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Finsler Steepest Descent with Applications to Piecewise-regular Curve Evolution

Nardi, Giacomo; Vialard, François-Xavier; Peyré, Gabriel; Charpiat, Guillaume (2013), Finsler Steepest Descent with Applications to Piecewise-regular Curve Evolution. https://basepub.dauphine.fr/handle/123456789/11606

Type
Document de travail / Working paper
Date
2013
Series title
Preprints Ceremade
Pages
42
Metadata
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Author(s)
Nardi, Giacomo
Vialard, François-Xavier
Peyré, Gabriel
Charpiat, Guillaume
Abstract (EN)
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., 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.
Subjects / Keywords
Curve evolution; gradient flow; shape registration; Finsler space

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