Lecture Notes in Statistics

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.