ELEC97093 (EE9FPN209) Wavelets, Representation Learning and their ApplicationsLecturer(s): Prof PierLuigi Dragotti Aims
Finding useful information in 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 their 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
play a pivotal role in many modern datadriven applications such as timeseries analysis, multimedia processing and applications in the biomedical domain. It is also at the heart of methods to learn the representation of a signal through data. The main aim of the course is to introduce students to wavelet theory. Students will learn about Hilbert spaces and the notions of signal approximation and projection. They will learn about orthogonal, biorthogonal and redundant representations and how to obtain some of these representations using filter banks. Students will learn how to design perfectreconstruction filter banks and how to relate these constructions to the multiresolution properties inherent to wavelets. The course will also cover some applications in which wavelets have been successful like image compression, image superresolution and in neuroscience. Finally, we will use the principles behind wavelet theory to understand representation learning in particular dictionary learning. Learning Outcomes
Knowledge and understanding
 understanding the fundamentals of wavelet theory and Hilbert Spaces  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 continuoustime bases for efficient signal analysis.  basic knowledge of image and video compression principles Dictionary learning 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. Timefrequency analysis. Multirate signal processing. Projections and approximations. Part II: DiscreteTime 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: ContinuousTime 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. 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, nonlinear approximation and compression. 9. Modern sampling theory: Shannon sampling theorem revisited, sampling parametric not bandlimited signals, multichannel sampling and image superresolution. Part V: Advanced topics 10. Dictionary Learning 11 Invertible Neural Networks and the lifting scheme Exam Duration: 3:00hrs 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): ELEC60010  Digital Signal Processing Course Homepage: http://www.commsp.ee.ic.ac.uk/~pld/Teaching/ Book List: Please see Module Reading list
