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Diffeomorphic B-spline vector fields with a tractable set of inequalities


Author: Michaël Sdika
Journal: Math. Comp. 88 (2019), 2827-2856
MSC (2010): Primary 65D07, 65D18, 37C05, 62H35, 68U10
DOI: https://doi.org/10.1090/mcom/3419
Published electronically: March 5, 2019
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Abstract: B-spline diffeomorphic vector fields are objects of great interest in image processing and analysis, more specifically for the registration of medical images. In this paper, several conditions on the B-spline coefficients ensuring that a given B-spline vector field is a diffeomorphism are proposed. Some properties of vector fields satisfying these conditions are established showing that they are not too restrictive while having a reasonable computational complexity. This work opens the way to the development of practical image registration algorithms in two and three dimensions whose unknowns would be such diffeomorphic B-spline vector fields.


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  • [1] Sylvain Arguillère, Emmanuel Trélat, Alain Trouvé, and Laurent Younes, Shape deformation analysis from the optimal control viewpoint, J. Math. Pures Appl. (9) 104 (2015), no. 1, 139–178 (English, with English and French summaries). MR 3350723, https://doi.org/10.1016/j.matpur.2015.02.004
  • [2] J. Ashburner, A fast diffeomorphic image registration algorithm, Neuroimage 38 (2007), no. 1, 95-113.
  • [3] J. Ashburner, C. Hutton, R. Frackowiak, I. Johnsrude, C. Price, K. Friston, et al., Identifying global anatomical differences: Deformation-based morphometry, Human Brain Mapping 6 (1998), no. 5-6, 348-357.
  • [4] B. B. Avants, C. L. Epstein, M. Grossman, and J. C. Gee, Symmetric diffeomorphic image registration with cross-correlation: Evaluating automated labeling of elderly and neurodegenerative brain, Medical Image Analysis 12 (2008), no. 1, 26-41.
  • [5] J. M. Ball, Global invertibility of Sobolev functions and the interpenetration of matter, Proc. Roy. Soc. Edinburgh Sect. A 88 (1981), no. 3-4, 315–328. MR 616782, https://doi.org/10.1017/S030821050002014X
  • [6] M. F. Beg, M. I. Miller, A. Trouvé, and L. Younes, Computing large deformation metric mappings via geodesic flows of diffeomorphisms, Internat. J. Computer Vision 61 (2005), no. 2, 139-157.
  • [7] Martin Burger, Jan Modersitzki, and Lars Ruthotto, A hyperelastic regularization energy for image registration, SIAM J. Sci. Comput. 35 (2013), no. 1, B132–B148. MR 3033063, https://doi.org/10.1137/110835955
  • [8] Y. Choi and S. Lee, Injectivity conditions of 2d and 3d uniform cubic b-spline functions, Graphical Models 62 (2000), no. 6, 411-427.
  • [9] S. Y. Chun and J. A. Fessler, A simple regularizer for B-spline nonrigid image registration that encourages local invertibility, IEEE J. Selected Topics in Signal Processing 3 (2009), 159-169.
  • [10] M. K. Chung, K. J. Worsley, T. Paus, C. Cherif, D. L. Collins, J. N. Giedd, J. L. Rapoport, and A. C. Evans, A unified statistical approach to deformation-based morphometry, NeuroImage 14 (2001), no. 3, 595-606.
  • [11] M. Droske and M. Rumpf, A variational approach to nonrigid morphological image registration, SIAM J. Appl. Math. 64 (2003/04), no. 2, 668–687. MR 2049668, https://doi.org/10.1137/S0036139902419528
  • [12] J. Hadamard, Sur les Correspondances Ponctuelles, Oeuvres, Editions du Centre Nationale de la Researche Scientifique, Paris, (1968), pp. 383-384.
  • [13] J. Kim, Intensity based image registration using robust similarity measure and constrained optimization: Applications for radiation therapy, Ph.D. thesis, University of Michigan, 2004.
  • [14] Dong C. Liu and Jorge Nocedal, On the limited memory BFGS method for large scale optimization, Math. Programming 45 (1989), no. 3, (Ser. B), 503–528. MR 1038245, https://doi.org/10.1007/BF01589116
