Stochastic metamorphosis in imaging science

DOI10.4310/AMSA.2018.V3.N1.A10zbMATH Open1478.68429arXiv1705.10149OpenAlexW2963953682WikidataQ130062877 ScholiaQ130062877MaRDI QIDQ1707644

D. D. Holm

Publication date: 3 April 2018

Published in: Annals of Mathematical Sciences and Applications (Search for Journal in Brave)

Abstract: In the pattern matching approach to imaging science, the process of emph{metamorphosis} in template matching with dynamical templates was introduced in cite{ty05b}. In cite{HoTrYo2009} the metamorphosis equations of cite{ty05b} were recast into the Euler-Poincar'e variational framework of cite{HoMaRa1998} and shown to contain the equations for a perfect complex fluid cite{Holm2002}. 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 cite{GBHR2013}. 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 emph{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 cite{holm2015variational,CrFlHo2017}.

Full work available at URL: https://arxiv.org/abs/1705.10149

Mathematics Subject Classification ID

Computing methodologies for image processing (68U10) PDEs in connection with fluid mechanics (35Q35) Dynamical systems in fluid mechanics, oceanography and meteorology (37N10) Pattern recognition, speech recognition (68T10) Applications of stochastic analysis (to PDEs, etc.) (60H30) PDEs with randomness, stochastic partial differential equations (35R60) Applications of variational problems in infinite-dimensional spaces to the sciences (58E50) Variational aspects of group actions in infinite-dimensional spaces (58E40)