Bjorn D. Jawerth

Image compression through wavelet transform coding

A novel 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 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, then wavelet-based methods are near-optimal within a larger class of stable transform-based, nonlinear methods of image compression. Based on previous experimental research it is argued that in most instances the error incurred in image compression should be measured in the integral sense instead of the mean-square sense. >

A discrete transform and decompositions of distribution spaces

Abstract We study a representation formula of the form ƒ = ∑ Q 〈ƒ, ϑ Q 〉ψ Q for a distribution ƒ on R n. This formula is obtained by discretizing and localizing a standard Littlewood-Paley decomposition. The map taking ƒ to the sequence {〈ƒ, ϑ Q 〉} Q , with Q running over the dyadic cubes in R n, is called the ϑ-transform. The functions ϑQ and ψQ have a particularly simple form. Moreover, most of the familiar distribution spaces (Lp-spaces, 1 R n. We also consider pointwise multipliers. For the characteristic function of a domain, we obtain boundedness results for a general class of domains which properly includes Lipschitz domains. Several interpolation methods are easily analyzed via the sequence spaces. For real interpolation, we obtain, among other things, an extension to the case p = 0. This in turn gives a new approach to the traditional atomic decomposition of Hardy spaces.

An overview of wavelet based multiresolution analyses

In this paper an overview of wavelet based multiresolution analyses is presented. First, the continuous wavelet transform in its simplest form is discussed. Then, the definition of a multiresolution analysis is given and how wavelets fit into it is shown. The authors take a closer look at orthogonal, biorthogonal and semiorthogonal wavelets. The fast wavelet transform, wavelets on an interval, multidimensional wavelets and wavelet packets are discussed. Several examples of wavelet families are introduced and compared. Finally, the essentials of two major applications are outlined: data compression and compression of linear operators.

Compressing still and moving images with wavelets

The wavelet transform has become a cutting-edge technology in image compression research. This article explains what wavelets are and provides a practical, nuts-and-bolts tutorial on wavelet-based compression that will help readers to understand and experiment with this important new technology.

Lattice Boltzmann Models for Anisotropic Diffusion of Images

The lattice Boltzmann method has attracted more and more attention as an alternative numerical scheme to traditional numerical methods for solving partial differential equations and modeling physical systems. The idea of the lattice Boltzmann method is to construct a simplified discrete microscopic dynamics to simulate the macroscopic model described by the partial differential equations. The use of the lattice Boltzmann method has allowed the study of a broad class of systems that would have been difficult by other means. The advantage of the lattice Boltzmann method is that it provides easily implemented fully parallel algorithms and the capability of handling complicated boundaries. In this paper, we present two lattice Boltzmann models for nonlinear anisotropic diffusion of images. We show that image feature selective diffusion (smoothing) can be achieved by making the relaxation parameter in the lattice Boltzmann equation be image feature and direction dependent. The models naturally lead to the numerical algorithms that are easy to implement. Experimental results on both synthetic and real images are described.

Wavelet-based sensor fusion

Sensor fusion can be performed either on the raw sensor output or after a segmentation step has been done. Our previous work has concentrated on neural network models for sensor fusion after segmentation. Although this method has been shown to be fast and reliable, there is still the overhead entailed from using entire images. The wavelet transform is a multiresolution method that is used to decompose images into detail and average channels. These channels maintain all of the image information and sensor fusion logic operations can be done within the wavelet coefficient space. In addition, image compression can be done within this same space for possible remote transmission. This paper examines sensor fusion within the wavelet coefficient space. The results of some experimental studies performed on the 1024 node NCUBE/10 at the University of South Carolina are also included.

Surface compression

DeVore, R.A., B. Jawerth and B.J. Lucier, Surface compression, Computer Aided Geometric Design 9 (1992) 219-239. We propose wavelet decompositions as a technique for compressing the number of a control parameters of surfaces that arise in Computer-Aided Geometric Design. In addition, we give a specific numerical algorithm for surface compression based on wavelet decompositions of surfaces into box splines.

Geometrical dimension versus smoothness

We study the relation between geometric dimension and smoothness, and give a precise characterization of the fractal dimension of the graph of a function in terms of smoothness classes of functions. We also express the fractal dimension in terms of different classical oscillation measures and in terms of wavelet expansions.

Wavelet probing for compression-based segmentation

In this paper we show how wavelets can be used for data segmentation. The basic idea is to split the data into smooth segments that can be compressed separately. A fast algorithm that uses wavelets on closed sets and wavelet probing is presented.

Local enhancement of compressed images

We develop a simple focusing technique for wavelet decompositions. This allows us to single out interesting parts of an image and obtain variable compression rates over the image. We also study similar techniques for image enhancement.