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The inverse source problem of the Helmholtz equation in an interior domain is investigated. We show the uniqueness and local stability, where the source consists of multiple point sources. An algebraic algorithm is proposed to identify the number, locations and intensities of the point sources from boundary measurements. Uniqueness and non-uniqueness results for some distributed sources are also established. The proposed method is verified numerically.

This paper proposes a method for reconstructing the positions, strengths, and number of point sources in a three-dimensional (3D) Poisson field from boundary measurements. Algebraic relations are obtained, based on multipole moments determined by the sources and data on the boundary of a domain. To solve for the source parameters with efficient use of data, we select the necessary number of equations from them in the following two ways: (1) the use of those starting from lower-degree multipole moments; and (2) the use of combined ones involving infinitely higher-degree multipole moments. We show that both methods are based on the projection of 3D sources onto a two-dimensional space: the xy-plane for the first one and the Riemann sphere which is set to contain the domain for the second one. We also show that they share the same fundamental equations which can be solved by a procedure proposed by El-Badia and Ha-Duong (2000 Inverse Problems 16 651–63). Numerical simulations show that projection onto the xy-plane is more appropriate for sources scattered in the middle of the domain, whereas projection onto the Riemann sphere is more appropriate for sources concentrated close to the boundary of the domain. We also give an appropriate method of measurement for the Riemann sphere projection.

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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