Multidimensional Compressed Sensing and their Applications by Cesar F. Caiafa,Andrzej Cichocki
Compressed Sensing (CS) comprises a set of relatively new techniques thatexploit the underlying structure of data sets allowing their reconstruction fromcompressed versions or incomplete information. CS reconstruction algorithmsare essentially non-linear, demanding heavy computation load and large storage memory, especially in the case of multidimensional signals. Excellent review papers discussing CS state-of-the-art theory and algorithms already existin the literature which mostly consider data sets in vector forms. In this article,we give an overview of existing techniques with special focus on the treatmentof multidimensional signals (tensors) and discuss recent trends that exploit thenatural multidimensional structure of signals (tensors) achieving simple and efﬁcient CS algorithms. The Kronecker structure of dictionaries is emphasizedand its equivalence to the Tucker tensor decomposition is exploited allowing usto use tensor tools and models for CS. Several examples based on real world multidimensional signals are presented illustrating common problems in signal processing such as: the recovery of signals from compressed measurements for MRI signals or for hyper-spectral imaging, and the tensor completion problem (multidimensional inpainting).
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