Journal of Mathematical Imaging and Vision

, Volume 59, Issue 3, pp 432–455 | Cite as

A Non-local Topology-Preserving Segmentation-Guided Registration Model

  • Noémie Debroux
  • Solène Ozeré
  • Carole Le Guyader


In this paper, we address the issue of designing a theoretically well-motivated segmentation-guided registration method capable of handling large and smooth deformations. The shapes to be matched are viewed as hyperelastic materials and more precisely as Saint Venant–Kirchhoff ones and are implicitly modeled by level set functions. These are driven in order to minimize a functional containing both a nonlinear-elasticity-based regularizer prescribing the nature of the deformation, and a criterion that forces the evolving shape to match intermediate topology-preserving segmentation results. Theoretical results encompassing existence of minimizers, existence of a weak viscosity solution of the related evolution problem and asymptotic results are given. The study is then complemented by the derivation of the discrete counterparts of the asymptotic results provided in the continuous domain. Both a pure quadratic penalization method and an augmented Lagrangian technique (involving a related dual problem) are investigated with convergence results.


Topology-preserving segmentation Registration Nonlinear elasticity Saint Venant–Kirchhoff material Quasi-convexity Relaxed problem Asymptotic behavior Weak viscosity solutions Nonconvex programming Augmented Lagrangian Duality with zero gap Supergradient method 



The project is co-financed by the European Union with the European regional development fund (ERDF, HN0002137) and by the Normandie Regional Council via the M2NUM project. The authors would like to thank Dr. Caroline Petitjean (LITIS, Université de Rouen, France) for providing the cardiac cycle MRI images.


