A Novel Formulation of Least-Squares Quasi-Conformal Parameterization

Abstract

Quasi-conformal parameterization is important in many CG and CAD applications. It is well-known that solving the Beltrami system {em with} natural boundary conditions gives a least-squares quasi-conformal map. In this research, we propose a novel formulation of least-squares quasi-conformal parameterization based on classical differential geometry tools and a formula of variational calculus. We found an interesting theoretical result such that the Beltrami system {em without} natural boundary conditions (fixed boundary conditions) also provides a least-squares quasi-conformal map. This result elucidates why the Beltrami system always generates a parameterization with some conformality even if we use fixed boundary conditions.