- journal article metadata
European Mathematical Society Publishing House
2016-09-19 17:05:04
Interfaces and Free Boundaries
Interfaces Free Bound.
IFB
1463-9963
1463-9971
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
10.4171/IFB
http://www.ems-ph.org/doi/10.4171/IFB
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
18
2016
1
Piecewise rigid curve deformation via a Finsler steepest descent
Guillaume
Charpiat
INRIA Saclay, Université Paris Sud, ORSAY CEDEX, FRANCE
Giacomo
Nardi
Université Paris IX - Paris Dauphine, PARIS CEDEX 16, FRANCE
Gabriel
Peyré
Université Paris IX - Paris Dauphine, PARIS CEDEX 16, FRANCE
François-Xavier
Vialard
Université Paris IX - Paris Dauphine, PARIS CEDEX 16, FRANCE
Curve evolution, Finsler space, gradient flow, shape registration
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.
Calculus of variations and optimal control; optimization
Numerical analysis
Computer science
1
44
10.4171/IFB/355
http://www.ems-ph.org/doi/10.4171/IFB/355