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

Chang-Yong Song; David Hyde

cs.CVMar 10, 2026

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

Authors: Chang-Yong Song, David Hyde

Organizations: Vanderbilt University, Nashville, TN, USA

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 & tables

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.

Condition	s $\to$ t	t $\to$ s	Two-sided	vs DCO
DCO (no reg.)	0.220	0.182	0.286	–
+ Repulsion ( $\lambda$ =0.01)	0.232	0.182	0.295	worse
+ Repulsion ( $\lambda$ =0.1)	0.250	0.182	0.309	worse
+ Rep. + Smooth	0.240	0.181	0.301	worse
+ Rep. + Smooth + Vol.	0.240	0.194	0.309	worst
DCD [ 49 ]	0.197	0.182	0.268	$\sim$ same

Table 1: Effect of adding local regularizers and density-aware re-weighting (DCD [ 49 ] ) to DCO. No condition improves t

\to

s coverage over pure DCO; the physics-only baseline outperforms all conditions.

Target	DCO	DCD [ 49 ]	Physics	DCO/Phys	DCD/Phys
Bunny	0.286	0.268	0.217	1.3 $\times$	1.2 $\times$
Duck	0.744	0.740	0.247	3.0 $\times$	3.0 $\times$
Cow	0.449	0.436	0.228	2.0 $\times$	1.9 $\times$
Teapot	0.209	0.199	0.230	0.9 $\times$	0.9 $\times$

Table 2: Generalization across four target shapes (Sphere source). DCO and DCD produce nearly identical results on every shape; the t

\to

s component is virtually unchanged by density re-weighting, confirming that the coverage failure is structural. The topologically complex dragon is analyzed separately in Section 5.4 .

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.

	s $\to$ t $\downarrow$		t $\to$ s $\downarrow$		Two-sided $\downarrow$
Target	Physics	Ours	Physics	Ours	Physics	Ours
Bunny	0.181	0.157	0.174	0.163	0.251	0.226
Duck	0.184	0.173	0.214	0.215	0.282	0.276
Cow	0.178	0.167	0.185	0.167	0.257	0.236
Teapot	0.180	0.148	0.190	0.181	0.262	0.234

Table 3: Chamfer distance at frame 39 for physics-only and our method (Sphere source). Our method improves s

\to

t on all four shapes (average

-

10.8%) and two-sided CD on all four.

	Hausdorff $\downarrow$		F1 ( $\tau{=}0.2$ ) $\uparrow$
Target	Physics	Ours	Physics	Ours
Bunny	0.963	0.580	0.757	0.851
Duck	0.930	0.646	0.626	0.643
Cow	0.715	0.672	0.733	0.819
Teapot	0.954	0.842	0.688	0.810

Table 4: Hausdorff distance (

\downarrow

) and F1-score (

\uparrow

\tau{=}0.2

) at frame 39 (Sphere source). Improvements are consistent across metrics, confirming that the Chamfer gap reduction reflects genuine geometric improvement.

Src $\downarrow$ / Tgt $\rightarrow$	Sphere	Bunny	Duck	Cow	Teapot
Sphere	—	0.251 / 0.226	0.282 / 0.276	0.257 / 0.236	0.262 / 0.234
Bunny	0.295 / 0.250	—	0.297 / 0.288	0.256 / 0.240	0.286 / 0.259
Duck	0.293 / 0.296	0.283 / 0.276	—	0.280 / 0.295	0.326 / 0.309
Cow	0.291 / 0.315	0.275 / 0.252	0.425 / 0.284	—	0.318 / 0.293
Teapot	0.226 / 0.213	0.257 / 0.252	0.355 / 0.312	0.237 / 0.259	—

Table 5: Pairwise morphing results (two-sided CD) in matrix format. Each cell shows Physics / Ours . Bold : improvement over physics-only. Our method improves two-sided CD on 16 of 20 directed pairs.

Method	s $\to$ t $\downarrow$	t $\to$ s $\downarrow$	Two-sided $\downarrow$	Hausdorff $\downarrow$	F1 $\uparrow$
DCO	0.609	0.255	0.660	2.695	0.466
DCD [ 49 ]	0.608	0.255	0.659	2.849	0.463
Physics-only	0.190	0.246	0.311	3.202	0.532
Ours (3 particles per cell)	0.160	0.296	0.336	3.251	0.493
Ours (4 particles per cell)	0.089	0.087	0.124	0.780	0.792

Table 6: Sphere

\to

dragon results. At the default resolution of 3 particles per cell, our method improves s

\to

t but worsens t

\to

s due to insufficient particle count. Increasing to 4 particles per cell resolves this limitation, achieving a 2.5

\times

improvement in two-sided CD over the physics-only baseline. Hausdorff distance and F1-score confirm the same trend.

