基于l1-al2最小化的部分支集已知的信号重建
Signal Reconstruction with Known Partial Support Based on l1-al2 Minimization
摘要: 压缩感知中测量矩阵的限制等距性在一定条件下可以确保重建稀疏信号。文章在l1-al2(0 < a≤1)最小化问题模型下，根据已知信号的先验支撑信息，利用测量矩阵的限制等距性，研究了三种噪声(高斯噪声、脉冲噪声、均匀噪声)情形下信号恢复的充分条件。这些条件直观地揭示了测量矩阵的限制等距性和信号恢复之间的密切关系。
Abstract: The restricted isometry property of the measurement matrix in compressed sensing can ensure the reconstruction of sparse signals under certain conditions. In this paper, the sufficient conditions for signal recovery under three kinds of noise (Gaussian noise, impulse noise and uniform noise) are studied according to the known prior support information of the signal and the restricted isometry property of the measurement matrix under l1-al2(0 < a≤1) minimization model. These conditions intuitively reveal the close relationship between the restricted isometry property of the measurement matrix and signal recovery.
文章引用：武思琪, 宋儒瑛. 基于l1-al2最小化的部分支集已知的信号重建[J]. 应用数学进展, 2022, 11(8): 6015-6028. https://doi.org/10.12677/AAM.2022.118634

参考文献

[1] Candès, E.J., Romberg, J.K. and Tao, T. (2006) Stable Signal Recovery from Incomplete and Inaccurate Measurements. Communications on Pure and Applied Mathematics, 59, 1207-1223. [Google Scholar] [CrossRef] 
[2] Donoho, D.L. (2006) Compressed Sensing. IEEE Transactions on Infor-mation Theory, 52, 1289-1306. [Google Scholar] [CrossRef] 
[3] Donoho, D.L., Elad, M. and Temlyakov, V.N. (2006) Stable Re-covery of Sparse over Complete Representations in the Presence of Noise. IEEE Transactions on Information Theory, 52, 6-18. [Google Scholar] [CrossRef] 
[4] Cai, T.T. and Zhang, A. (2014) Sparse Representation of a Polytope and Recovery of Sparse Signals and Low-Rank Matrices. IEEE Transactions on Information Theory, 60, 122-132. [Google Scholar] [CrossRef] 
[5] Zhang, R. and Li, S. (2018) A Proof of Conjecture on Restricted Isometry Property Constants . IEEE Transactions on Information Theory, 64, 1699-1705. [Google Scholar] [CrossRef] 
[6] Chartrand, R. (2007) Exact Reconstruction of Sparse Signals via Nonconvex Minimization. IEEE Signal Processing Letters, 14, 707-710. [Google Scholar] [CrossRef] 
[7] Chartrand, R. and Staneva, V. (2008) Restrictedisometry Properties and Nonconvex Compressive Sensing. Inverse Problems, 24, Article ID: 035020. [Google Scholar] [CrossRef] 
[8] Zhang, R. and Li, S. (2019) Optimal RIP Bounds for Sparse Signals Recovery via Minimization. Applied and Computational Harmonic Analysis, 47, 566-584. [Google Scholar] [CrossRef] 
[9] Ge, H.M., Wen, J.M. and Chen, W.G. (2018) The Null Space Property of the Truncated -Minimization. IEEE Signal Processing Letters, 25, 1261-1265. [Google Scholar] [CrossRef] 
[10] Lou, Y. and Yan, M. (2016) Fast Minimization via a Proximal Operator. Journal of Scientific Computing, 74, 767-785. [Google Scholar] [CrossRef] 
[11] Wang, W.D. and Wang, J.J. (2019) An Improved Sufficient Con-dition of Minimisation for Robust Signal Recovery. Electronics Letters, 55, 1199-1201. [Google Scholar] [CrossRef] 
[12] Wen, J., Weng, J., Tong, C., Ren, C. and Zhou, Z. (2019) Sparse Signal Recovery with Minimization of 1-Norm Minus 2-Norm. IEEE Transactions on Vehicular Technology, 68, 6847-6854. [Google Scholar] [CrossRef] 
[13] Yan, L., Shin, Y. and Xiu, D. (2017) Sparse Approximation Us-ing Minimization and Its Application to Stochastic Collocation. SIAM Journal on Scientific Computing, 39, A299-A254. [Google Scholar] [CrossRef] 
[14] Ge, H., Chen, W. and Ng, M.K. (2021) New Rip Analysis for Minimization Methods. SIAM Journal on Imaging Sciences, 14, 530-557. [Google Scholar] [CrossRef] 
[15] Lai, M.J., Xu, Y. and Yin, W. (2013) Improved Iteratively Reweighted Least Squares for Unconstrained Smoothed Minimization. SIAM Journal on Numerical Analysis, 51, 927-957. [Google Scholar] [CrossRef] 
[16] Li, P., Chen, W., Ge, H. and Ng, M.K. (2020) Minimization Methods for Signal and Image Reconstruction with Impulsive Noise Removal. Inverse Problems, 36, Article ID: 055009. [Google Scholar] [CrossRef] 
[17] Ge, H. and Li, P. (2022) The Dantzig Selector: Recovery of Signal via Minimization. Inverse Problems, 38, Article ID: 015006. [Google Scholar] [CrossRef] 
[18] Jacques, L. (2010) A Short Note on Compressed Sensing with Par-tially Known Signal Support. Signal Processing, 90, 3308-3312. [Google Scholar] [CrossRef] 
[19] Chen, W.G. and Li, Y.L. (2019) Recovery of Signals under the Condition on RIC and ROC via Prior Support Information. Applied and Computational Harmonic Analysis, 46, 417-430. [Google Scholar] [CrossRef] 
[20] Ince, T., Nacaroglu, A. and Watsuji, N. (2013) Noncon-vex Compressed Sensing with Partially Known Signal Support. Signal Processing, 93, 338-344. [Google Scholar] [CrossRef] 
[21] Blanchard, J.D. and Thompson, A. (2010) On Support Sizes of Restricted Isometry Constants. Applied and Computational Harmonic Analysis, 29, 382-390. [Google Scholar] [CrossRef] 
[22] Donoho, D.L. and Huo, X. (2001) Uncertainty Principles and Ideal Atomic Decomposition. IEEE Transactions on Information Theory, 47, 2845-2862. [Google Scholar] [CrossRef] 
[23] Yin, P., Lou, Y., He, Q. and Xin, J. (2015) Minimization of for Compressed Sensing. SIAM Journal on Scientific Computing, 37, A536-A563. [Google Scholar] [CrossRef] 

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