Efficient Simplical Mappings With Non-linear Control

Simplical mappings, i.e. mappings between simplical complex meshes, is a fundamental problem in digital geometry processing, where mesh parameterization and deformation are one of the most important simplical mappings. With the increasing development of digital geometry processing and related areas, new applications on these two simplical mappings are emerging day by day. And there are no lack of such problems with non-linear nature, which receive attentions from researchers. Therefore, how to model these problems and explore efficient algorithms conform to the trend of the development of digital geometry processing and become its important task.The dissertation focuses on some problems with non-linear control arisen from simplical mappings such as texture mapping, locally injective mapping, and volumetric parameteri-zation, and proposes corresponding modelling methods and solutions. The content of the dissertation is summarized as follows:· We presented a texture mapping method guided by an importance map to preserve the shape of the prominent content in the texture. Traditional texture mapping meth-ods mainly focus on how to decrease the distortions as well as how to fulfill the hard positional constraints without foldovers, and ignore the importance of the texture con-tent. We proposed a new problem of content-aware texture mapping and formulated it as a non-uniform-weighted parameterization whose distortions are weighted by im-portance values. The distortions are measured by an improved version of LSCM(Least Square Conformal Maps) measure, LSCM+, which is capable of greatly decreasing the appearance of shrunken and fold-over triangles besides the nice property of shape p-reservation. As the importance values and texture coordinates are dependent on each other in the formulation, the non-linear property of the parameterization is increased. To solve the parameterization, we iteratively update the weights on triangle faces and the texture coordinates. For the calculation of triangles’ weights, instead of adopting the brute force way that sums the importance values over the whole triangle, it uses an efficient method by transforming the area integral inside the triangle into a line in-tegral around its boundary. For the calculation of texture coordinates, it employs the ’L-M’(Levenberg-Marquardt) non-linear least square solver and rectify the non-PSD (Positive Semi-Defined) Hessian matrix of the energy. The method has superiority over the traditional texture mapping methods in controlling the distribution of the mapping distortions as well as preserving the shape of important content.· We proposed a remeshing-assisted optimization method for locally injective mappings. Existing methods on locally injective mappings are sorely geometry based, which solve the geometry problem on the fixed mesh domain. However, the prescribed tessellation may impose strong restrictions on the solution. As a consequence, the feasible region may be too small to contain an ideal solution, which leads to problems of slow conver-gence, poor solution, or even that no solution can be found. Our method integrates adaptive remeshing into interior point method based optimization solver. It updates the vertex positions via a parameter-free relaxation enhanced geometry optimization, and then uses edge-flip operations to reduce the residual and keep a reasonable condi-tion number for better convergence. For more robustness, when the iteration of interior point method terminates but leaves the positional constraints unsatisfied, it estimates the edges in the current tessellation that block vertices moving based on the conver-gence information of the optimization, and then splits neighbouring edges to break the restriction. The method has better performance than the solely geometric optimization approaches, especially for extreme deformations.·We introduced a stretch-minimizing volumetric parameterization method. Not many methods for volumetric parameterization guarantee bijectivity or local injectivity. Though recent work can produce such parameterizations with the bound control on angle dis-tortions, but may have large volume distortions. We extended the traditional stretch-distortion energy in surface parameterization to 3D and used it for volumetric parame-terization. The distortion energy naturally includes a barrier term which can effectively prevent tetrahedron elements from collapsing and foldover. Thus it saves the effort in other methods of formulating additional energies or constrains to ensure locally in-jectivity. It is then used for boundary constrained volumetric parameterization and optimized with a relaxation-enhanced solver. This is different from the conventional approach of surface parameterization where mesh nodes are optimized individually. Comparing to other volumetric parameterizations, the approach bears the advantages of stretch-minimizing method, being foldover-free and offering a good trade-off between angle and volume distortions.