Image restoration and decomposition using non-convex non-smooth regularisation and negative Hilbert–Sobolev norm

Abstract

A new model for image restoration and decomposition has been presented here. The proposed model applies the non-convex non-smooth regularisation and the Hilbert -Sobolev spaces of negative degree of differentiability to capture oscillatory patterns. The existence of a pseudosolution to the proposed model is proved. Moreover, two numerical algorithms for solving the minimisation problem are provided by applying the variable splitting and the penalty techniques. Finally, extended experiments on denoising, deblurring and decompositions of both real and synthetic images demonstrate the effectiveness and efficiency of the proposed numerical schemes.

TV+TV2Regularization with Nonconvex Sparseness-Inducing Penalty for Image Restoration

“…Another intriguing new application for the ℓ 1 sparseness-inducing penalty is compressive sensing (CS) [16][17][18] when A is the measurement or sampling operator in the optimization problem (3). In addition, it is worth mentioning the numerical results in [19,20] and theoretical works [21][22][23]. These results have justified that the robustness of restored images to noise and the nonsparsity can be increased by replacing ℓ 1 norm in (3) with the nonconvex nonsmooth ℓ (0 < < 1) quasinorm.…”

Parametric Blind Image Deblurring With Gradient Based Spectral Kurtosis Maximization

“…This is because unlike synthetic deblurring, real deblurring often suffers from noise and ringing effects. The blurred images may have been degraded by arbitrarily shaped PSFs that are complex and cannot be easily modelled (Gupta et al, 2010;Harmeling et al, 2011;Lu, 2012;Whyte et al, 2011;Yuquan et al, 2013).…”

Convex Image Denoising via Non-convex Regularization with Parameter Selection

“…Convex formulations benefit from convex optimization theory which leads to robust algorithms with guaranteed convergence. On the other hand, non-smooth, non-convex regularization has remarkable advantages over convex regularization for restoring images, in particular to restore high-quality piecewise constant images with neat edges [16], [22]. However, it may lead to challenging computation since it requires non-convex non-smooth minimization which, involving many minima, can often get stuck in shallow local minima.…”