Accelerated Proximal Iterative re-Weighted ${\mathrm{\ell }}_1$ Alternating Minimization for Image Deblurring

The quadratic penalty alternating minimization (AM) method is widely used for solving the convex ${\mathrm{\ell }}_1$ total variation (TV) image deblurring problem. However, quadratic penalty AM for solving the nonconvex nonsmooth ${\mathrm{\ell }}_p$, $0<p<1$ TV image deblurring problems is less studied. In this paper, we propose two algorithms, namely proximal iterative re-weighted ${\mathrm{\ell }}_1$ AM (PIRL1-AM) and its accelerated version, accelerated proximal iterative re-weighted ${\mathrm{\ell }}_1$ AM (APIRL1-AM) for solving the nonconvex nonsmooth ${\mathrm{\ell }}_p$ TV image deblurring problem. The proposed algorithms are derived from the proximal iterative re-weighted ${\mathrm{\ell }}_1$ (IRL1) algorithm and the proximal gradient algorithm. Numerical results show that PIRL1-AM is effective in retaining sharp edges in image deblurring while APIRL1-AM can further provide convergence speed up in terms of the number of algorithm iterations and computational time.