| In the early 80's, the convergence of research in engineering (signal processing), mathematics (harmonic analysis), and physics (quantum mechanics) gave birth to wavelet theory. At the same time, Gilles Deslauriers and Serge Dubuc (and others) came up with iterative interpolation. It is only later, in the early 90's, when wavelet theory became a mature subject, that the link between iterative interpolation and wavelets became obvious. Now, many researchers in Québec, Alberta, France, U.S.A., Israel, Scandinavia and Singapore are working on iterative interpolation for its applications in Computer Science (graphics) and in engineering (modelisation). However, few researchers have worked on building wavelets from iterative interpolation schemes. This is what we wanted to do.; On the one hand, we have discovered and shown that some well-known wavelets, the Cohen-Daubechies-Feauveau biorthogonal wavelets, are in fact, the derivatives of a certain family of functions (called fundamental) obtained by the iterative interpolation scheme. This same family of functions was adapted to the interval by Mongeau in the early 90's. Using these facts, we adapted these wavelets to the interval. We can then handle bounded regions of the real line.; Iterative interpolation can be easily casted into a multidimensional context. It was therefore natural to try to build multidimensional wavelets from some iterative interpolation schemes and, in order to apply them, it was necessary to be able to work, for example, on rectangular regions in the plane. We have done this in the last part of this thesis. We have included an example of an image analysis by means of these new wavelets. |
