Hyperbolic Chamfer Distance for Point Cloud Completion and Beyond

TL;DR

The paper tackles outlier sensitivity in Chamfer Distance for point cloud completion by introducing HyperCD, a loss that computes distances in hyperbolic space via $d(x,y)=\mathrm{arcosh}(1+\alpha\|x-y\|^2)$ with $\alpha>0$. This position-aware weighting preserves good matches while gradually refining poorer ones, yielding state-of-the-art results on PCN, ShapeNet-55/34, and ShapeNet-34, and extending effectively to single-view reconstruction and upsampling. The authors provide an efficient implementation with $O(|x|\,|y|)$-style operation counts comparable to Euclidean CD, offer theoretical insights into the monotonicity and gradient behavior via $h(x)=\mathrm{arcosh}(1+\alpha x^{\beta})$, and demonstrate practical benefits such as faster convergence of point correspondences. Overall, HyperCD offers a simple yet powerful hyperbolic-space alternative to standard CD, improving surface smoothness and detail preservation in diverse point-cloud generation tasks with broad potential impact for 3D vision pipelines.

Abstract

Chamfer Distance (CD) is widely used as a metric to quantify difference between two point clouds. In point cloud completion, Chamfer Distance (CD) is typically used as a loss function in deep learning frameworks. However, it is generally acknowledged within the field that Chamfer Distance (CD) is vulnerable to the presence of outliers, which can consequently lead to the convergence on suboptimal models. In divergence from the existing literature, which largely concentrates on resolving such concerns in the realm of Euclidean space, we put forth a notably uncomplicated yet potent metric specifically designed for point cloud completion tasks: {Hyperbolic Chamfer Distance (HyperCD)}. This metric conducts Chamfer Distance computations within the parameters of hyperbolic space. During the backpropagation process, HyperCD systematically allocates greater weight to matched point pairs exhibiting reduced Euclidean distances. This mechanism facilitates the preservation of accurate point pair matches while permitting the incremental adjustment of suboptimal matches, thereby contributing to enhanced point cloud completion outcomes. Moreover, measure the shape dissimilarity is not solely work for point cloud completion task, we further explore its applications in other generative related tasks, including single image reconstruction from point cloud, and upsampling. We demonstrate state-of-the-art performance on the point cloud completion benchmark datasets, PCN, ShapeNet-55, and ShapeNet-34, and show from visualization that HyperCD can significantly improve the surface smoothness, we also provide the provide experimental results beyond completion task.

Figures (6)

• Figure 1: Illustration of point matching in the (a) Euclidean space and (b) hyperbolic space. With the position-aware embeddings in hyperbolic space, the mismatched point pairs in Euclidean space may be corrected, leading to better completion performance.
• Figure 2: Illustration of (left) some distance metrics and (right) their corresponding gradients, where the dotted curves are used in $\ell_1$ and $\ell_2$-CD, the dash ones are used in density-aware Chamfer distance (DCD) wu2021densityaware, and the solid curves are special cases of our HyperCD.
• Figure 3: Illustration of the gradient weights using our HyperCD. All the numbers are normalized by $\frac{1}{\sqrt{2\alpha}}$.
• Figure 4: Visual comparison of point cloud completion results on PCN. Row-1: Inputs of incomplete point clouds. Row-2: Outputs of PointAttN with CD. Row-3: Outputs of PointAttN with DCD. Row-4: Outputs of PointAttN with HyperCD. Row-5: Ground truth.
• Figure 5: The L1-CD with different $\alpha$ and $lr$.

Theorems & Definitions (4)

• Proposition 1
• Proposition 2
• Proposition 3
• Proposition 4