Noise Reduction by Wavelet Thresholding

Wavelets is an important research area in statistics with applications to signal processing and compression. This book discusses a fundamental denoising technique called "wavelet shrinkage." The book will be a reference for researchers and graduate students.

Klappentext
This book discusses statistical applications of wavelet theory for use in signal and image processing. The emphasis is on smoothing by wavelet thresholding and extended methods. Wavelet thresholding is an example of non-linear and non-parametric smoothing. The first part discusses theoretical and practical issues concerned with minimum risk thresholding and fast threshold estimation, using generalized cross validation. The extensions in later chapters consider possibilities to exploit three key properties of wavelets in statistics: sparsity, multiresolution, and locality. The author discusses original contributions to problems of correlated noise, scale dependent processing, Bayesian algorithms with geometrical priors (Markov random fields), non-equispaced data, and many other extensions. The point of view lies on the bridge between statistics, signal and image processing, and approximation theory, and the book is accessible for researchers from all of these fields. Most of the material has in mind applications in signal or image processing, and signals and images are used extensively in the illustrations. Nevertheless, the algorithms are quite general in the sense that they could also serve in other regression problems. The book also pays attention to fast algorithms, and Matlab code reproducing many of the illustrations is available for free. Maarten Jansen received a Ph.D. in applied mathematics from the Katholieke Universiteit Leuven, Belgium, in 2000 and currently he is a postdoctoral fellow with the Belgian Foundation for Scientific Research (FWO). He has been a visiting researcher at several institutes, including Stanford University, Bristol University, and Rice University. TOC:Wavelets and wavelet thresholding.- The minimum mean squared error threshold.- Estimating the minimum MSE threshold.- Thresholding and GCV applicability in more realistic situations.- Bayesian correction with geometrical priors for image noise reduction.- Smoothing non-equidistantly spaced data using second generation wavelets and thresholding.

Zusammenfassung
Wavelet methods have become a widely spread tool in signal and image process­ ing tasks. This book deals with statistical applications, especially wavelet based smoothing. The methods described in this text are examples of non-linear and non­ parametric curve fitting. The book aims to contribute to the field both among statis­ ticians and in the application oriented world (including but not limited to signals and images). Although it also contains extensive analyses of some existing methods, it has no intention whatsoever to be a complete overview of the field: the text would show too much bias towards my own algorithms. I rather present new material and own insights in the questions involved with wavelet based noise reduction. On the other hand, the presented material does cover a whole range of methodologies, and in that sense, the book may serve as an introduction into the domain of wavelet smoothing. Throughout the text, three main properties show up ever again: sparsity, locality and multiresolution. Nearly all wavelet based methods exploit at least one of these properties in some or the other way. These notes present research results of the Belgian Programme on Interuniver­ sity Poles of Attraction, initiated by the Belgian State, Prime Minister's Office for Science, Technology and Culture. The scientific responsibility rests with me. My research was financed by a grant (1995 - 1999) from the Flemish Institute for the Promotion of Scientific and Technological Research in the Industry (IWT).

Inhalt

1. Introduction and overview.- 2. Wavelets and wavelet thresholding.- 3. The minimum mean squared error threshold.- 4. Estimating the minimum MSE threshold.- 5. Thresholding and GCV applicability in more realistic situations.- 6. Bayesian correction with geometrical priors for image noise reduction.- 7. Smoothing non-equidistantly spaced data using second generation wavelets and thresholding.