Remeshing By Tracing Flow Lines Of Scalar Fields

With the development of 3D scanning and other data acquisition techniques, polygonal meshes have become a de facto representation for 3D shapes. Nevertheless, in triangular meshes directly reconstructed from point clouds, face shape is generally bad and vertices are usually irregular. This makes it difficult to perform subsequent operations such as compression, shape computation, and structure analysis, and even leads to unstable numerical algorithms in some serious cases. Remeshing is an important way to improve mesh quality and to construct multi-resolution representation.Triangular meshes are widely used in 3D game, computer animation, and virtual reality. On the other hand, quadrilateral meshes are more favorable in applications such as parametric surface fitting, texturing, and FEA-based simulations. Both triangular remeshing and quadrilateral remeshing are studied in this paper but with emphasis on application of quadriangulation techniques.The constributions of this paper are as follows.1) A new discrete form is proposed for evaluation of Laplace–Beltrami operator over a quadrilateral mesh. It is derived by applying the discrete LBO for triangular meshes to all possible triangulations of the quadrilateral mesh, and therefore is named as mean Laplace–Beltrami operator (abbr. as MLBO). A symmetric MLBO is also explored.2) A triangular remeshing algorithm is constructed by tracing streamlines of scalar fields. It firstly establishes a Laplacian field on a given original mesh, and then generates two groups of streamlines with one group being a 60°rotational version of the other. Secondly, a rhombus-dominant mesh is constructed from the two groups of streamlines and then triangulated into a triangular base mesh. A multiresolution representation can be obtained by upsampling the original mesh through refining the base meshusing the streamline tracing again. Theoretically, triangles of the remeshing results should be close to be equilateral.. It has been approved by experiments.3) A quad mesh based multiresolution framework is established for given triangular meshes based on Morse theory. Firstly, a smooth Morse function is defined as the solution of a Laplacian equation with constraints in which critical points are either specified by user or exacted from an eigenfunction of the Laplacian matrix of the mesh.According to Morse theory, a coarse quad mesh can be produced by connecting the critical points of the function carefully. Finaly, a critical point exchange rule is designed to generate the structure of the finer quad mesh whose connectivity is also generated using stream line tracing. Parameterization is not required in the process as all resolution levels are created using the streamlines method.