Group Actions, Homeomorphisms, and Matching: A General Framework

@article{Miller2004GroupAH,
  title={Group Actions, Homeomorphisms, and Matching: A General Framework},
  author={Michael I. Miller and Laurent Younes},
  journal={International Journal of Computer Vision},
  year={2004},
  volume={41},
  pages={61-84}
}
  • M. Miller, L. Younes
  • Published 2004
  • Mathematics, Computer Science
  • International Journal of Computer Vision
This paper constructs metrics on the space of images I defined as orbits under group actions G. The groups studied include the finite dimensional matrix groups and their products, as well as the infinite dimensional diffeomorphisms examined in Trouvé (1999, Quaterly of Applied Math.) and Dupuis et al. (1998). Quaterly of Applied Math. Left-invariant metrics are defined on the product G × I thus allowing the generation of transformations of the background geometry as well as the image values… 
View on Springer
dam.brown.edu
346 Citations
Highly Influential Citations
14
Background Citations
117
Methods Citations
70
Results Citations
2

Figures from this paper

346 Citations

Citation Type

Citation Type

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.
Geodesic Shooting for Computational Anatomy
TLDR
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.
Exponential Barycenters of the Canonical Cartan Connection and Invariant Means on Lie Groups
When performing statistics on elements of sets that possess a particular geometric structure, it is desirable to respect this structure. For instance in a Lie group, it would be judicious to have a…
Metric Spaces of Shapes and Geometries Constructed from Set Parametrized Functions
In modelling, optimization, control, or identification problems with respect to a family of subsets of a fixed hold-all in \({\mathbb {R}}^N\), the nice vector space structure of the calculus of…
Metric registration of curves and surfaces using optimal control
Abstract This chapter presents an overview of recent developments in the analysis of shapes such as curves and surfaces through Riemannian metrics. We show that several constructions of metrics on…
Computing metamorphoses between discrete measures
Metamorphosis is a mathematical framework for diffeomorphic pattern matching in which one defines a distance on a space of images or shapes. In the case of image matching, this distance involves…
Morphing of Manifold-Valued Images Inspired by Discrete Geodesics in Image Spaces
TLDR
It is shown that the continuous manifold-valued functions are dense in $L^2(\Omega,\mathcal{H})$ and a space discrete model based on a finite difference approach on a staggered grid, where the linearized elastic potential in the regularizing term is focused on.
A Geometrically Constrained Manifold Embedding for an Extrinsic Gaussian Process
We introduce a new framework of local and adaptive manifold embedding for Gaussian regression. The proposed method, which can be generalized on any bounded domain in \(\mathbb {R}^n\), is used to…
Invariant higher-order variational problems: Reduction, geometry and applications
This thesis is centred around higher-order invariant variational problems defined on Lie groups. We are mainly motivated by applications in computational anatomy and quantum control, but the general…
Hybrid Riemannian Metrics for Diffeomorphic Shape Registration
TLDR
This quite simple approach inherits the advantages of both methods, both on the theoretical and experimental levels, and provide additional flexibility and modeling power, especially when dealing with complex configurations of shapes.
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 41 REFERENCES
On the Shape of Plane Images
TLDR
A definition of shape in terms of transformation groups, where the transformations need not be rigid and linear, but more structured than arbitrary homeomorphisms is given, and a limit theorem for the prior measures is presented.
Riemannian Geometry
THE recent physical interpretation of intrinsic differential geometry of spaces has stimulated the study of this subject. Riemann proposed the generalisation, to spaces of any order, of Gauss's…
Computable Elastic Distances Between Shapes
  • L. Younes
  • Computer Science, Mathematics
    SIAM J. Appl. Math.
  • 1998
TLDR
An elastic matching algorithm which is based on a true distance between intrinsic properties of the shapes, taking into account possible invariance to scaling or Euclidean transformations in the case they are required.
Deformable templates using large deformation kinematics
TLDR
Application of the method to intersubject registration of neuroanatomical structures illustrates the ability to account for local anatomical variability.
Ergodic Algorithms on Special Euclidean Groups for ATR
This paper describes a technique for estimating motions of rigid targets based on the deformable template representations of complex scenes. The efficient modeling of representations for…
Curve matching on brain surfaces using frenet distances
TLDR
This paper describes methods for diffeomorphic matching of curves on brain surfaces and defines fundus curves on the brain surfaces as extremal curvature lines generated using dynamic programming.
Uniform Distribution, Distance and Expectation Problems for Geometric Features Processing
  • X. Pennec, N. Ayache
  • Mathematics, Computer Science
    Journal of Mathematical Imaging and Vision
  • 2004
TLDR
The basic mathematical framework required to avoid paradoxes is established and three basic problems are analyzed: what is a random distribution of features, how to define a distance between features and what is the “mean feature” of a number of feature measurements.
Mapping of hyperelastic deformable templates using the finite element method
In the current work we integrate well established techniques from finite deformation continuum mechanics with concepts from pattern recognition and image processing to develop a new finite element…
Consistent Linear-Elastic Transformations for Image Matching
TLDR
It is shown that joint estimation of a consistent set of forward and reverse transformations constrained by linear-elasticity gives better registration results than using either constraint alone or none at all.
Hilbert-Schmidt Lower Bounds for Estimators on Matrix Lie Groups for ATR
TLDR
This paper studies the construction of minimum mean squared error estimators on the special orthogonal group, SO(n), for pose estimation, and shows the expected error associated with the HSE is shown to be a lower bound, called the Hilbert-Schmidt bound (HSB), on the error incurred by any other estimator.
...
1
2
3
4
5
...