Meshless quadrangulation by global parameterization

Point cloud is a basic description of discrete shape information. Parameterization of unorganized points is important for shape analysis and shape reconstruction of natural objects. In this paper we present a new algorithm for global parameterization of an unorganized point cloud and its application to the meshing of the cloud. Our method is guided by principal directions so as to preserve the intrinsic geometric properties. After initial estimation of principal directions, we develop a kNN(k-nearest neighbor) graph-based method to get a smooth direction field. Then the point cloud is cut to be topologically equivalent to a disk. The global parameterization is computed and its gradients align well with the guided direction field. A mixed integer solver is used to guarantee a seamless parameterization across the cut lines. The resultant parameterization can be used to triangulate and quadrangulate the point cloud simultaneously in a fully automatic manner, where the shape of the data is of any genus.

Graphical abstractResults of applying meshless quadrangulation to raw scan data of cup. The left picture shows raw data. In the middle, uv lines are displayed, the border is shown in black lines and singularities sare green points; the right figure demonstrates quad meshes of a half cup.Figure optionsDownload full-size imageDownload high-quality image (95 K)Download as PowerPoint slideHighlights► A kNN graph-based algorithm which construct smooth direction field on point cloud with main geometric feature captured. ► An extension of global parametrization method from triangle mesh to point cloud. ► Applications of this parametrization in triangle meshing and quad meshing.