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Wavelets from a Statistical Perspective



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ISBN 9781032200675
March 23, 2022 Forthcoming by Chapman and Hall/CRC
352 Pages 84 B/W Illustrations

 
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Book Description

Wavelets from a Statistical Perspective offers a modern, 2nd generation look on wavelets, far beyond the rigid setting of the equispaced, dyadic wavelets in the early days. With the methods of this book, based on the lifting scheme, researchers can set up a wavelet or another multiresolution analysis adapted to their data, ranging from images to scattered data or other irregularly spaced observations. Whereas classical wavelets stand a bit apart from other nonparametric methods, this book adds a multiscale touch to your spline, kernel or local polynomial smoothing procedure, thereby extending its applicability to nonlinear, nonparametric processing for piecewise smooth data.

One of the chapters of the book constructs B-spline wavelets on nonequispaced knots and multiscale local polynomial transforms.  In another chapter, the link between wavelets and Fourier analysis, ubiquitous in the classical approach, is explained, but without being inevitable. In further chapters the discrete wavelet transform is contrasted with the continuous version, the nondecimated (or maximal overlap) transform taking an intermediate position.  An important principle in designing a wavelet analysis through the lifting scheme is finding the right balance between bias and variance.  Bias and variance also play a crucial role in the nonparametric smoothing in a wavelet framework, in finding well working thresholds or other smoothing parameters. The numerous illustrations can be reproduced with the online available, accompanying software. The software and the exercises can also be used as a starting point in the further exploration of the material.

Table of Contents

Chapter 1 Wavelets: nonlinear processing in multiscale sparsity

Chapter 2 Wavelet building blocks

Chapter 3 Using lifting for the design of a wavelet transform

Chapter 4 Wavelet transforms from factored refinement schemes

Chapter 5 Dyadic wavelets

Chapter 6 Dyadic wavelet design in the frequency domain

Chapter 7 Design of dyadic wavelets

Chapter 8 Approximation in a wavelet basis

Chapter 9 Overcomplete wavelet transforms

Chapter 10 Two-dimensional wavelet transforms

Chapter 11 The multiscale local polynomial transform

Chapter 12 Estimation in a wavelet basis

Outlook

References

Subject Index

List of Recurrent symbols

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Author(s)

Biography

Maarten Jansen is professor at the Mathematics and Computer Science departments of the Université libre de Bruxelles.