When is overload distortion negligible in uniform scalar quantization?

Studies the asymptotic characteristics of uniform scalar quantizers that are optimal with respect to mean-squared error (MSE). When a symmetric source density with infinite support is sufficiently well behaved, the optimal step size /spl Delta//sub N/ for symmetric uniform scalar quantization decreases as 2/spl sigma/N/sup -1/V~/sup -1/(1/6N/sup 2/), where N is the number of quantization levels, /spl sigma//sup 2/ is the source variance and V~/sup -1/(/spl middot/) is the inverse of V~(y)=y/sup -1/ /spl int//sub y//sup /spl infin// P(/spl sigma//sup -1/X>x) dx. Equivalently, the optimal support length N/spl Delta//sub N/ increases as 2/spl sigma/V~/sup -1/(1/6N/sup 2/). Granular distortion is asymptotically well approximated by /spl Delta//sub N//sup 2//12, and the ratio of overload to granular distortion converges to a function of the limit /spl tau//spl equiv/lim/sub y/spl rarr//spl infin//y/sup -1/E[X|X>y], provided, as usually happens, that /spl tau/ exists. When it does, its value is related to the number of finite moments of the source density, an asymptotic formula for the overall distortion D/sub N/ is obtained, and /spl tau/=1 is both necessary and sufficient for the overall distortion to be asymptotically well approximated by /spl Delta//sub N//sup 2//12. Applying these results to the class of two-sided densities of the form b|x|/sup /spl beta//e(-/spl alpha/|x|/sup /spl alpha//), which includes Gaussian, Laplacian, Gamma, and generalized Gaussian, it is found that /spl tau/=1, that /spl Delta//sub N/ decreases as (ln N)/sup 1//spl alpha///N, that D/sub N/ is asymptotically well approximated by /spl Delta//sub N//sup 2//12 and decreases as (ln N)/sup 2//spl alpha///N/sup 2/, and that more accurate approximations to /spl Delta//sub N/ are possible. The results also apply to densities with one-sided infinite support, such as Rayleigh and Weibull, and to densities whose tails are asymptotically similar to those previously mentioned.

It is known that the generalized Lloyd method is applicable to locally optimal multiple description scalar quantizer (MDSQ) design. However, it remains unsettled when the resulting MDSQ is also globally optimal. We partially answer the above question by proving that for a fixed index assignment there is a unique locally optimal fixed-rate MDSQ of convex cells under Trushkin's sufficient conditions for the uniqueness of locally optimal fixed-rate single description scalar quantizer. This result holds for fixed-rate multiresolution scalar quantizer (MRSQ) of convex cells as well. Thus, the well-known log-concave probability density function (pdf) condition can be extended to the multiple description and multiresolution cases.

Asymptotic formulas are derived for the mean-squared error (MSE) distortion of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> -level uniform scalar quantizers designed to be MSE optimal for one density function, but applied to another, as <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> → ∞. These formulas, which are based on the Euler-Maclaurin formula, are then applied with generalized gamma, Bucklew-Gallagher, and Hui-Neuhoff density functions as the designed-for and applied-to densities. It is found that the mismatch between the designed-for and applied-to densities can disturb the delicate balance between granular and overload distortions in optimal quantization, with the result that, generally speaking, the granular or overload distortion dominates, respectively, depending on whether the applied-to density function has a lighter or heavier tail than the designed-for density. Specifically, in the case of generalized gamma densities, a variance mismatch makes overload distortion dominate for an applied-to source with a slightly larger variance, whereas a shape mismatch can tolerate a wider variance difference while retaining the dominance of the granular distortion. In addition, for the studied density functions, the Euler-Maclaurin approach is used to derive asymptotic formulas for the optimal quantizer step size in a simpler, more direct, way than previous approaches.

