# The Momentum Map Representation of Images

@article{Bruveris2011TheMM, title={The Momentum Map Representation of Images}, author={Martins Bruveris and François Gay‐Balmaz and Darryl D. Holm and Tudor S. Ratiu}, journal={Journal of Nonlinear Science}, year={2011}, volume={21}, pages={115-150} }

This paper discusses the mathematical framework for designing methods of Large Deformation Diffeomorphic Matching (LDM) for image registration in computational anatomy. After reviewing the geometrical framework of LDM image registration methods, we prove a theorem showing that these methods may be designed by using the actions of diffeomorphisms on the image data structure to define their associated momentum representations as (cotangent-lift) momentum maps. To illustrate its use, the momentum…

## 66 Citations

Geometry of diffeomorphism groups and shape matching

- Mathematics
- 2012

The large deformation matching (LDM) framework is a method for registration of images and other data structures, used in computational anatomy. We show how to reformulate the large deformation…

Diffeomorphic image matching with left-invariant metrics

- Mathematics
- 2015

The geometric approach to diffeomorphic image registration known as large deformation by diffeomorphic metric mapping (LDDMM) is based on a left action of diffeomorphisms on images, and a…

Reduction by Lie Group Symmetries in Diffeomorphic Image Registration and Deformation Modelling

- MathematicsSymmetry
- 2015

This work surveys the role of reduction by symmetry in the large deformation diffeomorphic metric mapping framework for registration of a variety of data types and describes these models in a common theoretical framework that draws on links between the registration problem and geometric mechanics.

Symmetries in LDDMM with higher order momentum distributions

- MathematicsArXiv
- 2013

This paper describes a tower of Lie groups which correspond to preserving $k$-th order jet-data and implies the existence of conserved momenta for the reduced system on $T^{\ast}Q^{(k)}$.

A Geometric Framework for Stochastic Shape Analysis

- MathematicsFound. Comput. Math.
- 2019

This work derives two approaches for inferring parameters of the stochastic model from landmark configurations observed at discrete time points and employs an expectation-maximization based algorithm using a Monte Carlo bridge sampling scheme to optimise the data likelihood.

Universitet A Geometric Framework for Stochastic Shape Analysis

- Mathematics
- 2018

We introduce a stochastic model of diffeomorphisms, whose action on a variety of data types descends to stochastic evolution of shapes, images and landmarks. The stochasticity is introduced in the…

Kernel Bundle Diffeomorphic Image Registration Using Stationary Velocity Fields and Wendland Basis Functions

- Computer ScienceIEEE Transactions on Medical Imaging
- 2016

Experimental results show that wKB-SVF is a robust, flexible registration framework that allows theoretically well-founded and computationally efficient multi-scale representation of deformations and is equally well-suited for both inter- and intra-subject image registration.

String Methods for Stochastic Image and Shape Matching

- Computer ScienceJournal of Mathematical Imaging and Vision
- 2018

A stochastic model compatible with the geometry of the LDDMM framework is applied and the stochastically version of the Beg algorithm is derived, which is compared with the string method and an expectation-maximization optimization of posterior likelihoods.

Gaussian diffeons for surface and image matching within a Lagrangian framework

- Computer Science
- 2014

Numerical schemes that can be used for surface and image matching and are based on representing the Eulerian velocity over a finite-dimensional basis that deforms over time are introduced and discussed.

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