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Stochastic Metamorphosis in Imaging Science

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VMVW03 - Flows, mappings and shapes

In the pattern matching approach to imaging science, the process of metamorphosis in template matching with dynamical templates was introduced in [7]. In [5] the metamorphosis equations of [7] were recast into the Euler-Poincar ́e variational framework of [4] and shown to contain the equations for a perfect complex fluid [3].   

This result related the data structure underlying the process of metamorphosis in image matching to the physical concept of order parameter in the theory of complex fluids [2]. In particular, it cast the concept of Lagrangian paths in imaging science as deterministically evolving curves in the space of diffeomorphisms acting on image data structure, expressed in Eulerian space. In contrast, the landmarks in the standard LDDMM approach are Lagrangian.  

For the sake of introducing an Eulerian uncertainty quantification approach in imaging science, we extend the method of metamorphosis to apply to image matching along stochastically evolving time dependent curves on the space of diffeomorphisms. The approach IS guided by recent progress in developing stochastic Lie transport models for uncertainty quantification in fluid dynamics in [6, 1].  

[1] D. O. Crisan, F. Flandoli, and D. D. Holm. Solution properties of a 3D stochastic Euler fluid equation. arXiv preprint arXiv:1704.06989, 2017. URL https://arxiv.org/abs/1704.06989. [2] F. Gay-Balmaz, D. D. Holm, and T. S. Ratiu. Geometric dynamics of optimization. Comm. in Math. Sciences, 11(1):163–231, 2013. [3] D. D. Holm. Euler-Poincaré dynamics of perfect complex fluids. In P. Newton, P. Holmes, and A. Weinstein, editors, Geometry, Mechanics, and Dynamics: in honor of the 60th birthday of Jerrold E. Marsden, pages 113–167. Springer, 2002. [4] D. D. Holm, J. E. Marsden, and T. S. Ratiu. The Euler–Poincar ́e equations and semidirect products with applications to continuum theories. Adv. in Math., 137:1–81, 1998. [5] D. D. Holm, A. Trouvé, and L. Younes. The Euler-Poincar ́e theory of metamorphosis. Quarterly of Applied Mathematics, 67:661–685, 2009. [6] Darryl D Holm. Variational principles for stochastic fluid dynamics. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 471(2176):20140963, 2015. [7] A. Trouvé and L. Younes. Metamorphoses through Lie group action. Found. Comp. Math., 173–198, 2005.

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