Discrete filtering of information over triangulated surfaces has proved very useful in computer graphics applications. This technique is based on diffusion equations and has been extensively applied to image processing, harmonic map regularization and texture generating, etc. [C. L. Bajaj and G. Xu, ACM Trans. Graph., 22 (2003), pp. 4–32], [C. Wu, J. Deng, and F. Chen, IEEE Trans. Vis. Comput. Graph., 14 (2008), pp. 666–679]. However, little has been done on analysis (especially quantitative analysis) of the behavior of these filtering procedures. Since in applications mesh surfaces can be of arbitrary topology and the filtering can be nonlinear and even anisotropic, the analysis of the quantitative behavior is a very difficult issue. In this paper, we first present the discrete linear, nonlinear, and anisotropic filtering schemes via discretizing diffusion equations with appropriately defined differential operators on triangulated surfaces, and then use concepts of discrete scale-spaces to describe these filtering procedures and analyze their properties respectively. Scale-space properties such as existence and uniqueness, continuous dependence on initial value, discrete semigroup property, grey level shift invariance and conservation of total grey level, information reduction (also known as topology simplification), and constant limit behavior have been proved. In particular, the information reduction property is analyzed by eigenvalue and eigenvector analysis of matrices. Different from the direct observation of the local filtering to the diffusion equations and other interpretation methods based on wholly global quantities such as energy and entropy, this viewpoint helps us understand the filtering both globally (information reduction as image components shrink) and locally (how the image component contributes to its shrink rate). With careful consideration of the correspondence between eigenvalues and eigenvectors and their features, differences between linear and nonlinear filtering, as well as between isotropic and anisotropic filtering, are discussed. We also get some stability results of the filtering schemes. Several examples are provided to illustrate the properties.

An algorithm is presented to compute the self-intersection curves of a rational ruled surface based on the theory of μ-bases. The algorithm starts by constructing the principal subresultants for a μ-basis of the rational ruled surface. The principal subresultant coefficients provide information about not only the parametric loci of the self-intersection curves, but also the orders of the self-intersection curves. Based on this observation, an efficient algorithm is provided to compute the parametric loci of the self-intersection curves as well as their corresponding orders. The isolated singular points of the rational ruled surface are also computed.

给出了求解代数曲面上两点之间测地线的一种算法。在算法中，把决定代数曲面测地线的微分方程组离散为一个非线性方程组，然后采用迭代数值方法求解。为此，给出了一种基于细分的初值生成方法。最后给出了一些数值算例用来验证算法的有效性。

In this paper, we introduce a new type of splines—polynomial splines over hierarchical T-meshes (called PHT-splines) to model geometric objects. PHT-splines are a generalization of B-splines over hierarchical T-meshes. We present the detailed construction process of spline basis functions over T-meshes which have the same important properties as B-splines do, such as nonnegativity, local support and partition of unity. As two fundamental operations, cross insertion and cross removal of PHT-splines are discussed. With the new splines, surface models can be constructed efficiently and adaptively to fit open or closed mesh models, where only linear systems of equations with a few unknowns are involved. With this approach, a NURBS surface can be efficiently simplified into a PHTspline which dramatically reduces the superfluous control points of the NURBS surface. Furthermore, PHT-splines allow for several important types of geometry processing in a natural and efficient manner, such as conversion of a PHT-spline into an assembly of tensor-product spline patches, and shape simplification of PHT-splines over a coarser T-mesh. PHT-splines not only inherit many good properties of Sederberg’s T-splines such as adaptivity and locality, but also extend T-splines in several aspects except that they are only C1 continuous. For example, PHT-splines are polynomial instead of rational; cross insertion/removal of PHT-splines is local and simple.

This paper discusses implicitization and parametrization of quadratic surfaces with one simple base point. The key point to fulfill the conversion between the implicit and the parametric form is to compute three linearly independent moving planes which we call the weak μ-basis of the quadratic surface. Beginning with the parametric form, it is easy to compute the weak μ-basis, and then to find its implicit equation. Inversion formulas can also be obtained easily from the weak μ-basis. For conversion from the implicit into the parametric form, we present a method based on the observation that there exists one self-intersection line on a quadratic surface with one base point. After computing the self-intersection line, we are able to derive the weak μ-basis, from which the parametric equation can be easily obtained. A method is also presented to compute the self-intersection line of a quadratic surface with one base point.