Fundamentals of Wavelets

Jaideva C. Goswami, PhD, is an Engineering Advisor at Schlumberger in Sugarland, Texas. He is also a former professor of Electronics and Communication Engineering at the Indian Institute of Technology, Kharagpur. Dr. Goswami has taught several short courses on wavelets and contributed to the Wiley Encyclopedia of Electrical and Electronics Engineering as well as Wiley Encyclopedia of RF and Microwave Engineering. He has many research papers and patents to his credit, and is a Fellow of IEEE. Andrew K. Chan, PhD, is on the faculty of Texas A&M University and is the coauthor of Wavelets in a Box and Wavelet Toolware. He is a Life Fellow of IEEE.

1.What is this book all about? 2. Mathematical Preliminary.

2.1 Linear Spaces.

2.2 Vectors and Vector Spaces.

2.3 Basis Functions, Orthogonality and Biothogonality.

2.4 Local Basis and Riesz Basis.

2.5 Discrete Linear Normed Space.

2.6 Approximation by Orthogonal Projection.

2.7 Matrix Algebra and Linear Transformation.

2.8 Digital Signals.

2.9 Exercises.

2.10 References.

3. Fourier Analysis.

3.1 Fourier Series.

3.2 Rectified Sine Wave.

3.3 Fourier Transform.

3.4 Properties of Fourier Transform.

3.5 Examples of Fourier Transform.

3.6 Poisson’s Sum and Partition of ZUnity.

3.7 Sampling Theorem.

3.8 Partial Sum and Gibb’s Phenomenon.

3.9 Fourier Analysis of Discrete-Time Signals.

3.10 Discrete Fourier Transform (DFT).

3.11 Exercise.

3.12 References.

4. Time-Frequency Analysis.

4.1 Window Function.

4.2 Short-Time Fourier Transform.

4.3 Discrete Short-Time Fourier Transform.

4.4 Discrete Gabor Representation.

4.5 Continuous Wavelet Transform.

4.6 Discrete Wavelet Transform.

4.7 Wavelet Series.

4.8 Interpretations of the Time-Frequency Plot.

4.9 Wigner-Ville Distribution.

4.10 Properties of Wigner-Ville Distribution.

4.11 Quadratic Superposition Principle.

4.12 Ambiguity Function.

4.13 Exercise.

4.14 Computer Programs.

4.15 References.

5. Multiresolution Anaylsis.

5.1 Multiresolution Spaces.

5.2 Orthogonal, Biothogonal, and Semiorthogonal Decomposition.

5.3 Two-Scale Relations.

5.4 Decomposition Relation.

5.5 Spline Functions and Properties.

5.6 Mapping a Function into MRA Space.

5.7 Exercise.

5.8 Computer Programs.

5.9 References.

6. Construction of Wavelets.

6.1 Necessary Ingredients for Wavelet Construction.

6.2 Construction of Semiorthogonal Spline Wavelets.

6.3 Construction of Orthonormal Wavelets.

6.4 Orthonormal Scaling Functions.

6.5 Construction of Biothogonal Wavelets.

6.6 Graphical Display of Wavelet.

6.7 Exercise.

6.8 Computer Programs.

6.9 References.

7. DWT and Filter Bank Algorithms.

7.1 Decimation and Interpolation.

7.2 Signal Representation in the Approximation Subspace.

7.3 Wavelet Decomposition Algorithm.

7.4 Reconstruction Algorithm.

7.5 Change of Bases.

7.6 Signal Reconstruction in Semiorthogonal Subspaces.

7.7 Examples.

7.8 Two-Channel Perfect Reconstruction Filter Bank.

7.9 Polyphase Representation for Filter Banks.

7.10 Comments on DWT and PR Filter Banks.

7.11 Exercise.

7.12 Computer Program.

7.13 References.

8. Special Topics in Wavelets and Algorithms.

8.1 Fast Integral Wavelet Transform.

8.2 Ridgelet Transform.

8.3 Curvelet Transform.

8.4 Complex Wavelets.

8.5 Lifting Wavelet transform.

8.6 References.

9. Digital Signal Processing Applications.

9.1 Wavelet Packet.

9.2 Wavelet-Packet Algorithms.

9.3 Thresholding.

9.4 Interference Suppression.

9.5 Faulty Bearing Signature Identification.

9.6 Two-Dimensional Wavelets and Wavelet Packets.

9.7 Edge Detection.

9.8 Image Compression.

9.9 Microcalcification Cluster Detection.