We propose diverse canonical parameterizations of 2D-curves. For instance, the arc-length parameterization is canonical in the sense that any open curve can be parameterized by arc-length in a unique way. We consider other natural parameterizations like the parameterization proportionnal to the curvature of the curve. Both aforementionned parameterizations are very natural and correspond to a natural physical movement: the arc-length parameterization corresponds to travelling along the curve at constant speed, whereas parameterization proportionnal to curvature corresponds to a constant-speed moving frame in SO(3). Many other canonical parameterizations are considered, interpolating between arc-length parameterization and curvature-length parameterization. The main idea is that to any strictly increasing function is associated a natural parameterization of 2D-curves, which gives an optimal sampling, and which can be used to compare unparameterized curves in a efficient and pertinent way. If time permits, the link to infinite-dimensional geometry will be explained. An application to point correspondence in medical imaging will be given.
Alice Barbora Tumpach is an Associate Professor in Mathematics since 2007 (University of Lille, France, currently on leave) and P.I. of a FWF Grant entitled "Banach Poisson-Lie groups and integrable systems" since 2021. She obtained a habilitation in mathematics in 2022, a PhD in mathematics in 2005 at Ecole Polytechnique, France, and spent two years at Ecole Polytechnique Fédérale de Lausanne as a Post-Doc. She started her studies in mathematics and physics and obtained a Bachelor in each of these specialities at ENS Paris. Her research interests include infinite dimensional geometry, Lie groups and applications to Shape Analysis. She is author and co-author of several publications in international journals (TPAMI, Communications in Mathematical Physics, Journal of Functional Analysis, Annales de l'Institut Fourier) in the above fields. She served as reviewer for many journals, including TPAMI, Mathematische Annalen, Journal of Mathematical Physics, Journal of Mathematical Analysis and Appplication, Journal of Differential Geometry...On the other hand, she has three children and loves chilli chocolate.
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