ℓ0TV: A Sparse Optimization Method for Impulse Noise Image Restoration

Ganzhao Yuan, Bernard Ghanem

Abstract: Total Variation (TV) is an effective and popular prior model in the field of regularization-based image processing. This paper focuses on total variation for removing impulse noise in image restoration. This type of noise frequently arises in data acquisition and transmission due to many reasons, e.g., a faulty sensor or analog-to-digital converter errors. Removing this noise is an important task in image restoration. State-of-the-art methods such as Adaptive Outlier Pursuit(AOP) <xref ref-type="bibr" rid="ref1">[1]</xref> , which is based on TV with <inline-formula><tex-math notation="LaTeX">$\ell _{02}$</tex-math></inline-formula> -norm data fidelity, only give sub-optimal performance. In this paper, we propose a new sparse optimization method, called <inline-formula><tex-math notation="LaTeX">$\ell _0TV$</tex-math></inline-formula> -PADMM, which solves the TV-based restoration problem with <inline-formula><tex-math notation="LaTeX">$\ell _0$</tex-math></inline-formula> -norm data fidelity. To effectively deal with the resulting non-convex non-smooth optimization problem, we first reformulate it as an equivalent biconvex Mathematical Program with Equilibrium Constraints (MPEC), and then solve it using a proximal Alternating Direction Method of Multipliers (PADMM). Our <inline-formula><tex-math notation="LaTeX">$\ell _0TV$</tex-math></inline-formula> -PADMM method finds a desirable solution to the original <inline-formula><tex-math notation="LaTeX">$\ell _0$</tex-math></inline-formula> -norm optimization problem and is proven to be convergent under mild conditions. We apply <inline-formula><tex-math notation="LaTeX">$\ell _0TV$</tex-math></inline-formula> -PADMM to the problems of image denoising and deblurring in the presence of impulse noise. Our extensive experiments demonstrate that <inline-formula><tex-math notation="LaTeX">$\ell _0TV$</tex-math></inline-formula> -PADMM outperforms state-of-the-art image restoration methods.