Denoising and enhancement of digital images : variational methods, integrodifferential equations, and wavelets

The topics of this thesis are methods for denoising, enhancement, and simplification of digital image data. Special emphasis lies on the relations and structural similarities between several classes of methods which are motivated from different contexts. In particular, one can distinguish the methods treated in this thesis in three classes: For variational approaches and partial differential equations, the notion of the derivative is the tool of choice to model regularity of the data and the desired result. A general framework for such approaches is proposed that involve all partial derivatives of a prescribed order and experimentally are capable of leading to piecewise polynomial approximations of the given data. The second class of methods uses wavelets to represent the data which makes it possible to understand the filtering as very simple pointwise application of a nonlinear function. To view these wavelets as derivatives of smoothing kernels is the basis for relating these methods to integrodifferential equations which are investigated here. In the third case, values of the image in a neighbourhood are averaged where the weights of this averaging can be adapted respecting different criteria. By refinement of the pixel grid and transfer to scaling limits, connections to partial differential equations become visible here, too. They are described in the framework explained before. Numerical aspects of the simplification of images are presented with respect to the NDS energy function, a unifying approach that allows to model many of the aforementioned methods. The behaviour of the filtering methods is documented with numerical examples.Gegenstand der vorliegenden Arbeit sind Verfahren zum Entrauschen, qualitativen Verbessern und Vereinfachen digitaler Bilddaten. Besonderes Augenmerk liegt dabei auf den Beziehungen und der strukturellen Ähnlichkeit zwischen unterschiedlich motivierten Verfahrensklassen. Insbesondere lassen sich die hier behandelten Methoden in drei Klassen einordnen: Bei den Variationsansätzen und partiellen Differentialgleichungen steht der Begriff der Ableitung im Mittelpunkt, um Regularität der Daten und des gewünschten Resultats zu modellieren. Hier wird ein einheitlicher Rahmen für solche Ansätze angegeben, die alle partiellen Ableitungen einer vorgegebenen Ordnung involvieren und experimentell auf stückweise polynomielle Approximationen der gegebenen Daten führen können. Die zweite Klasse von Methoden nutzt Wavelets zur Repräsentation von Daten, mit deren Hilfe sich Filterung als sehr einfache punktweise Anwendung einer nichtlinearen Funktion verstehen lässt. Diese Wavelets als Ableitungen von Glättungskernen aufzufassen bildet die Grundlage für die hier untersuchte Verbindung dieser Verfahren zu Integrodifferentialgleichungen. Im dritten Fall werden Werte des Bildes in einer Nachbarschaft gemittelt, wobei die Gewichtung bei dieser Mittelung adaptiv nach verschiedenen Kriterien angepasst werden kann. Durch Verfeinern des Pixelgitters und Übergang zu Skalierungslimites werden auch hier Verbindungen zu partiellen Differentialgleichungen sichtbar, die in den vorher dargestellten Rahmen eingeordnet werden. Numerische Aspekte beim Vereinfachen von Bildern werden anhand der NDS-Energiefunktion dargestellt, eines einheitlichen Ansatzes, mit dessen Hilfe sich viele der vorgenannten Methoden realisieren lassen. Das Verhalten der einzelnen Filtermethoden wird dabei jeweils durch numerische Beispiele dokumentiert

Methods for the atomistic simulation of ultrasmall semiconductor devices

As the feature sizes in VLSI technology shrink to less than 100 nm the effects due to the quantisation of electronic charge begin to emerge. There are a small number of carriers and impurities and the statistical variation in their number have significant effects on the threshold characteristics of the devices that hamper their large scale integration into future ULSI.The complex potential landscape arising from the Coulomb force, with its sharp localised peaks and troughs, faces problems due to band limiting in meshes and places heavy burdens on the integration techniques. A computationally efficient solution to the problem of band-limiting is presented and is shown to provide an accurate description of the electrostatics. This work also introduces a highly efficient and numerically stable multigrid solver, for Poisson's equation, that can cope with the complex potential distributions on large meshes.The study of ionised impurity scattering is used to validate these molecular dynamics simulations. Results have shown that the Brownian method - despite precluding the use of adaptive integration schemes - gives a good approximation to the standard results and has the advantage of smoothing away errors that can build up during the integration of motion and drives the system towards thermal equilibrium.The greatest hurdle to be cleared before these three-dimensional simulations can be practicable is the sheer computational effort that is required. The implementation of the problem on parallel architectures has been explored and discussed.The methods developed in this work are demonstrated through the simulation of an 80 nm dual-gate MESFET. The results were verified by comparing them with those from a commercial drift-diffusion simulator.The threshold behaviour of devices has been investigated through the study of the formation of conduction channels in blocks. The percolation threshold gives the point when conductive paths form across the gate barrier. The results from the FET simulation were found to be in agreement with the earlier studies on the blocks

A Novel Framework for Head Imaging with Electrical Impedance Tomography

Electrical Impedance Tomography (EIT) is a medical imaging technology with the potential to locate focal epilepsy, monitor patients with traumatic brain injury and diagnose stroke. EIT usually images conductivity changes in time or in frequency of the applied current and measured voltages. While it is nowadays clinically used for monitoring lung ventilation, its application in head imaging is complicated by the geometry of the head, containing tissues with strongly varying conductivities. The aim of this thesis is to provide a novel framework for EIT head imaging by addressing the requirements for higher modelling accuracy throughout the imaging process. An introduction to EIT, its applications for head imaging and the two main components of EIT image reconstructions is given in chapter 2. A procedure for generating more accurate head models is presented in chapter 3 and is used to evaluate, whether subject specifc head models are required for EIT imaging. To speed up simulations of current fow through the head and the computation of the Jacobian matrix required for image reconstructions, a fast parallel forward solver is implemented and validated in chapter 4. Stability of time-diference image reconstructions with respect to electrode modelling errors is addressed in chapter 5, followed by an evaluation of modelling error impacts on multi-frequency imaging in chapter 6. The fndings of chapters 5 and 6 are fnally combined in chapter 7 to recover electrode positions in multi-frequency stroke imaging, thereby reducing image artefacts and making stroke diagnosis with EIT feasible