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Journal of Mathematical Imaging and Vision

, Volume 60, Issue 9, pp 1482–1502 | Cite as

Measure-Valued Variational Models with Applications to Diffusion-Weighted Imaging

  • Thomas Vogt
  • Jan Lellmann
Article
  • 163 Downloads

Abstract

We develop a general mathematical framework for variational problems where the unknown function takes values in the space of probability measures on some metric space. We study weak and strong topologies and define a total variation seminorm for functions taking values in a Banach space. The seminorm penalizes jumps and is rotationally invariant under certain conditions. We prove existence of a minimizer for a class of variational problems based on this formulation of total variation and provide an example where uniqueness fails to hold. Employing the Kantorovich–Rubinstein transport norm from the theory of optimal transport, we propose a variational approach for the restoration of orientation distribution function-valued images, as commonly used in diffusion MRI. We demonstrate that the approach is numerically feasible on several data sets.

Keywords

Variational methods Total variation Measure theory Optimal transport Diffusion MRI Manifold-valued imaging 

References

  1. 1.
    Aganj, I., Lenglet, C., Sapiro, G.: ODF reconstruction in Q-ball imaging with solid angle consideration. In: Proceedings of the IEEE International Symposium on Biomed Imaging 2009, pp. 1398–1401 (2009)Google Scholar
  2. 2.
    Ahrens, C., Nealy, J., Pérez, F., van der Walt, S.: Sparse reproducing kernels for modeling fiber crossings in diffusion weighted imaging. In: Proceedings of the IEEE International Symposium on Biomed Imaging 2013, pp. 688–691 (2013)Google Scholar
  3. 3.
    Ambrosio, L.: Metric space valued functions of bounded variation. Ann. Sc. Norm. Super. Pisa Cl. Sci. IV. Ser. 17(3), 439–478 (1990)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Clarendon Press, Oxford (2000)zbMATHGoogle Scholar
  5. 5.
    Åström, F., Petra, S., Schmitzer, B., Schnörr, C.: Image labeling by assignment. J. Math. Imaging Vis. 58(2), 211–238 (2017)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Ball, J.: A version of the fundamental theorem for Young measures. In: PDEs and Continuum Models of Phase Transitions. Proceedings of an NSF-CNRS Joint Seminar Held in Nice, France, January 18–22, 1988, pp. 207–215 (1989)Google Scholar
  7. 7.
    Basser, P.J., Mattiello, J., LeBihan, D.: MR diffusion tensor spectroscopy and imaging. Biophys. J. 66(1), 259–267 (1994)CrossRefGoogle Scholar
  8. 8.
    Bačák, M., Bergmann, R., Steidl, G., Weinmann, A.: A second order non-smooth variational model for restoring manifold-valued images. SIAM J. Sci. Comput. 38(1), A567–A597 (2016)CrossRefGoogle Scholar
  9. 9.
    Becker, S., Tabelow, K., Voss, H.U., Anwander, A., Heidemann, R.M., Polzehl, J.: Position-orientation adaptive smoothing of diffusion weighted magnetic resonance data (POAS). Med. Image Anal. 16(6), 1142–1155 (2012)CrossRefGoogle Scholar
  10. 10.
    Bourbaki, N.: Integration. Springer, Berlin (2004)CrossRefGoogle Scholar
  11. 11.
    Callaghan, P.T.: Principles of Nuclear Magnetic Resonance Microscopy. Clarendon Press, Oxford (1991)Google Scholar
  12. 12.
    Canales-Rodríguez, E.J., Daducci, A., Sotiropoulos, S.N., Caruyer, E., Aja-Fernández, S., Radua, J., et al.: Spherical deconvolution of multichannel diffusion MRI data with non-Gaussian noise models and spatial regularization. PLoS ONE 10(10), 1–29 (2015)CrossRefGoogle Scholar
