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

A novel variational model for image registration using Gaussian curvature

Image registration is one important task in many image processing applications. It aims to align two or more images so that useful information can be extracted through comparison, combination or superposition. This is achieved by constructing an optimal trans- formation which ensures that the template image becomes similar to a given reference image. Although many models exist, designing a model capable of modelling large and smooth deformation field continues to pose a challenge. This paper proposes a novel variational model for image registration using the Gaussian curvature as a regulariser. The model is motivated by the surface restoration work in geometric processing [Elsey and Esedoglu, Multiscale Model. Simul., (2009), pp. 1549-1573]. An effective numerical solver is provided for the model using an augmented Lagrangian method. Numerical experiments can show that the new model outperforms three competing models based on, respectively, a linear curvature [Fischer and Modersitzki, J. Math. Imaging Vis., (2003), pp. 81- 85], the mean curvature [Chumchob, Chen and Brito, Multiscale Model. Simul., (2011), pp. 89-128] and the diffeomorphic demon model [Vercauteren at al., NeuroImage, (2009), pp. 61-72] in terms of robustness and accuracy.Comment: 23 pages, 5 figures. Key words: Image registration, Non-parametric image registration, Regularisation, Gaussian curvature, surface mappin

Multigrid methods and automatic segmentation: an application to CT images of the liver

We consider a segmentation problem which arises in medical imaging and liver surgery. The model problem is based on an active contour without edges technique formulated in a level set dictionary. Previous work indicates that a feasible solution can be obtained solving the gradient descent equation associated to the original minimization problem but the convergence of the algorithm is too slow for practical clinical purposes. Here, we study the implementation of multigrid methods to the elliptic problem and the numerical results are compared with the parabolic approach

A Multi-Grid Iterative Method for Photoacoustic Tomography

Inspired by the recent advances on minimizing nonsmooth or bound-constrained convex functions on models using varying degrees of fidelity, we propose a line search multigrid (MG) method for full-wave iterative image reconstruction in photoacoustic tomography (PAT) in heterogeneous media. To compute the search direction at each iteration, we decide between the gradient at the target level, or alternatively an approximate error correction at a coarser level, relying on some predefined criteria. To incorporate absorption and dispersion, we derive the analytical adjoint directly from the first-order acoustic wave system. The effectiveness of the proposed method is tested on a total-variation penalized Iterative Shrinkage Thresholding algorithm (ISTA) and its accelerated variant (FISTA), which have been used in many studies of image reconstruction in PAT. The results show the great potential of the proposed method in improving speed of iterative image reconstruction

An improved discontinuity-preserving image registration model and its fast algorithm

Recently, Chumchob–Chen(2010) proposed a discontinuity-preserving image registration model which was more flexible than those common techniques such as the diffusion and total variation based regularization techniques. However, each component of displacement field is regularized separately in this model and then the nonlinear diffusion processes resulting from the first variation of the discontinuity-preserving regularization do not enforce coupling between the primary components of the displacement field. Thus the discontinuity-preserving model may prevent to obtain a good registration in some situations, for example non-smooth registration problems with non-axis-aligned discontinuities. To utilize interdependence between the primary components of the deformation field for smooth and non-smooth registration problems, we propose an improved discontinuity-preserving image registration model in this paper, second we propose an idea of relaxed fixed point combining with Gauss–Newton scheme with Armijo's line search for solving the new model and further to combine with a multilevel method to achieve fast convergence. Numerical experiments not only confirm that our proposed method is efficient and stable, but also it can give more satisfying registration results according to image quality