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Relaxation and $\Gamma$-Convergence of Stretch Energy on Surface Homeomorphisms

Shu-Yung Liu

arXiv:2610.30214v1math.NAcs.NAmath.FA

TL;DR

Area-preserving simplicial parameterizations may not exist exactly, and it is unclear whether decreasing discrete area distortion corresponds to area-preserving homeomorphisms. The paper defines a relaxed stretch energy through pushforward area measures and lifts discrete maps to establish Γ-convergence. The minimal discrete energies converge to the continuum minimum, and limits of lifted discrete local minimizers in fixed-radius balls are area-preserving.

  • Problem

    It is unknown whether decreasing triangle-wise distortion in discrete parameterization yields area-preserving mappings corresponding to area-preserving surface homeomorphisms.

    Source: §1 · pp. 1–2

  • Method

    The paper defines relaxed stretch energy on homeomorphisms through pushforward area measures and constructs a lift from admissible discrete maps to homeomorphisms.

    Source: §1.1 · p. 3

  • Results

    Minimal discrete energies converge to the continuum minimum, and limits of lifted discrete local minimizers in fixed-radius balls are area-preserving.

    Source: §1.1 · pp. 3–4

  • Takeaways & Limitations

    The lift-based Γ-convergence relates discrete stretch energies to the relaxed continuum energy on homeomorphisms without requiring image-mesh shape regularity.

    Source: §1.1 · p. 3

    The analysis considers orientation-preserving homeomorphism limits, while the closest-point projection estimate for the image mesh is stated under a shape-regularity assumption.

    Source: §1.1 · p. 3, §4.1 · p. 15

Abstract

from arXiv · show

Area-preserving parameterization of triangular meshes is an essential task in geometry processing, yet an exactly area-preserving simplicial mapping may not exist in general. In this paper, we study area preservation of simplicial mappings in the limit of mesh refinement by establishing the $\Gamma$-convergence of the discrete stretch energy, a functional that measures their area distortion. We first introduce its limit functional, the relaxed stretch energy on surface homeomorphisms, defined through the pushforward of the area measure, and show that it is the relaxation of the stretch energy of diffeomorphisms with respect to uniform convergence. Then, to compare simplicial mappings with homeomorphisms, we construct a geometric lift that requires no shape-regularity assumption on the image mesh. Finally, we establish the $\Gamma$-convergence and show that the minimal discrete energies converge to the continuum minimum. Moreover, any homeomorphism arising as a limit of lifted discrete local minimizers in fixed-radius balls is area-preserving.

1 Introduction

Discrete area-distortion methods lack a clear connection to area-preserving homeomorphisms, especially because smoothing triangle-wise area ratios can introduce uncontrolled error. The paper relaxes stretch energy to homeomorphisms, constructs a lift for discrete maps, and establishes Γ-convergence with consequences for minima and local minimizers.

1 Introduction

  • Smoothing triangle-wise area ratios for differential-flow methods can have uncontrolled error and may destabilize algorithms.Piecewise-linear simplicial maps have constant Jacobian on each triangle.Source: §1 · p. 2
  • It remains unknown whether decreasing triangle-wise distortion yields area-preserving discrete maps corresponding to area-preserving surface homeomorphisms.The variational framework is effective in practice but lacks a connection to smooth-surface theory.Source: §1 · p. 2

1.1 Main results

  • The lift-based Γ-convergence connects discrete stretch energies to the relaxed energy on homeomorphisms, despite the maps belonging to different spaces.The lift requires no shape regularity on the image mesh.Source: §1 · p. 2, §1.1 · p. 3
  • Limits of lifted discrete local minimizers in fixed-radius balls minimize the relaxed energy there and are therefore area-preserving.The minimal discrete energies also converge to the continuum minimum |M|.Source: §1.1 · pp. 3–4

1.2 Outline of the proof

  • The proof establishes the liminf inequality using lower semicontinuity and lift-energy comparison, and the limsup inequality through two approximations.The approximations first interpolate a fixed diffeomorphism and then approach a homeomorphism with diffeomorphisms.Source: §1.2 · p. 4

2 Lower semicontinuity

The stretch energy on diffeomorphisms is not lower semicontinuous when extended by infinity to nonsmooth homeomorphic limits. Defining the relaxed energy through pushforward area measures yields lower semicontinuity under uniform convergence.

