Image compression through wavelet transform coding

Abstract

A new theory is introduced for analyzing image compression methods that are based on compression of wavelet decompositions. This theory precisely relates a) the rate of decay in the error between the original image and the compressed image (measured in one of a family of so-called L p norms) as the size of the compressed image representation increases (i.e., as the amount of compression decreases) to b) the smoothness of the image in certain smoothness classes called Besov spaces. Within this theory, the error incurred by the quantization of wavelet transform coefficients is explained. Several compression algorithms based on piecewise constant approximations are analyzed in some detail. It is shown that if pictures can be characterized by their membership in the smoothness classes considered here, then wavelet-based methods are near optimal within a larger class of stable (in a particular mathematical sense) transform-based, nonlinear methods of image compression. Based on previous experimental research on the spatial-frequencyintensity response of the human visual system, it is argued that in most instances the error incurred in image compression should be measured in the integral ( L ' ) sense instead of the mean-square ( L~) sense. Index Terms-Image compression, wavelets, smoothness of images, quantization.

Hybrid representations for audiophonic signal encoding

“…In addition, several families of functional spaces may be characterized by the speed of convergence of non linear wavelet approximations (we refer to [6,13], ... for a detailed mathematical presentation). Among them, the Besov spaces have received a particular attention in the context of image modeling, since they provide good models for functions with controllable "density" of singularities of a given strength.…”

Best Basis Selection for Approximation in L p

“…Also, much interest has centered around this form of approximation when the basis B consists of wavelets [9,11,26] or spline functions [8]. We also note that n-term approximation has found many interesting applications in image/signal processing [10,12], statistical estimation [16], and numerical methods for PDEs [3,4]. Given a basis B and a Banach space X, one of the central questions in n-term approximation is to characterize the set of functions which have a common rate of approximation.…”

Spherical Wavelets: Texture Processing

“…They have been used for applications such as image compression [4], image enhancement, feature detection [12], and noise removal [5]. Wavelets are computationally attractive as the associated transform is linear in the number of pixels.…”