Recovery of localised structure from signals with non-sparse components

We consider the recovery of localised structure from signals consisting of a piecewise constant structure and sparse components. A new algorithm is presented which aims to reconstruct signals of this type from a limited set of observed data. The algorithm is broken down into two subproblems which both involve minimisation of an $l_1$-regularised least squares problem. Numerical results are presented which demonstrate the effectiveness and efficiency of the proposed method. References M. V. Afonso, J. M. Bioucas-Dias, and M. A. T. Figueiredo. A Fast Algorithm for the Constrained Formulation of Compressive Image Reconstruction and Other Linear Inverse Problems. IEEE International Conference on Acoustics Speech and Signal Processing, pages 4034--4037, 2010. doi:10.1109/ICASSP.2010.5495758 R. L. Broughton, I. D. Coope, P. F. Renaud, and R. E. H. Tappenden. A Box Constrained Gradient Projection Algorithm for Compressed Sensing. Signal Processing, 91(8):1985--1992, August 2011. doi:10.1016/j.sigpro.2011.03.003 E. Candes, J. Romberg, and T. Tao. Robust Uncertainty Principles: Exact Signal Reconstruction from Highly Incomplete Frequency Information. IEEE Transactions on Information Theory, 52(2):489--509, 2004. doi:10.1109/TIT.2005.862083 E. J. Candes. Compressive Sampling. In Proceedings of the International Congress of Mathematics, pages 1433--1452, Madrid, Spain, 2006. European Mathematical Society. E. J. Candes and T. Tao. Near-Optimal Signal Recovery From Random Projections and Universal Encoding Strategies. IEEE Transactions on Information Theory, 52(12):5406--5425, December 2006. doi:10.1109/TIT.2006.885507 D. L. Donoho. Compressed Sensing. IEEE Transactions on Information Theory, 52(4):1289--1306, 2006. doi:10.1109/TIT.2006.871582 M. A. T. Figueiredo, R. D. Nowak, and S. J. Wright. Gradient Projection for Sparse Reconstruction: Application to Compressed Sensing and Other Inverse Problems. IEEE Journal of Selected Topics in Signal Processing, 1(4):586--597, December 2007. doi:10.1109/JSTSP.2007.910281 G. H. Golub and C. F. Van Loan. Matrix Computations. The Johns Hopkins University Press, 3 edition, 1996. A. Griffin and P. Tsakalides. Compressed sensing of audio signals using multiple sensors. Lausanne, Switzerland, 25--29 August 2008. EUSIPCO08. P. C. Hansen. {Rank-Deficient and Discrete Ill-Posed Problems}. SIAM, 1998. M. Lustig, D. Donoho, and J. Pauly. Sparse MRI: The Application of Compressed Sensing for Rapid MR Imaging. Magnetic Resonance in Imaging, 58(6):1182--1195, December 2007. doi:10.1002/mrm.21391 J. Provost and F. Lesage. The Application fo Compressed Sensing for Photo-Acoustic Tomography. IEEE Transactions on Medical Imaging, 28(4), April 2009. doi:10.1109/TMI.2008.2007825 J. L. Starck and J. Bobin. Astronomical Data Analysis and Sparsity: From Wavelets to Compressed Sensing. Proceedings of the IEEE, 98(6):1021--1030, 2010. doi:10.1109/JPROC.2009.2025663 H. Yu and G. Wang. Compressed sensing based interior tomography. Physics in Medicine and Biology, pages 2791--2805, 2009. doi:10.1088/0031-9155/54/9/01