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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).

Inhaltsverzeichnis

Frontmatter

1. Introduction and overview

Abstract
Thanks to the combination of a nice theoretical foundation and the promising applications, wavelets have become a popular tool in many research domains. In fact, wavelet theory combines many existing concepts into a global framework. This new theoretical basis reveals new insights and throws a new light on several domains of applications.
Maarten Jansen

2. Wavelets and wavelet thresholding

Abstract
Every theory starts from an idea. The wavelet idea is simple and clear. At a first confrontation, the mathematics that work out this idea might appear strange and difficult. Nevertheless, after a while, this theory leads to insight in the mechanism in wavelet based algorithms in a variety of applications.
Maarten Jansen

3. The minimum mean squared error threshold

Abstract
This chapter investigates the mean squared error as a criterion for selecting an optimal soft threshold. In applications like image processing, it is often objected that this expression of the error does not always correspond to a more subjective experience of quality. Our visual system, for instance, is much more sensitive to contrast than is expressed by a mean squared error. Nevertheless, even in the image processing world, definitions of signal-to-noise ratio, based on mean squared errors, are commonly used.
Maarten Jansen

4. Estimating the minimum MSE threshold

Abstract
The previous chapter has investigated the behavior of the minimum risk threshold. In practical problems, the mean square error function can never be evaluated exactly, because the uncorrupted coefficients are necessary to compute the error of the output. Therefore, we need to estimate this MSE function.
Maarten Jansen

5. Thresholding and GCV applicability in more realistic situations

Abstract
We have exploited the sparsity of a wavelet representation to motivate thresholding as a curve fitting method and to find GCV as an asymptotically optimal threshold assessment procedure. We explained that sparsity is a sort of smoothness and explained that wavelets are well suited to measure this concept of piecewise smoothness. We have seen that smooth reconstructions, e.g. using the universal threshold, show bias. In a context of image processing, bias is blur. On the other hand, the minimum risk threshold often leads to an output with many spurious structures, ‘blips’, which are remaining noise components.
Maarten Jansen

6. Bayesian correction with geometrical priors for image noise reduction

Abstract
Image processing is not merely a two-dimensional translation of traditional signal processing techniques. The two-dimensional character has some important consequences, such as the existence of line singularities, manifesting as edges. The observations explained in this section also provide the basis for the development of new types of basis functions, such as ridgelets [37].
Maarten Jansen

7. Smoothing non-equidistantly spaced data using second generation wavelets and thresholding

Abstract
A classical (first generation) wavelet transform assumes the input to be a regularly sampled signal. In most applications of digital signal processing or digital image processing, this assumption corresponds to reality. In many other applications however, data are not available on a regular grid, but rather as non-equidistant samples. Examples in this chapter illustrate what happens if we use classical wavelet transforms, pretending that the data are equispaced: the irregularity of the grid is reflected in the output.
Maarten Jansen

Backmatter

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