A Total Fractional-Order Variation Model for Image Restoration with Non-homogeneous Boundary Conditions and its Numerical Solution

To overcome the weakness of a total variation based model for image restoration, various high order (typically second order) regularization models have been proposed and studied recently. In this paper we analyze and test a fractional-order derivative based total $\alpha$-order variation model, which can outperform the currently popular high order regularization models. There exist several previous works using total $\alpha$-order variations for image restoration; however first no analysis is done yet and second all tested formulations, differing from each other, utilize the zero Dirichlet boundary conditions which are not realistic (while non-zero boundary conditions violate definitions of fractional-order derivatives). This paper first reviews some results of fractional-order derivatives and then analyzes the theoretical properties of the proposed total $\alpha$-order variational model rigorously. It then develops four algorithms for solving the variational problem, one based on the variational Split-Bregman idea and three based on direct solution of the discretise-optimization problem. Numerical experiments show that, in terms of restoration quality and solution efficiency, the proposed model can produce highly competitive results, for smooth images, to two established high order models: the mean curvature and the total generalized variation.Comment: 26 page

Image Structure-Preserving Denoising Based on Difference Curvature Driven Fractional Nonlinear Diffusion

The traditional integer-order partial differential equations and gradient regularization based image denoising techniques often suffer from staircase effect, speckle artifacts, and the loss of image contrast and texture details. To address these issues, in this paper, a difference curvature driven fractional anisotropic diffusion for image noise removal is presented, which uses two new techniques, fractional calculus and difference curvature, to describe the intensity variations in images. The fractional-order derivatives information of an image can deal well with the textures of the image and achieve a good tradeoff between eliminating speckle artifacts and restraining staircase effect. The difference curvature constructed by the second order derivatives along the direction of gradient of an image and perpendicular to the gradient can effectively distinguish between ramps and edges. Fourier transform technique is also proposed to compute the fractional-order derivative. Experimental results demonstrate that the proposed denoising model can avoid speckle artifacts and staircase effect and preserve important features such as curvy edges, straight edges, ramps, corners, and textures. They are obviously superior to those of traditional integral based methods. The experimental results also reveal that our proposed model yields a good visual effect and better values of MSSIM and PSNR