A new method for the restoration of images damaged by impulse noise is proposed. In the authors’ own words “In this article, we develop a variational model that associates with the non-convex function and the FOTV (fractional order total variation) regularization \(\dots\) for restoring blurred and impulse noisy images”. FOTV is defined as follows. Let V be the set of all possible images given an image restoration problem, a solution consists of finding \(u\in V\) such that \[ u=\operatorname{argmin}(\Vert\nabla^\alpha u\Vert_1+\lambda.\Vert Ku-f\Vert_1) \] where \(f\) is the observed (degraded) image, \(K\) is a blurring kernel with space invariance, \(\alpha\) is a rational number in \([1,2]\), \(\nabla^\alpha\) is the gradient operator of order \(\alpha\), \(\Vert\,\Vert_1\) is the \(l^1\) norm, and \(\lambda>0\) is a regularization parameter. The proposed model, NFOTV (non-convex FOTV), is a modified FOTV to solve the problem using an iterative algorithm, which is inspired by previous work by other authors. The aim is to find \(u\in V\) such that \[ u=\operatorname{argmin}(\omega^{l}\Vert\nabla^\alpha u\Vert_1+\lambda.\Vert Ku-f\Vert_1),\tag{1} \] where \(\omega\) is a weight function defined iteratively at the \(l\)-th iteration by \[ \omega^{l}=(p/(|\nabla^\alpha u^{l}|+\varepsilon)^{1-p})), \] with \(p\in (0,1)\) and \(\varepsilon>0\) to avoid division by zero in the definition of \(\omega^{l}\).
An algorithm for solving (1) is exposed in a very summarized but rigorous way. Also, its convergence is rigorously proved. To show the effectiveness of the new procedure, NFOTV is compared with five others already in use. A total of six real images are considered and degraded by imposing Gaussian and/or impulse noise. The different restoration techniques analyzed are then applied to the degraded images and the resulting images are compared with the original ones, both visually and numerically through the following three criteria: PSNR (peak signal-to-noise ratio), SSIM (structure similarity), and FSIM (feature similarity).
There are no notable visual differences between the different techniques or with the original images at the scale presented in the paper. It may be possible to detect visual differences working at a scale of 300% or larger. The numerical results for all criteria and all images show an advantage for the new technique, although these results are not very relevant because the precision with which they were achieved is not reported. In any case, the proposal seems promising and it would be useful to continue with more complete analyses to strengthen the superiority of the new technique.