Fast Optimal Transport Averaging of Neuroimaging Data

  • A. GramfortEmail author
  • G. Peyré
  • M. Cuturi
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9123)


Knowing how the Human brain is anatomically and functionally organized at the level of a group of healthy individuals or patients is the primary goal of neuroimaging research. Yet computing an average of brain imaging data defined over a voxel grid or a triangulation remains a challenge. Data are large, the geometry of the brain is complex and the between subjects variability leads to spatially or temporally non-overlapping effects of interest. To address the problem of variability, data are commonly smoothed before performing a linear group averaging. In this work we build on ideas originally introduced by Kantorovich [18] to propose a new algorithm that can average efficiently non-normalized data defined over arbitrary discrete domains using transportation metrics. We show how Kantorovich means can be linked to Wasserstein barycenters in order to take advantage of the entropic smoothing approach used by [7]. It leads to a smooth convex optimization problem and an algorithm with strong convergence guarantees. We illustrate the versatility of this tool and its empirical behavior on functional neuroimaging data, functional MRI and magnetoencephalography (MEG) source estimates, defined on voxel grids and triangulations of the folded cortical surface.


fMRI Data Optimal Transport Gaussian Smoothing Nonnegative Measure Voxel Grid 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



A. Gramfort was supported by the ANR grant THALAMEEG, ANR-14-NEUC-0002-01. M. Cuturi gratefully acknowledges the support of JSPS young researcher A grant 26700002, the gift of a K40 card from NVIDIA and fruitful discussions with K.R. Müller. The work of G. Peyré has been supported by the European Research Council (ERC project SIGMA-Vision).


