered source coding system. The dither is partly known to the lossless coding scheme. where @xA is the Gamma function [5], the GG-pdf is given by P ‘˜jx"j“ px@xY "Y ' YAa—e (8) where "Y 'PY are mean, variance, and shape parameter of the distribution, respectively.

IV. MINIMIZING THE ENTROPY—METHOD I In [3], it was shown that the entropy can be reduced by supplying the dither to the lossless encoding-decoding scheme. The effect of this method on the entropy is discussed in this section for uniform and generalized gaussian input distributions. The input signal is first modeled by a uniform pdf. The entropy is plotted in Fig. 2 as a function of the pdf width w relative to the quantization stepsize
X Note that the entropy only depends on this ratio. The entropy of the output of the quantizer in the undithered and dithered case are indicated with r@@ˆAA and r@@ˆ C hAA, respectively. The difference between these two curves, which is the increase of the entropy due to dithering, is also shown. Fig. 2 shows that this increase can amount up to 1.1 bps when the width of the Fig. 5. Loss in entropy due to the quantization of the dither relative to pdf equals
X r@@ˆ C hAA r@@ˆ C hAjhA (see text). Using the current dither value in the lossless encoder and decoder can yield a reduction of the entropy. The entropy in case the current in entropy due to dither can now be either positive or negative, which dither value is not used by the lossless coder equals r@@ˆ C hAAX implies that for some widths the coding efficiency can even improve In case the dither value is used, the entropy is given by I when using dither. The problem in a practical implementation when h r@@ˆ C hAjhAa r@@ˆ C #AApd@#A d# (9) using the dither in the entropy coder is the complexity increases. I Usually, the dither is amplitude discrete, i.e., d PfdHY


YdxIg V where the entropy r@@ˆ C #AA defines the entropy of the output with x large, say, P X The lossless encoder and decoder need to keep of the quantizer  given the value of the dither #X The entropy a table containing estimates of the probabilities of @ˆ CdA for each r@@ˆ C hAjhA is shown in Fig. 3. The increase in entropy due value of the dither value dY i.e., x tables in total. to dithering, given by r@@ˆ C hAjhA r@@ˆAA, is also shown A solution is to use a coarsely quantized version of the dither and amounts to 0.7 bps for a pdf width of
X Note that the increase instead of the true value of d; see Fig. 4. The dither is added to the

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(a) (b) Fig. 6. (a) Entropy loss due to dithering, Laplacian source. (b) Same, relative to the entropy when no dither was supplied. quantizer input, the result is quantized (Q) and entropy coded (C). The entropy coder has limited knowledge of the used dither value given by dH a H@dAX The decoder performs the reverse process. The H H H H p-level quantized dither d a  @dA is an element of fdHY


YdpIgX The entropy of the quantized signal now equals r@@ˆ C hAjH@hAA pI H H H a r@@ˆ C hAj @hAadiApd @diA (10) iaH where @iCIA@
apA@
aPA H Fig. 7. Dead zone quantizer. pd @diAa pd@#A d#X (11) @
apA@
aPA

