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

TL;DR

The work presents a variational, level-set–based approach to reconstruct implicit, watertight surfaces from point clouds, suitable as PDE computational domains. A semi-Lagrangian scheme on graded quadtrees/octrees is coupled with P1 constrained LS and CWENO constrained LS reconstructions to handle highly nonuniform grids while maintaining stability and accuracy. The method incorporates adaptive refinement, a narrow-band localization, a distance-function propagation scheme, reinitialization, and an energy-based stopping criterion, enabling robust topology changes and cavity detection in 2D and 3D, including real laser-scanned data. Numerical tests on synthetic and real datasets demonstrate accurate surface reconstruction, controllable smoothness, and resilience to missing data, highlighting the framework’s practicality for PDE analyses on complex geometries. The approach provides a scalable, parallelizable pipeline for generating high-fidelity implicit surfaces from unorganized point clouds, with potential applications in cultural heritage and predictive maintenance studies that rely on PDE-driven simulations.

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.

Figures (12)

• Figure 1: Example of quadrants with different neighbours. The main quadrants and their neighbours are depicted in red and orange, respectively.
• Figure 2: Propagation of the distance function. Left: the initialized quadrants before the propagation step are the one containing one or some of the points in $\mathcal{S}$. Center: quadrants with updated distance after one step of propagation. Right: quadrants with updated distance after two steps of propagation.
• Figure 3: Steps of the algorithm for the 2D square test. First row: the initial data and the first reconstruction. Second row: the reconstructions at the end of the second and third run. The zero isocontour of the level set function $\phi$ is always represented with a white line. All the data has been obtained with $\mathsf{P1}$ reconstruction, except from the last one that results from the $\mathsf{CWENO}$ version.
• Figure 4: Final isocontours obtained by setting different reconstruction operators in the square test with 3 runs. From left to right: only $\mathsf{P1}$, only $\mathsf{CWENO}$, the usual setting with $\mathsf{P1}$ in the first two runs and $\mathsf{CWENO}$ in the last one.
• Figure 5: Reconstruction from a heart-shaped point cloud. From left to right: the initial data and the final signed distance function. The zero isocontour of $\phi$ is depicted by the white line.

Theorems & Definitions (1)

• Definition 1