  1. 1.
    Ambrosio, L., Dal Maso, G.: A general chain rule for distributional derivatives. Proc. Am. Math. Soc. 108(3), 691–702 (1990)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    An, J.H., Chen, Y., Huang, F., Wilson, D., Geiser, E.: Medical Image Computing and Computer-Assisted Intervention—MICCAI 2005: 8th International Conference, Palm Springs, CA, USA, October 26–29, 2005. Proceedings, Part I, chap. A Variational PDE Based Level Set Method for a Simultaneous Segmentation and Non-Rigid Registration, pp. 286–293. Springer, Berlin (2005)Google Scholar
  3. 3.
    Ashburner, J., Friston, K.J.: Nonlinear spatial normalization using basis functions. Hum. Brain Mapp. 7(4), 254–266 (1999)CrossRefGoogle Scholar
  4. 4.
    Aubert, G., Kornprobst, P.: Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations. Applied Mathematical Sciences. Springer, New York (2001)MATHGoogle Scholar
  5. 5.
    Barles, G., Cardaliaguet, P., Ley, O., Monteillet, A.: Existence of weak solutions for general nonlocal and nonlinear second-order parabolic equations. Nonlinear Anal. Theory Methods Appl. 71(7–8), 2801–2810 (2009)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Beg, M., Miller, M., 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
  7. 7.
    Bourgoing, M.: Viscosity solutions of fully nonlinear second order parabolic equations with \({L}^1\) dependence in time and Neumann boundary conditions. Discret. Contin. Dyn. Syst. 21(3), 763–800 (2008)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Brezis, H.: Analyse fonctionelle. Théorie et Applications. Dunod, Paris (2005)Google Scholar
  9. 9.
    Broit, C.: Optimal registration of Deformed Images. Ph.D. thesis, Computer and Information Science, University of Pennsylvania (1981)Google Scholar
  10. 10.
    Burachik, R.S., Gasimov, R.N., Ismayilova, N.A., Kaya, C.Y.: On a modified subgradient algorithm for dual problems via sharp augmented Lagrangian. J. Global Optim. 34(1), 55–78 (2005). doi: 10.1007/s10898-005-3270-5 MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Burger, M., Modersitzki, J., Ruthotto, L.: A hyperelastic regularization energy for image registration. SIAM J. Sci. Comput. 35(1), B132–B148 (2013)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Caselles, V., Kimmel, R., Sapiro, G.: Geodesic active contours. Int. J. Comput. Vis. 22(1), 61–87 (1993)CrossRefMATHGoogle Scholar
  13. 13.
    Christensen, G., Rabbitt, R., Miller, M.: Deformable templates using large deformation kinematics. IEEE Trans. Image Process. 5(10), 1435–1447 (1996)CrossRefGoogle Scholar
  14. 14.
    Christensen, G.E.: Deformable shape models for anatomy. Ph.D. thesis, Washington University, Sever Institute of technology, USA (1994)Google Scholar
  15. 15.
    Ciarlet, P.: Elasticité Tridimensionnelle. Masson, Paris (1985)MATHGoogle Scholar
  16. 16.
    Ciarlet, P.: Mathematical Elasticity, Volume I: Three-Dimensional Elasticity. Amsterdam etc., North-Holland (1988)MATHGoogle Scholar
  17. 17.
    Clatz, O., Sermesant, M., Bondiau, P.Y., Delingette, H., Warfield, S.K., Malandain, G., Ayache, N.: Realistic simulation of the 3-D growth of brain tumors in MR images coupling diffusion with biomechanical deformation. IEEE Trans. Med. Imaging 24(10), 1334–1346 (2005)CrossRefGoogle Scholar
  18. 18.
    Crandall, M., Ishii, H., Lions, P.L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. 27, 1–67 (1992)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Dacorogna, B.: Direct Methods in the Calculus of Variations, 2nd edn. Springer, New York (2008)MATHGoogle Scholar
  20. 20.
    Davatzikos, C.: Spatial transformation and registration of brain images using elastically deformable models. Comput. Vis. Image Underst. 66(2), 207–222 (1997)CrossRefGoogle Scholar
  21. 21.
    Davis, M.H., Khotanzad, A., Flamig, D.P., Harms, S.E.: A physics-based coordinate transformation for 3-D image matching. IEEE Trans. Med. Imaging 16(3), 317–328 (1997)CrossRefGoogle Scholar
  22. 22.
    Demengel, F., Demengel, G., Erné, R.: Functional Spaces for the Theory of Elliptic Partial Differential Equations. Universitext. Springer, London (2012)CrossRefMATHGoogle Scholar
  23. 23.
    Derfoul, R., Le Guyader, C.: A relaxed problem of registration based on the Saint Venant–Kirchhoff material stored energy for the mapping of mouse brain gene expression data to a neuroanatomical mouse atlas. SIAM J. Imaging Sci. 7(4), 2175–2195 (2014)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Droske, M., Ring, W., Rumpf, M.: Mumford–Shah based registration: a comparison of a level set and a phase field approach. Comput. Vis. Sci. 12(3), 101–114 (2008)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Droske, M., Rumpf, M.: A variational approach to non-rigid morphological registration. SIAM J. Appl. Math. 64(2), 668–687 (2004)CrossRefMATHGoogle Scholar
  26. 26.
    Droske, M., Rumpf, M.: Multiscale joint segmentation and registration of image morphology. IEEE Trans. Pattern Anal. Mach. Intell. 29(12), 2181–2194 (2007)CrossRefGoogle Scholar
  27. 27.
    Fischer, B., Modersitzki, J.: Fast diffusion registration. AMS Contemp. Math. Inverse Probl. Image Anal. Med. Imaging 313, 11–129 (2002)MathSciNetMATHGoogle Scholar
  28. 28.
    Fischer, B., Modersitzki, J.: Curvature based image registration. J. Math. Imaging Vis. 18(1), 81–85 (2003)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Forcadel, N., Le Guyader, C.: A short time existence/uniqueness result for a nonlocal topology-preserving segmentation model. J. Differ. Equ. 253(3), 977–995 (2012)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Gasimov, R.N.: Augmented Lagrangian duality and nondifferentiable optimization methods in nonconvex programming. J. Global Optim. 24(2), 187–203 (2002). doi: 10.1023/A:1020261001771 MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Gooya, A., Pohl, K., Bilello, M., Cirillo, L., Biros, G., Melhem, E., Davatzikos, C.: GLISTR: glioma image segmentation and registration. IEEE Trans. Med. Imaging 31(10), 1941–1954 (2012)CrossRefGoogle Scholar
  32. 32.
