Statistical Edge Detection And UDF Learning For Shape Representation

TL;DR

Problem: learn high-fidelity unsigned distance functions (UDFs) to implicitly represent 3D surfaces. Approach: bias neural UDF training toward surface edges using a statistical edge detector that projects local neighborhoods onto an average plane, tests central symmetry with a Fréchet-centered KS statistic, and employs edge-aware sampling; evaluate via the Hausdorff distance $d_H$ between ground-truth and reconstructed surfaces. Contributions: (i) a robust Kolmogorov-Smirnov-based edge detector with Fréchet centering, (ii) an edge-focused data sampling strategy for UDF learning, and (iii) empirical demonstration of improved local edge accuracy and global surface reconstruction on ShapeNet data. Findings: edge-oversampling yields median reconstruction improvements around 15% across chairs, cars, tables, and airplanes, with larger gains near edges and robust performance across shapes. Significance: enables more expressive, data-efficient 3D shape representations and paves the way for integrating edge-aware sampling into broader implicit surface learning frameworks, including future work with DeepSDF-type models.

Abstract

In the field of computer vision, the numerical encoding of 3D surfaces is crucial. It is classical to represent surfaces with their Signed Distance Functions (SDFs) or Unsigned Distance Functions (UDFs). For tasks like representation learning, surface classification, or surface reconstruction, this function can be learned by a neural network, called Neural Distance Function. This network, and in particular its weights, may serve as a parametric and implicit representation for the surface. The network must represent the surface as accurately as possible. In this paper, we propose a method for learning UDFs that improves the fidelity of the obtained Neural UDF to the original 3D surface. The key idea of our method is to concentrate the learning effort of the Neural UDF on surface edges. More precisely, we show that sampling more training points around surface edges allows better local accuracy of the trained Neural UDF, and thus improves the global expressiveness of the Neural UDF in terms of Hausdorff distance. To detect surface edges, we propose a new statistical method based on the calculation of a $p$-value at each point on the surface. Our method is shown to detect surface edges more accurately than a commonly used local geometric descriptor.

Figures (17)

• Figure 1: DeepSDF model architecture. Each training input is the concatenation of a point $\bm{x_j^{(t)}}\in{\mathbb R}^3$ and a latent code $\bm{z_j}\in{\mathbb R}^L$ (the one referring to the envelope ${\mathcal{B}}_j$). The value of $L$ and the architecture of the network are hyperparameters of the model. The parameters optimized during the training step are both the latent codes $\bm{z_1},...,\bm{z_N}$ and the network's weights $\bm{\theta}$.
• Figure 2: Projection onto the local average plane. On the left, a quasi-plane surface (situation 0), and on the right, a pointed surface (situation 1). On the top row, the surface ${\mathcal{B}}$ is in transparent red color. Points (in grey) are uniformly sampled from the surface. We aim to know if the surface is folded or pointed in $\bm{x_0}\in{\mathcal{B}}$. First, we compute its $k$ nearest neighbors $\bm{x_1},...,\bm{x_k}$ (green points). Then the average plane of the neighboring points is computed (in transparent green color). On the bottom row, the plots represent the projection of $\bm{x_0},...,\bm{x_k}$, denoted $\bm{x_0'},...\bm{x_k'}$. We observe that in situation 0, the projection $\bm{x_0'}$ is in the middle of the neighbors' projections $\bm{x_1'},...\bm{x_k'}$. Whereas in situation 1, it is completely off-centered.
• Figure 3: Fréchet centering of polar angles in the average hyperplane. The four green plots on the left correspond to the projections of the points of four point clouds onto their mean hyperplanes. In each situation, the projection of the centroid point is represented by a red cross in the center. The green crosses represent the projections of the neighboring points. In all four situations, the projection of the centroid point (red cross) is off-centered with respect to the projections of the neighboring points (green crosses). These are therefore four surface edges and the obtained $p$-values are expected to be roughly the same. However, the four point clouds are positioned differently with respect to the polar reference axis $\bm{e_1}$ (corresponding to $\phi=0$ in polar coordinates). The distribution of polar angles $\phi_1,...,\phi_k$ is therefore different in the four situations (see the four associated histograms). This results in different $p$-values for each situation. By centering the angles around their Fréchet mean, we obtain the same distribution centered at 0, as in the resulting histogram on the right. Thus, we obtain the same $p$-value in all four situations, which makes our method agnostic to any polar reference axis.
• Figure 4: Neural UDF scheme for a watertight surface ${\mathcal{B}}$. The inputs of the network are tridimensional: they represents points in ${\mathbb R}^3$. The outputs are real values and the network is trained to fit the true UDF of the surface ${\mathcal{B}}$.
• Figure 5: Neural UDF training and evaluation: methodology scheme. Given a 3D surface ${\mathcal{B}}$ encoded as a mesh, we aim to train its Neural UDF and evaluate its precision.

Theorems & Definitions (6)

• Definition 1.1: Watertight surface
• Definition 1.2: Signed Distance Function
• Definition 1.3: Unsigned Distance Function
• Definition 1.4: Hausdorff distance
• Definition 1.5: Chamfer distance
• Definition 1.6: Wasserstein distance