2 Lower semicontinuity

  • Uniform limits can lose smoothness, so extending diffeomorphism stretch energy by infinity to other homeomorphisms is not lower semicontinuous.The illustrated sequence consists of area-preserving sphere maps whose limit is area-preserving but nonsmooth.Source: §2 · pp. 4–5, Figure 2.1 · p. 5
  • The relaxed energy inherits the area-preserving equality condition: equality holds exactly when the pushforward area measure equals the source area measure.The passage identifies this measure condition with area preservation.Source: §2 · p. 5
  • Uniform convergence of homeomorphisms implies uniform convergence of their inverses, which supports convergence arguments involving pushforward measures.The inverse-convergence result is stated for orientation-preserving homeomorphisms between the compact surfaces.Source: §2 · p. 6
  • The relaxed energy uses the pushforward area measure, whose weak convergence under uniform convergence of homeomorphisms supports lower semicontinuity.The paper establishes weak convergence of the area measures and lower semicontinuity of the relaxed energy.Source: §2 · pp. 5–6

3 Approximation by diffeomorphisms

The section constructs smooth diffeomorphisms approximating finite-energy homeomorphisms while controlling both displacement and area distortion. This approximation identifies the relaxed stretch energy and supports an area-preservation result for local minimizers when the surfaces have equal area.

3 Approximation by diffeomorphisms

  • Uniform smoothing alone does not control Jacobians or energy, so the construction combines smooth approximation with cellwise area correction.A local Moser flow realizes the cell-average area densities while keeping the map close to the original homeomorphism.Source: §3 · pp. 7, 8, 10
  • Cellwise averaging of the relaxed Jacobian converges in L2, providing the density approximation used in the construction.The proof controls the averaging error using continuous-function approximation and uniform continuity.Source: §3 · p. 8
  • Every finite-energy homeomorphism admits smooth diffeomorphisms converging uniformly with stretch energies converging to its relaxed energy.The approximation is obtained by shrinking the cells in the construction.Source: §3 · p. 11
  • Lower semicontinuity together with the smooth recovery sequence identifies the relaxed stretch energy as the relaxation of the diffeomorphism energy.The energy convergence follows from convergence of the Jacobians in L2.Source: §3 · p. 11
  • Local minimality of a homeomorphism implies area preservation when |M| = |N|, with ES(f) = |M|.The result extends the diffeomorphism conclusion to homeomorphisms; local minimality is also global.Source: §3 · pp. 12–13

4 Lifting discrete maps

This section constructs a lift from simplicial maps to homeomorphisms and estimates its error, including for non-shape-regular image meshes. It uses this framework to establish interpolation-based approximation and Γ-convergence of the lifted discrete stretch energy, with consequences for discrete minima and limits of local minimizers.

4 Lifting discrete maps

  • The section defines a lifting that transforms a simplicial map into a homeomorphism and estimates the error it produces.For closed surfaces, closest-point projections provide the transformation; surfaces with boundary require additional correction maps.Source: §4 · p. 14, §4.2 · p. 16

4.1 Closest-point projection

  • For meshes without shape regularity, closest-point projection has only one-sided control: for triangle τ with vertices on smooth surface N and diameter kh sufficiently small, 0 ≤ JπN|τ(y) ≤ 1 + Ck²h for a.e. y∈τ.The constant C depends only on N.Source: §4.1 · p. 15

4.2 Boundary correction

  • For meshes with boundary, the lifted boundary differs from the smooth boundary by O(h²) in position and O(h) tangentially, and correction gives |Jph − 1| ≤ C′′h².The correction map pushes the lifted boundary graph onto the smooth boundary.Source: §4.2 · pp. 16, 19, 20

4.3 Admissible maps and energy comparison

  • The lifting construction yields an energy comparison for admissible simplicial maps, with its constant independent of the map.Admissible image meshes need not be shape-regular; the source meshes satisfy Assumption 9.Source: §4.3 · pp. 20, 22

4.4 Approximation by interpolation

  • For an orientation-preserving diffeomorphism, its piecewise-linear interpolants are admissible for sufficiently small h and their lifts and energies converge to the original map and energy.The convergence is stated as dH(LhIhg, g) → 0 and ES(Ihg) → ES(g).Source: §4.4 · pp. 22–23

5.1 The main theorem

  • Theorem 18 addresses ESh and ES under Assumption 9, quantifying over every u and convergent sequence uh→u, and separately specifying an existence statement for each u.Source: §5.1 · p. 24

5.2 Local minimizers

  • Limits of discrete maps minimizing energy in fixed-radius balls minimize continuum energy there and, when |M| = |N|, preserve area.Their energies converge to the continuum minimum, which equals |M|.Source: §5.2 · pp. 25–26

5.3 Almost minimizers and perturbations

  • For equal-area surfaces, discrete minimum energies converge to the continuum minimum |M|, with discrete infima bounded above by |M| + Ch^2.This also supplies a bound on the area distortion of almost minimizers.Source: §5.3 · pp. 26–27
  • Almost minimizers have controlled area distortion, extending the diffeomorphism-space result of [21, Theorem 16].The supplied passages do not specify the distortion bound's formula.Source: §5.3 · p. 27
  • The Γ-convergence persists after adding any dH-continuous functional, including a landmark-mismatch penalty.The paper motivates this perturbation for surface registration with landmark matching.Source: §5.3 · p. 27
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