Metrics of Curves for Shape Analysis and Shape Optimization

Period of duration of course
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Course info
Number of course hours
20
Number of hours of lecturers of reference
20
CFU 3
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Modalità esame

Seminar, in-depth study, or numerical laboratory

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