Multiresolution representations and wavelets

TLDR: This dissertation develops a nonlinear multiresolution transform which translates when the signal is translated called the dyadic wavelet transform and studies the application of this signal representation to data compression in image coding, texture discrimination and fractal analysis.

Abstract: Multiresolution representations are very effective for analyzing the information in images. In this dissertation we develop such a representation for general purpose low-level processing in computer vision. We first study the properties of the operator which approximates a signal at a finite resolution. We show that the difference of information between the approximation of a signal at the resolutions 2$\sp{j+1}$ and 2$\sp{j}$ can be extracted by decomposing this signal on a wavelet orthonormal basis of ${\bf L}({\bf R}\sp{n}$). In ${\bf L}\sp2({\bf R})$, a wavelet orthonormal basis is a family of functions $\left\lbrack\sqrt{2\sp{j}}\ \psi(2\sp{j}x+n)\right\rbrack\sb{(j,n)\in{\rm Z}\sp2}$, which is built by dilating and translating a unique function $\psi(x)$, called a wavelet. This decomposition defines an orthogonal multiresolution representation called a wavelet representation. It is computed with a pyramidal algorithm of complexity n log(n). We study the application of this signal representation to data compression in image coding, texture discrimination and fractal analysis. The multiresolution approach to wavelets enables us to characterize the functions $\psi(x) \in {\bf L}\sp2({\bf R})$ which generate an orthonormal basis. The inconvenience of a linear multiresolution decomposition is that it does not provide a signal representation which translates when the signal translates. It is therefore difficult to develop pattern recognition algorithms from such representations. In the second part of the dissertation we introduce a nonlinear multiscale transform which translates when the signal is translated. This representation is based upon the zero-crossings and local energies of a multiscale transform called the dyadic wavelet transform. We experimentally show that this representation is complete and that we can reconstruct the original signal with an iterative algorithm. We study the mathematical properties of this decomposition and show that it is well adapted to computer vision. To illustrate the efficiency of this Energy Zero-Crossings representation, we have developed a coarse to find matching algorithm on stereo epipolar scan lines. While we stress the applications towards computer vision, wavelets are useful to analyze other types of signal such as speech and seismic-waves.