Surface reconstruction from point cloud using a semi-Lagrangian scheme with local interpolator

TL;DR

A level set method to reconstruct unknown surfaces from point clouds, without assuming that the connections between points are known, is proposed, with special attention to the localization of the method and to the development of fast algorithms that run in parallel, resulting in faster reconstruction and thus the opportunity to easily improve the resolution.

Abstract

We propose a level set method to reconstruct unknown surfaces from point clouds, without assuming that the connections between points are known. We consider a variational formulation with a curvature constraint that minimizes the surface area weighted by the distance of the surface from the point cloud. More precisely we solve an equivalent advection-diffusion equation that governs the evolution of an initial surface described implicitly by a level set function. Among all the possible representations, we aim to compute the signed distance function at least in the vicinity of the reconstructed surface. The numerical method for the approximation of the solution is based on a semi-Lagrangian scheme whose main novelty consists in its coupling with a local interpolator instead of a global one, with the aim of saving computational costs. In particular, we resort to a multi-linear interpolator and to a Weighted Essentially Non-oscillatory one, to improve the accuracy of the reconstruction. Special attention has been paid to the localization of the method and to the development of fast algorithms that run in parallel, resulting in faster reconstruction and thus the opportunity to easily improve the resolution. A preprocessing of the point cloud data is also proposed to set the parameters of the method. Numerical tests in two and three dimensions are presented to evaluate the quality of the approximated solution and the efficiency of the algorithm in terms of computational time.

Figures (15)

• Figure 1: Stencils of the two-dimensional $\mathsf{Q1}$ and $\mathsf{WENO}$ reconstructions. The red hatched region represents the cell $\Omega$ in which we compute the reconstruction. The multi-linear interpolator only requires the vertices of the cell $\Omega$, enclosed by the blue square on the left. On the right, the $\mathsf{WENO}$ interpolator involves the cell vertices and their first neighbours, enclosed in the orange square.
• Figure 2: Illustration of marking external points in parallel run with a $2\times2$ domain decomposition (grey lines). Left: initial state. Center: after the first sweep. Right: after the second sweep.
• Figure 3: On the left, the mask computed to detect the computational subgrid $\widetilde{\mathcal{G}}$. Active nodes are depicted in green, their first neighbours are depicted in light blue (they are inactive during the update, while they are active during the reinitialization step) and remaining inactive nodes are depicted in blue. On the right, the mask computed to detect the computational subgrid $\overline{\mathcal{G}}$ of the reinitialization. Colours are used in the same way, with in addition red nodes representing the nodes immediately close to the interface on which the signed distance function is computed explicitly with the one step procedure.
• Figure 4: Typical evolution of a front in 1d. Starting from the red line, the update makes the front move towards the data resulting in the blue line, sharper than the red one. Reinitialization (pink line) makes the computational narrow band move contextually with the front, fixing the zero level set. The dotted black line represents the last cutting step necessary to obtain the final update.
• Figure 5: Steps of the algorithm for the 2d circle case: on the left, the distance function is represented; the central panel shows the initial data and its zero level set (black line); the final data and its contour (black line) are represented on the right. Data represented here has been obtained with $\mathsf{WENO}$ reconstruction.