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Randomised approximation with Frames

Final Activity Report Summary - RAWF (Randomised approximation with frames)

This project investigated efficient algorithms for the recovery of digital signals (audio signals, images etc.) from incomplete data. As an important tool randomisation techniques are used. With these investigations the project contributed significantly to the mathematical foundations of the very recent and emerging field of compressive sensing. Potential applications are in signal and image processing, in particular, analogue to digital conversion, signal acquisition, sensor networks, medical imaging, radar, mobile communication and more.

The basic idea in compressive sensing is that many types of signals are sparse. This means that they can be represented with only a few significant terms with respect to a suitable basis of "simple signals", for instance "pure frequencies". Sometimes it is also adequate to assume sparsity with respect to a redundant frame, that is, a generalisation of a basis, which is more flexible. Hence, natural signals do not exhibit the full complexity of all theoretically possible signals, but can rather be described by a small number of significant features. Indeed, this is the reason why data compression standards such as JPEG or MP3 work so well.

In compressed sensing the astonishing fact was observed recently that it is not only possible to use sparsity of natural signals for data compression, but rather that is also possible to exploit the sparsity for measuring (acquiring) signals from a rather small number of measurements. Moreover, the sparse signals can be reconstructed by efficient algorithms. This is potentially useful when it is very expensive, very time-consuming or otherwise very difficult to take measurements of the signal of interest. Such measurements are usually taken at random in a certain sense.

The project contributed to the mathematical analysis of this phenomenon in various settings. As one of the main achievements the fellow showed that a sparse trigonometric polynomial (a superposition of a small number of "pure frequencies") can be recovered efficiently from a small number of random samples. As another achievement it could be shown that also signals that are sparse with respect to a redundant frame can be recovered from random measurements. Furthermore, motivated by data transmission problems in wireless communications, the projected investigated the recovery of an operator modelling the transmission channel from its action on a random transmit signal. Assuming sparsity of this operator in a certain sense, it could indeed be shown that such an efficient reconstruction is possible. This may potentially improve wireless data transmission systems such as mobile phones or WLAN.