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Wavelet Theory. An Elementary Approach with Applications - Product Image

Wavelet Theory. An Elementary Approach with Applications

  • ID: 2171917
  • November 2009
  • 480 Pages
  • John Wiley and Sons Ltd

A self-contained, elementary introduction to wavelet theory and applications

Exploring the growing relevance of wavelets in the field of mathematics, Wavelet Theory: An Elementary Approach with Applications provides an introduction to the topic, detailing the fundamental concepts and presenting its major impacts in the world beyond academia. Drawing on concepts from calculus and linear algebra, this book helps readers sharpen their mathematical proof writing and reading skills through interesting, real-world applications.

The book begins with a brief introduction to the fundamentals of complex numbers and the space of square-integrable functions. Next, Fourier series and the Fourier transform are presented as tools for understanding wavelet analysis and the study of wavelets in the transform domain. Subsequent chapters provide a comprehensive treatment of various types of wavelets and their related concepts, such as Haar spaces, multiresolution analysis, Daubechies wavelets, and biorthogonal wavelets. In addition, the authors include two chapters that carefully detail the transition from wavelet theory to the discrete wavelet transformations. To illustrate the relevance READ MORE >

Preface.

Acknowledgments.

1 The Complex Plane and the Space L2(R).

1.1 Complex Numbers and Basic Operations.

Problems.

1.2 The Space L2(R).

Problems.

1.3 Inner Products.

Problems.

1.4 Bases and Projections.

Problems.

2 Fourier Series and Fourier Transformations.

2.1 Euler's Formula and the Complex Exponential Function.

Problems.

2.2 Fourier Series.

Problems.

2.3 The Fourier Transform.

Problems.

2.4 Convolution and B-Splines.

Problems.

3 Haar Spaces.

3.1 The Haar Space V0.

Problems.

3.2 The General Haar Space Vj.

Problems.

3.3 The Haar Wavelet Space W0.

Problems.

3.4 The General Haar Wavelet Space Wj.

Problems.

3.5 Decomposition and Reconstruction.

Problems.

3.6 Summary.

4 The Discrete Haar Wavelet Transform and Applications.

4.1 The One-Dimensional Transformation.

Problems.

4.2 The Two-Dimensional Transformation.

Problems.

4.3 Edge Detection and Naive Image Compression.

5 Multiresolution Analysis.

5.1 Multiresolution Analysis.

Problems.

5.2 The View from the Transform Domain.

Problems.

5.3 Examples of Multiresolution Analyses.

Problems.

5.4 Summary.

6 Daubechies Scaling Functions and Wavelets.

6.1 Constructing the Daubechies Scaling Functions.

Problems.

6.2 The Cascade Algorithm.

Problems.

6.3 Orthogonal Translates, Coding and Projections.

Problems.

7 The Discrete Daubechies Transformation and Applications.

7.1 The Discrete Daubechies Wavelet Transform.

Problems.

7.2 Projections and Signal and Image Compression.

Problems.

7.3 Naive Image Segmentation.

Problems.

8 Biorthogonal Scaling Functions and Wavelets.

8.1 A Biorthogonal Example and Duality.

Problems.

8.2 Biorthogonality Conditions for Symbols and Wavelet Spaces.

Problems.

8.3 Biorthogonal Spline Filter Pairs and the CDF97 Filter Pair.

Problems.

8.4 Decomposition and Reconstruction.

Problems.

8.5 The Discrete Biorthogonal Wavelet Transformation.

Problems.

8.6 Riesz Basis Theory.

Problems.

9 Wavelet Packets.

9.1 Constructing Wavelet Packet Functions.

Problems.

9.2 Wavelet Packet Spaces.

Problems.

9.3 The Discrete Packet Transform and Best Basis Algorithm.

Problems.

9.4 The FBI Fingerprint Compression Standard.

Appendix A: Huffman Coding.

Problems.

References.

Topic Index.

Author Index.

"The book, putting emphasize on an analytic facet of wavelets, can be seen as complementary. to the previous Patrick J. Van Fleet's book, DiscreteWavelet Transformations: An Elementary. Approach with Applications, focused on their algebraic properties." (Zentralblatt MATH, 2011)

"Requiring only a prerequisite knowledge of calculus and linear algebra, Wavelet theory is an excellent book for courses in mathematics, engineering, and physics at the upper-undergraduate level. It is also a valuable resource for mathematicians, engineers, and scientists who wish to learn about wavelet theory on an elementary level." (Mathematical Reviews, 2011)

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