| Multiresolution analysis and wavelets provide useful and efficient tools for representing functions at multiple levels of detail. Wavelet representations have been used in a broad range of applications, including image compression, physical simulation, hierarchical optimization, and numerical analysis.; In this dissertation, I present a new class of wavelets, based on subdivision surfaces, that radically extends the class of representable functions. Whereas previous two-dimensional methods were restricted to functions defined on {dollar}IRsp2,{dollar} the subdivision wavelets developed here may be applied to functions defined on compact surfaces of arbitrary topological type.; The wavelets developed in this dissertation depend upon a generalization of the standard notion of multiresolution analysis. Rather than developing wavelets from scales and translates over regular domains, they are based on refinable scaling functions. Because these wavelets are built over subdivision surfaces, they may be used to model piecewise linear functions (such as polyhedral surfaces), tangent-plane smooth ({dollar}Csp1{dollar}) functions, or piecewise smooth functions with discontinuities.; I present several applications of this work, including smooth level-of-detail control for graphics rendering, compression of geometric models, and animation previewing. The resulting algorithms are shown to run quite efficiently in most cases.; As shapes in computer graphics and modeling become increasingly complex, the techniques developed here provide a powerful and efficient means to extract and simplify essential detail, at the same time eliminating redundant information. |
