Implicit reconstruction from point cloud: an adaptive level-set-based semi-Lagrangian method

TL;DR

This paper introduces an adaptive level-set semi-Lagrangian method on graded Cartesian grids for reconstructing surfaces from point clouds, enabling high-quality implicit shape representations suitable for PDE modeling.

Contribution

It presents a novel variational formulation combined with a semi-Lagrangian scheme and adaptive grid structures for efficient and accurate surface reconstruction from point clouds.

Findings

Effective reconstruction of complex shapes demonstrated in 2D and 3D tests.

Adaptive grids improve resolution near the surface, enhancing accuracy.

Method handles evolving topologies and complex geometries.

Abstract

We propose a level-set-based semi-Lagrangian method on graded adaptive Cartesian grids to address the problem of surface reconstruction from point clouds. The goal is to obtain an implicit, high-quality representation of real shapes that can subsequently serve as computational domain for partial differential equation models. The mathematical formulation is variational, incorporating a curvature constraint that minimizes the surface area while being weighted by the distance of the reconstructed surface from the input point cloud. Within the level set framework, this problem is reformulated as an advection-diffusion equation, which we solve using a semi-Lagrangian scheme coupled with a local high-order interpolator. Building on the features of the level set and semi-Lagrangian method, we use quadtree and octree data structures to represent the grid and generate a mesh with the finest resolution near the zero level set, i.e., the reconstructed surface interface. The complete surface reconstruction workflow is described, including localization and reinitialization techniques, as well as strategies to handle complex and evolving topologies. A broad set of numerical tests in two and three dimensions is presented to assess the effectiveness of the method.