Scalar Field Analysis over Point Cloud Data

Given a real-valued function f defined over some metric space $\mathbb{X}$, is it possible to recover some structural information about f from the sole information of its values at a finite set $L\subseteq\mathbb{X}$ of sample points, whose locations are only known through their pairwise distances in $\mathbb{X}$? We provide a positive answer to this question. More precisely, taking advantage of recent advances on the front of stability for persistence diagrams, we introduce a novel algebraic construction, based on a pair of nested families of simplicial complexes built on top of the point cloud L, from which the persistence diagram of f can be faithfully approximated. We derive from this construction a series of algorithms for the analysis of scalar fields from point cloud data. These algorithms are simple and easy to implement, they have reasonable complexities, and they come with theoretical guarantees. To illustrate the genericity and practicality of the approach, we also present some experimental results obtained in various applications, ranging from clustering to sensor networks.