Wavelets Analysis: A Concise Guide

his two-day course is based on a course taught at the Johns Hopkins University Engineering for Professionals Masters’ Degree program, designed to introduce the fundamentals of wavelet analysis to a wide audience of engineers, physicists, and applied mathematicians. It complements the ATI Wavelets: A Conceptual Practical Approach in providing more mathematical depth and detail required for a thorough understanding of the theory and implementation in any programming language (GUI computer code in IDL will be provided to participants).

The textbook Wavelets: A Concise Guide is provided to all attendees.

• Important mathematical structures of signal spaces: orthonormal bases and frames.
• Time, frequency, and scale localizing transforms: the windowed Fourier transform and the continuous wavelet transform, and their implementation.
• Multi-resolution analysis spaces, Haar and Shannon wavelet transforms. Orthogonal and biorthogonal wavelet transforms of compact support: implementation and applications.
• Orthogonal wavelet packets, their implementation, and the best basis algorithm.
• Wavelet transform implementation for 2D images and compression properties.

1. Mathematical structures of signal spaces. Review of important structures in function (signal) spaces required for analysis of signals, leading to orthogonal basis and frame representations and their inversion.
2. Linear time invariant systems. Review linear time invariant systems, convolutions and correlations, spectral factorization for finite length sequences, and perfect reconstruction quadrature mirror filters.
3. Time, frequency and scale localizing transforms. The windowed Fourier transform and the continuous wavelet transform (CWT). Implementation of the CWT.
4. The Harr and Shannon wavelets: two extreme examples of orthogonal wavelet transforms, and corresponding scaling and wavelet equations, and their description in terms of FIR and IIR interscale coefficients.
5. General properties of scaling and wavelet functions. The Haar and Shannon wavelets are seen to be special cases of a more general set of relations defining multi-resolution analysis subspaces that lead to orthogonal and biorthogonal wavelet representations of signals. These relations are examined in both time and frequency domains.
6. The Discrete Wavelet Transform (DWT). The orthogonal discrete wavelet transform applied to finite length sequences, implementation, denoising and thresholding. Implementation of the biorthogonal discrete wavelet transform to finite length sequences.
7. Wavelet Regularity and Solutions. Response of the orthogonal DWT to data discontinuities and wavelet regularity. The Daubechies orthogonal wavelets of compact support. Biorthogonal wavelets of compact support and algebraic methods to solve for them. The lifting scheme to construct biorthogonal wavelets of compact support.
8. Orthogonal Wavelet Packets and the Best Basis Algorithm. Orthogonal wavelet packets and their properties in the time and frequency domains. The minimum entropy best basis algorithm and its implementation.
9. The 2D Wavelet Transform. The DWT applied to 2D (image) data using the product representation, and implementation of the algorithm. Application of the 2D DWT to image compression and comparison with the DCT.

Dr. A. H. Najmi is a staff member of the Johns Hopkins University Applied Physics Laboratory, and a member of the faculty (Applied Physics and Electrical Engineering) of the Johns Hopkins Whiting School Engineering for Professionals Masters’ degree programs. Dr. Najmi holds the degrees of D.Phil. in theoretical physics from the university of Oxford, M.Math., M.A., and B.A. in mathematics from the university of Cambridge. He is the author of the textbook Wavelets: A Concise Guide (Johns Hopkins University Press, 2012).

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