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

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EMBEC & NBC 2017 (EMBEC 2017, NBC 2017)

Part of the book series: IFMBE Proceedings ((IFMBE,volume 65))

Abstract

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.

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References

  1. Baillet S., Mosher J. C., Leahy R. M. Electromagnetic brain mapping IEEE Signal Processing Magazine. 2001;18:14–30

    Google Scholar 

  2. Hämäläinen M. S., Ilmoniemi R. J. Interpreting magnetic fields of the brain: minimum norm estimates Med. Biol. Eng. Comput. 1994;32:35–42

    Google Scholar 

  3. Fuchs M., Wagner M., Wischmann H.-A. Linear and Nonlinear Current Density Reconstructions Journal of Clinical Neurophysiology. 1999;16:267–295

    Google Scholar 

  4. Burger M., Dirks H., Müller J. Inverse Problems in Imaging in Large Scale Inverse Problems (M. Cullen et al., ed.) De Gruyte 2013

    Google Scholar 

  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 1996

    Google Scholar 

  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;18:49–65

    Google Scholar 

  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 2000

    Google Scholar 

  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;52:1783–1800

    Google Scholar 

  9. Lin F.-H., Belliveau J. W., Dale A. M., Hämäläinen M. S. Distributed current estimates using cortical orientation constraints Human Brain Mapping. 2006;27:1–13

    Google Scholar 

  10. Pascual-Marqui R. D. Standardized low resolution brain electromagnetic tomography (sLORETA): technical report Methods Find. Exp. Clin. Pharmacol.. 2002;24 Suppl, D:5-12

    Google Scholar 

  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;42:726–738

    Google Scholar 

  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;61:1364–1382

    Google Scholar 

  13. Kaipio J. P., Somersalo E. Statistical and Computational Inverse Problems;160 of Applied Mathematical Series. Springer 2005

    Google Scholar 

  14. Koulouri A. Reconstruction of Bio-electric fields and Source Distributions in EEG Brain Imaging. Imperial College London 2015

    Google Scholar 

  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. 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;100:590–607

    Google Scholar 

  17. Boyd S. P., Vandenberghe L. Convex Optimization. Cambridge University Press 2004

    Google Scholar 

  18. Yin W., Osher S., Goldfarb D., Darbon J. Bregman iterative algorithms for l1-minimization with applications to compressed sensing SIAM J. Imaging Sci. 2008:143–168

    Google Scholar 

  19. M. Fuchs, M. Wagner, A. Wischmann H. Generalized minimum norm least squares reconstruction algorithms in ISBET Newsletter,(ISSN 0947-5133);5:8–11 1994

    Google Scholar 

  20. Rubner Y., Tomasi C., Guibas L.J. The Earth Mover’s Distance as a Metric for Image Retrieval IJCV. 2000;40:99–121

    Google Scholar 

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Author information

Authors and Affiliations

  1. Institute for Computational and Applied Mathematics, University of Münster, Münster, Germany

    A. Koulouri

  2. Institute for Biomagnetism and Biosignalanalysis, University of Münster, Münster, Germany

    V. Rimpiläinen

  3. Department of Electrical and Electronic Engineering, Imperial College London, London, UK

    M. Brookes

  4. Department of Mathematics, University of Auckland, Auckland, New Zealand

    J. P. Kaipio

Authors
  1. A. Koulouri
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  2. V. Rimpiläinen
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  3. M. Brookes
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  4. J. P. Kaipio
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Corresponding author

Correspondence to A. Koulouri .

Editor information

Editors and Affiliations

  1. Tampere University of Technology, BioMediTech Institute and Faculty of Biomedical Science and Engineering, Tampere, Finland

    Hannu Eskola

  2. Tampere University of Technology, BioMediTech Institute and Faculty of Biomedical Science and Engineering, Tampere, Etelä-Suomi, Finland

    Outi Väisänen

  3. Tampere University of Technology, BioMediTech Institute and Faculty of Biomedical Science and Engineering, Tampere, Etelä-Suomi, Finland

    Jari Viik

  4. Tampere University of Technology, BioMediTech Institute and Faculty of Biomedical Science and Engineering, Tampere, Etelä-Suomi, Finland

    Jari Hyttinen

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© 2018 Springer Nature Singapore Pte Ltd.

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Koulouri, A., Rimpiläinen, V., Brookes, M., Kaipio, J.P. (2018). Prior Variances and Depth Un-Biased Estimators in EEG Focal Source Imaging. In: Eskola, H., Väisänen, O., Viik, J., Hyttinen, J. (eds) EMBEC & NBC 2017. EMBEC NBC 2017 2017. IFMBE Proceedings, vol 65. Springer, Singapore. https://doi.org/10.1007/978-981-10-5122-7_9

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  • DOI: https://doi.org/10.1007/978-981-10-5122-7_9

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  • Publisher Name: Springer, Singapore

  • Print ISBN: 978-981-10-5121-0

  • Online ISBN: 978-981-10-5122-7

  • eBook Packages: EngineeringEngineering (R0)

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