Recovery of Undersampled Sparse Signals using Combined Smoothed $l_{0}-l_{1}$ Norm

Abstract

Compressed Sensing is a modern sampling technique that caught the attention of the scientific community due to its diverse applications. It accurately recovers sparse signals/images from fewer samples that are well below the Nyquist Criterion. The linear recovery of under-sampled sparse signals/images is not possible using $l_{2}$ -norm minimization. The $l_{2}$ norm based least squares linear recovery minimizes the total energy of the error resulting in a non-sparse solution. It is a proven fact that non-linear $l_{1}$ -norm minimization promotes sparsity in the solution, which is desired in sparse signal recovery, however, it increases the computational complexity of the algorithm. The principal method for sparse signals reconstruction is based on $l_{0}$ -norm minimization that looks for a feasible solution having a minimum number of non-zeros and finding the solution to the $l_{0}$ -norm minimization is a combinatorial problem. There are non-linear reconstruction techniques based on smoothed $l_{0}$ and $l_{1}$ norm approximations are proposed in the literature that are proven to be computationally efficient. In this research paper, the novel technique for recovery of a sparse signal is proposed, where $l_{0}$ and $l_{1}$ norm are combined to accurately recover compressively sampled sparse signals. The experimental results have shown that the proposed technique is more efficient than traditional smooth approximations of $l_{0}$ and $l_{1}$ norm.