    Gorthi, S., Duay, V., Bresson, X., Cuadra, M.B., Castro, F.J.S., Pollo, C., Allal, A.S., Thiran, J.P.: Active deformation fields: dense deformation field estimation for atlas-based segmentation using the active contour framework. Med. Image Anal. 15(6), 787–800 (2011)CrossRefGoogle Scholar
  33. 33.
    Haber, E., Heldmann, S., Modersitzki, J.: A computational framework for image-based constrained registration. Linear Algebra Its Appl. 431(3–4), 459–470 (2009). (Special Issue in honor of Henk van der Vorst)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Haber, E., Modersitzki, J.: Numerical methods for volume preserving image registration. Inverse Probl. 20(5), 1621–1638 (2004)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Haber, E., Modersitzki, J.: Image registration method with guaranteed displacement regularity. Int. J. Comput. Vis. 71(3), 361–372 (2007)CrossRefGoogle Scholar
  36. 36.
    Karaçali, B., Davatzikos, C.: Estimating topology preserving and smooth displacement fields. IEEE Trans. Med. Imaging 23(7), 868–880 (2004)CrossRefGoogle Scholar
  37. 37.
    Le Dret, H., Raoult, A.: The quasi-convex envelope of the Saint Venant–Kirchhoff stored energy function. Proc. R. Soc. Edinb. Sect. A Math. 125(6), 1179–1192 (1995)CrossRefMATHGoogle Scholar
  38. 38.
    Le Guyader, C., Vese, L.: Self-repelling snakes for topology-preserving segmentation models. IEEE Trans. Image Process. 17(5), 767–779 (2008)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Le Guyader, C., Vese, L.: A combined segmentation and registration framework with a nonlinear elasticity smoother. Comput. Vis. Image Underst. 115(12), 1689–1709 (2011)CrossRefGoogle Scholar
  40. 40.
    Lin, T., Le Guyader, C., Dinov, I., Thompson, P., Toga, A., Vese, L.: Gene expression data to mouse atlas registration using a nonlinear elasticity smoother and landmark points constraints. J. Sci. Comput. 50, 586–609 (2012)MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Lord, N., Ho, J., Vemuri, B., Eisenschenk, S.: Simultaneous registration and parcellation of bilateral hippocampal surface pairs for local asymmetry quantification. IEEE Trans. Med. Imaging 26(4), 471–478 (2007)CrossRefGoogle Scholar
  42. 42.
    Modersitzki, J.: Numerical Methods for Image Registration. Oxford University Press, Oxford (2004)MATHGoogle Scholar
  43. 43.
    Musse, O., Heitz, F., Armspach, J.P.: Topology preserving deformable image matching using constrained hierarchical parametric models. IEEE Trans. Image Process. 10(7), 1081–1093 (2001)CrossRefMATHGoogle Scholar
  44. 44.
    Negrón Marrero, P.: A numerical method for detecting singular minimizers of multidimensional problems in nonlinear elasticity. Numer. Math. 58, 135–144 (1990)MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Noblet, V., Heinrich, C., Heitz, F., Armspach, J.P.: 3-D deformable image registration: a topology preservation scheme based on hierarchical deformation models and interval analysis optimization. IEEE Trans. Image Process. 14(5), 553–566 (2005)MathSciNetCrossRefGoogle Scholar
  46. 46.
    Ozeré, S., Gout, C., Le Guyader, C.: Joint segmentation/registration model by shape alignment via weighted total variation minimization and nonlinear elasticity. SIAM J. Imaging Sci. 8(3), 1981–2020 (2015)MathSciNetCrossRefMATHGoogle Scholar
  47. 47.
    Ozeré, S., Le Guyader, C.: Scale Space and Variational Methods in Computer Vision: 5th International Conference, SSVM 2015, Lège-Cap Ferret, France, May 31–June 4, 2015, Proceedings, chap. Nonlocal Joint Segmentation Registration Model, pp. 348–359. Springer International Publishing, Cham (2015)Google Scholar
  48. 48.
    Ozeré, S., Le Guyader, C.: Topology preservation for image-registration-related deformation fields. Commun. Math. Sci. 13(5), 1135–1161 (2015)MathSciNetCrossRefMATHGoogle Scholar
  49. 49.
    Rabbitt, R., Weiss, J., Christensen, G., Miller, M.: Mapping of hyperelastic deformable templates using the finite element method. In: Proceedings SPIE, vol. 2573, pp. 252–265. SPIE (1995)Google Scholar
  50. 50.
    Rockafellar, R.T.: Lagrange multipliers and optimiality. SIAM 35, 183–238 (1993)CrossRefMATHGoogle Scholar
  51. 51.
    Rumpf, M., Wirth, B.: A nonlinear elastic shape averaging approach. SIAM J. Imaging Sci. 2(3), 800–833 (2009)MathSciNetCrossRefMATHGoogle Scholar
  52. 52.
    Sederberg, T., Parry, S.: Free-form deformation of solid geometric models. SIGGRAPH Comput. Graph. 20(4), 151–160 (1986)CrossRefGoogle Scholar
  53. 53.
    Sotiras, A., Davatzikos, C., Paragios, N.: Deformable medical image registration: a survey. IEEE Trans. Med. Imaging 32(7), 1153–1190 (2013)CrossRefGoogle Scholar
  54. 54.
    Vemuri, B., Ye, J., Chen, Y., Leonard, C.: Image Registration via level-set motion: applications to atlas-based segmentation. Med. Image Anal. 7(1), 1–20 (2003)CrossRefGoogle Scholar
  55. 55.
    Vese, L., Le Guyader, C.: Variational Methods in Image Processing. Chapman & Hall/CRC Mathematical and Computational Imaging Sciences Series. Taylor & Francis (2015)Google Scholar
  56. 56.
    Weickert, J., Kühne, G.: Geometric Level Set Methods in Imaging, Vision, and Graphics, chap. Fast Methods for Implicit Active Contour Models, pp. 43–57. Springer, New York (2003)Google Scholar
  57. 57.
    Wing-Sum, C.: Some discrete Poincaré-type inequalities. Int. J. Math. Math. Sci. 25(7), 479–488 (2001)MathSciNetCrossRefMATHGoogle Scholar
  58. 58.
    Yanovsky, I., Thompson, P.M., Osher, S., Leow, A.D.: Topology preserving log-unbiased nonlinear image registration: Theory and implementation. In: Proceedings IEEE Conference Computer Vision Pattern Recognition, pp. 1–8 (2007)Google Scholar
  59. 59.
    Yezzi, A., Zollei, L., Kapur, T.: A variational framework for joint segmentation and registration. In: Mathematical Methods in Biomedical Image Analysis, pp. 44–51. IEEE-MMBIA (2001)Google Scholar
  60. 60.
    Zagorchev, L., Goshtasby, A.: A comparative study of transformation functions for nonrigid image registration. IEEE Trans. Image Process. 15(3), 529–538 (2006)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Laboratoire de MathématiquesNormandie Université, INSA de RouenSaint-Etienne-du-Rouvray CedexFrance

Personalised recommendations