A combined first and second order variational approach for image reconstruction

In this paper we study a variational problem in the space of functions of bounded Hessian. Our model constitutes a straightforward higher-order extension of the well known ROF functional (total variation minimisation) to which we add a non-smooth second order regulariser. It combines convex functions of the total variation and the total variation of the first derivatives. In what follows, we prove existence and uniqueness of minimisers of the combined model and present the numerical solution of the corresponding discretised problem by employing the split Bregman method. The paper is furnished with applications of our model to image denoising, deblurring as well as image inpainting. The obtained numerical results are compared with results obtained from total generalised variation (TGV), infimal convolution and Euler's elastica, three other state of the art higher-order models. The numerical discussion confirms that the proposed higher-order model competes with models of its kind in avoiding the creation of undesirable artifacts and blocky-like structures in the reconstructed images -- a known disadvantage of the ROF model -- while being simple and efficiently numerically solvable.Comment: 34 pages, 89 figure

Optimisation for image processing

The main purpose of optimisation in image processing is to compensate for missing, corrupted image data, or to find good correspondences between input images. We note that image data essentially has infinite dimensionality that needs to be discretised at certain levels of resolution. Most image processing methods find a suboptimal solution, given the characteristics of the problem. While the general optimisation literature is vast, there does not seem to be an accepted universal method for all image problems. In this thesis, we consider three interrelated optimisation approaches to exploit problem structures of various relaxations to three common image processing problems: 1. The first approach to the image registration problem is based on the nonlinear programming model. Image registration is an ill-posed problem and suffers from many undesired local optima. In order to remove these unwanted solutions, certain regularisers or constraints are needed. In this thesis, prior knowledge of rigid structures of the images is included in the problem using linear and bilinear constraints. The aim is to match two images while maintaining the rigid structure of certain parts of the images. A sequential quadratic programming algorithm is used, employing dimensional reduction, to solve the resulting discretised constrained optimisation problem. We show that pre-processing of the constraints can reduce problem dimensionality. Experimental results demonstrate better performance of our proposed algorithm compare to the current methods. 2. The second approach is based on discrete Markov Random Fields (MRF). MRF has been successfully used in machine learning, artificial intelligence, image processing, including the image registration problem. In the discrete MRF model, the domain of the image problem is fixed (relaxed) to a certain range. Therefore, the optimal solution to the relaxed problem could be found in the predefined domain. The original discrete MRF is NP hard and relaxations are needed to obtain a suboptimal solution in polynomial time. One popular approach is the linear programming (LP) relaxation. However, the LP relaxation of MRF (LP-MRF) is excessively high dimensional and contains sophisticated constraints. Therefore, even one iteration of a standard LP solver (e.g. interior-point algorithm), may take too long to terminate. Dual decomposition technique has been used to formulate a convex-nondifferentiable dual LP-MRF that has geometrical advantages. This has led to the development of first order methods that take into account the MRF structure. The methods considered in this thesis for solving the dual LP-MRF are the projected subgradient and mirror descent using nonlinear weighted distance functions. An analysis of the convergence properties of the method is provided, along with improved convergence rate estimates. The experiments on synthetic data and an image segmentation problem show promising results. 3. The third approach employs a hierarchy of problem's models for computing the search directions. The first two approaches are specialised methods for image problems at a certain level of discretisation. As input images are infinite-dimensional, all computational methods require their discretisation at some levels. Clearly, high resolution images carry more information but they lead to very large scale and ill-posed optimisation problems. By contrast, although low level discretisation suffers from the loss of information, it benefits from low computational cost. In addition, a coarser representation of a fine image problem could be treated as a relaxation to the problem, i.e. the coarse problem is less ill-conditioned. Therefore, propagating a solution of a good coarse approximation to the fine problem could potentially improve the fine level. With the aim of utilising low level information within the high level process, we propose a multilevel optimisation method to solve the convex composite optimisation problem. This problem consists of the minimisation of the sum of a smooth convex function and a simple non-smooth convex function. The method iterates between fine and coarse levels of discretisation in the sense that the search direction is computed using information from either the gradient or a solution of the coarse model. We show that the proposed algorithm is a contraction on the optimal solution and demonstrate excellent performance on experiments with image restoration problems.Open Acces

Primal-dual active set methods for Allen-Cahn variational inequalities

This thesis aims to introduce and analyse a primal-dual active set strategy for solving Allen-Cahn variational inequalities. We consider the standard Allen-Cahn equation with non-local constraints and a vector-valued Allen-Cahn equation with and without non-local constraints. Existence and uniqueness results are derived in a formulation involving Lagrange multipliers for local and non-local constraints. Local Convergence is shown by interpreting the primal-dual active set approach as a semi-smooth Newton method. Properties of the method are discussed and several numerical simulations in two and three space dimensions demonstrate its efficiency. In the second part of the thesis various applications of the Allen-Cahn equation are discussed. The non-local Allen-Cahn equation can be coupled with an elasticity equation to solve problems in structural topology optimisation. The model can be extended to handle multiple structures by using the vector-valued Allen-Cahn variational inequality with non-local constraints. Since many applications of the Allen-Cahn equation involve evolution of interfaces in materials an important extension of the standard Allen-Cahn model is to allow materials to exhibit anisotropic behaviour. We introduce an anisotropic version of the Allen-Cahn variational inequality and we show that it is possible to apply the primal-dual active set strategy efficiently to this model. Finally, the Allen-Cahn model is applied to problems in image processing, such as segmentation, denoising and inpainting. The primal-dual active set method proves exible and reliable for all the applications considered in this thesis

Selective Image Segmentation Models and Fast Multigrid Methods

This thesis is concerned with developing robust and accurate variational selective image segmentation models along with fast multigrid methods to solve non-linear partial differential equations (PDEs). The first two major contributions are the development of new distance terms and new intensity fitting terms for selective image segmentation models. These give state-of-the-art segmentation results, with high robustness to the main parameters and to the user input. Therefore, these models are highly applicable to real-world applications such as segmenting single organs from medical scans. The final major contribution is to develop new novel non-standard smoothers for the non-linear full approximation scheme multigrid framework. Multigrid is an optimal O(N) iterative scheme when it converges. However, typically if we directly apply a multigrid solver to a non-linear problem, it will not converge. This is principally due to the ineffectiveness of the standard smoothing schemes such as Jacobi or Gauss-Seidel. We review the true reason that these smoothers are ineffective using local Fourier analysis and develop a smoother which is guaranteed to be effective. Experiments show that the smoother is effective and the algorithm converges as desired. These new non-standard smoothing schemes can be used to solve a whole class of non-linear PDEs quickly. This work also lays the groundwork in the development of a “black-box” non-linear multigrid solver which doesn’t require the degree of tuning that current multigrid algorithms do