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Incorporation of a Deformation Prior in Image Reconstruction

  • Barbara Gris
Article
  • 43 Downloads

Abstract

This article presents a method to incorporate a deformation prior in image reconstruction via the formalism of deformation modules. The framework of deformation modules allows to build diffeomorphic deformations that satisfy a given structure. The idea is to register a template image against the indirectly observed data via a modular deformation, incorporating this way the deformation prior in the reconstruction method. We show that this is a well-defined regularisation method (proving existence, stability and convergence) and present numerical examples of reconstruction from 2-D tomographic simulations and partially observed images.

Keywords

Image reconstruction Inverse problem Diffeomorphic deformation Deformation prior Image matching 

Notes

Acknowledgements

The work by Barbara Gris was supported by the Swedish Foundation for Strategic Research grant AM13-0049.

References

  1. 1.
    Abraham, I., Abraham, R., Bergounioux, M., Carlier, G.: Tomographic reconstruction from a few views: a multi-marginal optimal transport approach. Appl. Math. Optim. 75(1), 55–73 (2017)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Amit, Y., Grenander, U., Piccioni, M.: Structural image restoration through deformable templates. J. Am. Stat. Assoc. 86(414), 376–387 (1991)CrossRefGoogle Scholar
  3. 3.
    Arguillere, S.: Géométrie sous-riemannienne en dimension infinie et applications à l’analyse mathématique des formes. Ph.D. thesis, Paris 6 (2014)Google Scholar
  4. 4.
    Arguillere, S., Trélat, E., Trouvé, A., Younes, L.: Shape deformation analysis from the optimal control viewpoint. Journal de mathématiques pures et appliquées 104(1), 139–178 (2015)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Aronszajn, N.: Theory of reproducing kernels. Trans. Am. Math. Soc. 68(3), 337–404 (1950)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Arsigny, V., Commowick, O., Ayache, N., Pennec, X.: A fast and log-euclidean polyaffine framework for locally linear registration. J. Math. Imaging Vis. 33(2), 222–238 (2009)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Arsigny, V., Pennec, X., Ayache, N.: Polyrigid and polyaffine transformations: a novel geometrical tool to deal with non-rigid deformations-application to the registration of histological slices. Med. Image Anal. 9(6), 507–523 (2005)CrossRefGoogle Scholar
  8. 8.
    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)CrossRefGoogle Scholar
  9. 9.
    Benamou, J.-D., Carlier, G., Cuturi, M., Nenna, L., Peyré, G.: Iterative Bregman projections for regularized transportation problems. SIAM J. Sci. Comput. 37(2), A1111–A1138 (2015)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Berkels, B., Effland, A., Rumpf, M.: Time discrete geodesic paths in the space of images. SIAM J. Imaging Sci. 8(3), 1457–1488 (2015)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Blume, M., Martinez-Moller, A., Keil, A., Navab, N., Rafecas, M.: Joint reconstruction of image and motion in gated positron emission tomography. IEEE Trans. Med. Imaging 29(11), 1892–1906 (2010)CrossRefGoogle Scholar
  12. 12.
    Bruveris, M., Holm, D.D.: Geometry of image registration: the diffeomorphism group and momentum maps. In: Geometry, Mechanics, and Dynamics, pp. 19–56. Springer (2015)Google Scholar
  13. 13.
    Burger, M., Dirks, H., Frerking, L., Hauptmann, A., Helin, T., Siltanen, S.: A variational reconstruction method for undersampled dynamic X-ray tomography based on physical motion models. Inverse Probl. 33(12), 124008 (2017)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Chen, C., Öktem, O.: Indirect image registration with large diffeomorphic deformations. SIAM J. Imaging Sci. 11(1), 575–617 (2018)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Ehrhardt, J., Werner, R., Säring, D., Frenzel, T., Lu, W., Low, D., Handels, H.: An optical flow based method for improved reconstruction of 4D CT data sets acquired during free breathing. Med. Phys. 34(2), 711–721 (2007)CrossRefGoogle Scholar
  16. 16.
    Grenander, U., Srivastava, A., Saini, S.: A pattern-theoretic characterization of biological growth. IEEE Trans. Med. Imaging 26(5), 648–659 (2007)CrossRefGoogle Scholar
  17. 17.
    Gris, B.: Modular approach on shape spaces, sub-Riemannian geometry and computational anatomy. Ph.D. thesis, Université Paris-Saclay (2016)Google Scholar
  18. 18.
    Gris, B., Durrleman, S., Trouvé, A.: A sub-riemannian modular framework for diffeomorphism based analysis of shape ensembles. SIAM J. Imaging Sci. 11(1), 802–833 (2016)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Gris, B., Öktem, O.: Image reconstruction through metamorphosis. arXiv preprint arXiv:1806.01225v2 (2018)
