Contemporary sampling techniques and compressed sensing (Part III course)
This is a (non-examinable) graduate course on sampling theory and compressed sensing for use in signal processing and 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 field 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 focus on how to get compressed sensing to work in real life applications and is aimed at students and post docs who want to learn how compressed sensing can be used in their research.
References: The course will be based on slides and references to the books:
Compressed Sensing (Eldar, Kutyniok), CUP 2012,
A Mathematical Introduction to Compressive Sensing (Foucart, Rauhut), Birkhauser 2014
Week 1: Introduction to compressed sensing.
Week 2: Basic compressed sensing: RIP, null space property, uniform recovery, sparsity bases, lower bounds on measurementsLecture 1: Slides
Lecture 2: Slides
Reading list: Ch1 in EK. For discussions on random Bernoulli and Gaussian matrices, see Chapter 9 of FR.
For discussions on wavelets, see Chapter 6 and 9 (linear and non-linear approximation with wavelet) in "A wavelet tour of signal processing" by Stephane Mallat.
- Week 3: How to make the world sparse: Wavelets, curvelets, contourlets and shearlets
Lecture 3: Slides
Lecture: 1D numerical examples. Download Matlab code (zip archive). The SPGL1 solver by Ewout van den Berg and Michael Friedlander is included in the zip archive.
Reading list: Chapter 6 and 9 in "A wavelet tour of signal processing" by Stephane Mallat. Note that this book also has a section on curvelets.
- Week 4: Welcome to the world of multi-level/variable density sampling
Lecture 4: Slides
Lecture 5: Slides
- Section 4 of Generalized sampling: stable reconstructions, inverse problems and compressed sensing over the continuum
- Breaking the coherence barrier: A new theory for compressed sensing
- Week 5: Magnetic Resonance Imaging and the art of discretizing analog problems
Lecture 6: Slides
Lecture 7: Slides
- Section 4 of Generalized sampling: stable reconstructions, inverse problems and compressed sensing over the continuum.
- Relevant papers: M. Guerquin-Kern, M. Häberlin, K.P. Pruessmann, M. Unser, "A Fast Wavelet-Based Reconstruction Method for Magnetic Resonance Imaging," and M. Lustig, D. Donoho, J. Pauly, "Sparse MRI: The Application of Compressed Sensing for Rapid MR Imaging"
- Week 6: Real-world Applications
Lecture 8: 2D numerical examples (Matlab code to follow)
- How does a confocal microscope and fluorescence microscope work?
- V. Studer, J. Bobin, M. Chahid, H.S. Mousavi, E. Candes, M. Dahan Compressive fluorescence microscopy for biological and hyperspectral imaging