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Computing metamorphoses between discrete measures

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  • 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.
    Mathematics Subject Classification: Primary: 58E50; Secondary: 35S05, 68T10.


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