Isovolumetric Energy Minimization for Ball-Shaped Volume-Preserving Parameterizations of 3-Manifolds

TL;DR

This paper introduces a novel isovolumetric energy minimization approach for computing volume-preserving, ball-shaped parameterizations of 3-manifolds, with a new efficient algorithm and practical applications in shape registration.

Contribution

It formulates volume-preserving parameterizations as an IEM problem and develops a preconditioned nonlinear conjugate gradient algorithm with guaranteed convergence.

Findings

The proposed algorithm achieves higher accuracy than existing methods.

It significantly improves computational efficiency.

Applications demonstrate practical usefulness in shape registration.

Abstract

A volume-preserving parameterization is a bijective mapping that maps a 3-manifold onto a specified canonical domain that preserves the local volume. This paper formulates the computation of ball-shaped volume-preserving parameterizations as an isovolumetric energy minimization (IEM) problem with the boundary points constrained on a unit sphere. In addition, we develop a new preconditioned nonlinear conjugate gradient algorithm for solving the IEM problem with guaranteed theoretical convergence and significantly improved accuracy and computational efficiency compared to other state-of-the-art algorithms. Applications to solid shape registration and deformation are presented to highlight the usefulness of the proposed algorithm.