Surface Reconstruction from Point Clouds without Normals by Parametrizing the Gauss Formula

Abstract

We propose Parametric Gauss Reconstruction (PGR) for surface reconstruction from point clouds without normals. Our insight builds on the Gauss formula in potential theory, which represents the indicator function of a region as an integral over its boundary. By viewing surface normals and surface element areas as unknown parameters, the Gauss formula interprets the indicator as a member of some parametric function spaces. We can solve for the unknown parameters using the Gauss formula and simultaneously obtain the indicator function. Our method bypasses the need for accurate input normals as required by most existing non-data-driven methods, while also exhibiting superiority over data-driven methods since no training is needed. Moreover, by modifying the Gauss formula and employing regularization, PGR also adapts to difficult cases such as noisy inputs, thin structures, sparse or nonuniform points, for which accurate normal estimation becomes quite difficult. Our code is publicly available at https://github.com/jsnln/ParametricGaussRecon.

PPSurf: Combining Patches and Point Convolutions for Detailed Surface Reconstruction

“…We compare PPSurf to several recent data‐driven and non‐data‐driven reconstruction methods. PGR [LXSW22], Neural‐IMLS (IMLS) [WWW*22] and Shape as Points (SAP‐O) are non‐data‐driven methods that do not train on a large dataset and instead directly fit a surface to the input point cloud. Shape as Points also has a data‐driven variant (SAP) that uses a trained network.…”

“…Lin et al. proposed a parametric Gauss formula for reconstruction [LXSW22], which has quadratic complexity in memory leading to prohibitive costs for larger point clouds. VIPSS by Huang et al.…”

Few ‘Zero Level Set’-Shot Learning of Shape Signed Distance Functions in Feature Space

“…alpha shapes [5] Voronoi diagrams [1] or triangulation [9,41,59]. On the other hand, the input samples can be used to define an implicit function whose zero level set represents the target shape, using global smoothing priors [36,81,82] e.g. radial basis function [8] and Gaussian kernel fitting [62], local smoothing priors such as moving least squares [28,34,42,47], or by solving a boundary conditioned Poisson equation [33].…”

“…Alternatively, the input samples can contribute to defining an implicit function, with its zero level set representing the target shape. This is achieved through global smoothing priors [45,89,90], such as radial basis functions [11] and Gaussian kernel fitting [70], or local smoothing priors like moving least squares [31,40,51,56]. Another approach involves solving a boundary-conditioned Poisson equation [38].…”

Point normal orientation and surface reconstruction by incorporating isovalue constraints to Poisson equation

“…It is pointed out in Mullen et al (2010) that the two tasks are of nearly the same difficulties. Some remarkable unoriented reconstruction approaches have appeared in recent years such as PGR (Lin et al, 2022) and iPSR (Hou et al, 2022). In this work, we mainly focus on bridging orientation and reconstruction in the implicit space.…”