Mathematical methods for geometry reconstruction and shape analysis

Abstract

The last decades have witnessed an exponential growth in the amounts of data that are collected, stored, and shared worldwide daily. In many scientific disciplines, information consists of a set of sample points describing some phenomena. A natural question deals with geometry reconstruction, i.e., the recovery of shape through some representations. Approximation theory has traditionally focused on the case of data points sampled from smooth curves and surfaces; however, in most real-world scenarios, data is affected by noise and potentially other imperfections. To analyse and compare data automatically, shape descriptions are often preferred to representations: despite the latter being more complete than the former, they do not necessarily disclose any high-level information useful to discriminate between shapes. This thesis addresses a range of problems in geometry reconstruction and shape analysis, motivated in the light of real-world applications, including the need to identify geometric relationships in mechanical engineering, the approximation of noisy point clouds in life and earth sciences and the recognition of proteins from an ensemble of geometries in structural biology.