Simultaneous Detection of Multiple Magnetic Dipole Sources

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This article proposes a new method to simultaneously estimate the locations and magnetic moments of multiple magnetic dipole sources without the prior knowledge of the number of dipoles in the 3-D detection region. By initializing a large number of dipole sources evenly spaced in the detection region as potential candidates for the true dipoles, we introduce an indicator parameter for each dipole candidate such that its Sigmoid function is the probability that the candidate converges to a true dipole. A joint optimization is then formulated to minimize the mean square of the regularized error between the measured magnetic gradients and the calculated gradients from the estimated dipoles. The proposed nonlinear optimization is solved by the Levenberg–Marquardt algorithm, yielding the indicators and their corresponding dipole locations and magnetic moments. The implementation details are also provided, such as using multiple initialization schemes to avoid local minima, selection of measurement points and candidate locations to avoid the “high-wall effect,” and the need for preprocessing measurement data to avoid interference. Extensive simulations are conducted to investigate the effects of parameters, noise, and interference on the detection performance, and the results show that the proposed algorithm is robust in different scenarios as long as the total number of measurements is larger than the total number of unknowns in the optimization problem. When the false alarm rate is set at $5\times 10^{-2}$ , the proposed algorithm achieves $Recall$ of 0.91, 0.86, and 0.78 for the number of true dipoles being $N=2,4$ , and 6, respectively. The performance is robust against external interference and parameter selections.