Nonlocal Adaptive Biharmonic Regularizer for Image Restoration

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 L² 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.