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

TL;DR

This work introduces an isovolumetric energy minimization framework for computing ball-shaped volume-preserving parameterizations of simply connected 3-manifolds by optimizing interior and spherical-boundary mappings simultaneously. It derives explicit gradients, develops a preconditioned nonlinear conjugate gradient method with a problem-tailored preconditioner and a quadratic-step-length approximation, and proves global convergence under strong Wolfe conditions. The method yields superior accuracy and bijectivity compared to the state-of-the-art VSEM, demonstrated on diverse tetrahedral meshes and applications to shape registration and deformation of brain anatomy. The approach enables robust volume-preserving mappings and provides a practical tool for shape analysis and dissimilarity measurement in 3D geometry processing.

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.

Figures (10)

• Figure 1: An illustration of the dihedral angle between faces $f([v_{i},v_{k},v_{\ell}])$ and $f([v_{j},v_{\ell},v_{k}])$ in the tetrahedron $f(\tau)=[f_{i}, f_{j}, f_{k}, f_{\ell}]$.
• Figure 2: An illustration for the tetrahedron $f(\tau_\alpha)$ formed by the boundary face $f(\alpha)$ and the origin $\mathbf{0}_3$ of $\mathbb{R}^3$.
• Figure 3: The benchmark tetrahedral mesh models.
• Figure 4: Histograms of the local volume distortion $D_\mathrm{V}(f,\tau)$\ref{['eq:VolDist']} of the mappings computed by the IEM, Algorithm \ref{['alg:IEM']}.
• Figure 5: Relationship between the isovolumetric energy and the number of iterations of the IEM and VSEM algorithms, respectively.

Theorems & Definitions (14)

• Theorem 1
• proof
• Lemma 1
• proof
• Lemma 2: NoWr06
• Lemma 3
• proof
• Corollary 1
• proof
• Lemma 4: modified Zoutendijk's condition