Multidimensional compressed sensing and their applications

Abstract

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). Conflict of interest: The authors have declared no conflicts of interest for this article. are based on sampling analog signals at a high Nyquist rate followed by a compression coding stage keeping in memory only the essential information about signals, i.e. storing only the most significant coefficients. Compressed sensing (CS) theory suggests the compelling idea that the sampling process can be greatly simplified by taking only few informative measurements from which full data sets can be reconstructed almost perfectly. 2,3 CS theory has revolutionized signal processing, for instance, new imaging sensor paradigms were developed on the basis of CS, 4-8 and some classical image processing problems were approached using results of CS theory as in the case of denoising, 9-11 inpainting, 12-16 superresolution, 15,17-19 deblurring, 20-23 and others.Originally, CS theory was developed for digital signals, in particular for signals that are mapped to a one-dimensional (1D) array (vector). Powerful reconstruction techniques were developed for relatively small sized vectors which are measured by using random matrices. There are excellent review papers in the literature that cover the state-of-the-art Volume 3,

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