## EE4-45 Wavelets and Applications

Lecturer(s): Dr Pier-Luigi Dragotti

Aims:
Wavelets play a pivotal role in many modern research areas such as time-series analysis, image processing/compression and bioengineering just to name a few. The main aim of the course is to introduce students to wavelet theory. Students will understand the link between iterated filter banks and discrete or continuous-time wavelet bases.

Students will learn how to design perfect-reconstruction filter banks and how to relate these constructions to the multi-resolution properties inherent to wavelets. The course will also cover some applications in which wavelets have been successful like image compression.

Learning Outcomes:
Knowledge and understanding
- understanding the fundamentals of wavelet theory
- familiarity with the most commonly used wavelets (e.g Daubechies wavelets)
-understanding how to design perfect reconstruction filter banks
- understanding the link between design of filter banks and construction of discrete and continuous-time bases for efficient signal analysis.

Syllabus:
Part I: Introduction and Background
1. Motivation: Why wavelets, subband coding and multiresolution analysis? Mathematical background.
Hilbert spaces. Unitary operators. Review of Fourier theory. Continuous and discrete time signal
processing.
2. Time-frequency analysis. Multirate signal processing. Projections and approximations.

Part II: Discrete-Time Bases and Filter Banks
3. Elementary filter banks.
Analysis and design of filter banks. Spectral Factorization. Daubechies filters.
4. Orthogonal and biorthogonal filter banks.
Tree structured filter banks. Discrete wavelet transform. Multidimensional filter banks.

Part III: Continuous-Time Bases and Wavelets
5. Iterated filter banks. The Haar and Sinc cases.
The limit of iterated filter banks.
6. Wavelets from Filters. Construction of compactly supported wavelet bases.
Regularity. Approximation properties. Localization.
7. The idea of multiresolution. Multiresolution analysis. Haar as a basis for L2(R). The continuous
wavelet and short-time Fourier transform.

Part IV: Applications
8. Fundamentals of compression. Analysis and design of transform coding systems. Image Compression, the new compression standard (JPEG200) and the old standard. Why is the wavelet transform better than the discrete cosine transform? Advanced topics: Beyond JPEG2000, non-linear approximation and compression.
9. Modern sampling theory: Shannon sampling theorem revisited, sampling parametric not bandlimited signals, multichannel sampling and image super-resolution..

Assessment:

Coursework contribution: 25%

Term: Spring

Closed or Open Book (end of year exam): Closed

Coursework Requirement
To be announced

Oral Exam Required (as final assessment): N/A

Prerequisite: EE3-07 - Digital Signal Processing

Course Homepage: http://www.commsp.ee.ic.ac.uk/~pld/Teaching/