Recent advancements in visualizing deep neural networks provide insights into their structures and mesh extraction from Continuous Piecewise Affine (CPW A) functions.

For any two point sets $A,B \subset \mathbb{R}^d$ of size up to $n$, the Chamfer distance from $A$ to $B$ is defined as $\texttt{CH}(A,B)=\sum_{a \in A} \min_{b \in B} d_X(a,b)$, where $d_X$ is the underlying distance measure (e.g., the Euclidean or Manhattan distance). The Chamfer distance is a popular measure of dissimilarity between point clouds, used in many machine learning, computer vision, and graphics applications, and admits a straightforward $O(d n^2)$-time brute force algorithm. Further, Chamfer distance is often used as a proxy for the more computationally demanding Earth-Mover (Optimal Transport) Distance. However, the \emph{quadratic} dependence on $n$ in the running time makes the naive approach intractable for large datasets.We overcome this bottleneck and present the first $(1+\epsilon)$-approximate algorithm for estimating Chamfer distance with a near-linear running time. Specifically, our algorithm runs in time $O(nd \log (n)/\epsilon^2)$ and is implementable. Our experiments demonstrate that it is both accurate and fast on large high-dimensional datasets. We believe that our algorithm will open new avenues for analyzing large high-dimensional point clouds. We also give evidence that if the goal is to report a $(1+\epsilon)$-approximate mapping from $A$ to $B$ (as opposed to just its value), then any sub-quadratic time algorithm is unlikely to exist.

Chamfer Distance (CD) and Earth Mover's Distance (EMD) are two broadly adopted metrics for measuring the similarity between two point sets. However, CD is usually insensitive to mismatched local density, and EMD is usually dominated by global distribution while overlooks the fidelity of detailed structures. Besides, their unbounded value range induces a heavy influence from the outliers. These defects prevent them from providing a consistent evaluation. To tackle these problems, we propose a new similarity measure named Density-aware Chamfer Distance (DCD). It is derived from CD and benefits from several desirable properties: 1) it can detect disparity of density distributions and is thus a more intensive measure of similarity compared to CD; 2) it is stricter with detailed structures and significantly more computationally efficient than EMD; 3) the bounded value range encourages a more stable and reasonable evaluation over the whole test set.

The widespread adoption of depth sensors has substantially lowered the barrier to point-cloud acquisition. This letter proposes a semantic wireless transmission framework for three dimension (3D) point clouds built on Deep Joint Source - Channel Coding (DeepJSCC). Instead of sending raw features, the transmitter predicts combination weights over a receiver-side semantic orthogonal feature pool, enabling compact representations and robust reconstruction. A folding-based decoder deforms a 2D grid into 3D, enforcing manifold continuity while preserving geometric fidelity. Trained with Chamfer Distance (CD) and an orthogonality regularizer, the system is evaluated on ModelNet40 across varying Signal-to-Noise Ratios (SNRs) and bandwidths. Results show performance on par with SEmantic Point cloud Transmission (SEPT) at high bandwidth and clear gains in bandwidth-constrained regimes, with consistent improvements in both Peak Signal-to-Noise Ratio (PSNR) and CD. Ablation experiments confirm the benefits of orthogonalization and the folding prior.

Understanding and representing the structure of 3D objects in an unsupervised manner remains a core challenge in computer vision and graphics. Most existing unsupervised keypoint methods are not designed for unconditional generative settings, restricting their use in modern 3D generative pipelines; our formulation explicitly bridges this gap. W e present an unsupervised framework for learning spatially structured 3D keypoints from point cloud data. These key-points serve as a compact and interpretable representation that conditions an Elucidated Diffusion Model (EDM) to reconstruct the full shape. The learned keypoints exhibit repeatable spatial structure across object instances and support smooth interpolation in keypoint space, indicating that they capture geometric variation. Our method achieves strong performance across diverse object categories, yielding a 6 percentage-point improvement in keypoint consistency compared to prior approaches.