The relative entropy loss caused by applying only a 2-, 4-, or 8- the dead-zone quantizer is depicted in Fig. 7. Note that the resulting level quantized version of the dither to the coder instead of the quantization error will no longer be fully uncorrelated with the input full precision dither is shown in Fig. 5 for the uniform source signal. distribution. In the legend of this figure, these levels are denoted as In Fig. 8, the entropy and MSE/'P are plotted as a function H H H I@hAYP@hAYQ@hA, respectively. When the dither is quantized of 'a
for a Laplacian input and for the dead-zone widths to 2, 4, or 8 levels, respectively, only 40%, 20%, and 15% of the f
Y Q
Y P
Y S
Y Q
gX The relevant plots are indicated with “zero H P P obtainable coding improvement is lost. Note that I@dA corresponds subtraction.” The distortion increases monotonically with the dead- to the sign of the dither. The increase in entropy due to dithering is zone width, and the entropy decreases monotonically with the @ aIA plotted in Fig. 6(a) (solid curve) for a Laplacian pdf as a dead-zone width. The entropy is also plotted for the case that the 'a
' function of , where is the standard deviation of the distribution. dither is supplied to the lossless coder. Fig. 8(b) shows that applying 'a
X Note that the entropy only depends on the ratio The entropy the dither to the coder only reduces the entropy significantly if a loss amounts up to 0.52 bits per sample. Again, applying the dither uniform quantizer is used. to the lossless encoder and decoder reduces this loss significantly Fig. 8(a) shows that the MSE can become even larger than the (dashed curve). Even more important than the absolute loss is the signal power 'P for
b Q'X The distortion can be limited by not relative entropy loss. This is the increase in entropy due to dithering subtracting the dither from the quantizer output in case the quantizer relative to the undithered entropy, as is shown in Fig. 6(b). This output equals zero. Signals with a small standard deviation with
) 'X figure shows that the entropy increases excessively for respect to the quantizer step size are then quantized to zero. The When 'a
` HXQ, this increase in entropy is more than 100%. When resulting noise power will approximately equal the signal power that the dither is supplied to the encoder and the decoder, the relative P is smaller than
aIPX This is indicated with “no zero subtraction” entropy loss is decreased significantly, but it is still very large for in Fig. 8(a). The figure shows that the MSE no longer exceeds the 'a
` HXQX This is important when using dither in subband coding signal power for the deadzone quantizers considered. or in transform coding schemes since most of the transformed data The rate and distortion plots of Fig. 8 are combined to yield rate- is no longer quantized to zero. A method that reduces the entropy distortion curves of Fig. 9(a). In these rate-distortion plots, the dither significantly for small 'a
is discussed in the next section. is not supplied to the encoder and decoder, and the dither is not subtracted when the quantizer output equals zero. Curves for 31 V. MINIMIZING THE ENTROPY—METHOD II dead zones widths in between
and Q
are shown in Fig. 9(a). Another way of reducing the entropy is the usage of a dead-zone In Fig. 9(b), some of these curves are magnified. It follows from this quantizer [7] instead of a uniform quantizer. The transfer function of figure that the curves corresponding with different dead zones cross

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(a) (b) f
Y Q
Y P
Y S
Y Q
gX Fig. 8. (a) Distortion and (b) rate of a Laplacian source for dead-zone widths P P

(a) (b) Fig. 9. (a) Rate-distortion of a Laplacian source for several dead-zone widths. (b) Magnification of plot (a). each other. Therefore, the optimal dead-zone width depends on the entropy to be achieved. The dead-zone width that yields the lowest possible distortion at a given entropy was extracted from Fig. 9(a).  a I Y IY Q X This dependency is plotted in Fig. 10 for P P Note that  aIcorresponds to a Laplacian source. The optimal quantizer step size and dead-zone width can now be obtained in the following way. First, the standard deviation @'A and the peakedness @A of the source distribution are estimated [6]. Next, the optimal dead-zone width corresponding to the desired bit rate is read from Fig. 10. Then, the ratio 'a
is read from the solid curve in Fig. 8(b) corresponding with this dead-zone width and for the desired bit rate. Since the standard deviation is readily estimated, the quantizer step size can now be calculated from this ratio.

VI. EXPERIMENTS In [8], experimental results were presented. In the first place,  a I Y IY Q X improvements are reported when dithering the first subband in a four- Fig. 10. Optimal dead-zone as a function of the entropy for P P band audio subband system. In addition, noise shaping was used to shape the resulting flat noise floor conforming to a psychoacoustical hearing curve. Second, experiments have been done with the usage quantizer output to reduce the distortion. No significant improvements of dither in a video DCT system. Dead-zone quantizers were used in perceptual quality are reported, however, when using dither in this to reduce the entropy, and the dither was not subtracted for zero application.

