Fast Isometric Parametrization of 3D Triangular Mesh

Abstract

Abstract In this paper we describe a new mesh parametrization method that is bothcomputationally efficient and yields minimized distance errors. The methodhas four steps. First, the multidimensional scaling is used to locally flatteneach vertex. Second, an optimal method is used to compute the linear re-constructing weights of each vertex with respect to its neighbours. Thirdly,a spectral decomposition method is used to obtain initial 2D parametrizationcoordinates. Fourthly, we rotate and scale the initial coordinates to minimizethe distance errors. Examples are provided to show the effectiveness of thisparametrization method compared with alternatives. 1 Introduction Triangularmesh parametrizationaims to determine a 2D triangular mesh with its vertices,edges, and triangles corresponding to that of the original 3D triangular mesh, satisfyingan optimality criterion. The technique has been applied in a wide range of problemsin computer graphics and image processing, including texture mapping [12], morphing[8], and remeshing [5]. Extensive research has been undertaken into the theoretical issuesunderpinningthe methodand its practical application. For a tutorial and survey,the readeris referred to [4].A well-known parametrization method is that proposed by Floater [2]. It is a general-ization of the basic procedure originally proposed by Tutte [10] which was used to drawplanar graphs. The basic idea underpinningthis method is to use the vertex coordinates ofthe original 3D triangular mesh to compute reconstructing weights of each interior vertexwith respect to its neighbour vertices.These weights are subsequently used together withthe boundaryvertex coordinates on a plane to compute the interior vertex coordinates of a2D triangular mesh. A drawback of Floater’s parametrization method is that the boundaryvertex coordinates must be determined manually beforehand.Another parametrization method is that proposed by Zigelman et al. [12]. It first usesDijkstra algorithm to compute the geodesic distances between each pair of the vertices,and then uses multidimensional scaling (MDS) to determine the vertex coordinates on a2D plane. This method does not need the boundary vertex coordinates to be determined