Скачать или смотреть VMVW03 | Prof. Darryl Holm | Stochastic Metamorphosis in Imaging Science

VMVW03 | Prof. Darryl Holm | Stochastic Metamorphosis in Imaging Science

Speaker: Professor Darryl Holm (Imperial College London)
Date: 14th Dec 2017 - 16:00 to 17:00
Venue: INI Seminar Room 1
Title: Stochastic Metamorphosis in Imaging Science
Event: (VMVW03) Flows, mappings and shapes
Abstract: 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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