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In the field of computer graphics, maximal Poisson-disk sampling (MPS) is a fundamental research topic. An ideal sampling set should satisfy unbiased sampling property, minimal distance property, and maximal sampling property. In general, MPS is obtained by Dart Throwing, as we all know, the drawback of this method is unable to precisely control the number of samples. In view of the above problem, this work proposes a novelty algorithm that can precisely control the number of samples of two-dimensional radius-equal MPS, and satisfy other properties simultaneously. Unlike existing methods, the proposed method controls the number of samples by adjusting sampling radius. Firstly, according to user-specified the number of samples and sampling domain (closed polygon), initial samples are randomly obtained, then Delaunay triangulation is conducted, and taking as current sampling radius the shortest edge length of the triangulation. Secondly, iteratively removing the endpoint of global shortest edge with larger neighborhood-averaged edge length, and then using Dart Throwing to randomly insert it into gap region that is calculated at current sampling radius. By iteratively adjusting the position of points, the sampling radius increases gradually, finally, MPS with fixed number of sampling point can be achieved. Experimental results show that this method generates point sets with high quality, at the same time, ensures the excellent blue-noise property for MPS.

In this paper, we present a new algorithm for quad-dominant meshing of unorganized point clouds based on periodic global parameterization. Our meshing method is guided by principal directions so as to preserve the intrinsic geometric properties. We use local Delaunay triangulation to smooth the initial principal directions and adapt the global parameterization to point clouds. By optimizing the fairness measure we can find the two scalar functions whose gradients best align with the guided principal directions. To handle the redundant vertices in the iso-lines due to overlapped triangles, an approach is specially designed to clean the iso-lines. Our approach is fully automatic and applicable to a surface of arbitrary genus. We also show an application of our method in curve skeleton extraction from incomplete point cloud data.

This paper presents a general scheme to compute ridges on a smooth 2-manifold surface from the standpoint of a vector field. A ridge field is introduced. Starting with an initial ridge, which may or may not be umbilical, a ridge line is then traced by calculating an associated integral curve of this field in conjunction with a new projection procedure to prevent it from diverging. This projection is the first that can optimize a ridge guess to lie on a ridge line uniquely and accurately. In order to follow this scheme, we not only develop practical ridge formulae but also address their corresponding computational procedures for an analytical surface patch, especially for an implicit surface. In contrast to other existing methods, our new approach is mathematically sound and characterized by considering the full geometric structures and topological patterns of ridges on a generic smooth surface. The resulting ridges are accurate in the numerical sense and meet the requirement of high accuracy with complete topology. Although the objective of this paper is to develop a mathematically sound framework for ridges on a smooth surface, we give a comprehensive review of relevant works on both meshes and smooth surfaces for readers.