Impulsive noise rejection method for compressed measurement signal in compressed sensing

The Lorentzian norm of robust statistics is often applied in the reconstruction of the sparse signal from a compressed measurement signal in an impulsive noise environment. The optimization of the robust statistic function is iterative and usually requires complex parameter adjustments. In this article, the impulsive noise rejection for the compressed measurement signal with the design for image reconstruction is proposed. It is used as the preprocessing for any compressed sensing reconstruction given that the sparsified version of the signal is obtained by utilizing octave-tree discrete wavelet transform with db8 as the mother wavelet. The presence of impulsive noise is detected from the energy distribution of the reconstructed sparse signal. After the noise removal, the noise corrupted coefficients are estimated. The proposed method requires neither complex optimization nor complex parameter adjustments. The performance of the proposed method was evaluated on 60 images. The experimental results indicated that the proposed method effectively rejected the impulsive noise. Furthermore, at the same impulsive noise corruption level, the reconstruction with the proposed method as the preprocessing required much lower measurement rate than the model-based Lorentzian iterative hard thresholding.