Metrics of Curves for Shape Analysis and Shape Optimization

We will see the mathematics that stands behind some sections of Computer Vision, and in particular the so called “Shape Spaces theory”; we will address mostly the case in which the shape space is a space of closed immersed curves in the plane. To this end, we will consider this Shape Space of Immersed Curves as an infinite dimensional Differentiable Manifold; we will develop a convenient calculus; we will endow this manifold with some choices of Riemannian metrics that have been proposed in the current literature. These models justify the methods called active contours that are used for Shape Optimization; the active contour methods try to minimize a functional using a gradient descent approach; the functional is designed to achieve a task , such as image segmentation or tracking. These Riemannian Manifold models at the same time define some tools that are useful in Shape Analysis, such as “distance between two curves” or “geodetic of curves”. Time remaining, we will address some possible definitions of probabilities on spaces of curves.