Stable Backward Diffusion Models that Minimise Convex Energies

The inverse problem of backward diffusion is known to be ill-posed and highly unstable. Backward diffusion processes appear naturally in image enhancement and deblurring applications. It is therefore greatly desirable to establish a backward diffusion model which implements a smart stabilisation approach that can be used in combination with an easy to handle numerical scheme. So far, existing stabilisation strategies in literature require sophisticated numerics to solve the underlying initial value problem. We derive a class of space-discrete one-dimensional backward diffusion as gradient descent of energies where we gain stability by imposing range constraints. Interestingly, these energies are even convex. Furthermore, we establish a comprehensive theory for the time-continuous evolution and we show that stability carries over to a simple explicit time discretisation of our model. Finally, we confirm the stability and usefulness of our technique in experiments in which we enhance the contrast of digital greyscale and colour images

Stable Backward Diffusion Models that Minimise Convex Energies

The inverse problem of backward diffusion is known to be ill-posed and highly unstable. Backward diffusion processes appear naturally in image enhancement and deblurring applications. It is therefore greatly desirable to establish a backward diffusion model which implements a smart stabilisation approach that can be used in combination with an easy-to-handle numerical scheme. So far, existing stabilisation strategies in the literature require sophisticated numerics to solve the underlying initial value problem. We derive a class of space-discrete one-dimensional backward diffusion as gradient descent of energies where we gain stability by imposing range constraints. Interestingly, these energies are even convex. Furthermore, we establish a comprehensive theory for the time-continuous evolution and we show that stability carries over to a simple explicit time discretisation of our model. Finally, we confirm the stability and usefulness of our technique in experiments in which we enhance the contrast of digital greyscale and colour images

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