GPU-based Polygonization And Modeling For Implicit Surfaces

Implicit surface is one of the most important surface representations in computer graphics. It provides great advantages in inside/outside detection, complex topology rep-resentation, smooth objects merging etc., and thus it is commonly used in applications of geometry modeling and visualization. Among all research problems in implicit surfaces, two of them are very important:polygonization from implicit surfaces and reconstruc-tion of implicit surfaces from polygon meshes.Focusing mainly on these two problems, we first briefly introduce the research background and existing approaches for the implicit surface. Then we analyze the prob-lems in current researches and propose new solutions, including binary volume opti-mization, GPU-based polygonization for implicit surfaces, solid Fourier transform and its applications. The main contributions of this dissertation can be summarized as fol-lows:● We propose a new method for optimizing binary volume data. The novelty of this method is the adoption of "maximum a posteriori-Markov random field (MAP-MRF)" probability model that predicts the most probable value of a binary vol-ume, which is considered as optimal. Under this framework, a user can choose different prior knowledge and observation models to fit different situations, which makes our method flexible. We deduce a general formula, as well as formulae in several special cases, for optimizing binary volume data under the MAP-MRF framework. Our method can be applied in the visualization, smoothing, denoising and repairing for volume data.● We introduce a practical GPU-based approach to efficiently polygonize and opti-mize iso-surface meshes for implicit surfaces. Specifically, we design new schemes to maximally exploit the parallel features of the GPU hardware, by optimizing both the geometry (vertex position, vertex distribution, triangle shape, and trian-gle normal) and the topology (connectivity) aspects of a mesh. Our experimental results show that, besides significant improvement on the resultant mesh qual-ity, our GPU-based approach is approximately an order of magnitude faster than its CPU counterpart and faster than or comparable to other GPU iso-surface ex-traction methods. Furthermore, the achieved speedup becomes even higher if the resolution of the isosurface is increased.● We introduce a theory of the solid Fourier Transform (SFT). It transforms a solid, represented by a polygon mesh, to its frequency domain. Starting from processing a triangle, we calculate the transform by converting a volume integral into a surface integral. After that, we further extend it to a generalized version. We design an acceleration approach based on normal discretization that greatly improves the performance and makes the theory more practical.● Based on the theory of SFT, we introduce several applications in modeling and geometry processing. We propose a modeling method based on solid skeleton con-volution surfaces. The calculation of convolution can be transformed into to fre-quency domain by the Convolution Theorem. We propose a robust method for cre-ating 3-D mathematical morphology, by proving it can be achieved through solid skeleton convolution surfaces. We propose an implicit model repairing method for 3-D printing applications, which is able to repair polygon models with holes, self-intersections, inconsistent normals and non-manifold faces.