This paper studies a nonlocal biharmonic evolution equation with Dirichlet boundary condition that arises in image restoration. We prove the existence and uniqueness of solutions to the nonlocal problem by the variational method and show that the solutions of the nonlocal problem converge to the solution of the classical biharmonic equation with Dirichlet boundary condition if the nonlocal kernel is rescaled appropriately. The asymptotic behavior is discussed. Besides, we study the Navier problem by transforming it into a Dirichlet problem with a fixed point. The existence, uniqueness, convergence under the rescaling of the kernel, and asymptotic behavior of solutions to the Navier problem are discussed.

Image denoising has always been a challenging task. For performing this task, one of the most effective methods is based on variational PDE. Inspired by the LLT model, we first propose a new adaptive LLT model by adding a weighted function, and then we propose a class of fourth-order diffusion equations based on the new functional. Owing to the adaptive function, the new functional is better than the LLT model and other fourth-order models in terms of edge preservation. While generalizing the Euler-Lagrange equation of the new functional, we discuss a new fourth-order diffusion framework for image denoising. Different from those of other fourth-order diffusion models, the new diffusion coefficients depend on the first-order and second-order derivatives, which can preserve edges and smooth images, respectively. Regarding numerical implementations, we first design an explicit scheme for the proposed model. However, fourth-order diffusion equations require strict stability conditions, and the number of iterations needed is considerable. Consequently, we apply the fast explicit diffusion algorithm (FED) to the explicit scheme to reduce the time consumption of the proposed approach. Furthermore, the additive operator splitting (AOS) scheme is applied for the numerical implementation, and it is the most efficient among all of our algorithms. Finally, compared with other models, the new model exhibits superior effectiveness and efficiency.

The BM3D method achieves excellent denoising performance, but it has artificial effects and bias effects and its performance largely depends on the noise level parameter. To address this, we propose a hybrid BM3D and PDE method for non-parametric single image denoising. First, a non-local Perona- Malik (NLPM) filtering is proposed, and we prove its discontinuity maintaining, mean invariance, convergence, and local continuity. Based on these mathematical properties, an NLPM based noise level estimator (NLPM-NLE) is explored, which involves three steps: preprocessing by NLPM filtering, sample area selection, parameter estimation. And then, we advance a stable-BM3D (SBM3D) method with NLPM filtering to avoid artificial effects and bias effects. Finally, connecting the NLPM-NLE and SBM3D by merging the same part, we develop a non-parametric single image denoising (NPSID) method. Additionally, our proposed BM3D method with NLPM-NLE and the NPSID are compared with other blind denoising methods including PCA + BM3D, WTP + BM3D, and ESM + BM3D on real image denoising. Experiments show that the proposed non-parametric method can automatically and effectively remove noise and preserve details. (c) 2021 Elsevier B.V. All rights reserved.

In this paper, we propose a nonlocal adaptive biharmonic regularization term for image restoration, combining the advantages of fourth-order models (preserving slopes) and nonlocal methods (preserving textures). Besides the image deblurring and denoising, we apply the proposed nonlocal adaptive biharmonic regularizer to image inpainting, and a weight matrix normalization method is developed to cover the shortage of information loss of the nonlocal weight matrix and accelerate the inpainting process. The existence and uniqueness of the solution are proved. The mathematical property such as mean invariance is discussed. For the numerical solution, we employ the L-2 gradient descent and finite difference methods to design explicit and semi-implicit schemes. Numerical results for image restoration are shown on synthetic images, real images, and texture images. Comparisons with local fourth-order models, nonlocal second-order models, and other state-of-the-art methods are made, which help to illustrate the advantages of the proposed model.