  13. 13.
    Carothers, N.L.: Real Analysis. Cambridge University Press, Cambridge (2000)CrossRefGoogle Scholar
  14. 14.
    Chambolle, A., Caselles, V., Cremers, D., Novaga, M., Pock, T.: An introduction to total variation for image analysis. Theor. Found. Numer. Methods Sparse Recovery 9, 263–340 (2010)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Chambolle, A., Pock, T.: A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math. Imaging Vis. 40(1), 120–145 (2011)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Chambolle, A., Pock, T.: Total roto-translational variation. Technical Report arXiv:1709.09953, arXiv (2017)
  17. 17.
    Chan, T.F., Esedoglu, S.: Aspects of total variation regularized \(L^1\) function approximation. SIAM J. Appl. Math. 65(5), 1817–1837 (2005)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Chen, D., Mirebeau, J.M., Cohen, L.D.: Global minimum for a finsler elastica minimal path approach. Int. J. Comput. Vis. 122(3), 458–483 (2016).  https://doi.org/10.1007/s11263-016-0975-5 MathSciNetCrossRefGoogle Scholar
  19. 19.
    Clarke, F.: Functional Analysis, Calculus of Variations and Optimal Control. Springer, London (2013)CrossRefGoogle Scholar
  20. 20.
    Creusen, E., Duits, R., Vilanova, A., Florack, L.: Numerical schemes for linear and non-linear enhancement of DW-MRI. Numer. Math. Theor. Methods Appl. 6(1), 138–168 (2013)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Cuturi, M.: Sinkhorn distances: Lightspeed computation of optimal transport. In: Burges, C.J.C., Bottou, L., Welling, M., Ghahramani, Z., Weinberger, K.Q. (eds.) Advances in Neural Information Processing Systems 26, pp. 2292–2300. Curran Associates, Inc. (2013)Google Scholar
  22. 22.
    Dacorogna, B.: Direct Methods in the Calculus of Variations, 2nd edn. Springer, Berlin (2008)zbMATHGoogle Scholar
  23. 23.
    Daducci, A., et al.: Quantitative comparison of reconstruction methods for intra-voxel fiber recovery from diffusion MRI. IEEE Trans. Med. Imaging 33(2), 384–399 (2014)CrossRefGoogle Scholar
  24. 24.
    Daducci, A., Canales-Rodríguez, E.J., Descoteaux, M., Garyfallidis, E., Gur, Y., et al.: Quantitative comparison of reconstruction methods for intra-voxel fiber recovery from diffusion MRI. IEEE Trans. Med. Imaging 33(2), 384–399 (2014)CrossRefGoogle Scholar
  25. 25.
    Delputte, S., Dierckx, H., Fieremans, E., D’Asseler, Y., Achten, R., Lemahieu, I.: Postprocessing of brain white matter fiber orientation distribution functions. In: Proceedings of the IEEE International Symposium on Biomed Imaging 2007, pp. 784–787 (2007)Google Scholar
  26. 26.
    Descoteaux, M.: High angular resolution diffusion MRI: from local estimation to segmentation and tractography. Ph.D. thesis, University of Nice-Sophia Antipolis (2008)Google Scholar
  27. 27.
    Duchoň, M., Debiève, C.: Functions with bounded variation in locally convex space. Tatra Mt. Math. Publ. 49, 89–98 (2011)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Duits, R., Franken, E.: Left-invariant diffusions on the space of positions and orientations and their application to crossing-preserving smoothing of HARDI images. Int. J. Comput. Vis. 92(3), 231–264 (2011)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Duits, R., Haije, T.D., Creusen, E., Ghosh, A.: Morphological and linear scale spaces for fiber enhancement in DW-MRI. J. Math. Imaging Vis. 46(3), 326–368 (2012)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Duval, V., Aujol, J.F., Gousseau, Y.: The TVL1 model: a geometric point of view. Multiscale Model. Simul. 8(1), 154–189 (2009)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Ehricke, H.H., Otto, K.M., Klose, U.: Regularization of bending and crossing white matter fibers in MRI Q-ball fields. Magn. Reson. Imaging 29(7), 916–926 (2011)CrossRefGoogle Scholar
