This paper presents a novel algorithm to reconstruct parameters of a sufficient number of current dipoles that describe data (equivalent current dipoles, ECDs, hereafter) from radial/vector magnetoencephalography (MEG) with and without electroencephalography (EEG). We assume a three-compartment head model and arbitrary surfaces on which the MEG sensors and EEG electrodes are placed. Via the multipole expansion of the magnetic field, we obtain algebraic equations relating the dipole parameters to the vector MEG/EEG data. By solving them directly, without providing initial parameter guesses and computing forward solutions iteratively, the dipole positions and moments projected onto the xy-plane (equatorial plane) are reconstructed from a single time shot of the data. In addition, when the head layers and the sensor surfaces are spherically symmetric, we show that the required data reduce to radial MEG only. This clarifies the advantage of vector MEG/EEG measurements and algorithms for a generally-shaped head and sensor surfaces. In the numerical simulations, the centroids of the patch sources are well localized using vector/radial MEG measured on the upper hemisphere. By assuming the model order to be larger than the actual dipole number, the resultant spurious dipole is shown to have a much smaller strength magnetic moment (about 0.05 times smaller when the SNR = 16 dB), so that the number of ECDs is reasonably estimated. We consider that our direct method with greatly reduced computational cost can also be used to provide a good initial guess for conventional dipolar/multipolar fitting algorithms.

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Magnetoencephalography (MEG) is a measurement technique of the magnetic field outside the human brain, from which neural activities inside the brain are estimated. So far, we have proposed a direct method where the locations and moments of the equivalent current dipoles modeling the synchronous and focal neural activities are explicitly represented by the magnetic field on the closed surface which encloses the human brain. In this paper, we extend it to the case where the data is restricted on the partial boundary such as an upper hemisphere in the practical MEG. We use the super-resolution technique, which has been originally proposed for reconstruction of a function with a bounded support and missing spectral components. First, we estimate ECDs by the direct method using the data on the upper hemisphere only assuming that the data on the lower hemisphere which lacks the sensors to be zero. Second, using the estimated ECD parameters, we compute the magnetic field on the lower hemisphere. From the measured/computed data on the upper/lower hemisphere, the magnetic field on the closed surface is obtained. Then, we re-estimate ECDs using the direct method and repeat these processes. Convergence proof can be conducted using convex projections. Numerical experiments show that localization accuracy is sufficiently improved.