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MeshCone: Second-Order Cone Programming for Geometrically-Constrained Mesh Enhancement

Alexander Valverde

TL;DR

MeshCone tackles the problem of refining imperfect meshes by leveraging available reference geometry within a convex optimization framework. It formulates the refinement as a second-order cone program that jointly aligns vertex positions to a target distribution and enforces smoothness via edge-length constraints, solved with the Splitting Conic Solver to guarantee global optimality and stability. The work demonstrates strong quantitative improvements over unoptimized baselines (e.g., Laplacian smoothing) across diverse shapes, with sub-second inference times, and provides a principled baseline for geometry-aware mesh refinement that can integrate with template- and scan-based workflows. The findings suggest that convex optimization can offer reliable, interpretable refinements in production pipelines, while opening avenues for hybrid approaches that combine mathematical guarantees with data-driven priors.

Abstract

Modern mesh generation pipelines whether learning-based or classical often produce outputs requiring post-processing to achieve production-quality geometry. This work introduces MeshCone, a convex optimization framework for guided mesh refinement that leverages reference geometry to correct deformed or degraded meshes. We formulate the problem as a second-order cone program where vertex positions are optimized to align with target geometry while enforcing smoothness through convex edge-length regularization. MeshCone performs geometry-aware optimization that preserves fine details while correcting structural defects. We demonstrate robust performance across 56 diverse object categories from ShapeNet and ThreeDScans, achieving superior refinement quality compared to Laplacian smoothing and unoptimized baselines while maintaining sub-second inference times. MeshCone is particularly suited for applications where reference geometry is available, such as mesh-from-template workflows, scan-to-CAD alignment, and quality assurance in asset production pipelines.

MeshCone: Second-Order Cone Programming for Geometrically-Constrained Mesh Enhancement

TL;DR

MeshCone tackles the problem of refining imperfect meshes by leveraging available reference geometry within a convex optimization framework. It formulates the refinement as a second-order cone program that jointly aligns vertex positions to a target distribution and enforces smoothness via edge-length constraints, solved with the Splitting Conic Solver to guarantee global optimality and stability. The work demonstrates strong quantitative improvements over unoptimized baselines (e.g., Laplacian smoothing) across diverse shapes, with sub-second inference times, and provides a principled baseline for geometry-aware mesh refinement that can integrate with template- and scan-based workflows. The findings suggest that convex optimization can offer reliable, interpretable refinements in production pipelines, while opening avenues for hybrid approaches that combine mathematical guarantees with data-driven priors.

Abstract

Modern mesh generation pipelines whether learning-based or classical often produce outputs requiring post-processing to achieve production-quality geometry. This work introduces MeshCone, a convex optimization framework for guided mesh refinement that leverages reference geometry to correct deformed or degraded meshes. We formulate the problem as a second-order cone program where vertex positions are optimized to align with target geometry while enforcing smoothness through convex edge-length regularization. MeshCone performs geometry-aware optimization that preserves fine details while correcting structural defects. We demonstrate robust performance across 56 diverse object categories from ShapeNet and ThreeDScans, achieving superior refinement quality compared to Laplacian smoothing and unoptimized baselines while maintaining sub-second inference times. MeshCone is particularly suited for applications where reference geometry is available, such as mesh-from-template workflows, scan-to-CAD alignment, and quality assurance in asset production pipelines.

Paper Structure

This paper contains 24 sections, 16 equations, 6 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Surface Reconstruction Methods
  4. Mesh Generation
  5. Mesh Refinement
  6. Convex Optimization in 3D
  7. Method
  8. 3D Mesh Overview
  9. Convex Optimization
  10. Conic Solvers
  11. Mathematical Formulation
  12. Constraints
  13. Experiments
  14. Results
  15. Ablations
  16. ...and 9 more sections

Figures (6)

  • Figure 1: Overview of the MeshCone framework. Starting from a shapeless input mesh, we formulate the refinement as a second-order cone program with linear regularization and smoothness constraints, which is solved using the Splitting Conic Solver to produce a refined mesh.
  • Figure 2: Comparison of optimization landscapes: (left) a convex objective function with convex constraints showing a single global minimum, (right) a nonconvex saddle function with nonconvex constraints exhibiting multiple local minimums.
  • Figure 3: Second-Order Cone showing the constraint $\|\mathbf{x}\| \leq t$. The cone surface represents the feasible area boundary, with black points indicating valid solutions satisfying the constraint. For this example, $t$ has a maximum value of 3
  • Figure 4: Qualitative results on 16 meshes from ShapeNetv2
  • Figure 5: Quality comparison on three meshes: Ground Truth (left), TetWeave (middle left), Laplacian-smoothed TetWeave (middle right), and MeshCone (right).
  • ...and 1 more figures