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.

中文翻译：

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同时检测多个磁偶极子源

本文提出了一种新方法，可以在不了解 3-D 检测区域中偶极子数量的先验知识的情况下同时估计多个磁偶极子源的位置和磁矩。通过将检测区域中均匀分布的大量偶极子源初始化为真正偶极子的潜在候选者，我们为每个偶极子候选者引入了一个指示参数，使得其 Sigmoid 函数是候选者收敛到真正偶极子的概率。然后制定联合优化以最小化测量的磁梯度和估计的偶极子的计算梯度之间的正则化误差的均方。提出的非线性优化由 Levenberg-Marquardt 算法求解，产生指示器及其相应的偶极子位置和磁矩。还提供了实现细节，例如使用多种初始化方案来避免局部最小值，选择测量点和候选位置以避免“高墙效应”，以及需要对测量数据进行预处理以避免干扰。大量的仿真研究了参数、噪声和干扰对检测性能的影响，结果表明，只要测量的总数大于未知数的总数，该算法在不同场景下都是鲁棒的。优化问题。当误报率设置为 选择测量点和候选位置以避免“高墙效应”，并且需要对测量数据进行预处理以避免干扰。大量的仿真研究了参数、噪声和干扰对检测性能的影响，结果表明，只要测量的总数大于未知数的总数，该算法在不同场景下都是鲁棒的。优化问题。当误报率设置为 选择测量点和候选位置以避免“高墙效应”，并且需要对测量数据进行预处理以避免干扰。大量的仿真研究了参数、噪声和干扰对检测性能的影响，结果表明，只要测量的总数大于未知数的总数，该算法在不同场景下都是鲁棒的。优化问题。当误报率设置为 结果表明，只要测量的总数大于优化问题中未知数的总数，该算法在不同场景下都是鲁棒的。当误报率设置为 结果表明，只要测量的总数大于优化问题中未知数的总数，该算法在不同场景下都是鲁棒的。当误报率设置为 $5\乘以 10^{-2}$ ，所提出的算法实现 $召回$ 0.91、0.86 和 0.78 的真偶极数为 $N=2,4$ , 和 6，分别。性能强大，可抵抗外部干扰和参数选择。