# Stochastic Metamorphosis in Imaging Science

## TL;DR

This paper extends the metamorphosis framework in imaging science to include stochastic models, enabling uncertainty quantification in image matching by incorporating randomness into the evolution of diffeomorphic transformations.

## Contribution

It introduces a stochastic extension of the metamorphosis equations, linking image matching to stochastic fluid dynamics for uncertainty quantification.

## Key findings

- Develops a stochastic metamorphosis model for image matching
- Connects metamorphosis to complex fluid theory
- Provides a basis for uncertainty quantification in imaging

## 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}.