Electroencephalography (EEG) source imaging aims to reconstruct brain activity maps from the neuroelectric potential difference measured on the skull. To obtain the brain activity map, we need to solve an ill-posed and ill-conditioned inverse problem that requires regularization techniques to make the solution viable. When dealing with real-time applications, dimensionality reduction techniques can be used to reduce the computational load required to evaluate the numerical solution of the EEG inverse problem. To this end, in this paper we use the random dipole sampling method, in which a Monte Carlo technique is used to reduce the number of neural sources. This is equivalent to reducing the number of the unknowns in the inverse problem and can be seen as a first regularization step. Then, we solve the reduced EEG inverse problem with two popular inversion methods, the weighted Minimum Norm Estimate (wMNE) and the standardized LOw Resolution brain Electromagnetic TomogrAphy (sLORETA). The main result of this paper is the error estimates of the reconstructed activity map obtained with the randomized version of wMNE and sLORETA. Numerical experiments on synthetic EEG data demonstrate the effectiveness of the random dipole sampling method.

中文翻译：

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随机偶极子采样求解脑电图反问题

脑电图 (EEG) 源成像旨在根据头骨上测量的神经电位差重建大脑活动图。为了获得大脑活动图，我们需要解决不适定且病态的逆问题，需要正则化技术才能使解决方案可行。在处理实时应用时，可以使用降维技术来减少评估脑电图反问题的数值解所需的计算量。为此，在本文中，我们使用随机偶极子采样方法，其中使用蒙特卡罗技术来减少神经源的数量。这相当于减少反问题中未知数的数量，可以看作第一个正则化步骤。然后，我们用两种流行的反演方法解决简化脑电图反演问题，即加权最小范数估计（wMNE）和标准化低分辨率脑电磁断层扫描（sLORETA）。本文的主要结果是使用 wMNE 和 sLORETA 的随机版本获得的重建活动图的误差估计。合成脑电图数据的数值实验证明了随机偶极子采样方法的有效性。