Fast Mesh Interpolation and Mesh Expansion with Applications

A fast iterative method for constructing a smooth subdivision surface that interpolates the vertices of an arbitrary mesh is presented. The construction is done by iteratively adjusting vertices of the given mesh locally until control mesh of the required interpolating surface is reached. The new interpolation method has not only the simplicity of a local method, but also the capability of a global method in faithfully resembling the shape of a given mesh. The new method does not require solving a linear system, hence it can handle meshes with large number of vertices. 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 expanded as an infinite series of meshes (surfaces) which carry high and low frequency information of the given model. This mesh expansion scheme provides us with new approaches to some classic applications in computer graphics such as texture mapping, de-noising/smoothing/sharpening, and morphing. These new approaches are demonstrated in this paper and test results are included.