Fast Isometric Parametrization of 3D Triangular Mesh Xianfang Sun 12 and Edwin R. Hancock 2 1 School of Automation Science and Electrical Engineering Beihang University, Beijing 100083, P.R. China 2 Department of Computer Science, University of York York YO10 5DD, UK Abstract In this paper we describe a new mesh parametrization method that is both computationally efficient and yields minimized distance errors. The method has four steps. First, the multidimensional scaling is used to locally flatten each 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 parametrization coordinates. Fourthly, we rotate and scale the initial coordinates to minimize the distance errors. Examples are provided to show the effectiveness of this parametrization method compared with alternatives. 1 Introduction Triangular mesh parametrization aims to determine a 2D triangular mesh with its vertices, edges, and triangles corresponding to that of the original 3D triangular mesh, satisfying an optimality criterion. The technique has been applied in a wide range of problems in computer graphics and image processing, including texture mapping [12], morphing [8], and remeshing [5]. Extensive research has been undertaken into the theoretical issues underpinning the method and its practical application. For a tutorial and survey, the reader is 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 draw planar graphs. The basic idea underpinning this method is to use the vertex coordinates of the original 3D triangular mesh to compute reconstructing weights of each interior vertex with respect to its neighbour vertices.These weights are subsequently used together with the boundary vertex coordinates on a plane to compute the interior vertex coordinates of a 2D triangular mesh. A drawback of Floater’s parametrization method is that the boundary vertex coordinates must be determined manually beforehand. Another parametrization method is that proposed by Zigelman et al. [12]. It first uses Dijkstra algorithm to compute the geodesic distances between each pair of the vertices, and then uses multidimensional scaling (MDS) to determine the vertex coordinates on a 2D plane. This method does not need the boundary vertex coordinates to be determined BMVC 2005 doi:10.5244/C.19.11