Breaking the coherence barrier - A new theory for compressed sensing
Event details
| Date | 17.10.2014 |
| Hour | 14:15 |
| Speaker | Dr. Anders Hansen, University of Cambridge |
| Location | |
| Category | Conferences - Seminars |
Compressed sensing is based on the three pillars: sparsity, incoherence and uniform random subsampling. In addition, the concepts of uniform recovery and the Restricted Isometry Property (RIP) have had a great impact. Intriguingly, in an overwhelming number of inverse problems where compressed sensing is used or can be used (such as MRI, X-ray tomography, Electron microscopy, Reflection seismology etc.) these pillars are absent. Moreover, easy numerical tests reveal that with the successful sampling strategies used in practice one does not observe uniform recovery nor the RIP. In particular, none of the existing theory can explain the success of compressed sensing in a vast area where it is used. In this talk we will demonstrate how real world problems are not sparse, yet asymptotically sparse, coherent, yet asymptotically incoherent, and moreover, that uniform random subsampling yields highly suboptimal results.
Subsequently, we will introduce a new theory that aligns with the actual implementation of compressed sensing that is used in applications. This theory is based on asymptotic sparsity, asymptotic incoherence and multilevel sampling. This theory supports two intriguing phenomena observed in reality: 1. the success of compressed sensing is resolution dependent, 2. the optimal sampling strategy is signal structure dependent. The last point opens up for a whole new area of research, namely the quest for the optimal sampling strategies.
Finally, we will show that by using multilevel sampling, which exploits the structure of the signal, one can outperform random Gaussian/Bernoulli sampling even when the classical $l^1$ recovery algorithm is replaced by modified algorithms which aim to exploit structure such as model based or Bayesian compressed sensing or approximate message passing.
Bio: Dr. Hansen completed his PhD at Cambridge and then moved to Caltech as von Karman Fellow, before returning to Cambridge as a Research Fellow of Homerton College. His research is in computational mathematics and includes applied harmonic analysis, mathematical signal processing with emphasis on sampling theory and compressed sensing as well as spectral theory, computability theory, complexity theory and numerical analysis: a rapidly growing area of mathematics with enormous applications ranging from medical imaging to signal processing.
Subsequently, we will introduce a new theory that aligns with the actual implementation of compressed sensing that is used in applications. This theory is based on asymptotic sparsity, asymptotic incoherence and multilevel sampling. This theory supports two intriguing phenomena observed in reality: 1. the success of compressed sensing is resolution dependent, 2. the optimal sampling strategy is signal structure dependent. The last point opens up for a whole new area of research, namely the quest for the optimal sampling strategies.
Finally, we will show that by using multilevel sampling, which exploits the structure of the signal, one can outperform random Gaussian/Bernoulli sampling even when the classical $l^1$ recovery algorithm is replaced by modified algorithms which aim to exploit structure such as model based or Bayesian compressed sensing or approximate message passing.
Bio: Dr. Hansen completed his PhD at Cambridge and then moved to Caltech as von Karman Fellow, before returning to Cambridge as a Research Fellow of Homerton College. His research is in computational mathematics and includes applied harmonic analysis, mathematical signal processing with emphasis on sampling theory and compressed sensing as well as spectral theory, computability theory, complexity theory and numerical analysis: a rapidly growing area of mathematics with enormous applications ranging from medical imaging to signal processing.
Practical information
- General public
- Free
Organizer
- Ranieri Juri <[email protected]>
Contact
- Ranieri Juri <[email protected]>