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# Compressed Sensing

Compressed sensing is a novel research area, which was introduced in 2006, and since then has already become a key concept in various areas of applied mathematics, computer science, and electrical engineering. It surprisingly predicts that high-dimensional signals, which allow a sparse approximation [1] by a suitable basis or, more generally, a frame [2], can be recovered from what was previously considered highly incomplete linear measurements, by using efficient algorithms such as convex optimization.

Using a combination of techniques from geometric functional analysis, stochastics, and numerical analysis, many of the fundamental mathematical concepts enabling the surprising performance of compressed sensing could be clarified. However, key questions such as a tensor completion or non-linear extensions of compressed sensing remain wide open.

Already from the very beginning, the area profited from a fruitful interaction of applied mathematicians, computer scientists, and engineers, aiming to explore the potential of this new area for applications. As a result, a number of applications of compressed sensing at different technology readiness levels is available such as tomography, radar technology, ultrasonic location systems, super resolution microscopy, medical imaging, or night vision.

## Some of our Research Topics

- Development and analysis of approaches using compressed sensing for imaging science problems such as inpainting or feature extraction.
- Application of compressed sensing methods to fast data acquisition, for instance, in magnetic reconance tomography.
- Study of the interrelation of machine learning and compressed sensing.
- Application of compressed sensing to certain real-world problems in e.g. medicine (disease fingerprints in proteomics data) and telecommunication (Efficient Implementation of Massive MIMO).
- Investigation of non-standard situations of compressed sensing such as discrete or geometric input data.
- Working on first steps towards non-linear compressed sensing.

## Survey Papers

- H. Boche, R. Calderbank, G.
Kutyniok, and J. Vybiral.
**A Survey of Compressed Sensing.***Compressed Sensing and its Applications, 1-40, BirkhĂ¤user Boston, 2015.*

Download (PDF, 300,4 KB) [3] - G. Kutyniok.
**Compressed Sensing.***DMV Mitteilungen*Download (PDF, 669,0 KB) [4]**22**(2014), 24-29. - M. Davenport, M. Duarte,
Y. Eldar, and G. Kutyniok.

**Introduction to Compressed Sensing.**

*Compressed Sensing: Theory and Applications, 1-64, Cambridge University Press, 2012.*Download (PDF, 819,1 KB) [5]

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iok/Kutyniok/Papers/Chapter1.pdf

iok/Kutyniok/Papers/CompressedSensingDMV.pdf

iok/Kutyniok/Papers/IntroCS.pdf