This paper determines how the support regions of optimal and asymptotically optimal fixed-rate scalar quantizers (with respect to mean-squared error) depend on the number of quantization points N and the probability density of the variable being quantized. It shows that for asymptotic optimality it is necessary and sufficient that the support region grow fast enough that the outer (or overload) distortion decreases as o(1/N/sup 2/). Formulas are derived for the minimal support of asymptotically optimal quantizers for generalized gamma densities, including Gaussian and Laplacian. Interestingly, these turn out to be essentially the same as for the support of optimal fixed-rate uniform scalar quantizers. Heuristic arguments are then used to find closed-form estimates for the support of truly optimal quantizers for generalized gamma densities. These are found to be more accurate than the best prior estimates, as computed by numerical algorithms. They demonstrate that the support of an optimal quantizer is larger than the minimal asymptotically optimal support by a factor depending on the density but not N, and that the outer distortion of optimal quantizers decreases as 1/N/sup 3/.

This paper studies the asymptotic characteristics of optimum uniform scalar quantizers as N, the number of levels, becomes large. It is shown that the length of the support region increases as (ln N)/sup 1//spl alpha//, when applied to a random variable with a generalized Gaussian density of the form p(x)=ae(-b|x|/sup /spl alpha//). Moreover, the mean-squared error is asymptotically well approximated by /spl Delta//sup 2//12, where /spl Delta/ is the step size, and decreases as (ln N)/sup 2//spl alpha///N/sup 2/. >

In this paper a detail analysis of speech coding algorithm based on forward adaptive technique is carried out. We consider an algorithm that works on frame-by-frame basis, where a frame consists of a certain number of speech samples. Buffering frame-by-frame an estimation of the gain defined as squared root of the frame variance is enabled. The information about the gain (side information) and the code book of a nonadaptive quantizer, which is designed for the unit variance case of the input signal, are further used when designing an adaptive quantizer. In such a way better quantizer adaptation to the varying input statistics is provided. Observe that the goal of this paper is to investigate the preference that for the wide range of variance change could be achieved when implementing in the forward adaptive speech coding algorithm, the recently developed effective method for the Lloyd-Max's algorithm initialization, which provides optimal Lloyd-Max's quantizer performances for the unit variance case of the input signal. We destine to consider the speech coding algorithm based on forward adaptive technique since the backward adaptation provides SQNR (signal to quantization noise ratio) within 1 dB of the forward adaptation. We provide theoretical and experimental results (performances of our algorithm) which are compared with the optimal results. Additionally, we discuss the performances of speech coding schemes designed according to G. 711 standard and we point out the benefits that can be achieved by using our algorithm. Finally, in order to find better solution for implementation of the proposed algorithm in practice we consider the performances of our algorithm when log-uniform as well as uniform scalar quantizer are used for gain quantizing.

The matching pursuit algorithm has been successfully applied in many areas such as data compression and pattern recognition. The performance of matching pursuit is closely related to the selection of the dictionary. In this paper, we propose an algorithm to estimate the optimal dictionary distribution ratio and discuss the decay of the norm of residual signal in matching pursuit when the coefficients are quantized by a uniform scalar quantizer. It is proposed that if the approximation error E and the dimension of the space N are given, the optimal size of the dictionary and the optimal quantizer step should be obtained by minimizing the number of bits required to store the matching pursuit representation of any signal in the space to satisfy the error bound.

The statistical properties of the error in uniform scalar quantization have been analyzed by a number of authors in the past, and is a well-understood topic today. The analysis has also been extended to the case of dithered quantizers, and the advantages and limitations of dithering have been studied and well documented in the literature. Lattice vector quantization is a natural extension into multiple dimensions of the uniform scalar quantization. Accordingly, there is a natural extension of the analysis of the quantization error. It is the purpose of this paper to present this extension and to elaborate on some of the new aspects that come with multiple dimensions. We show that, analogous to the one-dimensional case, the quantization error vector can be rendered independent of the input in subtractive vector-dithering. In this case, the total mean square error is a function of only the underlying lattice and there are lattices that minimize this error. We give a necessary condition on such lattices. In nonsubtractive vector dithering, we show how to render moments of the error vector independent of the input by using appropriate dither random vectors. These results can readily be applied for the case of wide sense stationary (WSS) vector random processes, by use of iid dither sequences. We consider the problem of pre- and post-filtering around a dithered lattice quantifier, and show how these filters should be designed in order to minimize the overall quantization error in the mean square sense. For the special case where the WSS vector process is obtained by blocking a WSS scalar process, the optimum prefilter matrix reduces to the blocked version of the well-known scalar half-whitening filter.

