3D Shape Generation by Iterative Interpolation of Meshes with Arbitrary Topology Using General Catmull-Clark Subdivision Surfaces

Introduction: An algorithm for fast construction of a smooth subdivision surface that interpolates the vertices of an arbitrary input mesh M is presented. The central idea of the proposed algorithm is to find the inverse A of the matrix A that calculates the limit points of M for the chosen subdivision scheme. However, instead of a costly matrix computation process, a technique to calculate A indirectly by representing it as an infinite series of matrices is developed. With this infinite series of matrices, one can construct an infinite iterative series of control points, which converges to a mesh whose subdivision surface interpolates the given input mesh M. Most importantly, this infinite iterative series of control points can be calculated locally based on the chosen subdivision scheme. Hence, the matrices A and A do not have to be actually constructed. They are simply used in a theoretical derivation to obtain the iteration formula. The construction of the interpolation surface is done basically by iteratively adjusting vertices of the given mesh until some given error tolerance is reached. The concept of iterative interpolation has been presented in the literature before [5, 6]. The main differences between our algorithm and existing approaches are fivefold: 1. Our algorithm is the first one that derives the iterative equation from the perspective of computing A. 2. The existence, convergence and uniqueness of A are guaranteed (the proof will be provided in the complete paper). 3. Our iterative interpolation algorithm, converging at an exponential rate, is a local process and does not involve costly matrix computation. Hence the new method is very fast and can handle meshes with large number of vertices. 4. Our algorithm does not require fairing in the construction process because solution to the above interpolation process is unique. 5. Although only the general Catmull-Clark subdivision surface is used here for deriving the iterative algorithm, the idea of the proposed algorithm works for other popular subdivision schemes as well.