Manifold Repairing, Reconstruction and Denoising from Scattered Data in High-Dimension

TL;DR

This paper introduces R-MLOP, a novel method for repairing and reconstructing low-dimensional manifolds embedded in high-dimensional spaces from noisy, scattered data with holes, extending previous low-dimensional techniques.

Contribution

The paper proposes R-MLOP, an extension of MLOP, capable of repairing manifolds with holes in both low and high dimensions, with theoretical validation and practical demonstrations.

Findings

Effective in repairing multiple holes in manifolds

Works for various manifold topologies

Validates in both low and high-dimensional settings

Abstract

We consider a problem of great practical interest: the repairing and recovery of a low-dimensional manifold embedded in high-dimensional space from noisy scattered data. Suppose that we observe a point cloud sampled from the low-dimensional manifold, with noise, and let us assume that there are holes in the data. Can we recover missing information inside the holes? While in low-dimension the problem was extensively studied, manifold repairing in high dimension is still an open problem. We introduce a new approach, called Repairing Manifold Locally Optimal Projection (R-MLOP), that expands the MLOP method introduced by Faigenbaum-Golovin et al. in 2020, to cope with manifold repairing in low and high-dimensional cases. The proposed method can deal with multiple holes in a manifold. We prove the validity of the proposed method, and demonstrate the effectiveness of our approach by considering different manifold topologies, for single and multiple holes repairing, in low and high dimensions.