In this paper, we provide operational rate-distortion results for memoryless generalized Gaussian sources. Close approximations of the entropy are provided for these sources, after a uniform scalar quantization at low/high resolution. Asymptotic expressions of the distortion for an arbitrary p-th order error measure are also given. The resulting approximations at low/high bitrate of the operational rate-distortion function are thus compared with the Shannon optimal bound showing the overall good performance of uniform quantization rules.

We present a lossless coding procedure based on a recently introduced decorrelating scheme, where both intra- and interchannel redundancies are removed by lossless prediction. The resulting signals are scalar entropy coded. We show that for continuous-amplitude Gaussian sources discretized with uniform scalar quantizers, no suboptimality occurs in the proposed lossless coding scheme by using scalar instead of vector entropy coders, except from a term caused by the lossless constraint, which vanishes in the limit of small distortions. This lossless constraint is described in terms of excess bit rate. The proposed coder may be used either as a compressor, or as a scalable lossless coder. In this case, a multistage version of the lossless coder based on ADPCM-like lossless prediction loops allows one to transmit the data by means of substreams, which represent different resolution levels. We show that this multiresolution approach is slightly suboptimal in comparison with a single global compression because of the noise feedback created in the ADPCM-like loops, but not of the space filling loss of the scalar quantizers. We propose a strategy to fix the stepsizes of these quantizers so that the delivered rates approach some predetermined target rates.

In this paper, we propose a novel algorithm to design an optimal non-uniform scalar quantizer for distributed video coding, which aims at achieving a coding rate close to joint conditional entropy of the quantized video frames given the side information. Wyner-Ziv theory on source coding is employed as the basic coding principle and the asymmetric scenario is considered. In this algorithm, a probability distribution model, which considers the influence of the joint distribution of input source and side information to the coding performance, is established and used as the optimality condition firstly. Then, a modified Lloyd Max algorithm is used to design the scalar quantizer to give an optimal quantization for input source before coding. Experimental results show that compared to uniform scalar quantization, proposed algorithm can improve coding performance largely, especially at low bit rate

We propose a scheme for rate-constrained distributed nonparametric regression using a wireless sensor network. The scheme is universal across a wide range of sensor noise models, including unbounded and nonadditive noise; it has low complexity, requiring simple operations such as uniform scalar quantization with dither and message passing between neighboring nodes in the network, and attains minimax optimality for regression functions in common smoothness classes. We present theoretical results on the tradeoff between the compression rate, communication complexity of encoding, and the MSE and demonstrate empirical performance of the scheme using simulations.

We introduce an image coding algorithm based on the cortex transform, matching pursuit and perceptual pruning that selects and codes only the most perceptually relevant image components, as determined by a new image indistinguishability criterion that derives from a maximum likelihood paradigm. We demonstrate that this method selects image components that produce images with higher subjective quality than matching pursuit with the same number of components. We also show that our perceptual quantization rule makes a substantial gain over uniform scalar quantization. Finally, we present image coding results of high quality at low encoding rates.

This paper deals with the problem of image retrieval when the database is presented in a compressed form, by using typically the JPEG2000 encoding scheme based on wavelet transform followed by an uniform scalar quantization. The state-of-the-art method aims at applying a preprocessing step before the feature extraction to reduce the difference in the compression qualities between the images. Our contribution consists in extracting robust features directly from the quantized coefficients. More precisely, assuming that the unquantized coefficients within a subband have a Laplacian distribution, we propose to estimate the distribution parameter from the quantized coefficients. Then, the estimated parameters of the whole subbands are used to build a salient feature for the indexing process. Experimental results show that the proposed retrieval approach significantly improves the state-of-the-art one.

A novel optimal adaptive scalar quantization method is proposed, and its performance is investigated in the context of quantization for image compression. The idea is to perform frequent and complete (non-incremental) block-optimal quantizer redesign, but under a codebook constraint which assures that the overhead required to specify the new quantizer is small. The intuition is that a few levels are usually sufficient to accurately describe a small image block, provided these are chosen optimally and independently for each block. In addition to limiting overhead, the codebook constraint can be exploited to derive an efficient dynamic programming algorithm for block-optimal quantizer design. Some experimental results and comparisons are also included.