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An Optimal Transport-Based Restoration Method for Q-Ball Imaging

  • Thomas Vogt
  • Jan Lellmann
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10302)

Abstract

We propose a variational approach for edge-preserving total variation (TV)-based regularization of Q-ball data from high angular resolution diffusion imaging (HARDI). While total variation is among the most popular regularizers for variational problems, its application to orientation distribution functions (ODF), as they naturally arise in Q-ball imaging, is not straightforward. We propose to use an extension that specifically takes into account the metric on the underlying orientation space. The key idea is to write the difference quotients in the TV seminorm in terms of the Wasserstein statistical distance from optimal transport. We combine this regularizer with a matching Wasserstein data fidelity term. Using the Kantorovich-Rubinstein duality, the variational model can be formulated as a convex optimization problem that can be solved using a primal-dual algorithm. We demonstrate the effectiveness of the proposed framework on real and synthetic Q-ball data.

Keywords

Variational methods Total variation Q-ball imaging Wasserstein distance 

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 Biomedical Imaging, ISBI 2009, pp. 1398–1401 (2009)Google Scholar
  2. 2.
    Basser, P.J., Mattiello, J., LeBihan, D.: MR diffusion tensor spectroscopy and imaging. Biophys. J. 66(1), 259–267 (1994)CrossRefGoogle Scholar
  3. 3.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Chen, Y., Guo, W., Zeng, Q., Liu, Y.: A nonstandard smoothing in reconstruction of apparent diffusion coefficient profiles from diffusion weighted images. Inverse Prob. Imaging 2(2), 205–224 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cheng, J., Ghosh, A., Jiang, T., Deriche, R.: A Remannian framework for orientation distribution function computing. Med. Image Comput. Comput. Assist. Interv. 2009 12(1), 911–918 (2009)Google Scholar
  6. 6.
    Delputte, S., Dierckx, H., Fieremans, E., D’Asseler, Y., Achten, R., Lemahieu, I.: Postprocessing of brain white matter fiber orientation distribution functions. In: ISBI 2007, pp. 784–787 (2007)Google Scholar
  7. 7.
    Descoteaux, M., Angelino, E., Fitzgibbons, S., Deriche, R.: Apparent diffusion coefficients from high angular resolution diffusion imaging: estimation and applications. Magn. Reson. Med. 56(2), 395–410 (2006)CrossRefGoogle Scholar
  8. 8.
    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
  9. 9.
    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. Neuroinf. 8(8), 1–17 (2014)Google Scholar
  10. 10.
    Goh, A., Lenglet, C., Thompson, P., Vidal, R.: A nonparametric Riemannian framework for processing High Angular Resolution Diffusion Images (HARDI). In: CVPR 2009, pp. 2496–2503 (2009)Google Scholar
  11. 11.
    Goh, A., Lenglet, C., Thompson, P.M., Vidal, R.: A nonparametric Riemannian framework for processing high angular resolution diffusion images and its applications to ODF-based morphometry. Neuroimage 56(3), 1181–1201 (2011)CrossRefGoogle Scholar
  12. 12.
    Kantorovich, L.V., Rubinshten, G.Sh.: On a functional space and certain extremum problems. Dokl. Akad. Nauk SSSR 115, 1058–1061 (1957)Google Scholar
  13. 13.
    Kim, Y., Thompson, P.M., Vese, L.A.: HARDI data denoising using vectorial total variation and logarithmic barrier. Inverse Prob. Imaging 4(2), 273–310 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    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
  15. 15.
    Lellmann, J., Lorenz, D.A., Schönlieb, C., Valkonen, T.: Imaging with Kantorovich-Rubinstein discrepancy. SIAM J. Imaging Sci. 7(4), 2833–2859 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    McGraw, T., Vemuri, B., Ozarslan, E., Chen, Y., Mareci, T.: Variational denoising of diffusion weighted MRI. Inverse Prob. Imaging 3(4), 625–648 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Ncube, S., Srivastava, A.: A novel Riemannian metric for analyzing HARDI data. In: Proceedings of the SPIE 7962, Id. 79620Q (2011)Google Scholar
  18. 18.
    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
  19. 19.
    Reisert, M., Kellner, E., Kiselev, V.G.: About the geometry of asymmetric fiber orientation distributions. IEEE Trans. Med. Imaging 31(6), 1240–1249 (2012)CrossRefGoogle Scholar
  20. 20.
    Rokem, A., Yeatman, J., Pestilli, F., Wandell, B.: High angular resolution diffusion MRI. Stanford Digital Repository (2013). http://purl.stanford.edu/yx282xq2090
  21. 21.
    Srivastava, A., Jermyn, I.H., Joshi, S.H.: Riemannian analysis of probability density functions with applications in vision. In: CVPR 2007, pp. 1–8 (2007)Google Scholar
  22. 22.
    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
  23. 23.
    Tuch, D.S.: Q-ball imaging. Magn. Reson. Med. 52(6), 1358–1372 (2004)CrossRefGoogle Scholar
  24. 24.
    Villani, C.: Optimal Transport: Old and New. Grundlehren der mathematischen Wissenschaften, vol. 338. Springer, Berlin (2009)zbMATHGoogle Scholar
  25. 25.
    Weinmann, A., Demaret, L., Storath, M.J.: Mumford-Shah and potts regularization for manifold-valued data. J. Math. Imaging Vis. 55, 428 (2016)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

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

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