  32. 32.
    Fitschen, J.H., Laus, F., Schmitzer, B.: Optimal transport for manifold-valued images. In: 2017 Scale Space and Variational Methods in Computer Vision, pp. 460–472 (2017)Google Scholar
  33. 33.
    Fitschen, J.H., Laus, F., Steidl, G.: Transport between RGB images motivated by dynamic optimal transport. J. Math. Imaging Vis. 56(3), 409–429 (2016)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Garyfallidis, E., Brett, M., Amirbekian, B., Rokem, A., Van Der Walt, S., Descoteaux, M., Nimmo-Smith, I., Contributors, D.: Dipy, a library for the analysis of diffusion MRI data. Front. Neuroinform. 8(8), 1–17 (2014)Google Scholar
  35. 35.
    Goh, A., Lenglet, C., Thompson, P.M., Vidal, R.: Estimating orientation distribution functions with probability density constraints and spatial regularity. In: Medical Image Computing and Computer-Assisted Intervention—MICCAI 2009, pp. 877–885 (2009)CrossRefGoogle Scholar
  36. 36.
    Goldluecke, B., Strekalovskiy, E., Cremers, D.: The natural vectorial total variation which arises from geometric measure theory. SIAM J. Imaging Sci. 5(2), 537–563 (2012)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Goldstein, T., Esser, E., Baraniuk, R.: Adaptive primal dual optimization for image processing and learning. In: Proceedings of the 6th NIPS Workshop on Optimization for Machine Learning (2013)Google Scholar
  38. 38.
    Goldstein, T., Li, M., Yuan, X.: Adaptive primal-dual splitting methods for statistical learning and image processing. In: Cortes, C., Lawrence, N.D., Lee, D.D., Sugiyama, M., Garnett, R. (eds.) Advances in Neural Information Processing Systems, vol. 28, pp. 2089–2097. Curran Associates, Inc., New York (2015)Google Scholar
  39. 39.
    Goldstein, T., Li, M., Yuan, X., Esser, E., Baraniuk, R.: Adaptive primal-dual hybrid gradient methods for saddle-point problems. Technical Report arXiv:1305.0546v2, arXiv (2015)
  40. 40.
    Hewitt, E., Stromberg, K.: Real and Abstract Analysis. Springer, Berlin (1965)zbMATHGoogle Scholar
  41. 41.
    Hohage, T., Rügge, C.: A coherence enhancing penalty for diffusion MRI: regularizing property and discrete approximation. SIAM J. Imaging Sci. 8(3), 1874–1893 (2015)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Tulcea, A.I., Tulcea, C.I.: Topics in the Theory of Lifting. Springer, Berlin (1969)CrossRefGoogle Scholar
  43. 43.
    Kaden, E., Kruggel, F.: A reproducing kernel hilbert space approach for Q-ball imaging. IEEE Trans. Med. Imaging 30(11), 1877–1886 (2011)CrossRefGoogle Scholar
  44. 44.
    Kantorovich, L.V., Rubinshtein, G.S.: On a functional space and certain extremum problems. Dokl. Akad. Nauk SSSR 115, 1058–1061 (1957)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Karayumak, S.C., Özarslan, E., Unal, G.: Asymmetric orientation distribution functions (AODFs) revealing intravoxel geometry in diffusion MRI. Magn. Reson. Imaging 49, 145–158 (2018)CrossRefGoogle Scholar
  46. 46.
    Kezele, I., Descoteaux, M., Poupon, C., Abrial, P., Poupon, F., Mangin, J.F.: Multiresolution decomposition of HARDI and ODF profiles using spherical wavelets. In: Presented at the Workshop on Computational Diffusion MRI, MICCAI, New York, pp. 225–234 (2008)Google Scholar
  47. 47.
