ELEC70039 Wavelets, Representation Learning and their ApplicationsLecturer(s): Prof PierLuigi Dragotti Aims
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 datadriven applications such as timeseries analysis, multimedia processing and applications in the biomedical 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 perfectreconstruction filter banks and how to relate these constructions to the multiresolution properties inherent to wavelets. The module will also cover some applications in which wavelets have been successful like image compression, image superresolution and in neuroscience.
Learning Outcomes
Upon successful completion of this module, you will be able to: 1. design 2channel perfect reconstruction filterbanks  EAm 2. construct a basis and the dual basis for signals belonging to certain subspace  EAm 3. design wavelet transforms with specific properties like number of vanishing moments  EAm 4. compute orthogonal projections in shiftinvariant spaces  EAm
Syllabus
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; Timefrequency analysis; Multirate signal processing; Projections and approximations. Part 2 DiscreteTime 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 ContinuousTime 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; Nonlinear approximation and compression; Modern sampling theory: Shannon sampling theorem revisited; Sampling parametric not bandlimited signals; Multichannel sampling and image superresolution.
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: http://www.commsp.ee.ic.ac.uk/~pld/Teaching/ Book List: Please see Module Reading list
