Metrics of Curves for Shape Analysis and Shape Optimization
Prerequisiti
This course is proposed to PhD students, but may be apt
to last year undergraduates, since the presentations is
mostly self-contained.
Programma
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.
Obiettivi formativi
We will review some elements in Riemannian
Geometry, Functional Analysis, Global Analysis,
consequently we will explore some contemporary
research themes.
Riferimenti bibliografici
This class is bases on a 2008 CIME course. PDF notes are provided during the clas