cs.GRDec 11, 2024

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

Authors: Alexander Valverde

Organizations: University of California, Santa Cruz

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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.

Figures & tables

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 ∥x∥≤t\|\mathbf{x}\|\leq t . The cone surface represents the feasible area boundary, with black points indicating valid solutions satisfying the constraint. For this example, tt has a maximum value of 3
 Avg. time (s)Avg. itersSuccess rate
Convex refinement1.2225100%
Table 1: Solver statistics for ConvMesh optimization across 53 ShapeNet meshes.
MethodCD ↓\downarrowEMD ↓\downarrowHD ↓\downarrowNC ↑\uparrowCE ↓\downarrow
No Optimization (MC)0.00380.1250.3110.6860.062
No Optimization (PR)0.02260.2210.5940.5190.065
MeshCone (Ours)0.0030.0810.0360.9170.038
Table 2: Average geometric evaluation metrics across all methods. Sampled 5000 points for evaluation across 53 meshes.
Figure 4: Qualitative results on 16 meshes from ShapeNetv2
RepresentationMetricTetWeaveMeshCone
BeethovenAR ↓\downarrow2.0221.862
EMD ↓\downarrow0.0920.091
HD ↓\downarrow0.1650.160
ActaeonAR ↓\downarrow1.6961.161
EMD ↓\downarrow0.06740.0670
HD ↓\downarrow0.0950.092
Table 3: Comparison between raw TetWeave meshes and refined meshes after convex optimization
Figure 5: Quality comparison on three meshes: Ground Truth (left), TetWeave (middle left), Laplacian-smoothed TetWeave (middle right), and MeshCone (right).
λ\lambdaHD ↓\downarrowNC ↑\uparrowTime/mesh (s)
0.0010.03670.9171.12
0.010.03650.9181.22
0.10.0370.9151.23
Table 4: Effect of regularization weight λ\lambda on mesh quality. Fixed: δ=0.5\delta=0.5 , 53 meshes, 1000 iterations, 10000 sampled points.
δ\deltaHD ↓\downarrowNC ↑\uparrowTime/mesh (s)
0.0050.3300.6884.74
0.050.0750.8941.58
0.50.0380.9171.22
Table 5: Effect of edge constraint threshold δ\delta on mesh quality. Fixed: λ=0.1\lambda=0.1 , 53 meshes, 1000 iterations, 10000 sampled points.
Figure 6: Failure case of MeshCone on a specific mesh from a kitchen
Appendix figures & tables2 assets

Supplementary material from the paper’s appendix.

Appendix

Problem Size
Variables ( nn )40,169
Constraints ( mm )68,128
Cone Structure
Primal zero / dual free vars ( zz )20,418
Linear vars ( ll )9,542
SOC vars ( qq )38,168
Appendix
Table 6: SCS problem structure for a representative mesh.
IterPri. res.Dual res.GapObj.ScaleTime (s)
05.00×10−15.00\times 10^{-1}3.47×10−33.47\times 10^{-3}1.51×1011.51\times 10^{1}7.557.551.0×10−11.0\times 10^{-1}6.25×10−26.25\times 10^{-2}
251.00×10−81.00\times 10^{-8}8.61×10−128.61\times 10^{-12}1.51×10−71.51\times 10^{-7}3.773.771.0×10−11.0\times 10^{-1}1.05×10−11.05\times 10^{-1}
Appendix
Table 7: SCS convergence diagnostics for a representative mesh. Values shown for the first iteration and the iteration at convergence (iteration 25).
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