2三角域上Said-Ball基的PIA性质-中国图象图形学报.docVIP

中图法分类号：TP391 文献标识码：A 文章编号： 三角域上Said-Ball基的推广渐近迭代逼近 张莉1,2, 李园园1, 杨燕1, 檀结庆1,3 1.合肥工业大学数学学院, 合肥 230009； 2.亚利桑那州立大学，美国坦佩 85281； 3.合肥工业大学计算机学院, 合肥 230009 摘 要：目的：如果一组基函数是规范全正（Normalized Totally Positive, abbr. NTP）的，并且对应的配置矩阵是非奇异的，那么由它所生成的参数曲线或张量积曲面具有渐近迭代逼近（progressive iteration approximation, abbr. PIA）性质。为了进一步推广渐近迭代逼近性质的适用范围，本文提出对于一组基函数，如果其对应的配置矩阵不是全正的，那么该基函数也可能具有渐近迭代逼近性质。方法：提出的定理是以基函数具有渐近迭代逼近性质时其对应的配置矩阵所需满足的条件作为理论基础，建立了配置矩阵为严格对角占优或者广义严格对角占优矩阵与基函数具有渐近迭代逼近性质之间的联系。结果：配置矩阵为严格对角占优或者广义严格对角占优矩阵，则相应的三角曲面具有PIA性质或带权PIA性质，即广义PIA性质。数值试验验证了上述理论，并细致地分析了三角域上的低次Said-Ball基，指出了它们具有相应的广义PIA性质。结论：本文将渐近迭代逼近的适用范围推广到三角域上的一般混合基函数。类似三角域上Said-Ball基，本文算法亦可用于研究三角域上的其他各类广义Ball基的PIA性质。 关键词：渐近迭代逼近; 广义严格对角占优; Said-Ball基; 三角域 Generalized progressive iterative approximation for Said-Ball bases on triangular domains Zhang Li1,2, Li Yuanyuan1, Yang Yan1, Tan Jieqing1,3 1. College of Mathematics, Hefei University of Technology, Hefei, China 230009; 2. Computer Science, Arizona State University, Tempe, America 85281; 3. College of Computer and Information, Hefei University of Technology, Hefei, China 230009. Abstract: Objective: In the field of Computer Aided Design, a new data ?tting technique, progressive iterative approximation (abbr. PIA), has been proposed and attracts plenty of attention. By adjusting the control points iteratively, PIA method provides a straight forward way to generate a sequence of curves/surfaces with better and better precision for data fitting. The curve (tensor product surface) has the PIA property as long as the bases are normalized totally positive and the corresponding collocation matrix is nonsingular. In order to extend the applications of PIA property, our paper focuses on the triangular surfaces and the non-totally positive collocation matrices, and finds they may also possess PIA property under certain conditions. Method: The theory points out what kind of conditions are essential for a bases to satisfy PIA property. Given a set of triangular bases fu