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The Journal of Geometric Mechanics (JGM)
 

Computing metamorphoses between discrete measures

Pages: 131 - 150, Volume 5, Issue 1, March 2013      doi:10.3934/jgm.2013.5.131

 
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Casey L. Richardson - Center for Imaging Science, Johns Hopkins University, 3400 North Charles Street, Baltimore, MD 21218-2686, United States (email)
Laurent Younes - Center for Imaging Science and Department of Applied Mathematics and Statistics, Johns Hopkins University, 3400 North Charles Street, Baltimore, MD 21218-2686, United States (email)

Abstract: Metamorphosis is a mathematical framework for diffeomorphic pattern matching in which one defines a distance on a space of images or shapes. In the case of image matching, this distance involves computing the energetically optimal way in which one image can be morphed into the other, combining both smooth deformations and changes in the image intensity. In [12], Holm, Trouvé and Younes studied the metamorphosis of more singular deformable objects, in particular measures. In this paper, we present results on the analysis and computation of discrete measure metamorphosis, building upon the work in [12]. We show that, when matching sums of Dirac measures, minimizing evolutions can include other singular distributions, which complicates the numerical approximation of such solutions. We then present an Eulerian numerical scheme that accounts for these distributions, as well as some numerical experiments using this scheme.

Keywords:  Groups of diffeomorphisms, shape analysis, deformable templates, pseudo-differential operators.
Mathematics Subject Classification:  Primary: 58E50; Secondary: 35S05, 68T10.

Received: August 2012;      Revised: January 2013;      Available Online: April 2013.

 References