A unifying framework for $n$-dimensional quasi-conformal mappings

TL;DR

This paper introduces a comprehensive variational framework for computing n-dimensional quasi-conformal mappings, enabling applications like medical image registration and shape modeling with controlled distortions and geometric constraints.

Contribution

It develops a unifying variational model for n-dimensional quasi-conformal mappings that incorporates multiple distortions and constraints, with proven existence of solutions and efficient numerical algorithms.

Findings

Effective in 2D and 3D applications

Handles various deformation constraints

Improves mapping accuracy and control

Abstract

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.