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$n$-Dimensional Volumetric Stretch Energy Minimization for Volume-/Mass-Preserving Parameterizations

Zhong-Heng Tan, Tiexiang Li, Wen-Wei Lin, Shing-Tung Yau

TL;DR

This work introduces the $n$-dimensional volumetric stretch energy ($n$-VSE) as a principled objective for computing volume-/mass-preserving parameterizations of $n$-manifolds to the unit $n$-ball. It derives a continuous lower bound $E_V(f)\geq \frac{\nu(\\mathbb{B}^n)^2}{\mu(\\mathcal{M})}$, with equality iff the map is mass-preserving, and develops a discrete counterpart using a volumetric-stretch Laplacian with modified cotangent weights. The authors propose the $n$-VSEM algorithm, which splits the problem into boundary (spherical) and interior (ball) subproblems, solved via Newton-type methods and fixed-point iterations, respectively. Numerical experiments on ellipsoids and general meshes demonstrate high accuracy and robustness, with applications to registration and deformation that illustrate the method’s versatility in higher-dimensional geometry processing.

Abstract

In this paper, we develop an $n$ dimensional volumetric stretch energy ($n$-VSE) functional for the volume-/mass-preserving parameterization of the $n$-manifolds topologically equivalent to $n$-ball. The $n$-VSE has a lower bound and equal to it if and only if the map is volume-/mass-preserving. This motivates us to minimize the $n$-VSE to achieve the ideal volume-/mass-preserving parameterization. In the discrete case, we also guarantee the relation between the lower bound and the volume-/mass-preservation, and propose the spherical and ball volume-/mass-preserving parameterization algorithms. The numerical experiments indicate the accuracy and robustness of the proposed algorithms. The modified algorithms are applied to the manifold registration and deformation, showing the versatility of $n$-VSE.

$n$-Dimensional Volumetric Stretch Energy Minimization for Volume-/Mass-Preserving Parameterizations

TL;DR

This work introduces the -dimensional volumetric stretch energy (-VSE) as a principled objective for computing volume-/mass-preserving parameterizations of -manifolds to the unit -ball. It derives a continuous lower bound , with equality iff the map is mass-preserving, and develops a discrete counterpart using a volumetric-stretch Laplacian with modified cotangent weights. The authors propose the -VSEM algorithm, which splits the problem into boundary (spherical) and interior (ball) subproblems, solved via Newton-type methods and fixed-point iterations, respectively. Numerical experiments on ellipsoids and general meshes demonstrate high accuracy and robustness, with applications to registration and deformation that illustrate the method’s versatility in higher-dimensional geometry processing.

Abstract

In this paper, we develop an dimensional volumetric stretch energy (-VSE) functional for the volume-/mass-preserving parameterization of the -manifolds topologically equivalent to -ball. The -VSE has a lower bound and equal to it if and only if the map is volume-/mass-preserving. This motivates us to minimize the -VSE to achieve the ideal volume-/mass-preserving parameterization. In the discrete case, we also guarantee the relation between the lower bound and the volume-/mass-preservation, and propose the spherical and ball volume-/mass-preserving parameterization algorithms. The numerical experiments indicate the accuracy and robustness of the proposed algorithms. The modified algorithms are applied to the manifold registration and deformation, showing the versatility of -VSE.
Paper Structure (16 sections, 8 theorems, 92 equations, 3 figures, 2 tables, 4 algorithms)

This paper contains 16 sections, 8 theorems, 92 equations, 3 figures, 2 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. $n$-dimensional Volumetric Stretch Energy
  3. Discrete $n$-Manifold and $n$-Volumetric Stretch Energy
  4. Discrete $n$-Manifold
  5. Discrete Volumetric Stretch Energy
  6. Volumetric Stretch Energy Minimization
  7. Initial map for spherical parameterization
  8. Spherical $\varepsilon$-mass-preserving parameterization
  9. Ball $\varepsilon$-mass-preserving parameterization
  10. Numerical Experiments
  11. Ellipsoid case
  12. General cases
  13. Applications
  14. Registration
  15. Deformation
  16. ...and 1 more sections

Key Result

Theorem 1

Let $\mathcal{M}$ be an $n$-manifold to be topologically equivalent to $\mathcal{N}$, $\mu$ and $\nu$ be the measures defined on $\mathcal{M}$ and $\mathcal{N}$, $\rho_\mu$ and $\rho_\nu$ be the corresponding densities. For $n$-VSE $E_V(f)$ defined in def:EVfcontinue, let Then we have

Figures (3)

  • Figure 1: The area-preserving map $(r_f,\theta_f) = (r_v,\theta_v + kr_v)$ with the parameter $k = 0,1,3$.
  • Figure 2: The $3$ and $4$ dimensional benchmarks for the experiments. The first row is $3$ dimensional benchmarks and the second row is the projection along a direction of $4$ dimensional benchmarks.
  • Figure 3: Histograms of volume ratios $\delta + 1$ on the benchmarks.

Theorems & Definitions (23)

  • Remark 1
  • Theorem 1
  • proof
  • Definition 1
  • Theorem 2
  • proof
  • Remark 2
  • Definition 2
  • Definition 3
  • Lemma 1
  • ...and 13 more