  • [15] M. I. Miller, G. E. Christensen, Y. Amit, and U. Grenander, Mathematical textbook of deformable neuroanatomies, Proc. Nat. Acad. Sci. 90 (1993), no. 24, 11944.
  • [16] Richard S. Palais, Natural operations on differential forms, Trans. Amer. Math. Soc. 92 (1959), 125-141. MR 0116352, https://doi.org/10.2307/1993171
  • [17] J. Park, D. Metaxas, and L. Axel, Analysis of left ventricular wall motion based on volumetric deformable models and MRI-SPAMM, Medical Image Analysis 1 (1996), no. 1, 53-71.
  • [18] G. P. Penney, J. A. Schnabel, D. Rueckert, M. A. Viergever, and W. J. Niessen, Registration-based interpolation, Medical Imaging, IEEE Transactions 23 (2004), no. 7, 922-926.
  • [19] R. D. Rabbitt, J. A. Weiss, G. E. Christensen, and M. I. Miller, Mapping of hyperelastic deformable templates using the finite element method, Proceedings-SPIE the International Society for Optical Engineering, SPIE International Society for Optical, 1995, pp. 252-252.
  • [20] J. Schaerer, C. Casta, J. Pousin, and P. Clarysse, A dynamic elastic model for segmentation and tracking of the heart in mr image sequences, Medical Image Analysis 14 (2010), no. 6, 738-749.
  • [21] M. Sdika, A fast nonrigid image registration with constraints on the jacobian using large scale constrained optimization, Medical Imaging, IEEE Transactions 27 (Feb. 2008), no. 2, 271-281.
  • [22] Michaël Sdika, A sharp sufficient condition for B-spline vector field invertibility. Application to diffeomorphic registration and interslice interpolation, SIAM J. Imaging Sci. 6 (2013), no. 4, 2236–2257. MR 3129753, https://doi.org/10.1137/120879920
  • [23] Marius Staring, Stefan Klein, and Josien P. W. Pluim, A rigidity penalty term for nonrigid registration, Medical Physics 34 (2007), no. 11, 4098-4108.
  • [24] R. Stoica, J. Pousin, C. Casta, P. Croisille, Y.-M. Zhu, and P. Clarysse, Integrating fiber orientation constraint into a spatio-temporal fem model for heart borders and motion tracking in dynamic MRI, Statistical Atlases and Computational Models of the Heart, Imaging and Modelling Challenges, Springer, 2013, pp. 355-363.
  • [25] J. P. Thirion, Image matching as a diffusion process: An analogy with Maxwell's demons, Medical Image Analysis 2 (1998), no. 3, 243-260.
  • [26] A. Trouvé, Diffeomorphisms groups and pattern matching in image analysis, Internat. J. Computer Vision 28 (1998), no. 3, 213-221.
  • [27] Alain Trouvé, Action de groupe de dimension infinie et reconnaissance de formes, C. R. Acad. Sci. Paris Sér. I Math. 321 (1995), no. 8, 1031–1034 (French, with English and French summaries). MR 1360567
  • [28] M. Unser, A. Aldroubi, and M. Eden, B-spline signal processing: Part I - theory; and part II - efficient design and applications, IEEE Trans. Sig. Proc. 41 (1993), no. 2, 821-833 and 834-848.
  • [29] J. Vandemeulebroucke, S. Rit, J. Kybic, and P. C. D. Sarrut, Spatio-temporal motion estimation for respiratory-correlated imaging of the lungs, Framework 20 (2010), no. 21, 25.
  • [30] T. Vercauteren, X. Pennec, A. Perchant, and N. Ayache, Diffeomorphic demons: Efficient non-parametric image registration, NeuroImage 45 (2009), no. 1, Supp.1, S61-S72.
  • [31] S. Warfield, A. Robatino, J. Dengler, F. Jolesz, and R. Kikinis, Nonlinear registration and template driven segmentation, in Brain Warping (A. W. Toga, ed.), Academic Press, San Diego and London, 1999, pp. 67-84.

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Additional Information

Michaël Sdika
Affiliation: Université Lyon, INSA-Lyon, Université Claude Bernard Lyon 1, UJM-Saint Etienne, CNRS, Inserm, CREATIS UMR 5220, U1206, F-69000, Lyon, France
Email: michael.sdika@creatis.insa-lyon.fr

DOI: https://doi.org/10.1090/mcom/3419
Received by editor(s): June 23, 2017
Received by editor(s) in revised form: December 8, 2017, November 6, 2018, and November 25, 2018
Published electronically: March 5, 2019
Article copyright: © Copyright 2019 American Mathematical Society