Image restoration and reconstruction from blurry and noisy observation is known to be ill-posed. To stabilize the recovery, total variation (TV) regularization was introduced by Rudin, Osher and Fatemi in \cite{LIR92}, which has demonstrated superiority in preserving image edges. However, the nondifferentiability of TV makes the underlying optimization problems difficult to solve. In this paper, we propose to solve TV deconvolution problems by alternating direction method (ADM) — a variant of the classic augmented Lagrangian method for structured optimization. The main idea of our approach is to reformulate a TV problem as a linear equality constrained problem where the objective function is separable, and then minimize its augmented Lagrangian function using a Gauss-Seidel updates of both primal and dual variables. This ADM approach can be applied to both single- and multi-channel images with either Gaussian or impulsive noise, and permit cross-channel blurs when the underlying image has more than one channel. The per-iteration computational complexity of the algorithm is dominated by several fast Fourier transforms. We present extensive experimental results concerning different blurs and noise to compare with FTVd \cite{YJZ08,NW,JFYZ} — a state-of-the-art algorithm for TV image reconstruction, and the results indicate that the ADM approach is more stable and efficient.
Citation
TR0918, Department of Mathmatics, Nanjing University, August, 2009