Calculus of Variations and Geometric Measure Theory

A. C. G. Mennucci

Metrics of curves in shape optimization and analysis

created by mennucci on 21 Jan 2009
modified on 19 Jan 2015


chapter in a book

Inserted: 21 jan 2009
Last Updated: 19 jan 2015

Journal: Springer Lecture Notes in Mathematics
Volume: Level Set and PDE Based Reconstr
Pages: 205-319
Year: 2013
Doi: 10.1007/978-3-319-01712-9_4

CIME course on "Level set and PDE based reconstruction methods: applications to inverse problems and image processing", Cetraro, 2008


In these lecture notes we will explore the mathematics of the space of immersed curves, as is nowadays used in applications in computer vision. In this field, the space of curves is employed as a ``shape space''; for this reason, we will also define and study the space of geometric curves, that are immersed curves up to reparameterizations. To develop the usages of this space, we will consider the space of curves as an infinite dimensional differentiable manifold; we will then deploy an effective form of calculus and analysis, comprising tools such as a Riemannian metric, so as to be able to perform standard operations such as minimizing a goal functional by gradient descent, or computing the distance between two curves. Along this path of mathematics, we will also present some current literature results. (Another common and interesting example of ``shape spaces'' is the space of all compact subsets of $R^n$ --- we will briefly discuss this option as well, and relate it to the aforementioned theory).

Keywords: Computer Vision, Riemannian Geometry, space of curves, Image segmentation, shape space