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

Variational image registration by a total fractional-order variation model

In this paper, a new framework of nonlocal deformation in non-rigid image registration is presented. It is well known that many non-rigid image registration techniques may lead to unsteady deformation (e.g. not one to one) if the dissimilarity between the reference and template images is too large. We present a novel variational framework of the total fractional-order variation to derive the underlying fractional Euler-Lagrange equations and a numerical implementation combining the semi-implicit update and conjugate gradients (CG) solution to solve the nonlinear systems. Numerical experiments show that the new registration not only produces accurate and smooth solutions but also allows for a large rigid alignment, the evaluations of the new model demonstrate substantial improvements in accuracy and robustness over the conventional image registration approaches

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

PETROGENESIS AND TECTONIC IMPLICATION OF THE NEOPROTEROZOIC BONINITIC-THOLEIITIC LAVAS AND MAGNESIAN ANDESITES IN THE GORNY ALTAI TERRANE, NORTHWESTERN CENTRAL ASIAN OROGENIC BELT

The late Neoproterozoic tholeiitic-boninitic lavas and minor magnesian andesites cropping out in the Kurai Ridge, southeast of the Gorny Altai terrane, represent magmatic products of the nascent Kuznetsk-Altai intra-oceanic island arc southwest off the Siberian continent. Samples of these rocks can provide key information about sub-arc mantle and slab-mantle interaction during the early phase of ocean-ocean subduction.The late Neoproterozoic tholeiitic-boninitic lavas and minor magnesian andesites cropping out in the Kurai Ridge, southeast of the Gorny Altai terrane, represent magmatic products of the nascent Kuznetsk-Altai intra-oceanic island arc southwest off the Siberian continent. Samples of these rocks can provide key information about sub-arc mantle and slab-mantle interaction during the early phase of ocean-ocean subduction