# American Institute of Mathematical Sciences

March  2013, 5(1): 131-150. doi: 10.3934/jgm.2013.5.131

## Computing metamorphoses between discrete measures

 1 Center for Imaging Science, Johns Hopkins University, 3400 North Charles Street, Baltimore, MD 21218-2686, United States 2 Center for Imaging Science and Department of Applied Mathematics and Statistics, Johns Hopkins University, 3400 North Charles Street, Baltimore, MD 21218-2686, United States

Received  August 2012 Revised  January 2013 Published  April 2013

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.
Citation: Casey L. Richardson, Laurent Younes. Computing metamorphoses between discrete measures. Journal of Geometric Mechanics, 2013, 5 (1) : 131-150. doi: 10.3934/jgm.2013.5.131
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