Contemporary sampling techniques and compressed sensing (Part III course)
Lecturers: Anders Hansen and Bogdan Roman
Time, location: Tue & Thu 10pm, MR11
This is a (nonexaminable) graduate course on sampling theory and compressed sensing for use in signal processing and imaging. Compressed sensing is a theory of randomisation, sparsity and nonlinear 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,
The following papers will also be useful:
 On asymptotic structure in compressed sensing
 Breaking the coherence barrier: A new theory for compressed sensing

Week 1: Introduction to compressed sensing.
Intro lecture: Slides
Reading list: (1) Ch1 in FH, LMS lecture series by Emmanuel Candes, and "On asymptotic structure in compressed sensing" 
Week 2: Basic compressed sensing: RIP, null space property, uniform recovery, sparsity bases, lower bounds on measurements
Lecture 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 nonlinear 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 4: Slides
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 multilevel/variable density sampling
Lecture : Numerical examples
Lecture 5: Slides
Reading list: Section 4 of Generalized sampling: stable reconstructions, inverse problems and compressed sensing over the continuum and The quest for optimal sampling: computationally efficient, structureexploiting measurements for compressed sensing

Week 5: Magnetic Resonance Imaging and the art of discretizing analog problems
Lecture 6: Slides
Lecture 7: Slides
Reading list: Section 4 of Generalized sampling: stable reconstructions, inverse problems and compressed sensing over the continuum and The quest for optimal sampling: computationally efficient, structureexploiting measurements for compressed sensing 
Week 6: Fluorescence microscopy and Total Variation techniques
Lecture : Fluorescence microscopy and numerical examples
Lecture 8: Slides
Reading list:Total variation paper by Clarice Poon

Week 7: Electron Microscopy and Generalized Sampling
Lecture : Electron microscopy (by Rowan Leary) Slides
Lecture 9: Slides
Reading list: Generalized sampling: stable reconstructions, inverse problems and compressed sensing over the continuum and Sampling 50 years after Shannon 
Week 8: Shearlets and New Challenges in Compressed Sensing
Lecture : Shearlets (by Jackie Ma) Slides
Lecture 10: Slides
Reading list: Generalized sampling: stable reconstructions, inverse problems and compressed sensing over the continuum