Pattern Recognition 41 (2008) 1732 – 1743 www.elsevier.com/locate/pr Quasi-isometric parameterization for texture mapping  Xianfang Sun a , b, c , ∗ , Edwin R. Hancock b a School of Automation Science and Electrical Engineering, Beihang University, Beijing 100083, PR China b Department of Computer Science, University of York, York YO10 5DD, UK c School of Computer Science, Cardiff University, Cardiff CF24 3AA, UK Received 24 October 2006; received in revised form 3 July 2007; accepted 24 October 2007 Abstract In this paper, we present a new 3D triangular mesh parameterization method that is computationally efficient and yields minimized distance errors. The method has four steps. Firstly, multidimensional scaling (MDS) is used to flatten each submesh consisting of one vertex and its direct neighbours on the 3D triangular mesh. Secondly, an optimal method is used to compute the linear reconstructing weights of each vertex with respect to its neighbours. Thirdly, a spectral decomposition method is used to obtain initial 2D parameterization coordinates. Fourthly, the initial coordinates are rotated and scaled to minimize the distance errors. It is demonstrated that this method can be used for texture mapping. Analyses and examples show the effectiveness of this parameterization method compared with alternatives.  2007 Elsevier Ltd. All rights reserved. Keywords: Texture mapping; Surface parameterization; Isometric mapping; Locally linear embedding; Multidimensional scaling; Spectral decomposition 1. Introduction In computer graphics, textures are widely used to enhance the visual richness of images of natural and virtual 3D objects. There are two different types of methods for applying textures to decorate the surfaces of 3D objects—texture mapping and texture synthesis. Texture mapping maps a 2D texture image onto the surface of a 3D object, while texture synthesis directly computes the texture on a 3D surface using a texture sample as a reference. We will not discuss texture synthesis in this paper. Good reviews of this topic can be found in Refs. [1–4], and the references therein. Since the introduction of texture mapping by Catmull [5], numerous methods have appeared in the literature [6–9]. Because most 3D surfaces are not developable, texture distor- tions exist in the resulting images. Moreover, even if a 3D sur- face is developable, many methods still give rise to distortions. Hence, minimizing distortions is the main challenge in texture  This work was done while Xianfang Sun visited the University of York. ∗ Corresponding author. School of Computer Science, Cardiff University, Cardiff CF24 3AA, UK. Tel.: +44 29 20879355; fax: +44 29 20874598. E-mail addresses: xianfang.sun@cs.cardiff.ac.uk (X. Sun), erh@cs.york.ac.uk (E.R. Hancock). 0031-3203/$30.00  2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.patcog.2007.10.027 mapping. Much effort has been expended in an attempt to solve this problem [6,10,11]. One way of minimizing the distortions is to use optimization techniques for the mesh parameterization of 3D surface models [12]. The meshes of 3D surface models may have different con- nectivity structure, but the most commonly used meshes are triangular. The aim in parameterizing a 3D triangular mesh is to construct a 2D or a topologically simple 3D triangular mesh (such as a spherical 3D mesh) with its vertices, edges, and trian- gles corresponding to those of the original 3D triangular mesh, and satisfying an optimality criterion. Besides being used in texture mapping, parameterization techniques have also been applied in many other problems in computer graphics and im- age processing, including morphing [13] and remeshing [14]. Extensive research has been undertaken into the theoretical is- sues underpinning the method and its practical application. In this section, we will only review those papers directly related to our method. For a tutorial and survey on parameterization techniques, the reader is referred to Ref. [15]. Most parameterization methods deal with mapping between a 2D and 3D triangular mesh pair [12,16,17], and are therefore concerned with the specific case of planar parameterization. However, when a 3D mesh has a surface with a genus-zero