Research On Multiresolution Modeling Algorithms For Surfaces Of Arbitrary Topology

This dissertation presented the multiresolution geometric modeling algorithms based on the subdivision surfaces and/or the multiresolution surfaces. Subdivision surface has become an important and interesting research topic of the computer graphics recently. As extension of subdivision surface, the multiresolution surface is the fundamental element of digital geometric modeling, which includes the new computational representations and algorithms to meet the demands of complex computer graphics environments in the future. The contributions of this dissertation include: Subdivision has the ability to model arbitrary topological smooth surfaces owing to its characteristics of mesh refinement and the refinability of B-Spline. A Boolean operation, which is named as "Frame Boolean Operation", is proposed to construct the control mesh with the simple and basic geometric elements. It is an advancing geometric tool that can model complex surfaces much easier. A novel Interim Core Scheme (ICS) is proposed to construct a furcating object with multi-branches. The essential idea of the scheme is to construct a joint mesh that blends initial control meshes of the branching objects, and the smoothness of the resulting surfaces will only depend on the joint mesh and subdivision scheme applied. There is no closed mathematical form, or local parameterization patchwork for the subdivision surface. It is also well known that after subdivision, the mesh M j+1 of next level almost needs four times memory to store new vertices, edges, faces than the mesh M j does, so the memory needed for the storage will increase exponentially. This dissertation proposed a compact data structure based on the patches of triangle pair, which overcome these challenging difficulties caused by subdivision schemes: the memory cost has been reduced, a local and natural parameterization of the surface has been achieved, and it also enables the multiresolution detail editing. The subdivision/multiresolution surfaces can be of arbitrary topology; the characteristics make the problem of surface intersection more challengeable. The multiresolution numerical modeling method oriented to the digital geometry is developed, that can be used to solve the problem. The feasibility and advantages of the numerical modeling has been proved by its application of slicing operation on the subdivision surfaces. The resulting intersection point sequences can be transformed into multiresolution curves by the Lifting scheme, which can built the second-generation wavelets.