The lower bound of nonlocal gradient for non-convex and non-smooth image patches based regularization

In the inverse problem of image processing, we have witnessed that the non-convex and non-smooth regularizers can produce clearer image edges than convex ones such as total variation (TV). This fact can be explained by the uniform lower bound theory of the local gradient in non-convex and non-smooth regularization. In recent years, although it has been numerically shown that the nonlocal regularizers of various image patches based nonlocal methods can recover image textures well, we still desire a theoretical interpretation. To this end, we propose a non-convex non-smooth and block nonlocal (NNBN) regularization model based on image patches. By integrating the advantages of the non-convex and non-smooth potential function in the regularization term, the uniform lower bound theory of the image patches based nonlocal gradient is given. This approach partially explains why the proposed method can produce clearer image textures and edges. Compared to some classical regularization methods, such as total variation (TV), non-convex and non-smooth (NN) regularization, nonlocal total variation (NLTV) and block nonlocal total variation(BNLTV), our experimental results show that the proposed method improves restoration quality.

A second-order nonlocal regularized variational model for multiframe image super-resolution

In this paper, we propose a new nonlocal super-resolution (SR) model which is a combination of the nonlocal total variation (TV) regularization and the nonlocal p-Laplacian term (with p = 2). This choice is motivated by the success of the nonlocal TV term in preserving image edges and the efficiency of the nonlocal p-Laplacian term in preserving the image texture. To ensure the convergence of the proposed optimization SR problem, we prove the existence and uniqueness of a solution in a well-posed framework. In addition, to resolve the encountered minimization problem, we proposed a modified primal-dual algorithm and numerical results are also given to show the performance of the proposed approach.

Nonlocal Total Variation Using the First and Second Order Derivatives and Its Application to CT image Reconstruction

We propose a new class of nonlocal Total Variation (TV), in which the first derivative and the second derivative are mixed. Since most existing TV considers only the first-order derivative, it suffers from problems such as staircase artifacts and loss in smooth intensity changes for textures and low-contrast objects, which is a major limitation in improving image quality. The proposed nonlocal TV combines the first and second order derivatives to preserve smooth intensity changes well. Furthermore, to accelerate the iterative algorithm to minimize the cost function using the proposed nonlocal TV, we propose a proximal splitting based on Passty’s framework. We demonstrate that the proposed nonlocal TV method achieves adequate image quality both in sparse-view CT and low-dose CT, through simulation studies using a brain CT image with a very narrow contrast range for which it is rather difficult to preserve smooth intensity changes.