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In the sequel, we will discuss some of the applications, we are currently working on joint with collaborators from those application areas.

Electron Microscopy: This technology allows to image materials at atom level. One problem is as in magnetic resonance imaging the slow data acquisition process. One way to access the data is by taking point samples, which in case of undersampling leads to the inverse problem of inpainting. We already designed efficient approaches for inpainting, which now need to be suitably adapted to the specific types of image data.

Image and Video Compression: Today a major percentage of the data in internet traffic are images and videos. Hence improving current compression algorithms is a crucial task to prevent a collapse in the future. Our approach to this problem is to use applied harmonic analysis methods specifically adapted to images of video sequences.

Magnetic Resonance Imaging: Magnetic resonance imaging is a widely used unharmful method for diagnoses of diseases which are often otherwise not detectable such as myocarditis. However, in particular for young patients the duration of the data acquisition process during which they are not allowed to move is painfully long, up to 1 hour for the upper torso. To speed up the data acquisition, leading to an ill-posed inverse problem, we utilize compressed sensing type methods in combination with suitable systems from applied harmonic analysis such as shearlets.

Massive MIMO: Massive MIMO, i.e., very large scale multiuser multi-antenna technology, is widely expected to play a fundamental role in meeting the target performance oft he future generation of wireless/cellular networks, commonly indicated as 5G. The key idea is that by scaling up the number of jointly processed antennas at the infrastructure side (i.e., in the base stations), the wireless channel, notoriously affected by random propagation effects, converges to a deterministic limit in which the network behaves in a predictable and very desirable manner, where intra-cell multiuser interference can be nulled by precoding, and intra-cell interference can be easily controlled. However, a major obstacle in the implementation of Massive MIMO is represented by the very high complexity of the signal acquisition, requiring to demodulate and sample the output of hundreds of antennas, which we aim to attack by using methods from compressed sensing and sparse approximation.

Proteomics: One key problem in medicine is early diagnosis of, for instance, cancer as one of the most serious diseases. One approach is to analyze so-called proteomics data which is extracted from blood samples of patients and provides information of the set of proteins contained in the blood at a specific time. Since diseases affect the proteins, one aims to find disease fingerprints in the proteomics data to be able to distinguished diseased patients from healthy ones, thereby also allowing very diagnosis at a very early stage of the disease. The approach we pursue is to use 1-bit compressed sensing techniques as one main step to find a suitable (sparse) fingerprints for certain types of cancer.

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