The history of the theory and practice of quantization dates to 1948, although similar ideas had appeared in the literature as long ago as 1898. The fundamental role of quantization in modulation and analog-to-digital conversion was first recognized during the early development of pulsecode modulation systems, especially in the 1948 paper of Oliver, Pierce, and Shannon. Also in 1948, Bennett published the first high-resolution analysis of quantization and an exact analysis of quantization noise for Gaussian processes, and Shannon published the beginnings of rate distortion theory, which would provide a theory for quantization as analog-to-digital conversion and as data compression. Beginning with these three papers of fifty years ago, we trace the history of quantization from its origins through this decade, and we survey the fundamentals of the theory and many of the popular and promising techniques for quantization.

Recently, a new class of image coding algorithms coupling standard scalar quantization of frequency coefficients with tree-structured quantization (related to spatial structures) has attracted wide attention because its good performance appears to confirm the promised efficiencies of hierarchical representation [1, 2]. This paper addresses the problem of how spatial quantization modes and standard scalar quantization can be applied in a jointly optimal fashion in an image coder. We consider zerotree quantization (zeroing out tree-structured sets of wavelet coefficients) and the simplest form of scalar quantization (a single common uniform scalar quantizer applied to all non-zeroed coefficients), and we formalize the problem of optimizing their joint application and we develop an image coding algorithm for solving the resulting optimization problem. Despite the basic form of the two quantizers considered, the resulting algorithm demonstrates coding performance that is competitive (often...

In this paper we review some recent interactions between harmonic analysis and data compression. The story goes back of course to Shannon’s R(D) theory...

This paper introduces a new class of bases, called bandelet bases, which decompose the image along multiscale vectors that are elongated in the direction of a geometric flow. This geometric flow indicates directions in which the image grey levels have regular variations. The image decomposition in a bandelet basis is implemented with a fast subband filtering algorithm. Bandelet bases lead to optimal approximation rates for geometrically regular images. For image compression and noise removal applications, the geometric flow is optimized with fast algorithms, so that the resulting bandelet basis produces a minimum distortion. Comparisons are made with wavelet image compression and noise removal algorithms.

This paper addresses the problem of encoder optimization in a macroblock-based multi-mode video compression system. An efficient solution is proposed in which, for a given image region, the optimum combination of macroblock modes and the associated mode parameters are jointly selected so as to minimize the overall distortion for a given bit-rate budget. Conditions for optimizing the encoder operation are derived within a rate-constrained product code framework using a Lagrangian formulation. The instantaneous rate of the encoder is controlled by a single Lagrange multiplier that makes the method amenable to mobile wireless networks with time-varying capacity. When rate and distortion dependencies are introduced between adjacent blocks (as is the case when the motion vectors are differentially encoded and/or overlapped block motion compensation is employed), the ensuing encoder complexity is surmounted using dynamic programming. Due to the generic nature of the algorithm, it can be succ...

In 1996, the JPEGcommittee began to investigate possibilities for a new still image compression standard to serve current and future applications. This initiative, which was named JPEG2000, has resulted in a comprehensive standard (ISO 154447ITU-T Recommendation T.800) that is being issued in six parts. Part 1, in the same vein as the JPEG baseline system, is aimed at minimal complexity and maximal interchange and was issued as an International Standard at the end of 2000. Parts 2–6 define extensions to both the compression technology and the file format and are currently in various stages of development. In this paper, a technical description of Part 1 of the JPEG2000 standard is provided, and the rationale behind the selected technologies is explained. Although the JPEG2000 standard only specifies the decoder and the codesteam syntax, the discussion will span both encoder and decoder issues to provide a better

This paper is organized as follows. In the next section we review the construction of orthonormal windowed Fourier bases. This is followed in section III by a description of biorthogonal windowed Fourier bases. The new brushlet basis is given in section IV. A biorthogonal brushlet basis is presented in section V. In section VI we describe the image compression algorithm based on a brushlet expansion of the image. Results of experiments are presented in Section VII

IEEE Transactions on Image Processing This paper presents novel coding algorithms based on tree structured segmentation, which achieve the correct asymptotic rate-distortion (R-D) behavior for a simple class of signals, known as piecewise poly-nomials, by using an R-D based prune and join scheme. For the one dimensional (1-D) case, our scheme is based on binary tree segmentation of the signal. This scheme approximates the signal segments using polynomial models and utilizes an R-D optimal bit allocation strategy among the different signal segments. The scheme further encodes similar neighbors jointly to achieve the correct exponentially decaying R-D behavior � D(R) ∼ c02 −c1R � , thus improving over classic wavelet schemes. We also prove that the com-putational complexity of the scheme is of O (N log N). We then show the extension of this scheme to the two dimensional (2-D) case using a quadtree. This quadtree coding scheme also achieves an exponentially decaying R-D behavior, for the polygonal image model composed of a white polygon shaped object against a uniform black background, with low computational cost of O (N log N). Again, the key is an R-D opti-mized prune and join strategy. Finally, we conclude with numerical results, which show that the proposed quadtree coding scheme outperforms JPEG2000 by about 1 dB for real images, like cameraman, at low rates of around 0.15 bpp.

We extend our previous work on space-frequency quantization (SFQ) [1] for image coding from wavelet transforms to the more general wavelet packet transforms [2]. The resulting wavelet packet coder offers an universal transform coding framework within the constraints of filter bank structures by allowing joint transform and quantizer design without assuming a priori statistics of the input image. In other words, the new coder adaptively chooses the representation to suit the image and the quantization to suit the representation. Experimental results show that, for some image classes, our new coder gives excellent coding performance. 1 Introduction Recently, wavelet transforms have attracted considerable attention, especially with applications to image coding, due to their ability to provide attractive space-frequency resolution tradeoffs for natural images [3, 4]. In addition to conventional scalar (or vector) quantization strategies that are common in subband coding [5], the hierarchi...

this article is to look at recent wavelet advances from a signal processing perspective. In particular, approximation results are reviewed, and the implication on compression algorithms is discussed. New constructions and open problems are also addressed