    Kim, Y., Thompson, P.M., Vese, L.A.: HARDI data denoising using vectorial total variation and logarithmic barrier. Inverse Probl. Imaging 4(2), 273–310 (2010)MathSciNetCrossRefGoogle Scholar
  48. 48.
    Laude, E., Möllenhoff, T., Moeller, M., Lellmann, J., Cremers, D.: Sublabel-accurate convex relaxation of vectorial multilabel energies. In: Proceedings of the ECCV 2016 Part I, pp. 614–627 (2016)CrossRefGoogle Scholar
  49. 49.
    Lavenant, H.: Harmonic mappings valued in the Wasserstein space. Technical Report. arXiv:1712.07528, arXiv (2017)
  50. 50.
    Lee, J.M.: Riemannian Manifolds. An Introduction to Curvature. Springer, New York (1997)zbMATHGoogle Scholar
  51. 51.
    Lellmann, J., Lorenz, D.A., Schönlieb, C., Valkonen, T.: Imaging with Kantorovich–Rubinstein discrepancy. SIAM J. Imaging Sci. 7(4), 2833–2859 (2014)MathSciNetCrossRefGoogle Scholar
  52. 52.
    Lellmann, J., Strekalovskiy, E., Koetter, S., Cremers, D.: Total variation regularization for functions with values in a manifold. In: 2013 IEEE International Conference on Computer Vision, pp. 2944–2951 (2013)Google Scholar
  53. 53.
    McGraw, T., Vemuri, B., Ozarslan, E., Chen, Y., Mareci, T.: Variational denoising of diffusion weighted MRI. Inverse Probl. Imaging 3(4), 625–648 (2009)MathSciNetCrossRefGoogle Scholar
  54. 54.
    Meesters, S., Sanguinetti, G., Garyfallidis, E., Portegies, J., Duits, R.: Fast implementations of contextual PDE’s for HARDI data processing in DIPY. Technical Report, ISMRM 2016 Conference (2016)Google Scholar
  55. 55.
    Meesters, S., Sanguinetti, G., Garyfallidis, E., Portegies, J., Ossenblok, P., Duits, R.: Cleaning output of tractography via fiber to bundle coherence, a new open source implementation. Technical Report, Human Brain Mapping Conference (2016)Google Scholar
  56. 56.
    Michailovich, O.V., Rathi, Y.: On approximation of orientation distributions by means of spherical ridgelets. IEEE Trans. Image Process. 19(2), 461–477 (2010)MathSciNetCrossRefGoogle Scholar
  57. 57.
    Miranda, M.: Functions of bounded variation on "good" metric spaces. Journal de Mathématiques Pures et Appliquées 82(8), 975–1004 (2003)MathSciNetCrossRefGoogle Scholar
  58. 58.
    Mollenhoff, T., Laude, E., Moeller, M., Lellmann, J., Cremers, D.: Sublabel-accurate relaxation of nonconvex energies. In: The IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (2016)Google Scholar
  59. 59.
    MomayyezSiahkal, P., Siddiqi, K.: 3D stochastic completion fields for mapping connectivity in diffusion MRI. IEEE Trans. Pattern Anal. Mach. Intell. 35(4), 983–995 (2013)CrossRefGoogle Scholar
  60. 60.
    Ncube, S., Srivastava, A.: A novel Riemannian metric for analyzing HARDI data. In: Proceedings of the SPIE, p. 7962 (2011)Google Scholar
  61. 61.
    Ouyang, Y., Chen, Y., Wu, Y.: Vectorial total variation regularisation of orientation distribution functions in diffusion weighted MRI. Int. J. Bioinform. Res. Appl. 10(1), 110–127 (2014)CrossRefGoogle Scholar
  62. 62.
    Pock, T., Chambolle, A.: Diagonal preconditioning for first order primal-dual algorithms in convex optimization. In: 2011 International Conference on Computer Vision, Barcelona, pp. 1762–1769 (2011)Google Scholar
  63. 63.
    Portegies, J., Duits, R.: New exact and numerical solutions of the (convection–)diffusion kernels on SE(3). Differ. Geom. Appl. 53, 182–219 (2017)MathSciNetCrossRefGoogle Scholar
  64. 64.
    Portegies, J.M., Fick, R.H.J., Sanguinetti, G.R., Meesters, S.P.L., Girard, G., Duits, R.: Improving fiber alignment in HARDI by combining contextual PDE flow with constrained spherical deconvolution. PLOS ONE 10(10), e0138,122 (2015)CrossRefGoogle Scholar
