Signal and image denoising using inhomogeneous diffusion
Rahel Stichtenoth · DuEPublico (University of Duisburg-Essen) · 2008
A huge amount of data needs to be processed these days.In many fields one wishes to interpret given datasets, which are often corrupted by noise.The development of efficient methods of denoising therefore is a challenging area of research.The need also arises in connection with many applications, e.g.signal processing in measurement and control technique, medical image analysis, spectroscopy and sensors in digital cameras.This thesis is concerned with a new denoising method.We use a nonparametric approach where no prior information on the distribution of the data is assumed, and essentially focus on smooth datasets.In the first chapter we describe some nonparametric regression methods and discuss the problems concerning the selection of the smoothing parameters.In case of datasets with varying smoothness, estimators with a local smoothing parameter are preferred, naturally, to those with one global smoothing parameter.The smoothing parameter of the Nadaraya-Watson kernel estimator for instance can be localized.However it is not suitable for denoising two-dimensional datasets since it takes relatively long to compute it.Similar drawbacks of other known methods are pointed out in Chapter 1, to show that our method can be utilized with advantage.Indeed, not only the computing time is reduced considerably by the use of our method, but also smoother results can be obtained.We introduce the novel diffusion estimator fτ and its localized version fa in the second chapter for the one-dimensional setting.We give a brief description of the finite differences method, which we use to compute fτ and fa by solving particular differential equations numerically.The local smoothing parameter is selected with an iterative algorithm using the so called multiresolution criterion.In each iteration step, a statistical analysis of the residuals is made.The smoothing parameter is adapted such that eventually the residuals contain only the noise, which is to be removed.A balance between the smoothness of the solution and the closeness to the data has to be achieved.It is due to this iterative algorithm that the computational speed is significant.A numerical comparison of our method to other nonparametric regression methods is also presented at the end of Chapter 2. The third chapter is of main interest.It deals with the two-dimensional denoising problem.As the ingredients of our algorithm -the inhomogeneous diffusion process, its numerical solution and the choice of the smoothing parameter -are described in detail in the previous chapter, here the explanation is brief.In the two-dimensional setting, we additionally need a partition to be combined with the multiresolution criterion.This partition is also required to ensure reasonable computing time.Here we present two possible partitions, namely the partition into dyadic squares and the wedge partition.The results are compared, also to other i First of all, I would like to thank my supervisor Laurie Davies for introducing me to the interesting field of image processing and to the fascination of applied mathematics in general.I very much appreciate his excellent supervision and encouragement throughout my PhD studies.I would also like to thank the second supervisor Axel Munk for his willingness to review this thesis.I appreciate his careful reading and his useful comments.The working atmosphere in our group has always been very enjoyable, in particular I am grateful to my colleagues Monika Meise, Christian Höhenrieder, Evgeny Zoldin and Jan Kalina.I always felt that I was welcome to ask questions whenever I needed to.Special thanks goes to Monika Meise for her outstanding commitment.Here I would like to mention that the implementation for the wedge partition is