Researches Of Detail Editing And Symmetry Of Triangular Meshes

With the rapid development of geometric modeling techniques, the acquisition of3D geo-metrical data become increasingly easier.3D geometrical data are becoming a new type of medi-um after sound, image and video, and is widely applied in industry. The new type of data call for the corresponding processing methods, thus the digital geometry processing was born and has become one of focuses in computer graphics now. Compared with the traditional multimedia, the3D geometrical data are irregular, curved and don’t admit a consistent parameterization, thus the classical signal processing methods cannot be directly generalized to the digital geometry processing. This dissertation does some researches on detail editing and symmetry of triangular meshes in digital geometry processing, the main works include as follows:(1) We use the Laplacian coordinates to measure local details of surfaces, and propose a novel surface detail editing method based on filtering the Laplacian coordinates. This method is not only fast but also supports various kinds of surface detail editing. The surface detail editing method is finally converted to solving a linear system. Based on the pre-factorization of the coefficient matrix of the system and interactive substitutions during the detail editing, our approach is much faster than previous surface detail editing methods. Furthermore, compared with the previous methods of surface detail editing, our approach can get more types of surface detail editing, which not only include surface smoothing, sharp feature preserving denoising and detail enhancing, but also smoothing and enhancing of details belonging to specified frequencies or locations.(2) Empirical Mode Decomposition (EMD) is a powerful tool for analyzing non-linear and non-stationary signals, and has drawn a great deal of attentions in various areas. We generalize the classical EMD from Euclidean space to the setting of surfaces. The core of our general-ized EMD on surfaces is an envelope computation method form local extrema. This general-ized EMD can be effectively utilized in detail editing of scalar functions defined over surfaces and surfaces themselves in this paper. Since the generalized EMD on surfaces cannot preserve sharp features of surface, we still propose a feature preserving multi-scale surface decomposition based on the idea of extremal envelopes.(3) Based on the eigen-decomposition of Laplace-Beltrami operator, we construct a class of global intrinsic symmetry invariant functions on compact Riemannian manifolds, and give its rigid proof in theory. We also discretize the above general theories about global intrinsic symmetry invariant functions on2D manifold surfaces. Compared with the global intrinsic symmetry invariant functions on surfaces via geodesic distances, our global intrinsic symmetry invariant functions have the superiority in speed and robustness to local topology noises. Based on these global intrinsic symmetry invariant functions on surface, a class of symmetry factored embedding and distance is proposed and applied in the computation of symmetry orbits and symmetry aware segmentations in this paper.