A 3D point cloud completion task is to generate completed 3D objects given partial observations. Auto-encoder-based models suffer from poor generalization ability to untrained 3D data. Current diffusion-based models add isotropic noise with the same variance in three x, y, z axes. More importantly, these models ignore real-world anisotropic evolution properties of 3D particles from a non-equilibrium state to thermodynamic equilibrium in the real physical world due to the velocity and energy thermodynamics of the particles, leading to unstable completions of 3D object topology. This paper presents a novel physically-based anisotropic 3D diffusion model (3DDM) to address these issues. We also present derivations of our proposed forward and reverse processes and a loss function in closed form, thus reproducibility. The 3DDM contains anisotropic energy-aware forward and reverse processes with a novel anisotropic quadratic loss function. The forward process adds anisotropic 3D Gaussian noises per-axis and mimics the thermal non-equilibrium evolution towards Maxwellian equilibrium based on velocity and kinetic energy evolutions of 3D particles in the real physical space. The reverse process learns to denoise along per-axis and per-timestep anisotropically. The anisotropic quadratic loss function penalizes errors along certain axes, yielding a highly flexible and anisotropic reverse diffusion process and a physically realistic generative model. The 3DDM denoises along x, y, z axes with different velocities from the non-equilibrium evolution, achieving fewer than 20 diffusion steps and strong generalization to unseen 3D objects and real-world scenes that were not trained.

SUMMARY This paper presents a parallel algorithm for constructing Voronoi diagrams based on point‐set adaptive grouping. The binary tree splitting method is used to adaptively group the point set in the plane and construct sub‐Voronoi diagrams for each group. Given that the construction of Voronoi diagrams in each group consumes the majority of time and that construction within one group does not affect that in other groups, the use of a parallel algorithm is suitable. After constructing the sub‐Voronoi diagrams, we extracted the boundary points of the four sides of each sub‐group and used to construct boundary site Voronoi diagrams. Finally, the sub‐Voronoi diagrams containing each boundary point are merged with the corresponding boundary site Voronoi diagrams. This produces the desired Voronoi diagram. Experiments demonstrate the efficiency of this parallel algorithm, and its time complexity is calculated as a function of the size of the point set, the number of processors, the average number of points in each block, and the number of boundary points. Copyright © 2013 John Wiley & Sons, Ltd.