Higher Order Variational Methods for Noise Removal in Signals and Images Diploma Thesis

Preface Variational methods have become more and more important in image processing during the last years. They offer an intuitive way to understand the noise removal and image restoration process as minimisation of an energy functional. This energy functional provides the opportunity to compare the quality of two filtered versions of the same input image and thus can be seen as a quality measure. Variational image restoration methods in general yield high quality results. Non-linear approaches are known to suffer from the effect of turning smooth grey value changes in piecewise constant regions. This effect is called stair-casing. One can interpret stair-casing as the result of energy functionals that reward piecewise vanishing first derivatives which yields piecewise constant signals and images. A way to circumvent this would be to use higher derivatives in the penaliser terms: Piecewise vanishing second derivatives would lead to piecewise linear results that could better fit to smooth grey value changes. Higher derivatives could analogously yield piecewise polynomials of higher degree. There is a growing interest in higher order variational methods in the literature. Nielsen, Florack and Deriche (see [18]) have studied linear energy functionals of arbitrary order. They derive basic scale-space properties and give efficient filtering algorithms for the linear case. Greer and Bertozzi further investigate some theoretical properties of special fourth order evolution equations used in image restoration (see [10]). Other authors rather concentrate on the goal of showing the use of higher order methods for practical purposes. Chan, Marquina and Mulet (see [5]) study nonlinear energy functionals of second order and the related diffusion filtering with derivative orders up to four. Total variation methods with the second derivative leading to fourth order partial differential equations are used in [17] by Lysaker, Lundervold and Tai. Besides images they consider one-dimensional signal processing. You and Kaveh [34] also use fourth order PDEs for filtering. Numerical examples are presented which show that higher order methods can in practise be useful for image denoising. The goal of the present thesis is to investigate some theoretical and practical aspects of higher order variational methods. We use different penaliser functions and compare them in practical applications. To introduce variational methods some facts about first order methods are summarised in Chapter 1. In Chapter 2 we start with a continuous higher order energy functional and deduce necessary and sufficient conditions for a function to be a minimiser of this functional. …