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Numerische Mathematik

, Volume 116, Issue 3, pp 357–381 | Cite as

Derivatives with respect to metrics and applications: subgradient marching algorithm

  • F. BenmansourEmail author
  • G. Carlier
  • G. Peyré
  • F. Santambrogio
Article

Abstract

This paper introduces a subgradient descent algorithm to compute a Riemannian metric that minimizes an energy involving geodesic distances. The heart of the method is the Subgradient Marching Algorithm to compute the derivative of the geodesic distance with respect to the metric. The geodesic distance being a concave function of the metric, this algorithm computes an element of the subgradient in O(N 2 log(N)) operations on a discrete grid of N points. It performs a front propagation that computes a subgradient of a discrete geodesic distance. We show applications to landscape modeling and to traffic congestion. Both applications require the maximization of geodesic distances under convex constraints, and are solved by subgradient descent computed with our Subgradient Marching. We also show application to the inversion of travel time tomography, where the recovered metric is the local minimum of a non-convex variational problem involving geodesic distances.

Mathematics Subject Classification (2000)

Primary 65K10 65M32 65D25 Secondary 91-08 35F21 

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Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • F. Benmansour
    • 1
    • 2
    Email author
  • G. Carlier
    • 1
  • G. Peyré
    • 1
  • F. Santambrogio
    • 1
  1. 1.CEREMADE, UMR CNRS 7534Université Paris-DauphineParis Cedex 16France
  2. 2.Ecole Polytechnique Fédérale de Lausanne (EPFL)Computer Vision LaboratoryLausanneSwitzerland

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