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ELEC70039 Wavelets, Representation Learning and their Applications

Lecturer(s): Prof Pier-Luigi Dragotti


Finding useful information in a huge amount of data is as difficult as finding a needle in a haystack. The key insight of wavelet theory is that by finding alternative representations of signals, it is possible to extract the essential information in a fast and effective way. Wavelet theory provides the tools to find alternative representations of a signal and then to choose the representation which is more appropriate for the task at hand. Because of this flexibility, the wavelet transform plays a pivotal role in many modern data-driven applications such as time-series analysis, multimedia processing and applications in the bio-medical domain. The main aim of the module is to introduce you to wavelet theory. You will learn about Hilbert spaces and the notions of signal approximation and projection. You will learn about orthogonal, biorthogonal and redundant representations and how to obtain some of these representations using filter banks. You will learn how to design perfect-reconstruction filter banks and how to relate these constructions to the multi-resolution properties inherent to wavelets. The module will also cover some applications in which wavelets have been successful like image compression, image super-resolution and in neuroscience.

Learning Outcomes

Upon successful completion of this module, you will be able to: 1. design 2-channel perfect reconstruction filter-banks - EAm 2. construct a basis and the dual basis for signals belonging to certain sub-space - EAm 3. design wavelet transforms with specific properties like number of vanishing moments - EAm 4. compute orthogonal projections in shift-invariant spaces - EAm


Part 1 Introduction and Background: Motivation: Why wavelets, subband coding and multiresolution analysis?; Mathematical background; Hilbert spaces; Unitary operators; Review of Fourier theory; Continuous and discrete time signal processing; Time-frequency analysis; Multirate signal processing; Projections and approximations. Part 2 Discrete-Time Bases and Filter Banks: Elementary filter banks; Analysis and design of filter banks; Spectral Factorization; Daubechies filters; Orthogonal and biorthogonal filter banks; Tree structured filter banks; Discrete wavelet transform; Multidimensional filter banks. Part 3 Continuous-Time Bases and Wavelets: Iterated filter banks; The Haar and Sinc cases; The limit of iterated filter banks; Wavelets from Filters; Construction of compactly supported wavelet bases; Regularity; Approximation properties; Localization; The idea of multiresolution; Multiresolution analysis. Part 4 Applications: 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; Modern sampling theory: Shannon sampling theorem revisited; Sampling parametric not bandlimited signals; Multichannel sampling and image super-resolution.
Exam Duration: 3:00hrs
Exam contribution: 75%
Coursework contribution: 25%

Term: Autumn

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

Coursework Requirement:
         To be announced

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

Prerequisite module(s): None required

Course Homepage:

Book List:
Please see Module Reading list