Metrics of Curves in Shape Optimization and Analysis

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, which 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 (and a few examples of different “shape spaces”, including more than only curves).

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