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.