Title: Applications of Computational Conformal Geometry and Optimal Transport in CAD/CAE, Deep Generative Models and Medical Imaging

Abstract: In this dissertation, we will first introduce the application of conformal geometry in the CAD/CAE field. Specifically, we first discuss the application of harmonic foliation in weaving pattern design. Weaving and fabrication methods play important roles in the design and manufacturing of consumer products, toys, furniture, and architecture. One of the difficulties in weaving pattern design is to minimize the number of singularities, as well as preserve tensor-product structures in non-singular regions. Harmonic foliations, derived from holomorphic one-forms and quadratic differential forms, provide effective weaving patterns that meet these requirements: minimizes the number of singularities, and critical trajectories separate the shape into components that have tensor-product structures. Second, we introduce the application of conformal maps in topology optimization. Specifically, through conformal mappings, traditional level-set-based topology optimization techniques, which are performed on 2D planar domains, can be performed on freeform surfaces.  

In the second part, we will present optimal transport theory and its applications. Optimal transportation (OT) problem aims at finding the most economical way to transform one probability measure to the other, which plays a fundamental role in computer graphics, computer vision, machine learning, geometry processing and medical imaging. In this dissertation, First, we introduce the OT map-based deep generative model, AE-OT, which learns the distribution of a given dataset, and then generates infinitely many samples under that distribution. Compared to DNN based generator maps, this model could effectively mitigate mode collapse and mode mixture problems, which are caused by the singularities of distribution transport maps. Along this direction, we further analyze the properties of OT map singularities. Second, we introduce applications of OT maps under spherical settings in the field of medical imaging. Specifically, a signature shape method is proposed in the brain shape analysis of Alzheimer's disease patients. Signature shapes are computed from the intrinsic Alexandrov polyhedra of brain surfaces, which effectively captures area enlargement and shrinkage. Moreover, we discuss the hyperbolic power diagram and its relation to hyperbolic OT problems. An effective variational method that computes the hyperbolic OT map by solving a convex optimization is introduced.