A unifying framework for $n$-dimensional quasi-conformal mappings
This work addresses the need for effective mapping methods in higher-dimensional spaces for applications like medical imaging and computer graphics, though it appears incremental as it builds on existing quasi-conformal theory.
The authors tackled the problem of computing higher-dimensional quasi-conformal mappings by developing a unifying variational framework that integrates multiple constraints like distortion and landmarks, and they demonstrated its effectiveness in experiments with applications such as medical image registration and shape modeling.
With the advancement of computer technology, there is a surge of interest in effective mapping methods for objects in higher-dimensional spaces. To establish a one-to-one correspondence between objects, higher-dimensional quasi-conformal theory can be utilized for ensuring the bijectivity of the mappings. In addition, it is often desirable for the mappings to satisfy certain prescribed geometric constraints and possess low distortion in conformality or volume. In this work, we develop a unifying framework for computing $n$-dimensional quasi-conformal mappings. More specifically, we propose a variational model that integrates quasi-conformal distortion, volumetric distortion, landmark correspondence, intensity mismatch and volume prior information to handle a large variety of deformation problems. We further prove the existence of a minimizer for the proposed model and devise efficient numerical methods to solve the optimization problem. We demonstrate the effectiveness of the proposed framework using various experiments in two- and three-dimensions, with applications to medical image registration, adaptive remeshing and shape modeling.