Sparse signal reconstruction based on multiparameter approximation function with smoothed ℓ₀ norm

The smoothed ℓ₀ norm algorithm is a reconstruction algorithm in compressive sensing based on approximate smoothed ℓ₀ norm. It introduces a sequence of smoothed functions to approximate the ℓ₀ norm and approaches the solution using the specific iteration process with the steepest method. In order to choose an appropriate sequence of smoothed function and solve the optimization problem effectively, we employ approximate hyperbolic tangent multiparameter function as the approximation to the big "steep nature" in ℓ₀ norm. Simultaneously, we propose an algorithm based on minimizing a reweighted approximate ℓ₀ norm in the null space of the measurement matrix. The unconstrained optimization involved is performed by using a modified quasi-Newton algorithm. The numerical simulation results show that the proposed algorithms yield improved signal reconstruction quality and performance.