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A Unified Hyperelastic Joint Segmentation/Registration Model Based on Weighted Total Variation and Nonlocal Shape Descriptors

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Scale Space and Variational Methods in Computer Vision (SSVM 2017)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 10302))

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Abstract

In this paper, we address the issue of designing a unified variational model for joint segmentation and registration in which the shapes to be matched are viewed as hyperelastic materials, and more precisely, as Saint Venant-Kirchhoff ones. The dissimilarity measure relates local and global (or region-based) information, since relying on weighted total variation and on a nonlocal shape descriptor inspired by the Chan-Vese model for segmentation. Theoretical results emphasizing the mathematical and practical soundness of the model are provided, among which relaxation, existence of minimizers, analysis of two numerical methods of resolution, asymptotic results and a \(\varGamma \)-convergence property.

N. Debroux and C. Le Guyader—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.

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References

  1. An, J., Chen, Y., Huang, F., Wilson, D., Geiser, E.: A variational PDE based level set method for a simultaneous segmentation and non-rigid registration. In: Duncan, J.S., Gerig, G. (eds.) MICCAI 2005. LNCS, vol. 3749, pp. 286–293. Springer, Heidelberg (2005). doi:10.1007/11566465_36

    Chapter  Google Scholar 

  2. Aubert, G., Kornprobst, P.: Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations. Applied Mathematical Sciences. Springer, Heidelberg (2001)

    MATH  Google Scholar 

  3. Aubert, G., Kornprobst, P.: Can the nonlocal characterization of Sobolev spaces by Bourgain et al. be useful for solving variational problems? SIAM J. Numer. Anal. 47(2), 844–860 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. Baldi, A.: Weighted BV functions. Houst. J. Math. 27(3), 683–705 (2001)

    MathSciNet  MATH  Google Scholar 

  5. Boulanger, J., Elbau, P., Pontow, C., Scherzer, O.: Non-local functionals for imaging. In: Bauschke, H.H., Burachik, R.S., Combettes, P.L., Elser, V., Russell Luke, D., Wolkowicz, H. (eds.) Fixed-Point Algorithms for Inverse Problems in Science and Engineering, pp. 131–154. Springer, New York (2011)

    Chapter  Google Scholar 

  6. Bourgain, J., Brezis, H., Mironescu, P.: Another look at Sobolev spaces. In: Menaldi, J.L., Rofman, Sulem, A. (eds.) Optimal Control and Partial Differential Equations, in honour of Professor Alain Bensoussan’s 60th Birthday, pp. 439–455 (2001)

    Google Scholar 

  7. Bresson, X., Esedoḡlu, S., Vandergheynst, P., Thiran, J.P., Osher, S.: Fast global minimization of the active contour/snake model. J. Math. Imaging Vis. 68, 151–167 (2007)

    Article  MathSciNet  Google Scholar 

  8. Caselles, V., Kimmel, R., Sapiro, G.: Geodesic active contours. Int. J. Comput. Vis. 22(1), 61–87 (1993)

    Article  MATH  Google Scholar 

  9. Chan, T., Vese, L.: Active contours without edges. IEEE Trans. Image Process. 10(2), 266–277 (2001)

    Article  MATH  Google Scholar 

  10. Ciarlet, P.: Elasticité Tridimensionnelle. Masson, Paris (1985)

    MATH  Google Scholar 

  11. Dacorogna, B.: Direct Methods in the Calculus of Variations, 2nd edn. Springer, Heidelberg (2008)

    MATH  Google Scholar 

  12. Dávila, J.: On an open question about functions of bounded variation. Calc. Var. Partial Differ. Equ. 15(4), 519–527 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  13. Debroux, N., Le Guyader, C.: A joint segmentation/registration model based on a nonlocal characterization of weighted total variation and nonlocal shape descriptors, in preparation

    Google Scholar 

  14. Droske, M., Rumpf, M.: Multiscale joint segmentation and registration of image morphology. IEEE Trans. Pattern Anal. Mach. Intell. 29(12), 2181–2194 (2007)

    Article  Google Scholar 

  15. 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)

    Article  Google Scholar 

  16. Han, J., Berkels, B., Droske, M., Hornegger, J., Rumpf, M., Schaller, C., Scorzin, J., Urbach, H.: Mumford-Shah model for one-to-one edge matching. IEEE Trans. Image Process. 16(11), 2720–2732 (2007)

    Article  MathSciNet  Google Scholar 

  17. 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)

    Article  Google Scholar 

  18. 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)

    Article  Google Scholar 

  19. Mumford, D., Shah, J.: Optimal approximations by piecewise smooth functions and associated variational problems. Commun. Pure Appl. Anal. 42, 577–685 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  20. Negrón Marrero, P.: A numerical method for detecting singular minimizers of multidimensional problems in nonlinear elasticity. Numer. Math. 58, 135–144 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  21. 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)

    Article  MathSciNet  MATH  Google Scholar 

  22. Ozeré, S., Le Guyader, C.: A joint segmentation-registration framework based on weighted total variation and nonlinear elasticity principles. In: 2014 IEEE International Conference on Image Processing (ICIP), pp. 3552–3556 (2014)

    Google Scholar 

  23. Ponce, A.C.: A new approach to Sobolev spaces and connections to \(\varGamma \)-convergence. Calc. Var. Partial Differ. Equ. 19(3), 229–255 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  24. Rudin, L., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Phys. D 60(1–4), 259–268 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  25. Rumpf, M., Wirth, B.: A nonlinear elastic shape averaging approach. SIAM J. Imaging Sci. 2(3), 800–833 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  26. Sotiras, A., Davatzikos, C., Paragios, N.: Deformable medical image registration: a survey. IEEE Trans. Med. Imaging 32(7), 1153–1190 (2013)

    Article  Google Scholar 

  27. 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)

    Article  Google Scholar 

  28. Vese, L., Le Guyader, C.: Variational Methods in Image Processing. Chapman & Hall/CRC Mathematical and Computational Imaging Sciences Series. Taylor & Francis, Abingdon (2015)

    MATH  Google Scholar 

  29. Yezzi, A., Zollei, L., Kapur, T.: A variational framework for joint segmentation and registration. In: Math Methods Biomed Image Analysis, IEEE-MMBIA, pp. 44–51 (2001)

    Google Scholar 

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Authors and Affiliations

  1. Laboratoire de Mathématiques, Normandie Université, INSA de Rouen, 685 Avenue de l’Université, 76801, Saint-Etienne-du-Rouvray Cedex, France

    Noémie Debroux & Carole Le Guyader

Authors
  1. Noémie Debroux
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  2. Carole Le Guyader
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Correspondence to Carole Le Guyader .

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Editors and Affiliations

  1. University of Copenhagen , Copenhagen, Denmark

    François Lauze

  2. Technical University of Denmark , Kongens Lyngby, Denmark

    Yiqiu Dong

  3. Technical University of Denmark , Kongens Lyngby, Denmark

    Anders Bjorholm Dahl

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Debroux, N., Le Guyader, C. (2017). A Unified Hyperelastic Joint Segmentation/Registration Model Based on Weighted Total Variation and Nonlocal Shape Descriptors. In: Lauze, F., Dong, Y., Dahl, A. (eds) Scale Space and Variational Methods in Computer Vision. SSVM 2017. Lecture Notes in Computer Science(), vol 10302. Springer, Cham. https://doi.org/10.1007/978-3-319-58771-4_49

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  • DOI: https://doi.org/10.1007/978-3-319-58771-4_49

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-58770-7

  • Online ISBN: 978-3-319-58771-4

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