Ondelettes et applications en imagerie et en calcul de surfaces

This thesis is a theorical and numerical contribution to wavelet transform in image processing and surface computing. Group theory approach, multiresolution approach and lters banc approach of wavelets basis are reviewed. The wavelet transform applications concern image compression, discrete curve representation and surface approximation by radial functions. We begin by reviewing di erent image compression techniques. Graphically and numericallywe illustrate that wavelet transform, compared to other pyramidalmethods such as Burt and Adelson scheme performed better for image compression. We illustrate also that among hierarchical representations, edges are well preserved between scales by the wavelet representation. We develop an original multi-scale algorithm for synthesis and analysis of discrete curve. Our algorithm is applied to discrete curves or contours starting from its Freeman chain code, in order to obtain discrete approximations at several levels of resolution. The best results are obtained with Haar wavelet at minimal cost of time and memory. Last, after a review of interpolation problem by radial functions and a remainder about existence of solutions, we propose a new approach to solve the linear system which de nes the parameters of the problem. The approach is based on wavelet transform and allows the conversion of a dense matrix to a sparse one. The linear system can then be easily solved by an iterative procedure. We show that the approach is more e cient as the amount of data grows. Results of surface interpolation by thin plate spline and multiquadrics are presented. In the implementations developed in this thesis, we used three families of wavelets : spline wavelets, wavelets with compact support and biorthogonal wavelets. The graphical and numerical presentation of results illustrate the e ciency of the proposed approaches. Keys words : Wavelets, wavelet transform, group theory, multiresolution analysis, lters banc, image compression, pyramidal algorithm, multiscale representation, discret contour, interpolation, radial functions, thin plate spline, multiquadrics.