On advances in differential-geometric approaches for 2D and 3D shape analyses and activity recognition

In this paper we summarize recent advances in shape analysis and shape-based activity recognition problems with a focus on techniques that use tools from differential geometry and statistics. We start with general goals and challenges faced in shape analysis, followed by a summary of the basic ideas, strengths and limitations, and applications of different mathematical representations used in shape analyses of 2D and 3D objects. These representations include point sets, curves, surfaces, level sets, deformable templates, medial representations, and other feature-based methods. We discuss some common choices of Riemannian metrics and computational tools used for evaluating geodesic paths and geodesic distances for several of these shape representations. Then, we study the use of Riemannian frameworks in statistical modeling of variability within shape classes. Next, we turn to models and algorithms for activity analysis from various perspectives. We discuss how mathematical representations for human shape and its temporal evolutions in videos lead to analyses over certain special manifolds. We discuss the various choices of shape features, and parametric and non-parametric models for shape evolution, and how these choices lead to appropriate manifold-valued constraints. We discuss applications of these methods in gait-based biometrics, action recognition, and video summarization and indexing. For reader convenience, we also provide a short overview of the relevant tools from geometry and statistics on manifolds in the Appendix.