Joint sparse recovery using optimal combination of compressed sensing and array signal processing
The multiple measurement vector (MMV) problem addresses the identification of unknown input vectors that share common sparse support. Even though MMV problems have been traditionally addressed within the context of sensor array signal processing, the recent trend is to apply compressive sensing (CS) due to its capability to estimate sparse support even with an insufficient number of snapshots, in which case classical array signal processing fails. However, CS guarantees the accurate recovery in a probabilistic manner, which often shows inferior performance in the regime where the traditional array signal processing approaches succeed. The apparent dichotomy between the probabilistic CS and deterministic sensor array signal processing has not been fully understood. Recently, we developed a unified approach that unveils a missing link between CS and array signal processing. The new algorithm, which we call compressive MUSIC, identifies the parts of support using CS, after which the remaining supports are estimated using a novel generalized MUSIC criterion. Using a large system MMV model, we show that our compressive MUSIC requires a smaller number of sensor elements for accurate support recovery than the existing CS methods and that it can approach the optimal L0-bound with finite number of snapshots.
In a multiple measurement vector problem (MMV), where multiple signals share a common sparse support and are sampled by a common sensing matrix, we can expect joint sparsity to enable a further reduction in the number of required measurements. While a diversity gain from joint sparsity had been demonstrated earlier in the case of a convex relaxation method using an l1/l2 mixed norm penalty, only recently was it shown that similar diversity gain can be achieved by greedy algorithms if we combine greedy steps with a MUSIClike subspace criterion. However, the main limitation of these hybrid algorithms is that they often require a large number of snapshots or a high signal-to-noise ratio (SNR) for an accurate subspace as well as partial support estimation. One of the main contributions of this work is to show that the noise robustness of these algorithms can be significantly improved by allowing sequential subspace estimation and support filtering, even when the number of snapshots is insufficient. Numerical simulations show that a novel sequential compressive MUSIC (sequential CS-MUSIC) that combines the sequential subspace estimation and support filtering steps significantly outperforms the existing greedy algorithms and is quite comparable with computationally expensive state-of-art algorithms.
Software Requirements MATLAB (Mathworks, Natick, MA, http://www.mathworks.com).
This research was supported by the Korea Science and Engineering Foundation (KOSEF) grant funded by the Korea government (MEST) (No.2011-0000353).
-This software includes CS-MUSIC and sequential CS-MUSIC
Copyright (c) 2011, Jong Min Kim, Ok Kyun Lee and Jong Chul Ye, KAIST
Joint sparse recovery package is free; you can distribute it and/ or modify it under the terms of the GNU General Public License (GPL) as publishing by the Free Software Foundation, either version 3 of the License, or any later version. For the copy of the GNU General Public License, see (http://www.gnu.org/licenses/).
Additional Note from Authors
You can get the software freely - including source code - by downloading it here. We appreciate if you cite the following papers in producing the results using joint sparse recovery package.
 J. M. Kim, O. K. Lee, and J. C. Ye, Compressive MUSIC: Revisiting the Link Between Compressive Sensing and Array Signal Processing. IEEE Trans. Inf. Theory, vol. 58, no. 1, pp. 278-301, Jan 2012.
 J. M. Kim, O. K. Lee, and J. C. Ye, Compressive MUSIC with optimized partial support for joint sparse recovery. in Proc. IEEE Int. Symp. Inf. Theory (ISIT), 2011.
 J. M. Kim, O. K. Lee, and J. C. Ye, Improving Noise Robustness in Subspace-based Joint Sparse Recovery. IEEE Trans. Signal Process, vol. 60, no. 11, pp. 5799-5809, Nov 2012.
Download and Installation Instructions
In order to download NIRS-SPM software and sample data sets, registration is required. After uncompressing NIRS_SPM_v4_r1.zip, 'NIRS_SPM_v4_r1', 'batch_file', 'nfri_functions', 'Sample_data', and 'Documentation' directories will be created. Then, add 1) NIRS-SPM_v4_r1 directories with sub-folders, 2) spm5 directories with sub-folders in the MATLAB path.
Copyright (c) 2012, Sungho Tak and Jong Chul Ye, KAIST
NIRS-SPM is free software; you can distribute it and/ or modify it under the terms of the GNU General Public License (GPL) as publishing by the Free Software Foundation, either version 3 of the License, or any later version. For the copy of the GNU General Public License, see (http://www.gnu.org/licenses/).
Additional Note from Authors
You can get the software freely - including source code - by downloading it here. We appreciate if you cite the following papers in producing the results using NIRS-SPM toolbox.
 Ye, J.C., Tak, S., Jang, K.E., Jung, J.W., Jang, J.D., 2009. NIRS-SPM: Statistical parametric mapping for near-infrared spectroscopy. NeuroImage 44, 428-447.
 Jang, K.E., Tak, S., Jung, J.W., Jang, J.D., Jeong, Y., and Ye, J.C., 2009. Wavelet-MDL detrending for near infrared spectroscopy (NIRS). J. Biomed. Opt. 14, 1-13.
 Tak, S., Jang, J., Lee. K., and Ye, J.C., 2010. Quantification of CMRO2 without hypercapnia using simultaneous near-infrared spectroscopy and fMRI measurements. Phys. Med. Biol. 55, 3249-3269.
 Tak, S., Yoon, S. J., Jang, J., Yoo, K., Jeong, Y., and Ye, J.C., 2011. Quantitative analysis of hemodynamic and metabolic changes in subcortical vascular dementia using simultaneous near-infrared spectroscopy and fMRI measurements. NeuroImage 55, 176-184.
 Li, H., Tak, S., and Ye, J.C., 2012. Lipschitz-Killing curvature based expected Euler characteristics for p-value correction in fNIRS. J. Neurosci. Meth. 204, 61-67.