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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Baillet S., Mosher J. C., Leahy R. M. Electromagnetic brain mapping IEEE Signal Processing Magazine. 2001;18:14–30
Hämäläinen M. S., Ilmoniemi R. J. Interpreting magnetic fields of the brain: minimum norm estimates Med. Biol. Eng. Comput. 1994;32:35–42
Fuchs M., Wagner M., Wischmann H.-A. Linear and Nonlinear Current Density Reconstructions Journal of Clinical Neurophysiology. 1999;16:267–295
Burger M., Dirks H., Müller J. Inverse Problems in Imaging in Large Scale Inverse Problems (M. Cullen et al., ed.) De Gruyte 2013
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
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
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
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
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
Pascual-Marqui R. D. Standardized low resolution brain electromagnetic tomography (sLORETA): technical report Methods Find. Exp. Clin. Pharmacol.. 2002;24 Suppl, D:5-12
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
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
Kaipio J. P., Somersalo E. Statistical and Computational Inverse Problems;160 of Applied Mathematical Series. Springer 2005
Koulouri A. Reconstruction of Bio-electric fields and Source Distributions in EEG Brain Imaging. Imperial College London 2015
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
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
Boyd S. P., Vandenberghe L. Convex Optimization. Cambridge University Press 2004
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
M. Fuchs, M. Wagner, A. Wischmann H. Generalized minimum norm least squares reconstruction algorithms in ISBET Newsletter,(ISSN 0947-5133);5:8–11 1994
Rubner Y., Tomasi C., Guibas L.J. The Earth Mover’s Distance as a Metric for Image Retrieval IJCV. 2000;40:99–121
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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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