Surface Fairing Based On Biorthogonal Nonuniform B-spline Wavelets

The fairing of curves and surfaces is the basic and core of the CAGD, and it is an important problem in the design of the curves and surfaces. In recent years, the method of wavelet has been widely used in the fairing of curves and surfaces. The method of wavelet has the features of data compression and high efficiency, but most of the wavelet fairing algorithms are aim at uniform and quasi-uniform B-spline curves and surfaces. It cann't be used directly in nonuniform B-spline curves and surfaces. However, nonuniform B-spline is the basic of NURBS and the industry standard of the expression of curves and surfaces. This thesis mainly studies the fairing method of the nonuniform B-spline surfaces. The main works and contributions are summarized as follows:1.We use the biorthogonal nonuniform B-spline wavelets for the decomposition of B-spline surfaces. On the one hand, the global fairing of surfaces is realized through removing several levels of detail parts with given proportion rather than removing the whole detail parts, so it can preserve the shape of surfaces during fairing as much as possible. On the other hand, the local faring is realized by modifying the control points of detail parts. The modified control points of detail parts are related to the bad knots which are located through the analysis to the detail parts.2. The wavelet fairing algorithm is efficient with linear time complexity and can compresse data during fairing, but it cann't deal with some problems of constraints. Energy optimization is still widely used among a lot of the methods in the faring of B-spline curves and surfaces. Energy optimization can deal with the problems of constraints, and can be used in large flexibility and close B-spline curves and surfaces. So we present an approach that combines the wavelet fairing with energy optimization. The surface fairing approach we presented is realized through the decomposition of B-spline surface and energy optimization. The efficiency of the algorithm is improved by compressing data through the decomposition of B-spline surface, and fairing detail parts with boundary constraints is realized through the energy optimization.3.We study deeply into the surface fairing with boundary constraints. The algorithm we present can preserve the boundary of the surface during fairing. The algorithm is realized through the decomposition of B-spline surface, and surface fairing with boundary constraints is realized through preserving the control points of the detail parts boundary.