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This paper concerns artifact removal from multichannel EEG data. It has already been demonstrated that independent component analysis (ICA) can be an effective and applicable method for EEG de-noising. The goal of this paper is to propose a framework, based on ICA and wavelet denoising (WD), to improve the pre-processing of EEG signals. In particular we employ the concept of spatially constrained ICA (SCICA) to extract artifact-only independent components (ICs) from the given EEG data, use WD to remove any cerebral activity from extracted artifacts, and finally project back the artifacts to be subtracted from EEG signals to get clean EEG data. The main advantage of the proposed approach is faster computation, as all ICs are not identified. The computer experiments are carried out, which demonstrate the effectiveness of the proposed approach in removing focal artifacts that can be well separated by SCICA.