A Lennard-Jones Layer for Distribution Normalization

TL;DR

The paper tackles non-uniform point densities in 2D/3D point clouds and proposes a Lennard-Jones layer (LJL) that injects dissipative, nearest-neighbor LJ interactions to redistribute points into a blue-noise–like configuration while preserving global shape. The method treats points as particles within a pairwise interaction framework, updating positions with a decaying time step and gradient clipping, effectively decoupling into independent two-body subsystems and driving convergence to uniform sampling. It demonstrates the utility of LJL across 2D blue-noise synthesis, mesh-surface redistribution, and notably as a plug-in in pretrained generative and denoising networks to improve point distribution without retraining, achieving substantial distance-score gains with limited surface distortion. This approach offers a lightweight, general-purpose tool for distribution normalization in point-cloud workflows, with practical impact for graphics, vision, and 3D modeling pipelines.

Abstract

We introduce the Lennard-Jones layer (LJL) for the equalization of the density of 2D and 3D point clouds through systematically rearranging points without destroying their overall structure (distribution normalization). LJL simulates a dissipative process of repulsive and weakly attractive interactions between individual points by considering the nearest neighbor of each point at a given moment in time. This pushes the particles into a potential valley, reaching a well-defined stable configuration that approximates an equidistant sampling after the stabilization process. We apply LJLs to redistribute randomly generated point clouds into a randomized uniform distribution. Moreover, LJLs are embedded in the generation process of point cloud networks by adding them at later stages of the inference process. The improvements in 3D point cloud generation utilizing LJLs are evaluated qualitatively and quantitatively. Finally, we apply LJLs to improve the point distribution of a score-based 3D point cloud denoising network. In general, we demonstrate that LJLs are effective for distribution normalization which can be applied at negligible cost without retraining the given neural network.

Figures (22)

• Figure 1: An overview of the integration of LJLs into the inference process of well-trained generative models. A well-trained model generates a meaningful point cloud from random noise in a sequential way (bottom row). LJLs are inserted after certain intermediate-generation steps with damping step size $\Delta t_n$ and LJ potential parameters $\epsilon$ and $\sigma$ (top row). Embedding LJLs in generative models can improve the generation results by normalizing the point distribution.
• Figure 2: Illustration of the LJ potential and the LJL. The strength and position of the repulsive and weakly attractive zones in the LJ potential are controlled by the hyperparameters $\epsilon$ and $\sigma$, while $\Delta t$ controls the damping step size of the LJL. Applying the LJL to a point cloud leads to a more uniform distribution of the points.
• Figure 3: Influence of the parameter $\sigma$. The rectangle denotes the unit square in which 1024 points are initialized. (a) $\sigma=0.2\sigma'$ leads to clusters; (b) $\sigma=\sigma'$ distributes points well and keeps them close to the boundary; (c) $\sigma=10\sigma'$ spreads points out of the boundary extremely; (d) $\sigma=10\sigma'$ with fixed boundary conditions. $\sigma'$ denotes the estimated optimal value of $\sigma$.
• Figure 4: Spectral analysis of different blue noise generation methods.
• Figure 5: Redistribution of random 3D point cloud over a unit sphere. (a) Direct projection of random points. (b) Apply LJL before the projection as pre-processing. (c) Apply LJL after the projection as post-processing. (d) Apply projection in between LJL iterations.