Mind the Gap: Hole-Filling and Reconstruction of High-Dimensional Manifolds from Noisy Scattered Data

In this study, we consider the question of repairing and recovering a low-dimensional manifold embedded in high-dimensional space from noisy scattered data. Given a noisy point cloud sampled from a low-dimensional manifold, suppose that part of the scattered data is missing, which results in holes. In these settings, the main goal is to accurately and efficiently reconstruct the information within these gaps. While in three-dimensions the problem has been extensively studied, the challenge of reinstating missing information for high-dimensional manifolds remains open. In this paper, we propose a new approach named Manifold Repairing via Locally Optimal Projection (R-MLOP). The method is defined as a minimization problem with three terms. First, we leverage the spatial proximity to the holes to balance between denoising the data and preserving geometric continuity among points situated along the hole’s boundaries. In addition, a penalty term is added to guarantee a quasi-uniform sampling of the unknown manifold. We prove that the suggested solution recovers missing information inside the hole, with an approximation order that is controlled by the density of the given scattered data as well as the size of the amended hole. The effectiveness of our approach is demonstrated by considering different manifold topologies, for single and multiple-hole repairing, in low and high dimensions.