Implicit surface is one of the most important surface representations in computer graphics. It provides great advantages in inside/outside detection, complex topology representation, 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 reconstruction 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 problems in current researches and propose new solutions, including binary volume optimization, GPUbased polygonization for implicit surfaces, solid Fourier transform and its applications. The main contributions of this dissertation can be summarized as follows:â— We propose a new method for optimizing binary volume data. The novelty of this method is the adoption of "maximum a posterioriMarkov random field (MAPMRF)" probability model that predicts the most probable value of a binary volume, 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 MAPMRF framework. Our method can be applied in the visualization, smoothing, denoising and repairing for volume data.â— We introduce a practical GPUbased approach to efficiently polygonize and optimize isosurface 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 triangle normal) and the topology (connectivity) aspects of a mesh. Our experimental results show that, besides significant improvement on the resultant mesh quality, our GPUbased approach is approximately an order of magnitude faster than its CPU counterpart and faster than or comparable to other GPU isosurface extraction methods. Furthermore, the achieved speedup becomes even higher if the resolution of the isosurface is increased.â— 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.â— Based on the theory of SFT, we introduce several applications in modeling and geometry processing. We propose a modeling method based on solid skeleton convolution surfaces. The calculation of convolution can be transformed into to frequency domain by the Convolution Theorem. We propose a robust method for creating 3D mathematical morphology, by proving it can be achieved through solid skeleton convolution surfaces. We propose an implicit model repairing method for 3D printing applications, which is able to repair polygon models with holes, selfintersections, inconsistent normals and nonmanifold faces.
