Optimization problems associated withmanifold-valued curves with applications in computer vision

A commonly occurring need in many computer vision applications is the need to represent, compare, and manipulate manifold-valued curves, while allowing for enough flexibility to operate in resource constrained environments. We address these concerns in this chapter, by proposing a dictionary learning scheme that takes geometry and time into account, while performing better than the original data in applications such as activity recognition. We are able to do this with the use of the transport square-root velocity function, which provides an elastic representation for trajectories on Riemannian manifolds. Since these operations can be computationally very expensive, we also present a geometry-based symbolic approximation framework, as a result of which low-bandwidth transmission and accurate real-time analysis for recognition or searching through sequential data become fairly straightforward. We discuss the different optimization problems encountered in this context-learning a sparse representation for actions using extrinsic and intrinsic features, solving the registration problem between two Riemannian trajectories, and learning an optimal clustering scheme for symbolic approximation.