三角域上Said-Ball基的推广渐近迭代逼近

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。

Generalized progressive iterative approximation for Said-Ball bases on triangular domains

Objective In the field of computer aided design, a new data fitting technique, the progressive iterative approximation (PIA), has been proposed and attracts plenty of attention. By adjusting the control points iteratively, the PIA method provides a straightforward way to generate a sequences of curves/surfaces with better precision for data fitting. The curve (tensor product surface) has the PIA property as long as the bases are normalized completely positive and the corresponding collocation matrix is non-singular. In order to extend the scope of application of the PIA property, our paper focuses on the triangular surface and the non-totally positive collocation matrix. Furthermore, we assume that it may also possess the PIA property. Method The theory is based on certain conditions, which are essential for a basis to satisfy the PIA property. Given a set of triangular basis functions and its corresponding parametric values, we can obtain the collocation matrix of the triangular basis functions at the parametric values. Then, we get a new matrix, which is the result of the identity matrix subtracting the collocation matrix, and calculate the spectrum radius of the new matrix. If the value of the spectrum radius is less than 1, we call the triangular basis functions over a triangle domain having the PIA property. Given a collocation matrix, which is diagonally dominant or generalized diagonally, dominant and the elements of the matrix are positive number, then the real part of the eigenvalue of the collocation matrix is also a positive number. Our work proves that if the real part of the eigenvalue collocation matrix is a positive number, then the corresponding bases on triangular domain possess the PIA property (we call it as generalized PIA property). In the end, we build the relationship between diagonally dominant or generalized diagonally dominant matrix and the bases possess progressive iterative approximation property. Result If the collocation matrix is diagonally dominant or generalized diagonally dominant, the corresponding bases on triangular domain possess the PIA property (we call it as generalized PIA property). As we know, the Said-Ball bases have many good properties, such as the shape preserving property and the convex hull property. Since the Said-Ball bases are much better than Bernstein bases in recursive evaluation and degree elevation or reduction, it is worthwhile to develop the Said-Ball bases on triangle domain for free-form surface design. We take the Said-Ball bases as examples and find that the Said-Ball bases on triangular domain have the generalized PIA property. Numerical examples are given to demonstrate the correctness of our theory. Conclusion The main work of our paper is that we extend the scope of PIA property to the generalized blending basis over a triangle domain. The basic technique is to give a progressive iterative scheme of the triangle Said-Ball surfaces. Giving some scattered data points to form an initial control mesh, the limit surface can interpolate these original points by constructing an iterative sequence of the triangular Said-Ball patches. Moreover, the numerical examples of the triangular Said-Ball basis of low degree with the uniform or non-uniform parameters are given. Furthermore, we can generalize this method to the study of different kinds of generalized Ball bases in triangular domain.