Date of Award
Doctor of Philosophy (PhD)
Injective mapping of meshes is a fundamental yet long-standing problem in computer graphics. The problem is particularly challenging in the presence of constraints, as an injective initial map is often not available. In this dissertation, we propose methods for computing injective maps that satisfy constraints without the need for an injective initial map. Our key contribution is a family of novel energies that are smoothly defined for arbitrary maps (injective or non-injective) and that promote injectivity when minimized. We first introduce the Total Lifted Content (TLC) energy for mapping 2D and 3D meshes into target domains with constrained boundaries. Next, we present the Smooth Excess Area (SEA) energy for computing globally injective maps of triangle meshes with general positional constraints. Both TLC and SEA have desirable theoretical properties for injectivity and are simple to optimize using standard gradient-based solvers. The two methods proved highly successful in practice when tested on large-scale benchmarks for the injective mapping problem. Finally, we develop variants of these energy forms to reduce isometric distortion, which is often desirable in graphics applications. The proposed isometric variants retain the desirable traits and achieve a similar level of success in recovering injectivity as the original energies, but with significantly lower isometric distortion.
Ulugbek Kamilov, Brendan Juba, Quo-Shin Chi, Danny Kaufman,
Available for download on Wednesday, May 15, 2024