Multiresolution Representation Of B-spline Curves And Surfaces With Approximate Preservation Of Constrains

Multiresolution representation of curves and surfaces basing on B-spline wavelets is one of the most important directions of curve and surface modeling field. Using B-spline wavelet, a higher resolution B-spline curve or surface can be decomposed into a series of lower resolution curves or surfaces and the details, which makes us achieve local detail edition and integral control flexibly when dealing with complex geomitric shapes to satisfy geometric accuracy and save time.But most of studise and applications in this area are mainly aimed at uniform or quasi uniform B-spline wavelets which cannot be applied to the multiresolution modeling of non-uniform B-spline curves and surfaces directly. When representing a multi-resolution curve or surface, it is usually to filter out details of the higher resolution curve or surface by deleting knot vectors to obtain the approximation of curve or surface which is represented by less control points and lower resolution B-spline base.Because the geometric constraints to a B-spline curve or surface are presented by its control points and B-spline base, some geometric constraint characteristics of the higher resolution curve or surface cannot be maintained in the lower resolution curves or surfaces usually.This thesis mainly studies the multiresolution representation of curves and surfaces with approximate preservation of constrains.In the paper we firstly introduce the B-spline curves and surfaces and related properties as well as the method for the decomposition and reconstruction of non-uniform B-spline curves and surfaces by biorthogonal non-uniform B-spline wavelets.And we give a unified formula of geometric constraints for B-spline curves and surfaces.Then we propose two methods for the multi-resolution representation of B-spline curves and surfaces respectively that can approximately preserve the geomitric constrains.In the multi-resolution representation of curves, there are two steps to solve the problem how to approximately keep geometric constraint:firstly keep the knots relevant to the geometric constraints while delete of irrelevant knots selectively; secondly control the changes in those associated with constrains part of the energy functional.Aimed at the problem of the preservation of ROI in the multi-resolution representation to surfaces, three steps are utilized:firstly preserve the knot with maximum Gaussian curvature in the B-spline wavelet decomposition based on the property that Gaussian curvature of a point affects its neighboring areas structure; secondly preserve the doundary knots; finally propose an energy functional to control the wavelet decomposition to the surfaces and reconstruct part of detail surfaces corresponded to the control points of ROI to the high-resolution surface.Some examples are presented to illustrate the effectiveness of the proposed algorithm.