We consider image deblurring problem in the presence of impulsive noise. It is known that \emph{total variation} (TV) regularization with L1-norm penalized data fitting (TVL1 for short) works reasonably well only when the level of impulsive noise is relatively low. For high level impulsive noise, TVL1 works poorly. The reason is that all data, both corrupted and noise free, are equally penalized in data fitting, leading to insurmountable difficulty in balancing regularization and data fitting. In this paper, we propose to combine TV regularization with \emph{smoothly clipped absolute deviation} (SCAD) penalty for data fitting (TVSCAD for short). Our motivation is simply that data fitting should be enforced only when an observed data is not severely corrupted, while for those data more likely to be severely corrupted, less or even null penalization should be enforced. A \emph{difference of convex functions} algorithm is adopted to solve the nonconvex TVSCAD model, resulting in solving a sequence of TVL1-equivalent problems, each of which can then be solved efficiently by the alternating direction method of multipliers. Theoretically, we establish global convergence to a critical point of the nonconvex objective function. The R-linear and at-least-sublinear convergence rate results are derived for anisotropic and isotropic TV, respectively. Numerically, extensive experimental results are given to show that the TVSCAD model improves the TVL1 significantly, especially for cases with high level impulsive noise. In addition, the TVSCAD model is also compared with a recently proposed iteratively corrected TVL1 method to illustrate its promising performance.
Citation
Guoyong Gu, Suhong Jiang, and Junfeng Yang, A TVSCAD approach for image deblurring with impulsive noise, Research Report, 2017, Department of Mathematics, Nanjing University.