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.

Appendix figures & tables7 assets

Supplementary material from the paper’s appendix.

Appendix

Parameter	Symbol	Value
Grid
Grid spacing	$\Delta x$	1.0
Domain	$[\mathbf{x}_{\min},\mathbf{x}_{\max}]$	$[-16,16]^{3}$
Effective resolution		$32^{3}$ nodes
Material (Neo-Hookean)
First Lamé parameter	$\lambda$	38 889

Appendix

Table 7: MPM simulation parameters. All experiments share the same material model and grid configuration; only particle resolution varies.

Parameter	Symbol	Value
Gradient descent
Number of frames	$T$	40
Forward–backward passes		3
GD iterations per pass		1
Line search iterations		5
Initial step size	$\alpha_{0}$	0.01

Appendix

Table 8: Optimization and Chamfer schedule parameters.

Src $\downarrow$ / Tgt $\rightarrow$	Sphere	Bunny	Duck	Cow	Teapot
Sphere	—	0.181 / 0.157	0.184 / 0.173	0.178 / 0.167	0.180 / 0.148
Bunny	0.199 / 0.159	—	0.186 / 0.167	0.178 / 0.159	0.187 / 0.150
Duck	0.181 / 0.168	0.183 / 0.178	—	0.184 / 0.176	0.190 / 0.172
Cow	0.186 / 0.167	0.188 / 0.156	0.325 / 0.168	—	0.193 / 0.161
Teapot	0.176 / 0.146	0.178 / 0.165	0.273 / 0.176	0.178 / 0.169	—

Appendix

Table 9: Pairwise morphing results (s

\to

t CD) in matrix format. Each cell shows Physics / Ours . Bold : improvement over physics-only. Our method improves s

\to

t on all 20 directed pairs.

Method	Two-sided CD $\downarrow$	Coverage (%) $\uparrow$	Cluster Fraction (%) $\downarrow$
DCO	0.352	38	97
DCO + Repulsion	0.299	50	97
Shared-basis (Fourier, $K{=}12$ )	0.123	62	81

Appendix

Table 10: Quantitative results of the 2D collapse experiment (circle

\to

star).

Figure 6: Frame-wise source-to-target Chamfer distance across four target shapes. DCO and DCD converge to substantially worse plateaus on geometrically complex targets, while our method consistently refines the physics prior. On the nearly convex teapot, DCO achieves comparable s $\to$ t but at the cost of structural collapse.

Figure 7: Pairwise two-sided CD: Physics-only vs. Ours. Each of the 20 directed pairs is one point; marker shape indicates the source. Points below the diagonal (16/20) denote improvement by our method. The four regressions all involve convex targets (Sphere or Cow) where the physics baseline already achieves near-optimal coverage.

Configuration	s $\to$ t $\downarrow$	t $\to$ s $\downarrow$	Two-sided $\downarrow$
Full method (Ours)	0.157	0.163	0.226
Coupled schedule
Fixed $w_{\text{rev}}=1.0$ (no coupling)	0.158	0.176	0.237
Fixed $w_{\text{rev}}=0.5$	0.158	0.182	0.241
Clamp ratio $\kappa$
$\kappa=1$	0.162	0.184	0.245

Appendix

Table 11: Ablation study on Sphere

\to

bunny. Each row modifies one component from the full method. Our default configuration achieves the best two-sided CD.

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Non-rigid point cloud registration is central to soft-tissue shape analysis, but large deformations, noise, and outliers make correspondence estimation challenging. Most learning-based methods rely on local objectives such as Chamfer distance, which encourage point-wise proximity but do not constrain the global plausibility of the predicted deformation field. We address this limitation with DINE, a maximum a posteriori framework that augments distance-based registration with a learned statistical prior over displacement vector fields. DINE is applied to two registration backbones, Robust-DefReg and DefTransNet, using a two-stage strategy: a first-stage model is trained with Chamfer distance, its predicted deformation fields are used to estimate a prior, and the model is then refined with a combined distance and negative log-prior objective. We compare a full-field PCA Gaussian prior with a per-vector normalizing-flow prior. Experiments on DeformedTissue and SynBench show lower mean Chamfer distance under deformation and corruption. On DeformedTissue, DINE-PCA reduces Chamfer distance by approximately 27--69% relative to the corresponding Stage-1 backbone across deformation levels, and improves robustness by up to 66% for outliers and 83% for Gaussian noise. On SynBench, improvements are modest at the smallest deformation levels and reach approximately 59--79% from moderate to severe deformation. These results suggest that global deformation plausibility is an important constraint for reliable soft-tissue point cloud registration. (The code will be published soon.)

Abstract

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