Algorithms for User-Guided Surface Mappings
Computing mappings between spaces is a very general problem that appears in various forms in geometry processing. They can be used to provide descriptions or representations of shapes, or place shapes in correspondence. Their applications range from surface modeling and analysis to shape matching, morphing, attribute transfer and deformation. This thesis addresses two particular mapping problems that are of interest in the field, namely inter-surface maps and parameterizations. We focus on methods that are suitable for user-guided applications – we do not consider automatic methods, that do not leave space for the user to control the result. Existing meth- ods for the particular sub-problems that we are studying often either suffer from performance limitations, or cannot guarantee that the produced results align with the user’s intent; we improve upon the state of the art in both those respects. The first problem we study in this thesis is that of inter-surface mapping, with given sparse landmark point correspondences. We found that an efficient solution to this otherwise difficult topic emerges if one reformulates the mapping problem as a problem of finding affine combinations of points on the involved shapes. We extend the notion of standard Euclidean weighted averaging to 3D manifold shapes, and introduce a fast approximation that can be used to solve this problem much faster than the state of the art. We showcase applications of this approach in interactive attribute transfer between shapes. Next, we move on to the problem of surface parameterization. Here, we study the problem from the application point of view of surface remeshing; a popular way to generate a quadrilateral mesh for a given triangular mesh is to first compute a global parameterization, which is guided by a tangent vector field. This field then determines the directions of the quadrilateral edges on the output mesh. In order to design such a direction field, recent methods to tackle the problem are based on integer optimization problems, which often suffer from slow performance and local minima. We reformulate the problem in a way that the field design problem becomes a linear problem. We also add more flexibility by allowing for non- orthogonal directions. Still on the same problem of field-aligned surface parameterizations, we notice that the standard way of producing fields –namely, an optimization only focused iiion field smoothness– does not necessarily guarantee that the resulting quadrilateral meshing will be what the user intended in terms of edge directions. This is due to errors introduced in the post-processing of the field, during the later stages of the remeshing pipeline. This renders such fields suboptimal for user-guided meshing applications. We extend our efficient reformulation of the field design problem to generate fields that are guaranteed to not introduce such further errors, and thus make sure that the users obtain the expected results. Additionally, we allow users more flexible control, by supporting assignment of partial constraints for only some of the directions.