On the Structural Failure of Chamfer Distance in 3D Shape Optimization

TL;DR

The presence or absence of non-local coupling determines whether Chamfer optimization succeeds or collapses, and provides a practical design criterion for any pipeline that optimizes point-level distance metrics.

Abstract

Chamfer distance is the standard training loss for point cloud reconstruction, completion, and generation, yet directly optimizing it can produce worse Chamfer values than not optimizing it at all. We show that this paradoxical failure is gradient-structural. The per-point Chamfer gradient creates a many-to-one collapse that is the unique attractor of the forward term and cannot be resolved by any local regularizer, including repulsion, smoothness, and density-aware re-weighting. We derive a necessary condition for collapse suppression: coupling must propagate beyond local neighborhoods. In a controlled 2D setting, shared-basis deformation suppresses collapse by providing global coupling; in 3D shape morphing, a differentiable MPM prior instantiates the same principle, consistently reducing the Chamfer gap across 20 directed pairs with a 2.5$\times$ improvement on the topologically complex dragon. The presence or absence of non-local coupling determines whether Chamfer optimization succeeds or collapses. This provides a practical design criterion for any pipeline that optimizes point-level distance metrics.

Figures (7)

• Figure 1: Physics-guided Chamfer optimization on diverse source--target pairs. Each row morphs a different source shape (Heart, Cow, Duck, Sphere) into a letter target. The differentiable physics prior provides non-local coupling, instantiating the design principle derived in Corollary 1, producing physically valid trajectories that faithfully capture the target geometry despite large topological differences between source and target (e.g., the genus-1 torus-like mesh closing into C).
• Figure 2: Qualitative comparison on four target shapes. DCO and DCD converge to structurally degraded volumetric shapes on all targets. The physics-only baseline preserves global plausibility but leaves a residual geometric gap. Our coupled physics--Chamfer method improves alignment while maintaining physically plausible volumetric structure.
• Figure 3: 2D collapse experiment (circle$\to$star).Top: Per-point CD optimization (DCO) collapses source points onto star vertices. Middle: Adding local repulsion does not resolve collapse (Proposition 3). Bottom: Shared-basis (Fourier) deformation provides global coupling that suppresses collapse. Gray: target; colored: source at initial / mid / final steps. Right column: CD convergence.
• Figure 4: Sphere$\to$dragon: qualitative comparison. DCO and DCD produce severe surface collapse. Physics-only preserves volumetric structure but leaves a large geometric gap. Our method at 4 particles per cell closely matches the target geometry while maintaining physical validity.
• Figure 5: DCD vs. Ours on Cow$\to$Duck: trajectory and internal structure. Top two rows: morphing trajectory showing the target (yellow, leftmost). DCD dissolves the source structure into a noisy surface approximation; our method maintains volumetric coherence throughout. Bottom: cross-section at the mid-body plane. The target and our method show uniformly filled interiors, while DCD collapses particles onto the surface, leaving the interior hollow. 16.5% of DCD particles have nearest-neighbor distance $<$0.01 (effectively overlapping), compared to 0% for our method.

Theorems & Definitions (11)

• proposition 1
• proof
• remark 1
• proposition 2
• proof
• remark 2
• proposition 3
• proof
• remark 3
• corollary 1