Image restoration is an essential preprocessing step for many Image analysis applications. In any Image restoration techniques, keeping structure of the Image unchanged is very important. Such structure in an Image often corresponds to the region discontinuities and edges. The techniques based on partial differential equations, such as the heat equations, are receiving considerable attention in Image restoration. In this paper, a new algorithm is proposed for Image restoration using partial differential equations applied on neighbors longer than one pixel. To better preserve edges in this technique, slant edges are considered in addition to vertical and horizontal edges. A number of experiments have been preformed to evaluate performance of the proposed method and to compare its performance with the existing algorithms, wiener and median filters. The results express the considerable superiority of the proposed methods.

We propose a new approach for Image enhancement, denoising and restoration, using an anisotropic diffusion based on P-M model and L.V and al. equation, replacing the gradient by motion by mean curvature to detect noise direction for each degraded pixel locally, applying the gradient in Gaussian kernel term to restore the degraded pixels and adding a time term supporting the restoration process. For execution progress, the numerical discretization for the terms of PDE modeling (obtained by the approximation by difference finite volumes finite method, Taylor method and Simpsons improved method), concludes an algorithm treats noised Image regardless the noise type (salt-pepper or Gaussian or speckle) better than other filters based whether on anisotropic diffusion or total, shown in the experimental results (using MATLAB program), and demonstrated through PSNR and SSIM.

This paper presents a novel bilevel optimization approach for a nonlinear partial differential equation. The approach aims to enhance the quality of Image denoising  by estimating certain parameters within this equation. Our work deals with both analytical and numerical results. Analytically, we establish the existence of a solution to the bilevel optimization problem and apply the Alternating Direction Method of Multipliers algorithm to approximate this solution. Furthermore, the method fine-tunes the restoration process, effectively reducing noise while preserving crucial Image features. Finally, numerical results validate the performance of our method, surpassing traditional denoising approaches. This research makes an important contribution to Image restoration, paving the way for high-quality practical applications.

Non-linear Image processing operators give excellent results in a number of Image processing tasks such as restoration and object recognition. However they are frequently excluded from use in solutions because the system designer does not wish to introduce additional hardware or algorithms and because their design can appear to be ad hoc. In practice the median filter is often used though it is rarely optimal. This paper explains how various non-linear Image processing operators may be implemented on a basic linear Image processing system using only convolution and thresholding operations. The paper is aimed at Image processing system developers wishing to include some non-linear processing operators without introducing additional system capabilities such as extra hardware components or software toolboxes. It may also be of benefit to the interested reader wishing to learn more about non-linear operators and alternative methods of design and implementation. The non-linear tools include various components of mathematical morphology, median and weighted median operators and various order statistic filters. As well as describing novel algorithms for implementation within a linear system the paper also explains how the optimum filter parameters may be estimated for a given Image processing task. This novel approach is based on the weight monotonic property and is a direct rather than iterated method.

The conjugate gradient (CG) method is an optimization technique known for its rapid convergence; it has blossomed into significant developments and applications. Numerous variations of CG methods have emerged to en-hance computational efficiency and address real-world challenges. In this work, a novel conjugate gradient method is introduced to solve nonlinear unconstrained optimization problems. Based on the combination of PRP (Polak–Ribière–Polyak), HRM (Hamoda–Rivaie–Mamat) and NMFR (new modified Fletcher–Reeves) algorithms, our method produces a descent di-rection without depending on any line search. Moreover, it enjoys global convergence under mild assumptions and is applied successfully on various standard test problems as well as Image processing. The numerical results indicate that the proposed method outperforms several existing methods in terms of efficiency.

This paper presents a new hybrid conjugate gradient method for solving  nonlinear unconstrained optimization problems; it is based on a combination of $RMIL$  (Rivaie-Mustafa-Ismail-Leong)  and $hSM$  (hybrid Sulaiman- Mohammed) methods. The proposed algorithm enjoys the sufficient descent condition without depending on any line search; moreover, it is globally convergent under the usual and strong Wolfe line search assumptions.  The performance of the algorithm is demonstrated through numerical experiments on a set of 100 test functions from [1] and four Image restoration problems with two noise levels. The numerical comparisons with four existing methods show that the proposed method is promising and effective.