Curve And Surface Approximation Method Based On Biorthogonal Nonuniform B-Spline Wavelets

With the development of the computer science and testing technology, reverse engineering and CAD/CAGD technology become more intertwined. Reverse engineering is the integration of the digital technology for converting physical entity into CAD model, the reconstruction technology for geometric model and the manufacturing technology. Curve and surface fitting which is an important research direction in reverse engineering is widely used in various fields such as aircraft industry, automotive, toy industry, appliance industry and so on. Nonuniform B-spline curves and surfaces are common representations for free-form curves and surfaces. This thesis mainly studies the approximation methods of nonuniform B-spline curves and surfaces. The main works and contributions are summarized as follows:For the calculations of errors in approximating ordered data points using B-spline curves and surfaces, an error algorithm based on incremental method is proposed. This method mainly solves the calculations of projection distance in maximum norm error. Firstly, the equal step sampling points on B-spline curve segments which each data point parameter located in are obtained by incremental method based on tailor series expansion. Secondly, the shortest distances between data points and sampling points as approximate projection distances are calculated respectively for each data point. Then, the average error, maximum error and error vectors are obtained. The incremental method only involves addition operation except for the initialization step, so this algorithm is efficient. The algorithm is used in the curve and surface approximation algorithms proposed in this paper.For the expensive compution cost caused by the least square method in approximating ordered data points using B-spline curves, a curve approximation method based on biorthogonal nonuniform B-Spline wavelets is proposed. Firstly, the data points are approximated using a B-spline curve, which is generated by the least square method. Secondly, the error vectors are fitted using a detail curve, whose basis functions are biorthogonal nonuniform B-Spline wavelets. Thirdly, the new B-Spline curve is generated by adding the detail curve onto the original approximate B-spline curve. The process is iterative until the approximate B-spline curve within a given tolerance is obtained. The approach only computes additional linear systems and avoids computing original systems repeatedly. It is more efficient compared with the traditional least square method which is verified by some experimental results. In addition, the method provides a kind of multiresolution representation for B-spline approximating curve which shows a series of forms from coarse to fine for the more conventient post-processes.For the expensive compution cost caused by the least square surface approximation algorithm which computes linear systems repeatedly in approximating scanned data using B-spline surfaces, a surface fitting method is proposed using the curve approximation method based on biorthogonal nonuniform B-Spline wavelets. This approach only computes additional linear systems in fitting each row(column) of data points. It is also more efficient compared with the traditional least square method.