Abstract

Conformal© / Quasi-conformal (QC) geometry has a long history in pure mathematics, and is an active field in both modern geometry and modern physics. Recently, with the rapid development of 3D digital scanning technology, the demand for effective geometric processing and shape analysis is ever increasing. Computational conformal / quasi-conformal geometry plays an important role for these purposes. Applications can be found in different areas such as medical imaging, computer visions and computer graphics.

In this talk, I will first give an overview of how conformal geometry can be applied in medical imaging and computer graphics. Examples include brain registration and texture mapping, where the mappings are constructed to be as conformal as possible to reduce geometric distortions. In reality, most registrations and surface mappings involve non-conformal distortions, which require more general theories to study. A direct generalization of conformal mapping is quasiconformal mapping, where the mapping is allowed to have bounded conformality distortions. In the second part of my talk, theories of quasicoformal geometry and its applications will be presented. In particular, I will talk about how QC can be used for registration of biological surfaces, shape analysis, medical morphometry and the inpainting of surface diffeomorphism.