This thesis centers on modified stencils for the Finite-Difference Time-Domain method (FDTD), or Yee scheme, when modelling curved boundaries, obstacles and holes smaller than the discretization length.  The goal is to increase the accuracy while keeping the structure of the standard method, enabling improvements to existing implementations with minimal effort. We present an extension of a previously developed technique for consistent boundary approximation in the Yee scheme.  We consider both Maxwell's equations and the acoustic equations in three dimensions, which require separate treatment, unlike in two dimensions. The stability properties of coefficient modifications are essential for practical usability.  We present an analysis of the requirements for time-stable modifications, which we use to construct a simple and effective method for boundary approximations. The method starts from a predetermined staircase discretization of the boundary, requiring no further data on the underlying geometry that is being approximated. Not only is the standard staircasing of curved boundaries a poor approximation, it is inconsistent, giving rise to errors that do not disappear in the limit of small grid lengths. We analyze the standard staircase approximation by deriving exact solutions of the difference equations, including the staircase boundary. This facilitates a detailed error analysis, showing how staircasing affects amplitude, phase, frequency and attenuation of waves. To model obstacles and holes of smaller size than the grid length, we develop a numerical subgrid method based on locally modified stencils, where a highly resolved micro problem is used to generate effective coefficients for the Yee scheme at the macro scale. The implementations and analysis of the developed methods are validated through systematic numerical tests. / <p>QC 20120530</p>

Le modèle ROF (Rudin, Osher, Fatemi), introduit en 1992 en utilisant la variation totale comme terme de régularisation pour la restauration d'images, a fait l'objet de nombreuses recherches théoriques et numériques depuis. Dans cette thèse, nous présentons de nouveaux modèles inspirés de la variation totale mais construits par analogie avec une méthode de débruitage beaucoup plus récente et radicalement différente : les moyennes non locales (NL-means). Dans une première partie, nous transposons le modèle ROF dans un cadre bayésien, et montrons que l'estimateur associé à un risque quadratique (moyenne a posteriori) peut être calculé numériquement à l'aide d'un algorithme de type MCMC (Monte Carlo Markov Chain), dont la convergence est soigneusement contrôlée compte tenu de la dimension élevée de l'espace des images. Nous montrons que le débruiteur associé permet notamment d'éviter le phénomène de "staircasing", défaut bien connu du modèle ROF. Dans la deuxième partie, nous proposons tout d'abord une version localisée du modèle ROF et en analysons certains aspects : compromis biais-variance, EDP limite, pondération du voisinage, etc. Enfin, nous discutons le choix de la variation totale en tant que modèle a priori, en confrontant le point de vue géométrique (modèle ROF) au cadre statistique (modélisation bayésienne).

The focus of this thesis is variational image restoration techniques that involve novel non-smooth first-order gradient regularisers: Total Variation (TV) regularisation in image and data space for reconstruction of thin structures from PET data and regularisers given by an infimal-convolution of TV and $L^p$ seminorms for denoising images with piecewise affine structures. In the first part of this thesis, we present a novel variational model for PET reconstruction. During a PET scan, we encounter two different spaces: the sinogram space that consists of all the PET data collected from the detectors and the image space where the reconstruction of the unknown density is finally obtained. Unlike most of the state of the art reconstruction methods in which an appropriate regulariser is designed in the image space only, we introduce a new variational method incorporating regularisation in image and sinogram space. In particular, the corresponding minimisation problem is formed by a total variational regularisation on both the sinogram and the image and with a suitable weighted $L^2$ fidelity term, which serves as an approximation to the Poisson noise model for PET. We establish the well-posedness of this new model for functions of Bounded Variation (BV) and perform an error analysis through the notion of the Bregman distance. We examine analytically how TV regularisation on the sinogram affects the reconstructed image especially the boundaries of objects in the image. This analysis motivates the use of a combined regularisation principally for reconstructing images with thin structures. In the second part of this thesis we propose a first-order regulariser that is a combination of the total variation and $L^p$ seminorms with $1 < p \le \infty$. A well-posedness analysis is presented and a detailed study of the one dimensional model is performed by computing exact solutions for simple functions such as the step function and a piecewise affine function, for the regulariser with $p = 2$ and $p = 1$. We derive necessary and sufficient conditions for a pair in $BV \times L^p$ to be a solution for our proposed model and determine the structure of solutions dependent on the value of $p$. In the case $p = 2$, we show that the regulariser is equivalent to the Huber-type variant of total variation regularisation. Moreover, there is a certain class of one dimensional data functions for which the regularised solutions are equivalent to high-order regularisers such as the state of the art total generalised variation (TGV) model. The key assets of our regulariser are the elimination of the staircasing effect - a well-known disadvantage of total variation regularisation - the capability of obtaining piecewise affine structures for $p = 1$ and qualitatively comparable results to TGV. In addition, our first-order $TVL^p$ regulariser is capable of preserving spike-like structures that TGV is forced to smooth. The numerical solution of the proposed first-order model is in general computationally more efficient compared to high-order approaches.

In this thesis we study novel higher order total variation-based variational methods for digital image reconstruction. These methods are formulated in the context of Tikhonov regularisation. We focus on regularisation techniques in which the regulariser incorporates second order derivatives or a sophisticated combination of first and second order derivatives. The introduction of higher order derivatives in the regularisation process has been shown to be an advantage over the classical first order case, i.e., total variation regularisation, as classical artifacts such as the staircasing effect are significantly reduced or totally eliminated. Also in image inpainting the introduction of higher order derivatives in the regulariser turns out to be crucial to achieve interpolation across large gaps. First, we introduce, analyse and implement a combined first and second order regularisation method with applications in image denoising, deblurring and inpainting. The method, numerically realised by the split Bregman algorithm, is computationally efficient and capable of giving comparable results with total generalised variation (TGV), a state of the art higher order method. An additional experimental analysis is performed for image inpainting and an online demo is provided on the IPOL website (Image Processing Online). We also compute and study properties of exact solutions of the one dimensional total generalised variation problem with L^{2} data fitting term, for simple piecewise affine data functions, with or without jumps . This gives an insight on how this type of regularisation behaves and unravels the role of the TGV parameters. Finally, we introduce, study and analyse a novel non-local Hessian functional. We prove localisations of the non-local Hessian to the local analogue in several topologies and our analysis results in derivative-free characterisations of higher order Sobolev and BV spaces. An alternative formulation of a non-local Hessian functional is also introduced which is able to produce piecewise affine reconstructions in image denoising, outperforming TGV.

Les problèmes inverses consistent à retrouver une donnée qui a été transformée ou perturbée. Ils nécessitent une régularisation puisque mal posés. En traitement d'images, la variation totale en tant qu'outil de régularisation a l'avantage de préserver les discontinuités tout en créant des zones lisses, résultats établis dans cette thèse dans un cadre continu et pour des énergies générales. En outre, nous proposons et étudions une variante de la variation totale. Nous établissons une formulation duale qui nous permet de démontrer que cette variante coïncide avec la variation totale sur des ensembles de périmètre fini. Ces dernières années les méthodes non-locales exploitant les auto-similarités dans les images ont connu un succès particulier. Nous adaptons cette approche au problème de complétion de spectre pour des problèmes inverses généraux. La dernière partie est consacrée aux aspects algorithmiques inhérents à l'optimisation des énergies convexes considérées. Nous étudions la convergence et la complexité d'une famille récente d'algorithmes dits Primal-Dual.