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

Regularisation methods for imaging from electrical measurements

In Electrical Impedance Tomography the conductivity of an object is estimated from boundary measurements. An array of electrodes is attached to the surface of the object and current stimuli are applied via these electrodes. The resulting voltages are measured. The process of estimating the conductivity as a function of space inside the object from voltage measurements at the surface is called reconstruction. Mathematically the ElT reconstruction is a non linear inverse problem, the stable solution of which requires regularisation methods. Most common regularisation methods impose that the reconstructed image should be smooth. Such methods confer stability to the reconstruction process, but limit the capability of describing sharp variations in the sought parameter. In this thesis two new methods of regularisation are proposed. The first method, Gallssian anisotropic regularisation, enhances the reconstruction of sharp conductivity changes occurring at the interface between a contrasting object and the background. As such changes are step changes, reconstruction with traditional smoothing regularisation techniques is unsatisfactory. The Gaussian anisotropic filtering works by incorporating prior structural information. The approximate knowledge of the shapes of contrasts allows us to relax the smoothness in the direction normal to the expected boundary. The construction of Gaussian regularisation filters that express such directional properties on the basis of the structural information is discussed, and the results of numerical experiments are analysed. The method gives good results when the actual conductivity distribution is in accordance with the prior information. When the conductivity distribution violates the prior information the method is still capable of properly locating the regions of contrast. The second part of the thesis is concerned with regularisation via the total variation functional. This functional allows the reconstruction of discontinuous parameters. The properties of the functional are briefly introduced, and an application in inverse problems in image denoising is shown. As the functional is non-differentiable, numerical difficulties are encountered in its use. The aim is therefore to propose an efficient numerical implementation for application in ElT. Several well known optimisation methods arc analysed, as possible candidates, by theoretical considerations and by numerical experiments. Such methods are shown to be inefficient. The application of recent optimisation methods called primal- dual interior point methods is analysed be theoretical considerations and by numerical experiments, and an efficient and stable algorithm is developed. Numerical experiments demonstrate the capability of the algorithm in reconstructing sharp conductivity profiles