Joint methods in imaging based on diffuse image representations

This thesis deals with the application and the analysis of different variants of the Mumford-Shah model in the context of image processing. In this kind of models, a given function is approximated in a piecewise smooth or piecewise constant manner. Especially the numerical treatment of the discontinuities requires additional models that are also outlined in this work. The main part of this thesis is concerned with four different topics. Simultaneous edge detection and registration of two images: The image edges are detected with the Ambrosio-Tortorelli model, an approximation of the Mumford-Shah model that approximates the discontinuity set with a phase field, and the registration is based on these edges. The registration obtained by this model is fully symmetric in the sense that the same matching is obtained if the roles of the two input images are swapped. Detection of grain boundaries from atomic scale images of metals or metal alloys: This is an image processing problem from materials science where atomic scale images are obtained either experimentally for instance by transmission electron microscopy or by numerical simulation tools. Grains are homogenous material regions whose atomic lattice orientation differs from their surroundings. Based on a Mumford-Shah type functional, the grain boundaries are modeled as the discontinuity set of the lattice orientation. In addition to the grain boundaries, the model incorporates the extraction of a global elastic deformation of the atomic lattice. Numerically, the discontinuity set is modeled by a level set function following the approach by Chan and Vese. Joint motion estimation and restoration of motion-blurred video: A variational model for joint object detection, motion estimation and deblurring of consecutive video frames is proposed. For this purpose, a new motion blur model is developed that accurately describes the blur also close to the boundary of a moving object. Here, the video is assumed to consist of an object moving in front of a static background. The segmentation into object and background is handled by a Mumford-Shah type aspect of the proposed model. Convexification of the binary Mumford-Shah segmentation model: After considering the application of Mumford-Shah type models to tackle specific image processing problems in the previous topics, the Mumford-Shah model itself is studied more closely. Inspired by the work of Nikolova, Esedoglu and Chan, a method is developed that allows global minimization of the binary Mumford-Shah segmentation model by solving a convex, unconstrained optimization problem. In an outlook, segmentation of flowfields into piecewise affine regions using this convexification method is briefly discussed

Correspondence problems in computer vision : novel models, numerics, and applications

Correspondence problems like optic flow belong to the fundamental problems in computer vision. Here, one aims at finding correspondences between the pixels in two (or more) images. The correspondences are described by a displacement vector field that is often found by minimising an energy (cost) function. In this thesis, we present several contributions to the energy-based solution of correspondence problems: (i) We start by developing a robust data term with a high degree of invariance under illumination changes. Then, we design an anisotropic smoothness term that works complementary to the data term, thereby avoiding undesirable interference. Additionally, we propose a simple method for determining the optimal balance between the two terms. (ii) When discretising image derivatives that occur in our continuous models, we show that adapting one-sided upwind discretisations from the field of hyperbolic differential equations can be beneficial. To ensure a fast solution of the nonlinear system of equations that arises when minimising the energy, we use the recent fast explicit diffusion (FED) solver in an explicit gradient descent scheme. (iii) Finally, we present a novel application of modern optic flow methods where we align exposure series used in high dynamic range (HDR) imaging. Furthermore, we show how the alignment information can be used in a joint super-resolution and HDR method.Korrespondenzprobleme wie der optische Fluß, gehören zu den fundamentalen Problemen im Bereich des maschinellen Sehens (Computer Vision). Hierbei ist das Ziel, Korrespondenzen zwischen den Pixeln in zwei (oder mehreren) Bildern zu finden. Die Korrespondenzen werden durch ein Verschiebungsvektorfeld beschrieben, welches oft durch Minimierung einer Energiefunktion (Kostenfunktion) gefunden wird. In dieser Arbeit stellen wir mehrere Beiträge zur energiebasierten Lösung von Korrespondenzproblemen vor: (i) Wir beginnen mit der Entwicklung eines robusten Datenterms, der ein hohes Maß an Invarianz unter Beleuchtungsänderungen aufweißt. Danach entwickeln wir einen anisotropen Glattheitsterm, der komplementär zu dem Datenterm wirkt und deshalb keine unerwünschten Interferenzen erzeugt. Zusätzlich schlagen wir eine einfache Methode vor, die es erlaubt die optimale Balance zwischen den beiden Termen zu bestimmen. (ii) Im Zuge der Diskretisierung von Bildableitungen, die in unseren kontinuierlichen Modellen auftauchen, zeigen wir dass es hilfreich sein kann, einseitige upwind Diskretisierungen aus dem Bereich hyperbolischer Differentialgleichungen zu übernehmen. Um eine schnelle Lösung des nichtlinearen Gleichungssystems, dass bei der Minimierung der Energie auftaucht, zu gewährleisten, nutzen wir den kürzlich vorgestellten fast explicit diffusion (FED) Löser im Rahmen eines expliziten Gradientenabstiegsschemas. (iii) Schließlich stellen wir eine neue Anwendung von modernen optischen Flußmethoden vor, bei der Belichtungsreihen für high dynamic range (HDR) Bildgebung registriert werden. Außerdem zeigen wir, wie diese Registrierungsinformation in einer kombinierten super-resolution und HDR Methode genutzt werden kann

Local Fourier analysis for saddle-point problems

The numerical solution of saddle-point problems has attracted considerable interest in recent years, due to their indefiniteness and often poor spectral properties that make efficient solution difficult. While much research already exists, developing efficient algorithms remains challenging. Researchers have applied finite-difference, finite element, and finite-volume approaches successfully to discretize saddle-point problems, and block preconditioners and monolithic multigrid methods have been proposed for the resulting systems. However, there is still much to understand. Magnetohydrodynamics (MHD) models the flow of a charged fluid, or plasma, in the presence of electromagnetic fields. Often, the discretization and linearization of MHD leads to a saddle-point system. We present vector-potential formulations of MHD and a theoretical analysis of the existence and uniqueness of solutions of both the continuum two-dimensional resistive MHD model and its discretization. Local Fourier analysis (LFA) is a commonly used tool for the analysis of multigrid and other multilevel algorithms. We first adapt LFA to analyse the properties of multigrid methods for both finite-difference and finite-element discretizations of the Stokes equations, leading to saddle-point systems. Monolithic multigrid methods, based on distributive, Braess-Sarazin, and Uzawa relaxation are discussed. From this LFA, optimal parameters are proposed for these multigrid solvers. Numerical experiments are presented to validate our theoretical results. A modified two-level LFA is proposed for high-order finite-element methods for the Lapalce problem, curing the failure of classical LFA smoothing analysis in this setting and providing a reliable way to estimate actual multigrid performance. Finally, we extend LFA to analyze the balancing domain decomposition by constraints (BDDC) algorithm, using a new choice of basis for the space of Fourier harmonics that greatly simplifies the application of LFA. Improved performance is obtained for some two- and three-level variants

The RBF-FD method: developments and applications

Radial Basis Function (RBF) methods have become a truly meshless alternative for the interpolation of multidimensional scattered data and the solution of partial differencial equations (PDEs) on irregular domains. The dependence on the distance between centers makes RBF methods conceptually simple and easy to implement in anys dimension or shape of the domain. There are two different formulations for the solution of PDEs: the global RBF method and the local RBF method. In the global RBF formulation, the approximate solution is computed in the functional space spanned by a set of translated RBFs. The coordinates of the solution in this space are obtained by collocation. This formulation yields dense differentiation matrices which are spectrally convergent independently of the distribution of RBF centers. The principal drawback is that, as the overall number of centers increases, the condition number of the collocation matrices also increases, what restricts the applicability of the method in practical problems ....