Recent Advances in Mathematics and Technology by Serge Dos Santos & Mostafa Maslouhi & Kasso A. Okoudjou

Author:Serge Dos Santos & Mostafa Maslouhi & Kasso A. Okoudjou
Language: eng
Format: epub
ISBN: 9783030352028
Publisher: Springer International Publishing

5.2 Notations and Background

Let I be the unit interval and let be the set of square integrable functions from I to . Let denotes a parametrized curve in satisfying: (1) η is absolutely continuous and (2) . Note that absolute continuity is equivalent to requiring that exists for almost t ∈ I, that is summable, and that . In this section we are interested in studying the geometry of shapes using the q-function representation of continuous curves in . It is shown that it is an efficient representation for analyzing shapes of curves [3]. Furthermore, it is the representation in which the elastic metric reduces to a simple L 2 metric and the space of unit length curves becomes the unit Hilbert sphere.

Now, we shall take a finite set of parametrized curves η 0, …, η n satisfying all conditions and consider the shape representation of each η i, i = 0, …, n by their q-functions q 0, …, q n. Let denote the space of these shapes. Indeed, is an infinite-dimensional manifold with a Riemannian structure on it as will be detailed next. The first goal is to compute the geodesic deformation vector field between any two parametrized curves on .

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