Geodesic
Piecewise rigid curve deformation via a Finsler steepest descent
Guillaume Charpiat1 ; Giacomo Nardi2 ; Gabriel Peyré2 ; François-Xavier Vialard2
1INRIA Saclay, Université Paris Sud, Orsay, France
2Université Paris IX - Paris Dauphine, France
Interfaces and free boundaries, Tome 18 (2016) no. 1, pp. 1-44

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DOI

This paper introduces a novel steepest descent flow in Banach spaces. This extends previous works on generalized gradient descent, notably the work of Charpiat et al. [15], to the setting of Finsler metrics. Such a generalized gradient allows one to take into account a prior on deformations (e.g., piecewise rigid) in order to favor some specific evolutions. We define a Finsler gradient descent method to minimize a functional defined on a Banach space and we prove a convergence theorem for such a method. In particular, we show that the use of non-Hilbertian norms on Banach spaces is useful to study non-convex optimization problems where the geometry of the space might play a crucial role to avoid poor local minima.We show some applications to the curve matching problem. In particular, we characterize piecewise rigid deformations on the space of curves and we study several models to perform piecewise rigid evolution of curves.
DOI : 10.4171/ifb/355
Classification : 49-XX, 65-XX, 68-XX
Mots-clés : Curve evolution, Finsler space, gradient flow, shape registration

Guillaume Charpiat  1   ; Giacomo Nardi  2   ; Gabriel Peyré  2   ; François-Xavier Vialard  2

1 INRIA Saclay, Université Paris Sud, Orsay, France
2 Université Paris IX - Paris Dauphine, France 
Guillaume Charpiat; Giacomo Nardi; Gabriel Peyré; François-Xavier Vialard. Piecewise rigid curve deformation via a Finsler steepest descent. Interfaces and free boundaries, Tome 18 (2016) no. 1, pp. 1-44. doi: 10.4171/ifb/355
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     title = {Piecewise rigid curve deformation via a {Finsler} steepest descent},
     journal = {Interfaces and free boundaries},
     pages = {1--44},
     year = {2016},
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