Discrete connections for geometry processing

Abstract

Connections provide a way to compare local quantities defined at different points of a geometric space. This thesis develops a discrete theory of connections that naturally leads to practical, efficient numerical algorithms for geometry processing. Our formulation is motivated by real-world applications where meshes may be noisy or coarsely discretized. Further, because our discrete framework closely parallels the smooth theory, we can draw upon a huge wealth of existing knowledge to develop and interpret mesh processing algorithms. The main contribution of this thesis is a new algorithm for computing trivial connections on discrete surfaces that are as smooth as possible everywhere but on a set of isolated singularities of given index. A connection is represented via an angle associated with each dual edge, i.e., a discrete angle-valued 1-form. These angles are determined by the solution to a linear system, and are globally optimal in the sense that they describe the trivial connection closest to Levi-Civita among all solutions with the prescribed set of singularities. Relative to previous methods our algorithm is surprisingly simple, and can be implemented using standard operations from mesh processing and linear algebra. The solution can be used to construct rotationally symmetric direction fields with a prescribed set of singularities and directional constraints, which are essential in applications such as quadrilateral remeshing and texture synthesis.