Abstract-Multiresolution representations are very effective for ana-lyzing the information content of images. We study the properties of the operator which approximates a signal at a given resolution. We show that the difference of information between the approximation of a signal at the resolutions 2 ’ + ’ and 2jcan be extracted by decomposing this signal on a wavelet orthonormal basis of L*(R”). In LL(R), a wavelet orthonormal basis is a family of functions ( @ w (2’ ~-n)),,,“jEZt, which is built by dilating and translating a unique function t+r (xl. This decomposition defines an orthogonal multiresolution rep-resentation called a wavelet representation. It is computed with a py-ramidal algorithm based on convolutions with quadrature mirror lil-ters. For images, the wavelet representation differentiates several spatial orientations. We study the application of this representation to data compression in image coding, texture discrimination and fractal analysis. Index Terms-Coding, fractals, multiresolution pyramids, quadra-ture mirror filters, texture discrimination, wavelet transform.

Oriented filters are useful in many early vision and image processing tasks. One often needs to apply the same filter, rotated to different angles under adaptive control, or wishes to calculate the filter response at various orientations. We present an efficient architecture to synthesize filters of arbitrary orientations from linear combinations of basis filters, allowing one to adaptively "steer" a filter to any orientation, and to determine analytically the filter output as a function of orientation.

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.

Orthogonal wavelet transforms have recently become a popular representation for multiscale signal and image analysis. One of the major drawbacks of these representations is their lack of translation invariance: the content of wavelet subbands is unstable under translations of the input signal. Wavelet transforms are also unstable with respect to dilations of the input signal, and in two dimensions, rotations of the input signal. We formalize these problems by defining a type of translation invariance that we call "shiftability". In the spatial domain, shiftability corresponds to a lack of aliasing; thus, the conditions under which the property holds are specified by the sampling theorem. Shiftability may also be considered in the context of other domains, particularly orientation and scale. We explore "jointly shiftable" transforms that are simultaneously shiftable in more than one domain. Two examples of jointly shiftable transforms are designed and implemented: a one-dimensional tran...

ABSTRACT. This paper is essentially tutorial in nature. We show how any discrete wavelet transform or two band subband filtering with finite filters can be decomposed into a finite sequence of simple filter-ing steps, which we call lifting steps but that are also known as ladder structures. This decomposition corresponds to a factorization of the polyphase matrix of the wavelet or subband filters into elementary matrices. That such a factorization is possible is well-known to algebraists (and expressed by the formula); it is also used in linear systems theory in the electrical engineering community. We present here a self-contained derivation, building the decomposition from basic principles such as the Euclidean algorithm, with a focus on applying it to wavelet filtering. This factorization provides an alternative for the lattice factorization, with the advantage that it can also be used in the biorthogonal, i.e, non-unitary case. Like the lattice factorization, the decomposition presented here asymptotically re-duces the computational complexity of the transform by a factor two. It has other applications, such as the possibility of defining a wavelet-like transform that maps integers to integers. 1.

We develop a statistical characterization of natural images in the wavelet transform domain. This characterization describes the joint statistics between pairs of subband coefficients at adjacent spatial locations, orientations, and scales. We observe that the raw coefficients are nearly decorrelated, but their magnitudes are highly correlated. A linear magnitude predictor coupled with both multiplicative and additive uncertainties accounts for the joint coefficient statistics of a wide variety of images including photographic images, graphical images, and medical images. In order to directly demonstrate the power of this model, we construct an image coder called EPWIC (Embedded Predictive Wavelet Image Coder), in which subband coefficients are encoded one bitplane at a time using a non-adaptive arithmetic encoder that utilizes probabilities calculated from the model. Bitplanes are ordered using a greedy algorithm that considers the MSE reduction per encoded bit. The decoder uses the statistical model to predict coefficient values based on the bits it has received. The rate-distortion performance of the coder compares favorably with the current best image coders in the literature. 1

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...

In just a few years the "Internet Multicast Backbone", or MBone, has risen from a small, research curiosity to a large scale and widely used communications infrastructure. A driving force behind this growth was our development of multipoint audio, video, and shared whiteboard conferencing applications that are now used daily by the large and growing MBone community. Because these real-time media are transmitted at a uniform rate to all the receivers in the network, the source must either run below the bottleneck rate or overload portions of the multicast distribution tree. In this dissertation, we propose a solution to this problem by moving the burden of rate-adaptation from the source to the receivers with a scheme we call Receiver-driven Layered Multicast, or RLM. In RLM, a source distr...

In this paper a new image transformation fitted to reversible (lossless) image compression is presented. It uses a simple pyramid multiresolution scheme which is enhanced via predictive coding. The new transformation is similar to the subband decomposition, but it uses only integer operations. The number of bits required to represent the transformed image is kept small though careful scaling and truncations. The lossless coding compression rates are smaller than those obtained with predictive coding of equivalent complexity. It is also shown that the new transform can be effectively used, with the same coding algorithm, for both lossless and lossy compression. When used for lossy compression, its rate-distortion function is comparable to other efficient lossy compression methods.