Second order self-adaptive dynamical system for sparse signal reconstruction and applications to image recovery. (English) Zbl 07701098
Summary: In this article, we consider sparse signal reconstruction problems by an alternative second order self-adaptive dynamical system. By split feasibility problem of sparse signal reconstruction, we introduce a new second order self-adaptive dynamical system. Then, we prove that the proposed system has a unique solution under reasonable conditions. Furthermore, it is shown that the corresponding orbit of the system always converges. Finally, all kinds of numerical results on synthetic data and data from practical problems verify the efficiency of the proposed approach.
MSC:
| 90C30 | Nonlinear programming |
| 47J25 | Iterative procedures involving nonlinear operators |
| 90C25 | Convex programming |
Software:
CoSaMPReferences:
| [1] | Candés, E.; Romberg, J.; Tao, T., Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information, IEEE Trans. Inf. Theory, 52, 2, 489-509 (2006) · Zbl 1231.94017 |
| [2] | Li, J.; Zhou, W., Bregman linearized reweighted alternating minimization for robust sparse recovery, Signal Process., 188, 108194 (2021) |
| [3] | Bruckstein, A. M.; Donoho, D. L.; Elad, M., From sparse solutions of systems of equations to sparse modeling of signals and images, SIAM Rev., 51, 1, 34-81 (2009) · Zbl 1178.68619 |
| [4] | Tan, M.; Tsang, I. W.; Wang, L., Matching pursuit LASSO part II: applications and sparse recovery over batch signals, IEEE Trans. Signal Process., 63, 3, 742-753 (2014) · Zbl 1394.94579 |
| [5] | Byrne, C., A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Probl., 20, 1, 103-120 (2003) · Zbl 1051.65067 |
| [6] | Cands, E. J.; Wakin, M. B., An introduction to compressive sampling, IEEE Signal Process. Mag., 25, 2, 21-30 (2008) |
| [7] | Zhang, H.; Dong, Y.; Fan, Q., Wavelet frame based Poisson noise removal and image deblurring, Signal Process., 137, 363-372 (2017) |
| [8] | Candes, E.; Tao, T., The Dantzig selector: statistical estimation when pis much larger than n, Ann. Stat., 35, 6, 2313-2351 (2007) · Zbl 1139.62019 |
| [9] | Donoho, D. L., For most large underdetermined systems of linear equations the minimal norm solution is also the sparsest solution, Commun. Pure Appl. Math., 59, 6, 797-829 (2006) · Zbl 1113.15004 |
| [10] | Tsaig, Y.; Donoho, D. L., Breakdown of equivalence between the minimal \(l_1\)-norm solution and the sparsest solution, Signal Process., 86, 3, 533-548 (2006) · Zbl 1163.94398 |
| [11] | Liu, Y.; Hu, J., A neural network for \(l_1 - l_2\) minimization based on scaled gradient projection: application to compressed sensing, Neurocomputing, 173, 3, 988-993 (2016) |
| [12] | Tropp, J. A.; Gilbert, A. C., Signal recovery from random measurements via orthogonal matching pursuit, IEEE Trans. Inf. Theory, 53, 12, 4655-4666 (2007) · Zbl 1288.94022 |
| [13] | Needell, D.; Tropp, J. A., CosaMP: iterative signal recovery from incomplete and inaccurate samples, Appl. Comput. Harmon. Anal., 26, 3, 301-321 (2009) · Zbl 1163.94003 |
| [14] | Tropp, J. A., Just relax: convex programming methods for identifying sparse signals in noise, IEEE Trans. Inf. Theory, 52, 3, 1030-1051 (2006) · Zbl 1288.94025 |
| [15] | Byrne, C., Iterative oblique projection onto convex sets and the split feasibility problem, Inverse Probl., 18, 2, 441-453 (2002) · Zbl 0996.65048 |
| [16] | Yang, Q., The relaxed CQ algorithm solving the split feasibility problem, Inverse Probl., 20, 4, 1261-1266 (2004) · Zbl 1066.65047 |
| [17] | Zhao, J.; Yang, Q., Self-adaptive projection methods for the multiple-sets split feasibility problem, Inverse Probl., 27, 3, 035009 (2011) · Zbl 1215.65115 |
| [18] | Polyak, B. T., Some methods of speeding up the convergence of iteration methods, USSR Comput. Math. Math. Phys., 4, 5, 1-17 (1964) · Zbl 0147.35301 |
| [19] | Sahu, D.; Cho, Y.; Dong, Q.; Kashyap, M.; Li, X., Inertial relaxed CQ algorithms for solving a split feasibility problem in Hilbert spaces, Numer. Algorithms, 87, 1075-1095 (2021) · Zbl 1471.65044 |
| [20] | Tan, Z.; Hu, R.; Zhu, M.; Fang, P., A dynamical system method for solving the split convex feasibility problem, J. Ind. Manag. Optim., 13, 5, 2989-3011 (2021) · Zbl 1476.65106 |
| [21] | Boţ, R. I.; Csetnek, E. R., Second order forward-backward dynamical systems for monotone inclusion problems, SIAM J. Control Optim., 54, 3, 1423-1443 (2016) · Zbl 1339.34070 |
| [22] | Zarantonello, E. H., Projections on convex sets in Hilbert space and spectral theory, Contributions to Nonlinear Functional Analysis (1971), Academic Press: Academic Press New York · Zbl 0281.47043 |
| [23] | Boţ, R. I.; Csetnek, E. R., Convergence rates for forward-backward dynamical systems associated with strongly monotone inclusions, J. Math. Anal. Appl., 457, 2, 1135-1152 (2018) · Zbl 1394.37114 |
| [24] | Alvarez, F., On the minimizing property of a second order dissipative system in Hilbert spaces, SIAM J. Control Optim., 38, 4, 1102-1119 (2000) · Zbl 0954.34053 |
| [25] | Antipin, A. S., Minimization of convex functions on convex sets by means of differential equations, Differ. Equ., 30, 9, 1365-1375 (1994) · Zbl 0852.49021 |
| [26] | Zhuang, Y.; Che, H.; Chen, H., A linearly convergent algorithm without prior knowledge of operator norms for solving \(\ell_1 - \ell_2\) minimization, Appl. Math. Lett., 125, 107717 (2022) · Zbl 1503.90096 |
| [27] | Haraux, A., Systémes Dynamiques Dissipatifs et Applications (1991), Elsevier Masson · Zbl 0726.58001 |
| [28] | Abbas, B.; Attouch, H., Dynamical systems and forward-backward algorithms associated with the sum of a convex subdifferential and a monotone cocoercive operator, Optimization, 64, 10, 2223-2252 (2015) · Zbl 1345.34115 |
| [29] | Bauschke, H. H.; Combettes, P. L., Convex Analysis and Monotone Operator Theory in Hilbert Spaces (2011), Springer: Springer New York · Zbl 1218.47001 |
| [30] | López, G.; Martn-Márquez, V.; Wang, F.; Xu, H., Solving the split feasibility problem without prior knowledge of matrix norms, Inverse Probl., 28, 8, 374-389 (2012) · Zbl 1262.90193 |
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