Electroencephalogram (EEG) signals reflect brain activity across different brain states, characterized by distinct frequency distributions. Through multifractal analysis tools, we investigate the scaling behaviour of different classes of EEG signals and artifacts. We show that brain states associated to sleep and general anaesthesia are not in general characterized by scale invariance. The lack of scale invariance motivates the development of artifact removal algorithms capable of operating independently at each scale. We examine here the properties of the wavelet quantile normalization algorithm, a recently introduced adaptive method for real-time correction of transient artifacts in EEG signals. We establish general results regarding the regularization properties of the WQN algorithm, showing how it can eliminate singularities introduced by artefacts, and we compare it to traditional thresholding algorithms. Furthermore, we show that the algorithm performance is independent of the wavelet basis. We finally examine its continuity and boundedness properties and illustrate its distinctive non-local action on the wavelet coefficients through pathological examples.

翻译：脑电图（EEG）信号通过不同的频率分布特征反映不同脑状态下的神经活动。我们利用多重分形分析工具，研究了不同类别脑电信号及伪迹的标度行为。研究表明，与睡眠及全身麻醉相关的脑状态通常不具备尺度不变性特征。这种尺度不变性的缺失促使我们开发能够在各尺度上独立运行的伪迹去除算法。本文深入研究了小波分位数归一化算法——一种近年来提出的用于实时校正脑电信号中瞬态伪迹的自适应方法。我们建立了关于WQN算法正则化特性的普适性结论，阐明了其消除伪迹引入奇异性特征的机制，并与传统阈值算法进行了对比。进一步研究表明，该算法的性能与小波基的选择无关。最后，我们分析了算法的连续性与有界性特征，并通过典型病理案例展示了其对小波系数独特的非局域作用效果。