  65. 65.
    Prčkovska, V., Andorrà, M., Villoslada, P., Martinez-Heras, E., Duits, R., Fortin, D., Rodrigues, P., Descoteaux, M.: Contextual diffusion image post-processing aids clinical applications. In: Hotz, I., Schultz, T. (eds.) Visualization and Processing of Higher Order Descriptors for Multi-Valued Data, pp. 353–377. Springer, Berlin (2015)CrossRefGoogle Scholar
  66. 66.
    Reisert, M., Kellner, E., Kiselev, V.: About the geometry of asymmetric fiber orientation distributions. IEEE Trans. Med. Imaging 31(6), 1240–1249 (2012)CrossRefGoogle Scholar
  67. 67.
    Reisert, M., Skibbe, H.: Fiber continuity based spherical deconvolution in spherical harmonic domain. In: Medical Image Computing and Computer-Assisted Intervention—MICCAI 2013, pp. 493–500. Springer, Berlin (2013)CrossRefGoogle Scholar
  68. 68.
    Rokem, A., Yeatman, J., Pestilli, F., Wandell, B.: High angular resolution diffusion MRI. Stanford Digital Repository (2013). http://purl.stanford.edu/yx282xq2090. Accessed 20 Sept 2017
  69. 69.
    Skibbe, H., Reisert, M.: Spherical tensor algebra: a toolkit for 3d image processing. J. Math. Imaging Vis. 58(3), 349–381 (2017)MathSciNetCrossRefGoogle Scholar
  70. 70.
    Srivastava, A., Jermyn, I.H., Joshi, S.H.: Riemannian analysis of probability density functions with applications in vision. In: CVPR ’07, pp. 1–8 (2007)Google Scholar
  71. 71.
    Stejskal, E., Tanner, J.: Spin diffusion measurements: spin echos in the presence of a time-dependent field gradient. J. Chem. Phys. 42, 288–292 (1965)CrossRefGoogle Scholar
  72. 72.
    Tax, C.M.W., Jeurissen, B., Vos, S.B., Viergever, M.A., Leemans, A.: Recursive calibration of the fiber response function for spherical deconvolution of diffusion MRI data. NeuroImage 86, 67–80 (2014)CrossRefGoogle Scholar
  73. 73.
    Tournier, J.D., Calamante, F., Connelly, A.: Robust determination of the fibre orientation distribution in diffusion MRI: non-negativity constrained super-resolved spherical deconvolution. NeuroImage 35(4), 1459–1472 (2007)CrossRefGoogle Scholar
  74. 74.
    Tournier, J.D., Calamante, F., Gadian, D., Connelly, A.: Direct estimation of the fibre orientation density function from diffusion-weighted MRI data using spherical deconvolution. NeuroImage 23(3), 1176–1185 (2004)CrossRefGoogle Scholar
  75. 75.
    Tuch, D.S.: Q-ball imaging. Magn. Reson. Med. 52(6), 1358–1372 (2004)CrossRefGoogle Scholar
  76. 76.
    Tuch, D.S., Reese, T.G., Wiegell, M.R., Makris, N., Belliveau, J.W., Wedeen, V.J.: High angular resolution diffusion imaging reveals intravoxel white matter fiber heterogeneity. Magn. Reson. Med. 48(4), 577–582 (2002)CrossRefGoogle Scholar
  77. 77.
    Villani, C.: Optimal Transport. Old and New. Springer, Berlin (2009)CrossRefGoogle Scholar
  78. 78.
    Vogt, T., Lellmann, J.: An optimal transport-based restoration method for Q-ball imaging. In: 2017 Scale Space and Variational Methods in Computer Vision, pp. 271–282 (2017)Google Scholar
  79. 79.
    Weaver, N.: Lipschitz Algebras. World Scientific, Singapore (1999)CrossRefGoogle Scholar
  80. 80.
    Weinmann, A., Demaret, L., Storath, M.J.: Mumford–Shah and Potts regularization for manifold-valued data. J. Math. Imaging Vis. 55(3), 428–445 (2016)MathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Institute of Mathematics and Image Computing (MIC)University of LübeckLübeckGermany

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