Robust Formulation for Solving Underdetermined Random Linear System of Equations Via Admm

Compressive sensing (CS) is a well-known theory which allowshighly efficient data acquisition schemes and has received much attention in diverse applications such as medical imaging, and hyperspectral imaging. Many algorithms have been proposed in the literature for solving the ill-posed inverse problem present in CS based on the sparsity of the signal of interest. Traditionally, these algorithms minimize a cost function that is a combination between an ℓ1 and ℓp norms with 1 < p < ∞. Specifically, the non-smooth ℓ1-norm has been used as a convex relaxation of the ℓ0 pseudo-norm to promote sparsity, and the ℓp-norm has been commonly used as the data fidelity term. However, in many applications, the measurements can be very noisy, and using an ℓp-norm as a data fit term becomes useless to reduce the effect of the noise in order to obtain the desired signal. Therefore, this paper proposes a new algorithm that minimizes a combination of an ℓ1 and ℓ∞ norms to solve the inverse problem present in CS. In this case, the ℓ∞ works as the data fidelity term which leads to a more robust algorithm against the noise. The proposed method requires less number of iterations to solve the CS inverse problem compared with the state-of-the-art-algorithms.