User:Mim.cis/Beg's LDDMM for Dense Image Matching

Beg's LDDMM for Dense Image Matching

In CA the geodesics and their coordinates are generated by solving inexact matching problems calculating the geodesic flows of diffeomorphisms from their start point onto targets. Inexact matching has been examined in many cases and have come to be called Large Deformation Diffeomorphic Metric Mapping (LDDMM) originally solved for dense image matching by Faisal Beg for his PhD at Johns Hopkins University.[1] It was the first example in Computational Anatomy where a numerical code had been created whose fixed points satisfy the necessary conditions for geodesic shortest paths solving the Euler equatoin and minimizing the dense image matching problem for which Dupuis, Grenander and Miller[2] had derived the necessary Sobolev condition for existence of solutions of geodesic flows of diffeomorphisms in image matching. These methods have been extended to landmarks without registration via measure matching,[3] curves,[4] surfaces,[5][6] dense vector[7] and tensor [8] imagery, and varifolds removing orientation.[9]

The endpoint condition which Beg solved for dense image matching with action and endpoint deviation measured via the squared-error metric. In the dense image seting, the Eulerian momentum has a density so that the Euler equation for geodesic flows of diffeomorphisms in Computational Anatomy has a classical solution.

Dense image matching

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Control Problem : The dense image matching problem satisfies the Euler equation with boundary condition at time t=1:

 

 

The conservation equation implies that with the necessary fixed endpoint condition, the initial condition on the momentum is given by

  .
Dense image matching illustrates one of the two extremes of  , the momentum having a vector density pointwise function, so that   for

  a vector function. For dense images the action   implies we will requires the variation of the inverse   with respect to   for the chain rule calculation   . This requires the identity   following from the identity  . This is for such function spaces the generalization of the classic matrix perturbation of the inverse.

LDDMM for image via perturbation of the vector fields

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The original large deformation diffeomorphic metric mapping (LDDMM) algorithms of Beg, Miller, Trouve, Younes took variations with respect to the vector field parameterization of the group, since   are in a vector spaces. Variations satisfy the necessary optimality conditions..

  • Beg solved the Control Problem for dense image matching maximizing with respect to the velocity field; as did Joshi for Landmark matching.

 

necessary conditions become   where  ..

The perturbation in the vector field requires the identity   which implies   Take the variation in the vector fields   using the chain rule   which gives the first variation

 .
Category:Diffeomorphisms

References

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  1. ^ "Computing Large Deformation Metric Mappings via Geodesic Flows of Diffeomorphisms". ResearchGate. doi:10.1023/B:VISI.0000043755.93987.aa. Retrieved 2015-11-22.
  2. ^ "Variational Problems on Flows of Diffeomorphisms for Image Matching". ResearchGate. Retrieved 2016-02-13.
  3. ^ "L.: Diffeomorphic matching of distributions: A new approach for unlabelled point-sets and sub-manifolds matching". ResearchGate. doi:10.1109/CVPR.2004.1315234. Retrieved 2015-11-25.
  4. ^ Glaunès, Joan; Qiu, Anqi; Miller, Michael I.; Younes, Laurent (2008-12-01). "Large Deformation Diffeomorphic Metric Curve Mapping". International journal of computer vision. 80 (3): 317–336. doi:10.1007/s11263-008-0141-9. ISSN 0920-5691. PMC 2858418. PMID 20419045.
  5. ^ Cite error: The named reference Vaillant 1149–1159 was invoked but never defined (see the help page).
  6. ^ Vaillant, Marc; Glaunès, Joan (2005-01-01). "Surface matching via currents". Proceedings of Information Processing in Medical Imaging (IPMI 2005), number 3565 in Lecture Notes in Computer Science: 381–392.
  7. ^ Cao, Yan; Miller, M.I.; Winslow, R.L.; Younes, L. (2005-10-01). "Large deformation diffeomorphic metric mapping of fiber orientations". Tenth IEEE International Conference on Computer Vision, 2005. ICCV 2005. 2: 1379–1386 Vol. 2. doi:10.1109/ICCV.2005.132.
  8. ^ Cao, Yan; Miller, M.I.; Winslow, R.L.; Younes, L. (2005-09-01). "Large deformation diffeomorphic metric mapping of vector fields". IEEE Transactions on Medical Imaging. 24 (9): 1216–1230. doi:10.1109/TMI.2005.853923. ISSN 0278-0062. PMC 2848689. PMID 17427733.
  9. ^ Charon, N.; Trouvé, A. (2013-01-01). "The Varifold Representation of Nonoriented Shapes for Diffeomorphic Registration". SIAM Journal on Imaging Sciences. 6 (4): 2547–2580. doi:10.1137/130918885.
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