Contemporary Sampling Techniques and Compressed Sensing - (L24) A. Hansen



Contemporary Sampling Techniques and Compressed Sensing - (L24) A. Hansen

This is a graduate course on sampling theory and compressed sensing for use in signal processing and medical imaging. Compressed sensing is a theory of randomisation, sparsity and non-linear optimisation techniques that breaks traditional barriers in sampling theory. Since its introduction in 2004 the field has exploded and is rapidly growing and changing. Thus, we will take the word contemporary quite literally and emphasise the latest developments, however, no previous knowledge of the eld is assumed. Although the main focus will be on compressed sensing, it will be presented in the general framework of sampling theory. The course will also present related areas of sampling theory such as generalised sampling and sampling at a finite rate of innovation.

Although the course will be rather mathematical, it will be fairly self contained, and applications will be emphasised (in particular, signal processing and Magnetic Resonance Imaging (MRI)). Students from other disciplines than mathematics are encouraged to participate.

Desirable Previous Knowledge
Sampling theory and compressed sensing require a mix of mathematical tools from approximation theory, harmonic analysis, linear algebra, functional analysis, optimisation and probability theory. The course will contain discussions of both finite-dimensional and infinite-dimensional/analog signal models and thus linear algebra, Fourier analysis and functional analysis (at least basic Hilbert space theory) are important. The course will be self-contained, but students are encouraged to refresh their memories on properties of the Fourier transform as well as basic Hilbert space theory. Some basic knowledge of wavelets is useful as well as very basic probability (Bernstein's inequality, Hoe ding's inequality).
Introductory Reading
For a quick and dense review of basic Fourier analysis and functional analysis chapters 5 and 8 of "Real Analysis" (Folland) are good choices. For an introductory exposition to Hilbert space theory one may use "An Introduction to Hilbert Space" (Young). And for a review of wavelets see chapters 1 and 2 of "A First Course on Wavelets" (Hernandez, Weiss). The course will cover some of the chapters of "Compressed Sensing" (Eldar, Kutyniok), so to get a feeling about the topic one may consult chapter 1 as a start.

1. Eldar, Y and Kutyniok, G., Compressed Sensing, CUP
2. Folland, G. B., Real Analysis, Wiley.
3. Hernandez, E. and Weiss, G., A First Course on Wavelets, CRC
4. Young, N., An Introduction to Hilbert Space, CUP

Reading to complement course material
1. Adcock, B and Hansen, A., Stable reconstructions in Hilbert spaces and the resolution of the Gibbs phenomenon, Appl. Comp. Harm. Anal., 32 (2012)
2. Candès, E., Romberg, J. and Tao, T., Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information, IEEE Trans. Inform. Theory 52 (2006)
3. Donoho, D., Compressed sensing, IEEE Trans. Inform. Theory 52 (2006)
4. Körner, T. W., Fourier Analysis, CUP
5. Reed, M. and Simon, B., Functional Analysis, Elsevier

 

 

 

 


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