Sparse signal reconstruction of compressively sampled signals using smoothed ℓ0-norm

Compressed Sensing is a novel sampling technique that can be used to faithfully recover sparse signals from fewer measurements than those proposed by the Nyquist theorem. A simple and intuitive measure of sparsity in a signal is ℓ0-norm. However, the ℓ0-norm function does not satisfy all the axiomatic properties of a true mathematical norm. The discrete and discontinuous nature of ℓ0-norm poses many challenges in its applications to recover sparse signals from their subsampled measurements. This paper presents, a novel mathematical function that can be used to closely approximate the ℓ0-norm. The proposed function is smooth and differentiable that allows gradient based algorithms to be used in the reconstruction of sparse signals. We use the proposed approximation along with steepest ascent method to develop a complete sparse signal recovery algorithm for the compressed sensing framework. Experimental results have shown that the proposed recovery algorithm outperforms the conventional SL0 method in terms of reconstruction accuracy such as Mean Square Error (MSE) and Signal-to-Noise Ratio (SNR).