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

Combining Total Variation and Nonlocal Means Regularization for Edge Preserving Image Deconvolution

We propose a new edge preserving image deconvolution model by combining total variation and nonlocal means regularization. Natural images exhibit an high degree of redundancy. Using this redundancy, the nonlocal means regularization strategy is a good technique for detail preserving image restoration. In order to further improve the visual quality of the nonlocal means based algorithm, total variation is introduced to the model to better preserve edges. Then an efficient alternating minimization procedure is used to solve the model. Numerical experiments illustrate the effectiveness of the proposed algorithm

Automatic regularization parameter selection for the total variation mixed noise image restoration framework

Image restoration consists in recovering a high quality image estimate based only on observations. This is considered an ill-posed inverse problem, which implies non-unique unstable solutions. Regularization methods allow the introduction of constraints in such problems and assure a stable and unique solution. One of these methods is Total Variation, which has been broadly applied in signal processing tasks such as image denoising, image deconvolution, and image inpainting for multiple noise scenarios. Total Variation features a regularization parameter which defines the solution regularization impact, a crucial step towards its high quality level. Therefore, an optimal selection of the regularization parameter is required. Furthermore, while the classic Total Variation applies its constraint to the entire image, there are multiple scenarios in which this approach is not the most adequate. Defining different regularization levels to different image elements benefits such cases. In this work, an optimal regularization parameter selection framework for Total Variation image restoration is proposed. It covers two noise scenarios: Impulse noise and Impulse over Gaussian Additive noise. A broad study of the state of the art, which covers noise estimation algorithms, risk estimation methods, and Total Variation numerical solutions, is included. In order to approach the optimal parameter estimation problem, several adaptations are proposed in order to create a local-fashioned regularization which requires no a-priori information about the noise level. Quality and performance results, which include the work covered in two recently published articles, show the effectivity of the proposed regularization parameter selection and a great improvement over the global regularization framework, which attains a high quality reconstruction comparable with the state of the art algorithms.Tesi

A new augmented Lagrangian primal dual algorithm for elastica regularization

Regularization is a key element of variational models in image processing. To overcome the weakness of models based on total variation, various high order (typically second order) regularization models have been proposed and studied recently. Among these, Euler's elastica energy based regularizer is perhaps the most interesting in terms of both mathematical and physical justifications. More importantly its success has been proven in applications; however it has been a major challenge to develop fast and effective algorithms. In this paper we propose a new idea for deriving a primal dual algorithm, based on Legendre–Fenchel transformations, for representing the elastica regularizer. Combined with an augmented Lagrangian for-mulation, we are able to derive an equivalent unconstrained optimization that has fewer variables to work with than previous works based on splitting methods. We shall present our algorithms for both the image restoration problem and the image segmentation model. The idea applies to other models where the elastica regularizer is required. Numerical experiments show that the proposed method can produce highly competitive results with better efficiency. </jats:p