Multidimensional compressed sensing and their applications

Compressed sensing (CS) comprises a set of relatively new techniques that exploit the underlying structure of data sets allowing their reconstruction from compressed versions or incomplete information. CS reconstruction algorithms are essentially nonlinear, demanding heavy computation overhead and large storage memory, especially in the case of multidimensional signals. Excellent review papers discussing CS state-of-the-art theory and algorithms already exist in the literature, which mostly consider data sets in vector forms. In this paper, we give an overview of existing techniques with special focus on the treatment of multidimensional signals (tensors). We discuss recent trends that exploit the natural multidimensional structure of signals (tensors) achieving simple and efficient CS algorithms. The Kronecker structure of dictionaries is emphasized and its equivalence to the Tucker tensor decomposition is exploited allowing us to 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 magnetic resonance imaging (MRI) signals or for hyper-spectral imaging, and the tensor completion problem (multidimensional inpainting). © 2013 John Wiley & Sons, Ltd.