Second generation wavelets and applications pdf

While this research went on and still goes on , it was soon realized that the concept of time-scale analysis is closely related to the sort of signal processing that happened in so-called Laplacian pyramids. In spite of this immediate success, the classical discrete wavelet transform is somehow limited. A typical example of such a limitation is the assumption that the input has to be a regularly observed sampled signal, where the number of observations is a power of two.

Nevertheless, it seems more interesting if we can incorporate the grid structure and the interval boundaries into the actual construction of the multiresolution analysis. The construction of second-generation wavelets is based on the lifting scheme.

This includes not only the abovementioned irregularly observed data and data on an interval, but nearly any type of structure. One could now think, for instance, about a multiresolution analysis of DNA-molecule structures, large networks, surfaces in computer graphics applications and so on.

Even for common images, it pays off using the lifting scheme, as was illustrated by the new compression standard in JPEG, which makes use of lifting. Lifting also allows one to add a multiscale element to previously developed vi Preface methods.

In statistics, for instance, lifting can be used in combination with existing smoothing methods, such as spline smoothing. Another extension, equally easy to construct with the lifting scheme, are nonlinear and data-adaptive multiscale analyses.

A well known example of a nonlinear multiresolution decomposition is the integer wavelet transform that maps an array of integers to another array of integers.

Data-adaptive multiresolution transforms are extensively discussed further in this book. While the concept of this edgeadaptive multiscale decomposition is quite easy to understand, the resulting convergence rate compares to that of more complicated approaches, such as curvelets, contourlets, edgelets and so on. While people in signal processing are used to illustrations and diagrams, mathematicians prefer clear formulas to think and reason about.

Samples in signal processing correspond to observations in statistics, while a sample in statistical literature is mostly the entire set of observations. Signals in signal processing are functions in mathematics. Yet, we believe that doing so is a key to opening the doors of many papers that otherwise remain unread. As for the mathematical depth of our approach, our point of view is to provide a guide for the exploration of the literature, rather than a fully elaborated mathematical treatment of all aspects of second-generation wavelets.

We therefore omit some of the full, rigourous proofs if they would take too much space and attention. We hope to compose a clear general overview on the topic of second-generation wavelets. Writing and even thinking this book would be a much harder job without the presence of our families. I, Maarten Jansen, would like to express my gratitute to my wife, Gerda Claeskens, of course for her support and patience, but also for her keen curiosity and interest in my work.

Her remarks after careful proofreading my chapters made a substantial improvement in readability. As she is writing a book herself now, I am sure that, if she shows the same enthusiasm there as well, that book is going to be another piece of her work she and I!

Last but not least Preface vii I have to make my apologies to my daughter Sacha for being a less playful dad in the last year. After this last sentence many formulas will be replaced by playtime. Contents 1. Second-generation Wavelets. Nonlinear and Adaptive Lifting. Numerical Condition. Applications of Nonlinear Lifting in Imaging.

Although most readers of this book will be familiar with at least one of these publications, we will start with a wrap up of the wavelet transform as it has been used for the past two decades. Using these transforms we can also write 1. One can think of functions that are compactly supported, like a real window that slides along the signals s.

Mathematically speaking, a desirable property of a wavelet is that an inversion formula exists, i. Condition 1. By polarization, 1. Second Generation Wavelets and Applications introduces "second generation wavelets" and the lifting transform that can be used to apply the traditional benefits of wavelets into a wide range of new areas in signal processing, data processing and computer graphics. This book details the mathematical fundamentals of the lifting transform and illustrates the latest applications of the transform in signal and image processing, numerical analysis, scattering data smoothing and rendering of computer images.

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Second Generation Wavelets and Applications. Authors view affiliations Maarten Jansen Patrick Oonincx. Front Matter Pages i-x. This solution allows to define the second generation wavelet transformation for multiresolution analysis of Umar Faruq 1, Dr. Ramanaiah 2, Dr. Heil, C. Jansen, M. Skip to content. Author : Maarten H.