Skip to main content
Log in

String Methods for Stochastic Image and Shape Matching

  • Published:
Journal of Mathematical Imaging and Vision Aims and scope Submit manuscript

Abstract

Matching of images and analysis of shape differences is traditionally pursued by energy minimization of paths of deformations acting to match the shape objects. In the large deformation diffeomorphic metric mapping (LDDMM) framework, iterative gradient descents on the matching functional lead to matching algorithms informally known as Beg algorithms. When stochasticity is introduced to model stochastic variability of shapes and to provide more realistic models of observed shape data, the corresponding matching problem can be solved with a stochastic Beg algorithm, similar to the finite-temperature string method used in rare event sampling. In this paper, we apply a stochastic model compatible with the geometry of the LDDMM framework to obtain a stochastic model of images and we derive the stochastic version of the Beg algorithm which we compare with the string method and an expectation-maximization optimization of posterior likelihoods. The algorithm and its use for statistical inference is tested on stochastic LDDMM landmarks and images.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

References

  1. Arnaudon, A., Holm, D.D., Pai, A., Sommer, S.: A stochastic large deformation model for computational anatomy. In: Information Processing for Medical Imaging (IPMI) (2017)

  2. Arnaudon, A., Holm, D.D., Sommer, S.: A geometric framework for stochastic shape analysis. Accept. Found. Comput. Math. arXiv:1703.09971 (2018)

  3. Arnaudon, A., De Castro, A.L., Holm, D.D.: Noise and dissipation on coadjoint orbits. J. Nonlinear Sci. 28(1), 91–145 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  4. Beg, M.F., Miller, M.I., Trouvé, A., Younes, L.: Computing large deformation metric mappings via geodesic flows of diffeomorphisms. Int. J. Comput. Vis. 61(2), 139–157 (2005)

    Article  Google Scholar 

  5. Bruveris, M., Gay-Balmaz, F., Holm, D.D., Ratiu, T.S.: The momentum map representation of images. J. Nonlinear Sci. 21(1), 115–150 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Delyon, B., Lavielle, M., Moulines, E.: Convergence of a stochastic approximation version of the EM algorithm. Ann. Stat. 27(1), 94–128 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  7. Dempster, A.P., Laird, N.M., Rubin, D.B.: Maximum likelihood from incomplete data via the EM algorithm. J. R. Stat. Soc. B 39(1), 1–38 (1977)

    MathSciNet  MATH  Google Scholar 

  8. Fréchet, M.: Les élèments aléatoires de nature quelconque dans un espace distancié. Ann. Inst. H. Poincaré 10, 215–310 (1948)

    MathSciNet  MATH  Google Scholar 

  9. Hart, G., Zach, C., Niethammer, M.: An optimal control approach for deformable registration. IEEE (2009). https://doi.org/10.1109/CVPRW.2009.5204344

    Google Scholar 

  10. Hastie, T., Stuetzle, W.: Principal curves. J. Am. Stat. Assoc. 84(406), 502–516 (1989). https://doi.org/10.1080/01621459.1989.10478797

    Article  MathSciNet  MATH  Google Scholar 

  11. Holm, D.D., Marsden, J.E.: Momentum maps and measure-valued solutions (peakons, filaments, and sheets) for the epdiff equation. In: The Breadth of Symplectic and Poisson Geometry. Springer, pp. 203–235 (2005)

  12. Holm, D.D.: Geometric Mechanics—Part I: Dynamics and Symmetry, 2, edition edn. Imperial College Press, London (2011)

    Book  MATH  Google Scholar 

  13. Holm, D.D.: Variational principles for stochastic fluid dynamics. Proc. R. Soc. Lond. A Math. Phys. Eng. Sci. 471(2176), 20140963 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  14. Holm, D.D., Tyranowski, T.M.: Variational principles for stochastic soliton dynamics. Proc. R. Soc. A 472(2187), 20150827 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kühnel, L., Arnaudon, A., Sommer, S.: Differential geometry and stochastic dynamics with deep learning numerics. arXiv:1712.08364 (2017)

  16. Marsland, S., Shardlow, T.: Langevin Equations for Landmark Image Registration with Uncertainty. SIAM Journal on Imaging Sciences pp. 782–807 (2017). 10.1137/16M1079282

  17. Øksendal, B.: Stochastic Differential Equations: An Introduction with Applications. Springer, Berlin (2003)

    Book  MATH  Google Scholar 

  18. Stegmann, M.B., Fisker, R., Ersbll, B.K.: Extending and applying active appearance models for automated, high precision segmentation in different image modalities. In: Proc. 12th Scandinavian Conference on Image Analysis—SCIA 2001, Bergen, Norway pp. 90–97 (2001)

  19. Tibshirani, R.: Principal curves revisited. Stat. Comput. 2(4), 183–190 (1992). https://doi.org/10.1007/BF01889678

    Article  Google Scholar 

  20. Trouvé, A., Vialard, F.X.: Shape splines and stochastic shape evolutions: a second order point of view. Q. Appl. Math. 70(2), 219–251 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  21. Vanden-Eijnden, E., Venturoli, M.: Revisiting the finite temperature string method for the calculation of reaction tubes and free energies. J. Chem. Phys. 130(19), 194103 (2009). https://doi.org/10.1063/1.3130083

    Article  Google Scholar 

  22. Vialard, F.X.: Extension to infinite dimensions of a stochastic second-order model associated with shape splines. Stoch. Process. Appl. 123(6), 2110–2157 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  23. Weinan, E., Ren, W., Vanden-Eijnden, E.: String method for the study of rare events. Phys. Rev. B 66(5), 052301 (2002). https://doi.org/10.1103/PhysRevB.66.052301

    Google Scholar 

  24. Weinan, E., Ren, W., Vanden-Eijnden, E.: Finite temperature string method for the study of rare events. J. Phys. Chem. B 109(14), 6688–6693 (2005). https://doi.org/10.1021/jp0455430

    Article  Google Scholar 

  25. Younes, L.: Shapes and Diffeomorphisms. Springer, Berlin (2010)

    Book  MATH  Google Scholar 

Download references

Acknowledgements

Alexis Arnaudon acknowledges partial support from an Imperial College London Roth Award and the EPSRC through award EP/N014529/1 funding the EPSRC Centre for Mathematics of Precision Healthcare. Alexis Arnaudon and Darryl Holm are partially supported by the European Research Council Advanced Grant 267382 FCCA held by Darryl Holm. Darryl Holm is also grateful for support from EPSRC Grant EP/N023781/1. Stefan Sommer is partially supported by the CSGB Centre for Stochastic Geometry and Advanced Bioimaging funded by a grant from the Villum Foundation. We thank the anonymous reviewers for insightful comments that have substantially improved the manuscript.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Stefan Sommer.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Arnaudon, A., Holm, D. & Sommer, S. String Methods for Stochastic Image and Shape Matching. J Math Imaging Vis 60, 953–967 (2018). https://doi.org/10.1007/s10851-018-0823-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10851-018-0823-z

Keywords

Navigation