Theoretical foundations for 1-D shock filtering

While shock filters are popular morphological image enhancement methods, no well-posedness theory is available for their corresponding partial differential equations (PDEs). By analysing the dynamical system of ordinary differential equations that results from a space discretisation of a PDE for 1-D shock filtering, we derive an analytical solution and prove well-posedness. We show that the results carry over to the fully discrete case when an explicit time discretisation is applied. Finally we establish an equivalence result between discrete shock filtering and local mode filtering

Stabilizing discontinuous Galerkin methods using Dafermos' entropy rate criterion

A novel approach for the stabilization of the discontinuous Galerkin method based on the Dafermos entropy rate crition is presented. The approach is centered around the efficient solution of linear or nonlinear optimization problems in every timestep as a correction to the basic discontinuous Galerkin scheme. The thereby enforced Dafermos criterion results in improved stability compared to the basic method while retaining the order of the method in numerical experiments. Further modification of the optimization problem allows also to enforce classical entropy inequalities for the scheme. The proposed stabilization is therefore an alternative to flux-differencing, finite-volume subcells, artificial viscosity, modal filtering, and other shock capturing procedures

Stabilizing Discontinuous Galerkin Methods Using Dafermos' Entropy Rate Criterion: II -- Systems of Conservation Laws and Entropy Inequality Predictors

A novel approach for the stabilization of the Discontinuous Galerkin method based on the Dafermos entropy rate crition is presented. First, estimates for the maximal possible entropy dissipation rate of a weak solution are derived. Second, families of conservative Hilbert-Schmidt operators are identified to dissipate entropy. Steering these operators using the bounds on the entropy dissipation results in high-order accurate shock-capturing DG schemes for the Euler equations, satisfying the entropy rate criterion and an entropy inequality

Nonlocal evolutions in image processing

The main topic of this thesis is to study a general framework which encompasses a wide class of nonlocal filters. For that matter, we introduce a general initial value problem, defined in terms of integro-differential equations and following the work of Weickert, we impose a set of basic assumptions that turn it into a well-posed model and develop nonlocal scale-space theory. Moreover, we go one step further and consider the consequences of relaxing some of this initial set of requirements. With each particular modification of the initial requirements, we obtain a particular framework which encompasses a more specific, yet wide, family of nonlocal processes.Das Hauptthema dieser Arbeit ist die Untersuchung eines allgemeinen Rahmens für eine breite Klasse nichtlokaler Filter. Zuerst führen wir ein Modell ein, das auf Integro-Differentialgleichungen basiert. Wir ergänzen es mit einer Reihe von Grundannahmen, die es uns ermöglichen, eine nichtlokale Skalenraumtheorie zu entwickeln, wie in [1]. Außerdem betrachten wir die Konsequenzen der Abschwächung einiger dieser Annahmen. Wir stellen verschiedene Beispiele für die mit jeder Relaxation erhaltenen Prozesse vor

Modeling Artificial Boundary Conditions for Compressible Flow

We review artificial boundary conditions (BCs) for simulation of inflow, outflow, and far-field (radiation) problems, with an emphasis on techniques suitable for compressible turbulent shear flows. BCs based on linearization near the boundary are usually appropriate for inflow and radiation problems. A variety of accurate techniques have been developed for this case, but some robustness and implementation issues remain. At an outflow boundary, the linearized BCs are usually not accurate enough. Various ad hoc models have been proposed for the nonlinear case, including absorbing layers and fringe methods. We discuss these techniques and suggest directions for future modeling efforts

Diffusion-inspired shrinkage functions and stability results for wavelet denoising

We study the connections between discrete 1-D schemes for non-linear diffusion and shift-invariant Haar wavelet shrinkage. We show that one step of a (stabilised) explicit discretisation of nonlinear diffusion can be expressed in terms of wavelet shrinkage on a single spatial level. This equivalence allows a fruitful exchange of ideas between the two fields. In this paper we derive new wavelet shrinkage functions from existing diffusivity functions, and identify some previously used shrinkage functions as corresponding to well known diffusivities. We demonstrate experimentally that some of the diffusion-inspired shrinkage functions are among the best for translation-invariant multiscale wavelet denoising. Moreover, by transferring stability notions from diffusion filtering to wavelet shrinkage, we derive conditions on the shrinkage function that ensure that shift invariant single-level Haar wavelet shrinkage is maximum-minimum stable, monotonicity preserving, and variation diminishing

Time-integration schemes for the finite element dynamic Signorini problem

International audienceThe discretization of the dynamic Signorini problem with finite elements in space and a time-stepping scheme is not straightforward. Consequently a large variety of methods for this problem have been designed over the last two decades. Up to date, no systematic comparison of such methods has been performed. The aim of the present work is to classify and compare them. For each method, we discuss the presence of spurious oscillations and the energy conservation. For explicit approaches, the stability condition on the time step is also discussed. Numerical simulations on two 1D benchmark problems with analytical solutions illustrate the properties of the different methods. Most of the discretizations considered herein can be found in the literature, but the semi-explicit modified mass method is new and features, in our opinion, several attractive properties

Anisotropic Diffusion Partial Differential Equations in Multi-Channel Image Processing : Framework and Applications

We review recent methods based on diffusion PDE's (Partial Differential Equations) for the purpose of multi-channel image regularization. Such methods have the ability to smooth multi-channel images anisotropically and can preserve then image contours while removing noise or other undesired local artifacts. We point out the pros and cons of the existing equations, providing at each time a local geometric interpretation of the corresponding processes. We focus then on an alternate and generic tensor-driven formulation, able to regularize images while specifically taking the curvatures of local image structures into account. This particular diffusion PDE variant is actually well suited for the preservation of thin structures and gives regularization results where important image features can be particularly well preserved compared to its competitors. A direct link between this curvature-preserving equation and a continuous formulation of the Line Integral Convolution technique (Cabral and Leedom, 1993) is demonstrated. It allows the design of a very fast and stable numerical scheme which implements the multi-valued regularization method by successive integrations of the pixel values along curved integral lines. Besides, the proposed implementation, based on a fourth-order Runge Kutta numerical integration, can be applied with a subpixel accuracy and preserves then thin image structures much better than classical finite-differences discretizations, usually chosen to implement PDE-based diffusions. We finally illustrate the efficiency of this diffusion PDE's for multi-channel image regularization - in terms of speed and visual quality - with various applications and results on color images, including image denoising, inpainting and edge-preserving interpolation