Multiresolution Editing Of Curves Based On Non-Uniform B-Spline Wavelets

In the filed of computer-aided geometric design and computer graphics, non-uniform rational B-spline has become a standard to design and describe a variety of shape of the complex curves and surfaces, because it gives a unified mathematical description of free-form curves and surfaces. After the initial NURBS curves and surfaces are constructed, they often need to be modified or smoothed in order to meet the user's design requirements. Reposition the control points and changing the weights are two basic approaches of modifying the shape of curves, but they are both local methods, that is, the change in shape occurs only in local area after each modification. However, in some cases we hope to modify the curves globally, and sometimes we hope to put the detail of one curve onto another one, obviously, it can be implemented by the traditional methods conveniently. Finkelstein proposed multi-resolution editing of curves based on quasi-uniform B-spline wavelets, which provided a new way of solving this problem. In this paper, we extend Finkelstein's work to arbitrary non-uniform B-spline wavelets. Our main contribution includes:1. The curve multi-resolution editing based on non-uniform B-spline wavelets is studied. The non-uniform B-spline wavelets decomposition of curves is implemented, upon this, two curve multi-resolution editing methods are proposed and implemented: 1) While preserving the details of the curve unchanged, the control points of the approximation curve are modified; 2) While preserving the approximation curve is not changed, the details of the curve are modified. The proposed methods increase the facility and flexibility of curve modification tools.2. The application of wavelets in the smoothing of the curves is studied, a new smoothing method of curve based on non-uniform B-spline wavelets is proposed. First, the knots which need to be removed are determined according to the criterion of smoothness such that the knot vector of the approximation curve is obtained; Then,the decomposition of non-uniform B-spline wavelets is done to the original curve and the smoothed curve is obtained. Also, the error control is considered in the paper. The experiments show the algorithm is simple, practical and easy to carry out, the smoothness of the curve is improved.