Signal Recovery from Highly Incomplete Data

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A fundamental problem in science and technology concerns the recovery of an object---a digital signal or image---from incomplete measurements. The examples of such situations are numerous, ranging from the sampling of continuous signals in signal processing, to the measurement of the two-dimensional frequency spectrum of an image as in biomedical imaging. In Magnetic Resonance Imaging (MRI) for instance, one would like to reconstruct high-resolution images from heavily undersampled frequency data as this would allow image acquisition speeds far beyond those offered by current technologies. In all these applications, there are many more unknown signal values than available observations, a situation which a priori seems desperately hopeless.In many instances, the object we wish to recover is known to be structured in the sense that it is sparse or compressible. This means that the unknown object depends upon a smaller number of unknown parameters, with only a few significant entries in some fixed representation. This premise radically changes the problem, making the search for solutions feasible. The research involves a systematic effort to exploit and extend a mathematical breakthrough which shows that it is surprisingly possible to reconstruct such signalsaccurately, and sometimes even exactly, from a limited number of measurements. There are three main outcomes: the development of a coherent and comprehensive knowledge of what can and cannot beexpected from reconstruction strategies based upon incomplete information; the development of flexible and convenient algorithms able to handle large scale problems; the deployment of the resulting new concepts and tools into targeted applications. The initial applicative focus is in the field of Magnetic Resonance angiography, and on the design of a brand new generation of encoding schemes.