Abstract
In the seminal paper E. Tadmor, S. Nezzar and L. Vese, A multiscale image representation using hierarchical \((BV, L^2)\) decompositions, Multiscale Model. Simul., 2(4), 554–579, (2004), the authors introduce a multiscale image decomposition model providing a hierarchical decomposition of a given image into the sum of scale-varying components. In line with this framework, we extend the approach to the case of registration, task which consists of mapping salient features of an image onto the corresponding ones in another, the underlying goal being to obtain such a kind of hierarchical decomposition of the deformation relating the two considered images (—from the coarser one that encodes the main structural/geometrical deformation, to the more refined one—). To achieve this goal, we introduce a functional minimisation problem in a hyperelasticity setting by viewing the shapes to be matched as Ogden materials. This approach is complemented by hard constraints on the \(L^{\infty }\)-norm of both the Jacobian and its inverse, ensuring that the deformation is a bi-Lipschitz homeomorphism. Theoretical results emphasising the mathematical soundness of the model are provided, among which the existence of minimisers, a \(\varGamma \)-convergence result and an analysis of a suitable numerical algorithm, along with numerical simulations demonstrating the ability of the model to produce accurate hierarchical representations of deformations.
L. Vese acknowledges support from the National Science Foundation under Grant # 2012868. This project was co-financed by the European Union with the European regional development fund (ERDF, 8P03390/18E01750/18P02733) and by the Haute-Normandie Régional Council via the M2SINUM project.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Ambrosio, L., Bertrand, J.: DC calculus. Math. Zeitschrift 288(3), 1037–1080 (2018)
Athavale, P., Xu, R., Radau, P., Nachman, A., Wright, G.A.: Multiscale properties of weighted total variation flow with applications to denoising and registration. Med. Image Anal. 23(1), 28–42 (2015)
Ball, J.M.: Global invertibility of Sobolev functions and the interpenetration of matter. P. Roy. Soc. Edin. A 88(3–4), 315–328 (1981)
Bennet, R., Sharpley, R.: Interpolation of Operators. Academic Press, New York (1988)
Bergh, J., Löfström, J.: Interpolation Spaces: An Introduction. Springer, Heidelberg (1976)
Ciarlet, P.: Elasticité Tridimensionnelle. Masson (1985)
Debroux, N., Le Guyader, C.: A joint segmentation/registration model based on a nonlocal characterization of weighted total variation and nonlocal shape descriptors. SIAM J. Imaging Sci. 11(2), 957–990 (2018)
Debroux, N., et al.: A variational model dedicated to joint segmentation, registration, and atlas generation for shape analysis. SIAM J. Imaging Sci. 13(1), 351–380 (2020)
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 (2018)
Lam, K.C., Ng, T.C., Lui, L.M.: Multiscale representation of deformation via beltrami coefficients. Multiscale Model. Simul. 15(2), 864–891 (2017)
Meyer, Y.: Oscillating Patterns in Image Processing and Nonlinear Evolution Equations: The Fifteenth Dean Jacqueline B. American Mathematical Society, Lewis Memorial Lectures (2001)
Modersitzki, J.: Numerical Methods for Image Registration. Oxford University Press, Oxford (2004)
Modersitzki, J.: FAIR: Flexible Algorithms for Image Registration. SIAM (2009)
Modin, K., Nachman, A., Rondi, L.: A multiscale theory for image registration and nonlinear inverse problems. Adv. Math. 346, 1009–1066 (2019)
Negrón Marrero, P.: A numerical method for detecting singular minimizers of multidimensional problems in nonlinear elasticity. Numer. Math. 58, 135–144 (1990)
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)
Paquin, D., Levy, D., Schreibmann, E., Xing, L.: Multiscale image registration. Math. Biosci. Eng. 3(2), 389–418 (2006)
Paquin, D., Levy, D., Xing, L.: Hybrid multiscale landmark and deformable image registration. Math. Biosci. Eng. 4(4), 711–737 (2007)
Paquin, D., Levy, D., Xing, L.: Multiscale deformable registration of noisy medical images. Math. Biosci. Eng. 5(1), 125–144 (2008)
Risser, L., Vialard, F.X., Wolz, R., Murgasova, M., Holm, D.D., Rueckert, D.: Simultaneous multi-scale registration using large deformation diffeomorphic metric mapping. IEEE T. Med. Imaging 30(10), 1746–1759 (2011)
von Siebenthal, M., Székely, G., Gamper, U., Boesiger, P., Lomax, A., Cattin, P.: 4D MR imaging of respiratory organ motion and its variability. Phys. Med. Biol. 52(6), 1547 (2007)
Sommer, S., Lauze, F., Nielsen, M., Pennec, X.: Sparse multi-scale diffeomorphic registration: the kernel bundle framework. J. Math. Imaging Vis. 46, 292–308 (2013)
Sotiras, A., Davatzikos, C., Paragios, N.: Deformable medical image registration: a survey. IEEE Trans. Med. Imaging 32(7), 1153–1190 (2013)
Tadmor, E., Nezzar, S., Vese, L.: A multiscale image representation using hierarchical \(({BV},{L}^2)\) decompositions. Multiscale Model. Simul. 2(4), 554–579 (2004)
Vercauteren, T., Pennec, X., Perchant, A., Ayache, N.: Diffeomorphic demons: efficient non-parametric image registration. NeuroImage 45, S61–72 (2008)
Vese, L., Le Guyader, C.: Variational Methods in Image Processing. Chapman & Hall/CRC Mathematical and Computational Imaging Sciences Series, Taylor & Francis (2015)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 Springer Nature Switzerland AG
About this paper
Cite this paper
Debroux, N., Le Guyader, C., Vese, L.A. (2021). Multiscale Registration. In: Elmoataz, A., Fadili, J., Quéau, Y., Rabin, J., Simon, L. (eds) Scale Space and Variational Methods in Computer Vision. SSVM 2021. Lecture Notes in Computer Science(), vol 12679. Springer, Cham. https://doi.org/10.1007/978-3-030-75549-2_10
Download citation
DOI: https://doi.org/10.1007/978-3-030-75549-2_10
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-75548-5
Online ISBN: 978-3-030-75549-2
eBook Packages: Computer ScienceComputer Science (R0)