Geometry of diffeomorphism groups and shape matching

@inproceedings{Bruveris2012GeometryOD,
  title={Geometry of diffeomorphism groups and shape matching},
  author={Martins Bruveris},
  year={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 matching framework for registration in a geometric way. The general framework also allows to generalize the large deformation matching framework to include multiple scales by using the iterated semidirect product of groups. An important ingredient in the LDM framework is the choice of a suitable… 
Sub-Riemannian Landmark Matching and its interpretation as residual neural networks
TLDR
Sub-Riemannian landmark matching is demonstrated to have connections with neural networks, in particular the interpretation of residual neural networks as time discretizations of continuous control problems, which allows shape analysis practitioners to think about neural networks in terms of shape analysis, thereby providing a bridge between the two shapes.
About simple variational splines from the Hamiltonian viewpoint
In this paper, we study simple splines on a Riemannian manifold $Q$ from the point of view of the Pontryagin maximum principle (PMP) in optimal control theory. The control problem consists in finding
Geometry of Image Registration: The Diffeomorphism Group and Momentum Maps
These lecture notes explain the geometry and discuss some of the analytical questions underlying image registration within the framework of large deformation diffeomorphic metric mapping (LDDMM) used
GLOBAL ESTIMATES AND BLOW-UP CRITERIA FOR THE GENERALIZED HUNTER-SAXTON SYSTEM
The generalized, two-component Hunter-Saxton system comprises several well-known models of fluid dynamics and serves as a tool for the study of one-dimensional fluid convection and stretching. In

References

SHOWING 1-10 OF 92 REFERENCES
Elastic Geodesic Paths in Shape Space of Parameterized Surfaces
TLDR
A novel Riemannian framework for shape analysis of parameterized surfaces provides efficient algorithms for computing geodesic paths which, in turn, are important for comparing, matching, and deforming surfaces.
The Momentum Map Representation of Images
TLDR
A theorem is proved showing that Large Deformation Diffeomorphic Matching 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.
Riemannian Geometries on Spaces of Plane Curves
We study some Riemannian metrics on the space of regular smooth curves in the plane, viewed as the orbit space of maps from the circle to the plane modulo the group of diffeomorphisms of the circle,
Diffeomorphic 3D Image Registration via Geodesic Shooting Using an Efficient Adjoint Calculation
TLDR
The key contribution of this work is to provide an accurate estimation of the so-called initial momentum, which is a scalar function encoding the optimal deformation between two images through the Hamiltonian equations of geodesics.
Large Deformation Diffeomorphic Metric Curve Mapping
TLDR
A discretized version of the matching criterion for curves is presented, in which discrete sequences of points along the curve are represented by vector-valued functionals, which gives a convenient and practical way to define a matching functional for curves.
Sectional Curvature in Terms of the Cometric, with Applications to the Riemannian Manifolds of Landmarks
TLDR
This paper fully explore the case of geodesics on which only two points have nonzero momenta and compute the sectional curvatures of 2-planes spanned by the tangents to such geodesic, and gives insight into the geometry of the full manifolds of landmarks.
A Continuum Mechanical Approach to Geodesics in Shape Space
TLDR
The proposed shape metric is derived from a continuum mechanical notion of viscous dissipation and implemented via a level set representation of shapes, and a finite element approximation is employed as spatial discretization both for the pairwise matching deformations and for the level set representations.
A New Riemannian Setting for Surface Registration
We present a new approach for matching regular surfaces in a Riemannian setting. We use a Sobolev type metric on deformation vector fields which form the tangent bundle to the space of surfaces. In
Transport of Relational Structures in Groups of Diffeomorphisms
TLDR
This paper discusses two main options for translating the relative variation of one shape with respect to another in a template centered representation, based on the Riemannian metric and the coadjoint transport.
...
1
2
3
4
5
...