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Non-Nested Multiresolution Analysis and its applications to scientific visualization.

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Abstract:
A flexible construction of second generation wavelets allowing two distinct interpretations is presented. The first one, called the subdivision framework, focuses on the well-known relationship between subdivision schemes and multiresolution analysis. The second one, called the non-nested framework, introduces so-called approximating spaces, which play the role of the traditional scaling spaces but with the nestedness property relaxed.
The first part of the manuscript describes the algebraic construction of the multiresolution framework, as well as several analytical results, most of them related to the non-nested framework. In particular, various techniques for the design of the analysis or synthesis operators are presented and discussed.
The second part of the thesis is devoted to applications. The non-nested framework is used to design a multiresolution description for piecewise constant or linear functions defined over irregular planar or spherical triangulations, thus enabling progressive visualization of huge such datasets. Decomposition and reconstruction algorithms are presented in detail, especially those aspects related with their effective implementation, which turns out to be significantly more complicated than in the classical case. Traditional wavelet-based applications such that data compression or level-of-detail editing are also extended to these functions. On top of that, the use of the non-nested framework to deal with the analysis and reconstruction of functions defined over 3D meshes in a decimation-based multiresolution representation is also discussed.
Eventually, it is shown on several occasions how the non-nested framework allows an unified approach of wavelet-based and decimation-based algorithms, which are two techniques usually opposed.