Geodesic-Fan Based Mesh Parameterization in Shoe CAD System

Summary Parameterization of a 3D mesh is a fundamental problem in a variety of applications such as shoe CAD system and geometric modeling. There are two major paradigms in mesh parameterization: energy functional minimization and the convex combination approach. In general, the convex combination approach is widely used because of simple concept and fast computation. However, the approach has some problems such as fixed boundary and high distortion near the boundary. In this paper, we present an extension of the approach based on the geodesic-fan representation and mean value coordinates, which resolves the drawbacks of the convex combination approach. Moreover, we apply our result to a 3D shoe CAD system in order to resolve the practical problems of industry. Key words: Parameterization, Mean Value Coordinates 1. Introduction Parameterization of a 3D triangular mesh is a fundamental problem in various applications of computer graphics and computer animation. Mesh parameterization of 3D surface is a one-to-one mapping without any fold-over. This special property provides an elegant and robust solution to various problems in geometric modeling and computer graphics, such as texture mapping, mesh morphing, and smooth surface fitting [1-8]. In general, the mesh cannot be flattened over a plane without distortion. The main purpose of research on mesh parameterization is to minimize distortion. Especially, such distortion minimization is very important problem in the practical applications such as shoe cad system [9,10]. There exist two major approaches in mesh parameterization. One is energy functional minimization and the other is convex combination approach. For the first approach, several methods have been developed to define and minimize an energy functional that measures distortion in the embedded mesh. Maillot et al. proposed a method to minimize a norm of the Green–Lagrange deformation tensor based on elasticity theory [5]. Eck et al. minimized the metric dispersion of harmonic imbedding instead of [1]. A non-deformation criterion (i.e., Dirichlet energy per parameter area) is introduced in [11] with extrapolating capabilities. One of the most widespread methods for parameterization of 3D mesh surface with the topology of a disk is proposed by Floater [12]. In general, this approach is called as convex combination approach because it is based on the convex combination of neighborhood vertices. The convex combination approach is an extension of the barycentric mapping approach proposed by Tutte [13]. This approach obtains parameterization by fixing the boundary vertices of a 3D mesh onto a 2D convex polygon and solving a linear system to determine the 2D embedded positions of the interior vertices. The linear system is constructed by representing each interior vertex as a convex combination of its neighborhood. In this approach, the major problem concerns how to determine the coefficients of the convex combination for each interior vertex. Floater proposed shape preserving parameterization, where the coefficients are determined by using conformal mapping and barycentric coordinates [12]. The harmonic embedding [1] is also a special case of this approach, except that the coefficients may be negative. Fig.1. Shoe CAD system: (a) 3D shoe last (b) 2D parameterization of the last