Gromov–Hausdorff Approximation of Filamentary Structures Using Reeb-Type Graphs

Abstract

In many real-world applications, data appear to be sampled around 1-dimensional filamentary structures that can be seen as topological metric graphs. In this paper, we address the metric reconstruction problem of such filamentary structures from data sampled around them. We prove that they can be approximated with respect to the Gromov---Hausdorff distance by well-chosen Reeb graphs (and some of their variants) and provide an efficient and easy-to-implement algorithm to compute such approximations in almost linear time. We illustrate the performance of our algorithm on a few datasets.