Reasoning About Shape in Complex Datasets: Geometry, Structure and Semantics

In recent years, acquisition and modelling of 3D data has gained a significant boost due to the availability of commodity devices. Digital 3D shape models are becoming a key component in many industrial, entertainment and scientific sectors. Consequently, large collections of 3D data are nowadays available both in the public (e.g., on the Internet) as well as in private domains. Analyzing, classifying, and querying such 3D data collections are becoming topics of increasing interest in the computer vision, pattern recognition, computer graphics and digital geometry processing communities. The purpose of this tutorial is to introduce the fundamental mathematical tools for the analysis of collections of 3D models and overview the state-of-the-art techniques. We will first introduce some of the main challenges in shape analysis, underlining the role of Mathematics in the identification of the geometry, structure and semantics of a shape. We then overview the mathematical concepts that root on Differential Geometry and Topology and show examples about surface correspondence, retrieval and attribute transfer, to demonstrate how the surveyed concepts have been exploited in recent research works. We will overview the fundamental tools for the analysis of the variability in 3D shape collections. We will review statistical shape analysis techniques, outlining the potential of statistical analysis on non-linear manifolds. Finally, we discuss the potential of structural shape analysis to achieve a smart and semantic representation of a digital object. We conclude the tutorial with an overview of some (classical and non-classical) applications where 3D shape analysis plays a central role.