The Momentum Map Representation of Images

  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},
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… 
Geometry of diffeomorphism groups and shape matching
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
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
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.
Splines for diffeomorphisms
Symmetries in LDDMM with higher order momentum distributions
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
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
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
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
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
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.


Large deformation diffeomorphic metric mapping of vector fields
This paper proposes a method to match diffusion tensor magnetic resonance images (DT-MRIs) through the large deformation diffeomorphic metric mapping of vector fields, focusing on the fiber
On the metrics and euler-lagrange equations of computational anatomy.
Current experimental results from the Toga & Thompson group in growth, the Van Essen group in macaque and human cortex mapping, and the Csernansky group in hippocampus mapping for neuropsychiatric studies in aging and schizophrenia are shown.
Geodesic Shooting for Computational Anatomy
It is shown that this momentum can be also used for describing a deformation of given visual structures, like points, contours or images, and that, it has the same dimension as the described object, as a consequence of the normal momentum constraint the authors introduce.
Symmetric Data Attachment Terms for Large Deformation Image Registration
This paper proposes two novel cost functions in the large deformation diffeomorphic framework that are inverse consistent and implemented these cost functions, and presents experimental results to validate their inverse consistent property and registration accuracy.
Strategies for Data Reorientation during Non-rigid Warps of Diffusion Tensor Images
Three methods for the estimation of an appropriate reorientation of the data from the local displacement field, which describes the image transformation, are presented and tested and indicate that the best matches are obtained from a re orientation strategy that takes into account the effects of local shearing on the data as well as the rigid rotational component of the displacement.
Local Geometry of Deformable Templates
This paper provides a rigorous and general construction of this infinite dimensional "shape manifold" on which a Riemannian metric is placed and uses this to provide a geometrically founded linear approximation of the deformations of shapes in the neighborhood of a given template.
Spatial transformations of diffusion tensor magnetic resonance images
One method, the preservation of principal direction algorithm, which takes into account shearing, stretching and rigid rotation, is shown to be the most effective and improve the consistency between registered and target images over naive warping algorithms.
Landmark matching via large deformation diffeomorphisms
Conditions for the existence of solutions in the space of diffeomorphisms are established, with a gradient algorithm provided for generating the optimal flow solving the minimum problem.
Soliton dynamics in computational anatomy
Diffeomorphisms Groups and Pattern Matching in Image Analysis
  • A. Trouvé
  • Mathematics
    International Journal of Computer Vision
  • 2004
This paper constructs a distance between deformations defined through a metric given the cost of infinitesimal deformations, and proposes a numerical scheme to solve a variational problem involving this distance and leading to a sub-optimal gradient pattern matching.