Surface Matching via Currents

@article{Vaillant2005SurfaceMV,
  title={Surface Matching via Currents},
  author={Marc Vaillant and Joan Alexis Glaun{\`e}s},
  journal={Information processing in medical imaging : proceedings of the ... conference},
  year={2005},
  volume={19},
  pages={
          381-92
        }
}
  • M. Vaillant, J. Glaunès
  • Published 10 July 2005
  • Mathematics, Medicine, Computer Science
  • Information processing in medical imaging : proceedings of the ... conference
We present a new method for computing an optimal deformation between two arbitrary surfaces embedded in Euclidean 3-dimensional space. Our main contribution is in building a norm on the space of surfaces via representation by currents of geometric measure theory. Currents are an appropriate choice for representations because they inherit natural transformation properties from differential forms. We impose a Hilbert space structure on currents, whose norm gives a convenient and practical way to… 
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References

SHOWING 1-10 OF 31 REFERENCES
Statistics on diffeomorphisms via tangent space representations
TLDR
An algorithm for solving the variational problem with respect to the initial momentum is derived and principal component analysis (PCA) is demonstrated in this setting with three-dimensional face and hippocampus databases.
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.
Landmark Matching via Large Deformation Diffeomorphisms on the Sphere
This paper presents a methodology and algorithm for generating diffeomorphisms of the sphere onto itself, given the displacements of a finite set of template landmarks. Deformation maps are
Geodesic Interpolating Splines
TLDR
This method is based on spline interpolation, and on recent techniques developed for the estimation of flows of diffeomorphisms, and provides a Riemannian distance on sets of landmarks (with fixed cardinality), which can be defined intrinsically, without refering to diffEomorphisms.
3D Brain surface matching based on geodesics and local geometry
TLDR
A new approach for brain surface matching by determining the correspondence of 3D point sets between pairs of surfaces using a combination of geodesic distance and surface curvature is presented.
Discrete exterior calculus
This thesis presents the beginnings of a theory of discrete exterior calculus (DEC). Our approach is to develop DEC using only discrete combinatorial and geometric operations on a simplicial complex
Restricted delaunay triangulations and normal cycle
TLDR
A definition of the curvature tensor for polyhedral surfaces is derived in a very simple and new formula that yields an efficient and reliable curvature estimation algorithm.
Diffeomorphic matching of distributions: a new approach for unlabelled point-sets and sub-manifolds matching
  • J. Glaunès, A. Trouvé, L. Younes
  • Computer Science
    Proceedings of the 2004 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2004. CVPR 2004.
  • 2004
TLDR
A new algorithm to compare two arbitrary unlabelled sets of points, and it is shown that it behaves properly in limit of continuous distributions on sub-manifolds and may apply to various matching problems, such as curve or surface matching, or mixings of landmark and curve data.
Diffeomorphic matching of distributions: a new approach for unlabelled point-sets and sub-manifolds matching
In the paper, we study the problem of optimal matching of two generalized functions (distributions) via a diffeomorphic transformation of the ambient space. In the particular case of discrete
On the metrics and euler-lagrange equations of computational anatomy.
TLDR
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.
...
1
2
3
4
...