Annals of Mathematical Sciences and Applications

Volume 3 (2018)

Number 1

Special issue in honor of Professor David Mumford, dedicated to the memory of Jennifer Mumford

Guest Editors: Stuart Geman, David Gu, Stanley Osher, Chi-Wang Shu, Yang Wang, and Shing-Tung Yau

Stochastic metamorphosis in imaging science

Pages: 309 – 335

DOI: http://dx.doi.org/10.4310/AMSA.2018.v3.n1.a10

Author

Darryl D. Holm (Department of Mathematics, Imperial College London, United Kingdom)

Abstract

In the pattern matching approach to imaging science, the process of metamorphosis in template matching with dynamical templates was introduced in [31]. In [17] the metamorphosis equations of [31] were recast into the Euler–Poincaré variational framework of [16] and shown to contain the equations for a perfect complex fluid [14]. 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 [12]. 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 will be guided by recent progress in developing stochastic Lie transport models for uncertainty quantification in fluid dynamics in [19, 8].

Full Text (PDF format)

I am grateful to A. Trouvé and L. Younes for their collaboration in developing the Euler–Poincaré description of metamorphosis. I am also grateful to F. Gay-Balmaz, T. S. Ratiu and C. Tronci for many illuminating collaborations in complex fluids and other topics in geometric mechanics during the course of our long friendships. Finally, I am also grateful to M. I. Miller and D. Mumford for their encouragement over the years to pursue the role of EPDiff in imaging science. During the present work the author was partially supported by the European Research Council Advanced Grant 267382 FCCA and the EPSRC Grant EP/N023781/1.

Received 5 June 2017

Published 27 March 2018