Ripples in Mathematics

1. Front Matter

   Pages I-IX

   PDF

2. Introduction

   • Arne Jensen, Anders la Cour-Harbo

   Pages 1-5

3. A First Example

   • Arne Jensen, Anders la Cour-Harbo

   Pages 7-10

4. The Discrete Wavelet Transform via Lifting

   • Arne Jensen, Anders la Cour-Harbo

   Pages 11-24

5. Analysis of Synthetic Signals

   • Arne Jensen, Anders la Cour-Harbo

   Pages 25-35

6. Interpretation

   • Arne Jensen, Anders la Cour-Harbo

   Pages 37-50

7. Two Dimensional Transforms

   • Arne Jensen, Anders la Cour-Harbo

   Pages 51-60

8. Lifting and Filters I

   • Arne Jensen, Anders la Cour-Harbo

   Pages 61-86

9. Wavelet Packets

   • Arne Jensen, Anders la Cour-Harbo

   Pages 87-98

10. The Time-Frequency Plane

    • Arne Jensen, Anders la Cour-Harbo

    Pages 99-126

11. Finite Signals

    • Arne Jensen, Anders la Cour-Harbo

    Pages 127-150

12. Implementation

    • Arne Jensen, Anders la Cour-Harbo

    Pages 151-187

13. Lifting and Filters II

    • Arne Jensen, Anders la Cour-Harbo

    Pages 189-209

14. Wavelets in Matlab

    • Arne Jensen, Anders la Cour-Harbo

    Pages 211-232

15. Applications and Outlook

    • Arne Jensen, Anders la Cour-Harbo

    Pages 233-237

16. Back Matter

    Pages 239-246

    PDF

Yet another book on wavelets. There are many books on wavelets available, written for readers with different backgrounds. But the topic is becoming ever more important in mainstream signal processing, since the new JPEG2000 standard is based on wavelet techniques. Wavelet techniques are also impor­ tant in the MPEG-4 standard. So we thought that there might be room for yet another book on wavelets. This one is limited in scope, since it only covers the discrete wavelet trans­ form, which is central in modern digital signal processing. The presentation is based on the lifting technique discovered by W. Sweldens in 1994. Due to a result by I. Daubechies and W. Sweldens from 1996 this approach covers the same class of discrete wavelet transforms as the one based on two channel filter banks with perfect reconstruction. The goal of this book is to enable readers, with modest backgrounds in mathematics, signal analysis, and programming, to understand wavelet based techniques in signal analysis, and perhaps to enable them to apply such methods to real world problems. The book started as a set of lecture notes, written in Danish, for a group of teachers of signal analysis at Danish Engineering Colleges. The material has also been presented to groups of engineers working in industry, and used in mathematics courses at Aalborg University.

• Matrix
• Signal
• Wavelet
• algorithm
• analysis
• electrical engineering
• linear algebra
• signal analysis
• matrix theory

From the reviews:

"This book is a very well-written introduction to discrete wavelet transforms, very convenient for students in electrical engineering, computer science, and applied mathematics. It is based on the lifting approach to discrete wavelet transforms ... . MATLAB is used as the computational environment for examples and implementations of discrete wavelet transforms." (Manfred Tasche, Zentralblatt MATH, Vol. 989 (14), 2002)

"This is an algorithm based, completely elementary introduction to the discrete wavelet transform (DWT) and wavelet packet transform, easy to read and easy to understand, well suited for an introductory course on wavelets for undergraduate students of applied sciences or mathematics. ... Implementations and examples using basic Matlab (TM) as well as the public domain ubi-wave wavelet toolbox help to further a deeper understanding of the algorithms." (C. Cenker, Internationale Mathematische Nachrichten, Vol. 56 (189), 2002)

"This book gives an introduction to the discrete wavelet transform and some of its applications. It is based on a novel approach to discrete wavelets called lifting. ... MATLAB is used as the computational environment for examples and implementation of transforms. The book is well suited for undergraduate mathematics and electrical engineering students and engineers in industry." (ETDE Energy Database, December, 2001)

• Department of Mathematical Sciences, Aalborg University, Aalborg, Denmark

  Arne Jensen

• Department of Control Engineering, Aalborg University, Aalborg, Denmark

  Anders Cour-Harbo

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