Implicit Reconstruction from Point Cloud: An Adaptive Level Set-Based Semi-Lagrangian Method

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 and the point cloud data. 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.