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Total Variation (TV) is an effective and popular prior model in the field ofregularization-based image processing. This paper focuses on total variationfor removing impulse noise in image restoration. This type of noise frequentlyarises in data acquisition and transmission due to many reasons, e.g. a faultysensor or analog-to-digital converter errors. Removing this noise is animportant task in image restoration. State-of-the-art methods such as AdaptiveOutlier Pursuit(AOP) \cite{yan2013restoration}, which is based on TV with$\ell_{02}$-norm data fidelity, only give sub-optimal performance. In thispaper, we propose a new sparse optimization method, called $\ell_0TV$-PADMM,which solves the TV-based restoration problem with $\ell_0$-norm data fidelity.To effectively deal with the resulting non-convex non-smooth optimizationproblem, we first reformulate it as an equivalent biconvex Mathematical Programwith Equilibrium Constraints (MPEC), and then solve it using a proximalAlternating Direction Method of Multipliers (PADMM). Our $\ell_0TV$-PADMMmethod finds a desirable solution to the original $\ell_0$-norm optimizationproblem and is proven to be convergent under mild conditions. We apply$\ell_0TV$-PADMM to the problems of image denoising and deblurring in thepresence of impulse noise. Our extensive experiments demonstrate that$\ell_0TV$-PADMM outperforms state-of-the-art image restoration methods.

L0TV: A Sparse Optimization Method for Impulse Noise Image Restoration