  1. 1.
    Agueh, M., Carlier, G.: Barycenters in the Wasserstein space. SIAM J. Math. Anal. 43(2), 904–924 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Benamou, J.D.: Numerical resolution of an unbalanced mass transport problem. ESAIM. Math. Model. Numer. Anal. 37(5), 851–868 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Benamou, J.D., Carlier, G., Cuturi, M., Nenna, L., Peyré, G.: Iterative bregman projections for regularized transportation problems. arXiv preprint arXiv:1412.5154 (2014)
  4. 4.
    Bertsimas, D., Tsitsiklis, J.: Introduction to linear optimization. Athena Scientific Belmont, Boston (1997)Google Scholar
  5. 5.
    Bonneel, N., Rabin, J., Peyré, G., Pfister, H.: Sliced and radon wasserstein barycenters of measures. J. Math. Imaging Vis. 51(1), 1–24 (2014)Google Scholar
  6. 6.
    Cuturi, M.: Sinkhorn distances: lightspeed computation of optimal transport. Adv. Neural Inf. Process. Sys. 26, 2292–2300 (2013)Google Scholar
  7. 7.
    Cuturi, M., Doucet, A.: Fast computation of wasserstein barycenters. In: Proceedings of the 31st International Conference on Machine Learning (ICML-14) (2014)Google Scholar
  8. 8.
    Cuturi, M., Peyré, G., Rolet, A.: A smoothed dual approach for variational wasserstein problems. arXiv preprint arXiv:1503.02533 (2015)
  9. 9.
    Dale, A., Liu, A., Fischl, B., Buckner, R.: Dynamic statistical parametric neurotechnique mapping: combining fMRI and MEG for high-resolution imaging of cortical activity. Neuron 26, 55–67 (2000)CrossRefGoogle Scholar
  10. 10.
    Descoteaux, M., Deriche, R., Knosche, T., Anwander, A.: Deterministic and probabilistic tractography based on complex fibre orientation distributions. IEEE Trans. Med. Imaging 28(2), 269–286 (2009)CrossRefGoogle Scholar
  11. 11.
    Durrleman, S., Prastawa, M., Charon, N., Korenberg, J.R., Joshi, S., Gerig, G., Trouvé, A.: Morphometry of anatomical shape complexes with dense deformations and sparse parameters. NeuroImage 101, 35–49 (2014)CrossRefGoogle Scholar
  12. 12.
    Gramfort, A., Luessi, M., Larson, E., Engemann, D., Strohmeier, D., Brodbeck, C., Parkkonen, L., Hämäläinen, M.: MNE software for processing MEG and EEG data. NeuroImage 86, 446–460 (2014)CrossRefGoogle Scholar
  13. 13.
    Gramfort, A., Strohmeier, D., Haueisen, J., Hämäläinen, M., Kowalski, M.: Time-frequency mixed-norm estimates: Sparse M/EEG imaging with non-stationary source activations. NeuroImage 70, 410–422 (2013)CrossRefGoogle Scholar
  14. 14.
    Guittet, K.: Extended kantorovich norms: a tool for optimization. Technical repot 4402, INRIA (2002)Google Scholar
  15. 15.
    Hanin, L.: An extension of the kantorovich norm. Contemp. Math 226, 113–130 (1999)MathSciNetGoogle Scholar
  16. 16.
    Henson, R.N., Wakeman, D.G., Litvak, V., Friston, K.J.: A parametric empirical bayesian framework for the EEG/MEG inverse problem: generative models for multisubject and multimodal integration. Front. Hum. Neuro. 5(76), 141–153 (2011)Google Scholar
  17. 17.
    Joshi, S., Davis, B., Jomier, M., Gerig, G.: Unbiased diffeomorphic atlas construction for computational anatomy. NeuroImage 23, 151–160 (2004)CrossRefGoogle Scholar
  18. 18.
    Kantorovich, L., Rubinshtein, G.: On a space of totally additive functions, vestn. Vestn Lening. Univ. 13, 52–59 (1958)zbMATHGoogle Scholar
  19. 19.
    Kanwisher, N., Mcdermott, J., Chun, M.M.: The fusiform face area: a module in human extrastriate cortex specialized for face perception. J. Neurosci. 17, 4302–4311 (1997)Google Scholar
  20. 20.
    Pele, O., Werman, M.: A linear time histogram metric for improved SIFT matching. In: Forsyth, D., Torr, P., Zisserman, A. (eds.) ECCV 2008, Part III. LNCS, vol. 5304, pp. 495–508. Springer, Heidelberg (2008) CrossRefGoogle Scholar
  21. 21.
    Pinel, P., Thirion, B., Meriaux, S., Jobert, A., Serres, J., Le Bihan, D., Poline, J., Dehaene, S.: Fast reproducible identification and large-scale databasing of individual functional cognitive networks. BMC neuroscience 8, 91 (2007)CrossRefGoogle Scholar
  22. 22.
    Rabin, J., Peyré, G., Delon, J., Bernot, M.: Wasserstein barycenter and its application to texture mixing. In: Bruckstein, A.M., ter Haar Romeny, B.M., Bronstein, A.M., Bronstein, M.M. (eds.) SSVM 2011. LNCS, vol. 6667, pp. 435–446. Springer, Heidelberg (2012) CrossRefGoogle Scholar
  23. 23.
    Rubner, Y., Guibas, L., Tomasi, C.: The earth movers distance, multi-dimensional scaling, and color-based image retrieval. In: Proceedings of the ARPA Image Understanding Workshop, pp. 661–668 (1997)Google Scholar
  24. 24.
    Scherg, M., Von Cramon, D.: Two bilateral sources of the late AEP as identified by a spatio-temporal dipole model. Electroencephalogr. Clin. Neurophysiol. 62(1), 32–44 (1985)CrossRefGoogle Scholar
  25. 25.
    Thirion, B., Pinel, P., Mériaux, S., Roche, A., Dehaene, S., Poline, J.B.: Analysis of a large fMRI cohort: statistical and methodological issues for group analyses. NeuroImage 35(1), 105–120 (2007)CrossRefGoogle Scholar
  26. 26.
    Villani, C.: Optimal transport: Old and New. Springer, Heidelberg (2009) CrossRefGoogle Scholar
  27. 27.
    Wipf, D., Ramirez, R., Palmer, J., Makeig, S., Rao, B.: Analysis of empirical bayesian methods for neuroelectromagnetic source localization. In: Proceedings of the Neural Information Processing Systems (NIPS) (2007)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Institut Mines-Télécom, Telecom ParisTechCNRS LTCIParisFrance
  2. 2.NeuroSpinCEA SaclayGif-sur-YvetteCedex France
  3. 3.CNRS and CEREMADEUniversité Paris-DauphineParisFrance
  4. 4.Graduate School of InformaticsKyoto UniversityKyotoJapan

Personalised recommendations