Mesh Processing Based On Variational Approaches

As the development of3D scanning technology and the improvement of com-puter performance, mesh surface processing has been an active research direction in computer graphics. Typical problems include mesh reconstruction, denoising, segmentation, simplification, parameterization and mesh editing etc. Mesh surface processing methods have been well developed in the past decades and many classical and efficient algorithms have been proposed. These problems play a more and more important role in computer graphics, industrial manufacturing and reverse engineer-ing. In the thesis, we focus on mesh denoising, mesh segmentation and mesh simpli-fication. We propose several variational methods for these problems, overcoming the drawbacks of existing methods, such as failing preserving sharp features, instability of mesh segmentation algorithms, not semantic enough of segmentation results and so on. The contents include the following several parts.In Chapter1, we first briefly introduce the relevant knowledge of mesh surface processing and total variation. Then, we discuss about mesh denoising, mesh seg-mentation and mesh simplification from research background, related work and our contributions. b Finally, we present the structure arrangement of the thesis.In Chapter2, we first define some notation. Because piecewise linear space has many problems in some applications in computer graphics, we define piecewise constant space and associated differential operators. Furthermore, we give the rig-orous definition of total variation norm on meshes. Then, we compare piecewise constant space with piecewise linear space from various aspects and comparison re-sults demonstrate that piecewise constant space outperforms piecewise linear space in some applications in computer graphics.In Chapter3, we study feature preserving variational mesh denoising problem. Mesh surface denoising is a fundamental problem in geometry processing. The main challenge is to remove noise while preserving sharp features (such as edges and cor-ners) and preventing generating false edges. Based on the good edge preserving property of total variation (TV) and piecewise constant function space. An effective feature preserving mesh denoising method will then be presented by combining TV and piecewise constant function space, and provide two estimated formulae for com- puting parameters automatically. It is proved that, the solution of the variational problem is, in some sense continuously dependent on its parameter. To solve the variational problem, the thesis propose an efficient iterative algorithm based on vari-able splitting and augmented Lagrangian method. At last, the proposed denoising method is discussed and compared to several typical existing methods in various aspects. Experimental results show that our method outperforms all the compared methods for both CAD and non-CAD meshes at reasonable costs. It can preserve different levels of features well, and prevent from generating false edges in most cases.In Chapter4, the paper discusses part-type mesh segmentation. Most of ex-isting segmentation methods either depend on initial values severely, or segmenta-tion results are not semantic enough. This paper presents a new stable, effective, and triangle-based variational method. The Chapter first define four new Lapla-cian matrices by incorporating high order information of the meshes. Compared to the classical Laplacian matrices, these new matrices can reveal more correct se-mantic information. Considering that triangles are usually distributed on a mesh non-uniformly, we propose a triangle-based variational segmentation model to di-rectly cluster triangles, by using piecewise constant function space and the classical Mumford-Shah model. This variational problem is solved by an efficient iterative algorithm based on alternative minimization and augmented Lagrangian method. Since directly clustering triangles instead of vertices, our variational model gener-ates real region-based segmentation results. Due to the non-convexity of the vari-ational problem, we give an effective initialization technique for making our model stable. In addition, we introduce a simple and efficient user interaction technique to improve the segmentation results when needed. Finally, experiments demonstrate the advantages such as stability and effectiveness of our method. It outperforms competitive segmentation methods when evaluated on the Princeton Segmentation Benchmark.In Chapter5, we discuss surface-type mesh segmentation and piecewise constant surface mesh simplification. According to the discussion of the preceding chapters, we find that mesh denoising and mesh segmentation both belong to different levels of mesh simplification method. Based on the good properties of piecewise constant space and the similarity between mesh denoising and mesh simplification, the Chap-ter first proposes a surface-type mesh segmentation method. Then, we propose a new mesh approximation method by piecewise constant surface based on the vari-ational mesh denoising method proposed in Chapter3. Experimental results show that both our surface-type mesh segmentation method and our piecewise constant approximation method can produce quite satisfying results.In Chapter6, we summarize the main content of this thesis and put forward some prospects for the future work.