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 Document Version: Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication: • A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. 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If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, please follow below link for the End User Agreement: www.tue.nl/taverne Take down policy If you believe that this document breaches copyright please contact us at: [email protected] providing details and we will investigate your claim. Download date: 26. Sep. 2021 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 45, NO. 8, AUGUST 1997 2097 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 Y YhYY 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@~ A (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 iX 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@IaPA 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 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 C h is greater than the entropy x I @xAa C of X 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@@ C hAA ! r@@AAX (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 1053–587X/97$10.00 1997 IEEE Authorized licensed use limited to: Eindhoven University of Technology. Downloaded on July 14,2010 at 07:20:30 UTC from IEEE Xplore. Restrictions apply. 2098 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 45, NO. 8, AUGUST 1997 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 Ya 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 ' YAae (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@@AA and r@@ C hAA, respectively.