Compressed Sensing

Compressed sensing has gained a significant amount of attention since its introduction by Donoho, Candes, Tao and coworkes in 2006, see [Don06, CRT06]. These works show that it is possible to reconstruct discrete signals or images from a very limited number of random measurements. This is a significant improvement over Shannon's sampling theorem [Sha49], which states that stable reconstruction of signals is only possible when the measurements are performed at the so called Nyquist rate, which is twice the maximum frequency of the signal. In the recent years, compressed sensing has been proposed to speed up the measurement process in various medical imaging devices such as MRI, see [LDSP08]. Another prominent example is the single pixel camera proposed in [Bar08] to circumvent the use of expensive high resolution sensors in digital photography.

In our research, we develop compressed sensing techniques for tomographic image reconstruction problems and other imaging problems. The goal is to obtain high quality reconstructions from limited measurements and therefore speeding up the measurement process and to increase the robustness of the reconstruction process with respect to the noise. For example in [SKBBH15], we propose a novel compressed sensing scheme for speeding up data acquisition in photoacoustic tomography.

References:

[CRT06] Emmanuel Candes, Justin Romberg, and Terence Tao. Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information. IEEE Transactions on Information Theory 52, pages 489–509, 2006. 

[Bar08] Richard G Baraniuk. Single-pixel imaging via compressive sampling. IEEE Signal Processing Magazine, 2008. 

[Don06] David L Donoho. Compressed sensing. IEEE Transactions on Information Theory, 52(4):1289–1306, 2006. 

[LDSP08] Michael Lustig, David L Donoho, Juan M Santos, and John M Pauly. Compressed sensing MRI. Signal Processing Magazine, IEEE, 25(2):72–82, 2008. 

[Sha49] Claude E. Shannon. Communication in the presence of noise. Proc. I.R.E., 37, 1949. 

[SKBBH15] M. Sandbichler, F. Krahmer, T. Berer, P. Burgholzer, M. Haltmeier, A Novel Compressed Sensing Scheme for Photoacoustic Tomography, Submitted, 2015