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VII. CONCLUSIONS Frequency Rate Estimation at High SNR Dithering is a technique that renders the quantization error white, uniformly distributed, and independent of the input signal. Care must Peter Handel¨ and Petr Tichavsky´ be taken, however, when dithering signals with a small standard deviation with respect to the quantizer step size. Both entropy and Abstract—The problem of estimating the frequency rate-of-change of distortion can increase excessively in this case. This increase in complex-valued frequency-modulated signals from noisy observations is entropy can be reduced by using the dither values in the lossless considered. The performance of four related estimators is studied, both encoder and decoder. Furthermore, a dead-zone quantizer can be analytically and by means of simulations, and their relationship to the used. In this case, the resulting error will, however, no longer be fully estimators proposed by Djuric/Kay´ and Lang/Musicus is established. uncorrelated with the input signal. The increase in distortion, which occurs for sources with a small standard deviation compared with I. INTRODUCTION the quantizer step size, can be reduced by not subtracting the dither In the signal processing literature, several estimators have been when the quantizer output equals zero. A method for the calculation proposed for estimation of the frequency rate-of-change of complex- of the optimal quantizer step size and dead-zone width is presented valued linear frequency-modulated signals from noisy discrete time for a dithered quantizer. observations. An early reference is [1], and a comprehensive survey of instantaneous frequency estimators is given in [2]. Recent work in- cludes FFT-based methods that utilize the polynomial phase transform ACKNOWLEDGMENT [3], [4] and methods that utilize the so-called Tretter approximation The authors would like to thank J. Ritzerfeld and P. Sommen of [5], including the estimators in [6]–[8] and methods based on fourth- the Eindhoven University of Technology for their support. order sample moments, [9]. There is a strong relationship between frequency rate estimation and the classical problem of estimating the frequency of a sinusoidal REFERENCES signal embedded in additive noise, and existing methods for the latter problem may often be modified in order to solve the former; see, for [1] L. G. Roberts, “Picture coding using pseudo-random noise,” IRE Trans. Inform. Theory, vol. 8, pp. 145–154, Feb. 1962. example, [7], where the technique of [5] is employed for estimation [2] S. P. Lipshitz et al., “Quantization and dither: A theoretical survey,” J. of frequency rate. The aim of this correspondence is twofold. First, Audio Eng. Soc., vol. 40, pp. 355–375, May 1992. the linear predictor and weighted linear predictor frequency estima- [3] R. Zamir and M. Feder, “Rate distortion performance in coding band- tors in [10] are generalized to frequency rate estimation, and their limited sources by sampling and dithered quantization,” IEEE Trans. performances are characterized. In addition, the generalization of the Inform. Theory, vol. 41, pp. 141–154, Jan. 1995. [4] T. M. Cover and J. A. Thomas, Elements of Information Theory. New phase averager and the weighted phase averager [10] are considered. York: Wiley, 1991, 2nd ed. Second, it is shown that the weighted phase averager coincides with [5] M. Abramowith and I. A. Stegun, Eds., Handbook of Mathematical the Gauss–Markov estimator in [6] and (after a proper transformation) Functions. New York: Dover, 1965. coincides with the estimator in [7]. Thus, a relationship with the [6] K. Sharifi and A. Leon-Garcia, “Estimation of shape parameter for generalized gaussian distributions in subband decompositions of video,” estimators in [6] and [7] is established. IEEE Trans. Circuits Syst. Video Technol., vol. 5, pp. 52–56, Feb. 1995. Consider [7] N. S. Jayant and P. Noll. Digital Coding of Waveforms. Englewood i0 Cliffs, NJ: Prentice-Hall, 1984. x– a s– C v–Y–aIY


YxY s– aee (1) [8] D. W. E. Schobben, “Dither and data compression,” M.S. thesis, Tech. i" Univ. Eindhoven, Eindhoven, The Netherlands, Sept. 1995. where e a jeje is a complex-valued amplitude. The noise v– is zero mean complex-valued white Gaussian with variance 'PX The P real and imaginary parts of v– are independent with variances ' aP, respectively. Further  0 aP% f–C –P – P (2) where f is the normalized frequency, and  is the frequency rate of change. The parameters @eY fY Y 'PA are all unknown, but here, the frequency rate is the only parameter of interest. For an unambiguous estimation of , it is required that  P @HXSY HXSAX

Manuscript received February 12, 1996; revised April 7, 1997. This work was supported by the Swedish Research Council for Engineering Sciences under Contract 282-96-88 and by the Grant Agency of the Academy of Sciences of the Czech Republic through Grant K 1075601. The associate editor coordinating the review of this paper and approving it for publication was Dr. Andreas Spanias. P. Handel¨ is with the Signal Processing Laboratory, Tampere University of Technology, Tampere, Finland, on leave from the Research and Development Division, Ericsson Radio Systems AB, Stockholm, Sweden. P. Tichavsky´ is with the Institute of Information Theory and Automation, Academy of Sciences of the Czech Republic, Prague, Czech Republic. Publisher Item Identifier S 1053-587X(97)05795-4.

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