Fast Mesh interpolation and Mesh Decomposition with Applications

A new approach for constructing a smooth subdivision surface to interpolate the vertices of an arbitrary mesh is presented. The construction process does require setting up neither any linear systems, nor any matrix computation, but is simply done by iteratively moving vertices of the given mesh locally until control mesh of the required interpolating surface is reached. The new interpolation method has the simplicity of a local method in effectively dealing with meshes of a large number of vertices. It also has the capability of a global method in faithfully resembling the shape of a given mesh. Furthermore, the new method is fast and does not require a fairing step in the construction process because the iterative process converges to a unique solution at an exponential rate. Another important result of this work is, with the new iterative process, each mesh (surface) can be decomposed into a sum of simpler meshes (surfaces) which carry high-and low-frequency information of the given model. This mesh decomposition scheme provides us with new approaches to some classic applications in computer graphics such as texture mapping, denoising/smoothing/sharpening, and morphing. These new approaches are demonstrated in this paper and test results are included.