  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.
