Prior Variances and Depth Un-Biased Estimators in EEG Focal Source Imaging

  • A. KoulouriEmail author
  • V. Rimpiläinen
  • M. Brookes
  • J. P. Kaipio
Conference paper
Part of the IFMBE Proceedings book series (IFMBE, volume 65)


In electroencephalography (EEG) source imaging, the inverse source estimates are depth biased in such a way that their maxima are often close to the sensors. This depth bias can be quantified by inspecting the statistics (mean and covariance) of these estimates. In this paper, we find weighting factors within a Bayesian framework for the used \(\ell _1/\ell _2\) sparsity prior that the resulting maximum a posterior (MAP) estimates do not favour any particular source location. Due to the lack of an analytical expression for the MAP estimate when this sparsity prior is used, we solve the weights indirectly. First, we calculate the Gaussian prior variances that lead to depth un-biased maximum a posterior (MAP) estimates. Subsequently, we approximate the corresponding weight factors in the sparsity prior based on the solved Gaussian prior variances. Finally, we reconstruct focal source configurations using the sparsity prior with the proposed weights and two other commonly used choices of weights that can be found in literature.


Electroencephalography sparsity prior Gaussian prior Bayesian inverse problems depth bias 


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  1. 1.
    Baillet S., Mosher J. C., Leahy R. M. Electromagnetic brain mapping IEEE Signal Processing Magazine. 2001;18:14–30Google Scholar
  2. 2.
    Hämäläinen M. S., Ilmoniemi R. J. Interpreting magnetic fields of the brain: minimum norm estimates Med. Biol. Eng. Comput. 1994;32:35–42Google Scholar
  3. 3.
    Fuchs M., Wagner M., Wischmann H.-A. Linear and Nonlinear Current Density Reconstructions Journal of Clinical Neurophysiology. 1999;16:267–295Google Scholar
  4. 4.
    Burger M., Dirks H., Müller J. Inverse Problems in Imaging in Large Scale Inverse Problems (M. Cullen et al., ed.) De Gruyte 2013Google Scholar
  5. 5.
    T. Köhler et. al. Depth normalization in MEG/EEG current density imaging in Proc. 18th Ann. Int. Conf. IEEE Eng. Med. Biol. Soc;2:812–813 1996Google Scholar
  6. 6.
    Pascual-Marqui R. D., Michel C. M., Lehmann D. Low resolution electromagnetic tomography: a new method for localizing electrical activity in the brain Int. J. Psychophysiol. 1994;18:49–65Google Scholar
  7. 7.
    M. Wagner et. al. Current Density Reconstructions Using the L1 Norm in Biomag 96: Vol. 1/Vol. 2, Proc. Biomagnetism (C. J. Aine et. al., ed.):393–396Springer 2000Google Scholar
  8. 8.
    Palmero-Soler E., Dolan K., Hadamschek V., Tass P.A. swLORETA: a novel approach to robust source localization and synchronization tomography Phys. Med. Biol. 2007;52:1783–1800Google Scholar
  9. 9.
    Lin F.-H., Belliveau J. W., Dale A. M., Hämäläinen M. S. Distributed current estimates using cortical orientation constraints Human Brain Mapping. 2006;27:1–13Google Scholar
  10. 10.
    Pascual-Marqui R. D. Standardized low resolution brain electromagnetic tomography (sLORETA): technical report Methods Find. Exp. Clin. Pharmacol.. 2002;24 Suppl, D:5-12Google Scholar
  11. 11.
    Haufe S., Nikulin V. V., Ziehe A., Müller K.-R., Nolte G. Combining sparsity and rotational invariance in EEG/MEG source reconstruction. NeuroImage. 2008;42:726–738Google Scholar
  12. 12.
    Lucka F., Pursiainen S., Burger M., Wolters C. H. Hierarchical Bayesian inference for the EEG inverse problem using realistic FE head models: Depth localization and source separation for focal primary currents NeuroImage. 2012;61:1364–1382Google Scholar
  13. 13.
    Kaipio J. P., Somersalo E. Statistical and Computational Inverse Problems;160 of Applied Mathematical Series. Springer 2005Google Scholar
  14. 14.
    Koulouri A. Reconstruction of Bio-electric fields and Source Distributions in EEG Brain Imaging. Imperial College London 2015Google Scholar
  15. 15.
    Pascual-Marqui R. D.. Discrete, 3D distributed, linear imaging methods of electric neuronal activity. Part 1: exact, zero error localization [math-ph]arXiv:0710.3341 [math-ph]. 2007
  16. 16.
    Vorwerk J., Cho J.-H., Rampp S., Hamer H., Knösche T. R., Wolters C. H. A guideline for head volume conductor modeling in EEG and MEG NeuroImage. 2014;100:590–607Google Scholar
  17. 17.
    Boyd S. P., Vandenberghe L. Convex Optimization. Cambridge University Press 2004Google Scholar
  18. 18.
    Yin W., Osher S., Goldfarb D., Darbon J. Bregman iterative algorithms for l1-minimization with applications to compressed sensing SIAM J. Imaging Sci. 2008:143–168Google Scholar
  19. 19.
    M. Fuchs, M. Wagner, A. Wischmann H. Generalized minimum norm least squares reconstruction algorithms in ISBET Newsletter,(ISSN 0947-5133);5:8–11 1994Google Scholar
  20. 20.
    Rubner Y., Tomasi C., Guibas L.J. The Earth Mover’s Distance as a Metric for Image Retrieval IJCV. 2000;40:99–121Google Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  • A. Koulouri
    • 1
    Email author
  • V. Rimpiläinen
    • 2
  • M. Brookes
    • 3
  • J. P. Kaipio
    • 4
  1. 1.Institute for Computational and Applied Mathematics, University of MünsterMünsterGermany
  2. 2.Institute for Biomagnetism and Biosignalanalysis, University of MünsterMünsterGermany
  3. 3.Department of Electrical and Electronic Engineering, Imperial College LondonLondonUK
  4. 4.Department of Mathematics, University of AucklandAucklandNew Zealand

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