An Extended Reweighted <inline-formula><math display="inline"><semantics><mrow><msub><mo>ℓ</mo><mn>1</mn></msub></mrow></semantics></math></inline-formula> Minimization Algorithm for Image Restoration

Abstract

This paper proposes an effective extended reweighted ℓ1 minimization algorithm (ERMA) to solve the basis pursuit problem minu∈Rn{||u||1:Au=f} in compressed sensing, where A∈Rm×n, m≪n. The fast algorithm is based on linearized Bregman iteration with soft thresholding operator and generalized inverse iteration. At the same time, it also combines the iterative reweighted strategy that is used to solve minu∈Rn{||u||pp:Au=f} problem, with the weight ωi(u,p)=(ε+|ui|2)p/2−1. Numerical experiments show that this ℓ1 minimization persistently performs better than other methods. Especially when p=0, the restored signal by the algorithm has the highest signal to noise ratio. Additionally, this approach has no effect on workload or calculation time when matrix A is ill-conditioned.

Keywords

• extended reweighted <i>ℓ</i>₁ minimization
• fast algorithm
• image restoration
• iterative generalized inverse