A non-convex denoising model for impulse and Gaussian noise mixture removing using bi-level parameter identification

Abstract

<p style='text-indent:20px;'>We propose a new variational framework to remove a mixture of Gaussian and impulse noise from images. This framework is based on a non-convex PDE-constrained with a fractional-order operator. The non-convex norm is applied to the impulse component controlled by a weighted parameter <inline-formula><tex-math id="M1">\begin{document}$ \gamma $\end{document}</tex-math></inline-formula>, which depends on the level of the impulse noise and image feature. Furthermore, the fractional operator is used to preserve image texture and edges. In a first part, we study the theoretical properties of the proposed PDE-constrained, and we show some well-posdnees results. In a second part, after having demonstrated how to numerically find a minimizer, a proximal linearized algorithm combined with a Primal-Dual approach is introduced. Moreover, a bi-level optimization framework with a projected gradient algorithm is proposed in order to automatically select the parameter <inline-formula><tex-math id="M2">\begin{document}$ \gamma $\end{document}</tex-math></inline-formula>. Denoising tests confirm that the non-convex term and learned parameter <inline-formula><tex-math id="M3">\begin{document}$ \gamma $\end{document}</tex-math></inline-formula> lead in general to an improved reconstruction when compared to results of convex norm and other competitive denoising methods. Finally, we show extensive denoising experiments on various images and noise intensities and we report conventional numerical results which confirm the validity of the non-convex PDE-constrained, its analysis and also the proposed bi-level optimization with learning data.</p>

On a Variational Problem with a Nonstandard Growth Functional and Its Applications to Image Processing

“…Recently, the non-convex functions have been performed as a fidelity term and have achieved a considerable success in removing complex non-smooth noise [25,53]. In particular, the following concave variant of fidelity term has been proposed in [1]:…”

“…As for the data fidelity term, in contrast to the quadratic data-fitting term in the ROF model (3) (see also (1)), where it is specialized for zero-mean Gaussian noise, we take this term in L 1 -norm just in order to apply the model to other types of noise, especially when the noise is impulsive or contains gross outliers. Following this way, we are going to increase the noise robustness of the model (10) albeit it makes such variational problem completely non-smooth and, hence, significantly more difficult from a minimization point of view.…”

“…There is a wide choice of the regularization term proposed in the literature to ensure the well-posedness of the denoising problem (1). In the most cases, this term can be represented as follows:…”

A non-convex PDE-constrained denoising model for impulse and Gaussian noise mixture reduction

“…For this reason alternative optimization strategies with respect to convex optimization must be employed. Nevertheless, many algorithms are developed, dissjointly for different area, to deal with non-smooth non-convex optimization [1,3,25,40]. Otherwise, to reduce the undesirable staircase effect, high order PDEs have been employed [20,22,44] in the denoising context.…”

Fictitious Controls and Approximation of an Optimal Control Problem for Perona-Malik Equation

“…Let us mention that control problems for the non-smoothed Perona-Malik equation have received very little attention in the literature. Formulating the control problem (1.9)-(1.13) for the nonlinear equation of the porous medium type is mainly motivated by the observation that this statement can be successfully applied to the image processing, in particular, to the reduction of mixture of Gaussian and impulse noise with keeping safe the image contours and texture (see, for instance, [2] and the references therein). On the other hand, the novelty of this problem is that we involve into optimization the nonlinear equation with rather special type (non-convex and non-coercive) of non-linearity.…”