Multidimensional compressed sensing and their applications
Article first published online: 18 OCT 2013
© 2013 John Wiley & Sons, Ltd.
Wiley Interdisciplinary Reviews: Data Mining and Knowledge Discovery
Volume 3, Issue 6, pages 355–380, November/December 2013
How to Cite
Caiafa, C. F. and Cichocki, A. (2013), Multidimensional compressed sensing and their applications. WIREs Data Mining Knowl Discov, 3: 355–380. doi: 10.1002/widm.1108
- Issue published online: 28 OCT 2013
- Article first published online: 18 OCT 2013
- Manuscript Accepted: 10 SEP 2013
- Manuscript Revised: 1 JUL 2013
- Manuscript Received: 7 MAR 2013
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). WIREs Data Mining Knowl Discov 2013, 3:355–380. doi: 10.1002/widm.1108
Conflict of interest: The authors have declared no conflicts of interest for this article.
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