One dimensional (graph-like) spaces arise naturally in many fields,both in modeling natural phenomena, and in understanding abstract procedures and simulations. They are the underlying structures for modeling road and river networks, blood vessels, tree roots' systems, and particle trajectories, to name a few.

Learning these one dimensional spaces becomes an important problem under the more general rubric of space inference. However, there has been only limited work on obtaining a general-purpose algorithm to automatically extract skeleton graph structures.

We present an algorithm that reconstructs one dimensional spaces from a potentially noisy point cloud, by bringing in a topological concept called the Reeb graph to extract skeleton graphs. Our algorithm is simple, efficient, and easy to use. We demonstrate the generality and effectiveness of our algorithm via several applications in both low and high dimensions.

We focus on one application in particular: features reconstruction for a special class of singular surfaces. We discuss in some detail how the above algorithm, combined with recent results from spectral theory, allows for a novel algorithm to recover and reconstruct sharp features.