  20. 20.
    Haber, E., Modersitzki, J.: A multilevel method for image registration. SIAM J. Sci. Comput. 27(5), 1594–1607 (2006)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Hahn, B.N.: Motion estimation and compensation strategies in dynamic computerized tomography. Sens. Imaging 18(1), 10 (2017)CrossRefGoogle Scholar
  22. 22.
    Hinkle, J., Szegedi, M., Wang, B., Salter, B., Joshi, S.: 4D CT image reconstruction with diffeomorphic motion model. Med. Image Anal. 16(6), 1307–1316 (2012)CrossRefGoogle Scholar
  23. 23.
    Isola, A., Ziegler, A., Koehler, T., Niessen, W., Grass, M.: Motion-compensated iterative cone-beam CT image reconstruction with adapted blobs as basis functions. Phys. Med. Biol. 53(23), 6777 (2008)CrossRefGoogle Scholar
  24. 24.
    Karlsson, J., Ringh, A.: Generalized Sinkhorn iterations for regularizing inverse problems using optimal mass transport. SIAM J. Imaging Sci. 10(4), 1935–1962 (2017)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Katsevich, A.: An accurate approximate algorithm for motion compensation in two-dimensional tomography. Inverse Probl. 26(6), 065007 (2010)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Lu, W., Mackie, T.R.: Tomographic motion detection and correction directly in sinogram space. Phys. Med. Biol. 47(8), 1267 (2002)CrossRefGoogle Scholar
  27. 27.
    Mair, B.A., Gilland, D.R., Sun, J.: Estimation of images and nonrigid deformations in gated emission CT. IEEE Trans. Med. Imaging 25(9), 1130–1144 (2006)CrossRefGoogle Scholar
  28. 28.
    McLeod, K., Sermesant, M., Beerbaum, P., Pennec, X.: Spatio-temporal tensor decomposition of a polyaffine motion model for a better analysis of pathological left ventricular dynamics. IEEE Trans. Med. Imaging 34(7), 1562–1575 (2015)CrossRefGoogle Scholar
  29. 29.
    Modersitzki, J.: Numerical Methods for Image Registration. Oxford University Press, Oxford (2004)zbMATHGoogle Scholar
  30. 30.
    Neumayer, S., Persch, J., Steidl, G.: Regularization of inverse problems via time discrete geodesics in image spaces. arXiv preprint arXiv:1805.06362 (2018)
  31. 31.
    Oektem, O., Chen, C., Domanic, N.O., Ravikumar, P., Bajaj, C.: Shape-based image reconstruction using linearized deformations. Inverse Probl. 33(3), 035004 (2017)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Portman, N.: The modelling of biological growth: a pattern theoretic approach. Ph.D. thesis, University of Waterloo (2009)Google Scholar
  33. 33.
    Reyes, M., Malandain, G., Koulibaly, P.M., González-Ballester, M.A., Darcourt, J.: Model-based respiratory motion compensation for emission tomography image reconstruction. Phys. Med. Biol. 52(12), 3579 (2007)CrossRefGoogle Scholar
  34. 34.
    Rit, S., Wolthaus, J., van Herk, M., Sonke, J.-J.: On-the-fly motion-compensated cone-beam ct using an a priori motion model. In: International Conference on Medical Image Computing and Computer-Assisted Intervention, pp. 729–736. Springer (2008)Google Scholar
  35. 35.
    Ritchie, C.J., Hsieh, J., Gard, M.F., Godwin, J.D., Kim, Y., Crawford, C.R.: Predictive respiratory gating: a new method to reduce motion artifacts on CT scans. Radiology 190(3), 847–852 (1994)CrossRefGoogle Scholar
  36. 36.
    Rohé, M.-M., Duchateau, N., Sermesant, M., Pennec, X.: Combination of polyaffine transformations and supervised learning for the automatic diagnosis of LV infarct. In: International Workshop on Statistical Atlases and Computational Models of the Heart, pp. 190–198. Springer (2015)Google Scholar
  37. 37.
    Seiler, C., Pennec, X., Reyes, M.: Capturing the multiscale anatomical shape variability with polyaffine transformation trees. Med. Image Anal. 16(7), 1371–1384 (2012)CrossRefGoogle Scholar
  38. 38.
    Srivastava, A., Saini, S., Ding, Z., Grenander, U.: Maximum-likelihood estimation of biological growth variables. In: Energy Minimization Methods in Computer Vision and Pattern Recognition, pp. 107–118. Springer (2005)Google Scholar
  39. 39.
    Trouvé, A., Younes, L.: Metamorphoses through lie group action. Found. Comput. Math. 5(2), 173–198 (2005)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Van Eyndhoven, G., Sijbers, J., Batenburg, J.: Combined motion estimation and reconstruction in tomography. In European Conference on Computer Vision, pp. 12–21. Springer (2012)Google Scholar
  41. 41.
    Younes, L.: Shapes and Diffeomorphisms, 171st edn. Springer, New York (2010)CrossRefGoogle Scholar
  42. 42.
    Younes, L.: Constrained diffeomorphic shape evolution. Found. Comput. Math. 12(3), 295–325 (2012)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.LJLL - Laboratoire Jacques-Louis LionsUPMCParisFrance

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