无约束的$\ell_{2,1}$-分析法重构冗余紧框架下分块稀疏信号的条件

doi:10.12341/jssms21065

系统科学与数学 ›› 2022, Vol. 42 ›› Issue (3): 509-527.DOI: 10.12341/jssms21065

无约束的$\ell_{2,1}$-分析法重构冗余紧框架下分块稀疏信号的条件

刘洋铄, 刘宏宇, 葛焕敏

北京体育大学体育工程学院, 北京 100084

收稿日期:2021-02-02 修回日期:2021-07-17 出版日期:2022-04-20 发布日期:2022-04-20
通讯作者: 葛焕敏,Email:gehuanmin@163.com.
基金资助:
国家自然科学基金（11901037），北京体育大学大学生创新创业训练计划项目（2018）资助课题.

PDF ( 71 ) 引用

刘洋铄, 刘宏宇, 葛焕敏. 无约束的$\ell_{2,1}$-分析法重构冗余紧框架下分块稀疏信号的条件[J]. 系统科学与数学, 2022, 42(3): 509-527.

LIU Yangshuo, LIU Hongyu, GE Huanmin. The Condition for the Recovery of Block Sparse Signal Based on Redundant Tight Frame via the Unconstrained $\ell_{2,1}$-Analysis Method[J]. Journal of Systems Science and Mathematical Sciences, 2022, 42(3): 509-527.

The Condition for the Recovery of Block Sparse Signal Based on Redundant Tight Frame via the Unconstrained $\ell_{2,1}$-Analysis Method

LIU Yangshuo, LIU Hongyu, GE Huanmin

School of Sports Engineering, Beijing Sport University, Beijing 100084

Received:2021-02-02 Revised:2021-07-17 Online:2022-04-20 Published:2022-04-20

文章主要利用分块稀疏信号的凸分解技术分析无约束的$\ell_{2，1}$-分析模型，建立无约束的$\ell_{2，1}$-分析法重构冗余紧框架下分块稀疏信号的条件，其条件基于紧框架下的限制等距性质.首先，利用分块稀疏信号的凸分解技术建立两个重要技术引理.其次，基于发展的两个技术引理建立无约束的$\ell_{2，1}$-分析法恢复冗余紧框架下分块稀疏信号新的恢复条件，其条件基于紧框架下的限制等距性质，改进了现存最好的恢复条件.最后，设计数值实验，说明无约束的$\ell_{2，1}$-分析法重构冗余紧框架下分块稀疏信号的性能.

关键词： 压缩感知, 分块稀疏信号, 无约束的$\ell_{2,1}$-分析法, 紧框架下的块RIP条件

In this paper, we mainly apply the convex decomposition of block sparse signals to analyse the unconstrained $\ell_{2,1}$-analysis model and develop the condition for the recovery of block sparse signals based on redundant tight frames via the unconstrained $\ell_{2,1}$-analysis method, which is based on restricted isometry property under tight frame. We first develop two significant lemmas based on the convex decomposition theory. Second, we build the weak condition based on restricted isometry property under tight frame for the recovery of block sparse signals based on redundant tight frames via the unconstrained $\ell_{2,1}$-analysis method. Last, numerical experiments is established to verify the recovery performance of the unconstrained $\ell_{2,1}$-analysis method.

Key words: Compressed sensing, block sparse signal, the unconstrained $\ell_{2,1}$-analysis method, block RIP condition based on tight frame

MR(2010)主题分类:

分享此文:

[1] Candès E J, Romberg J K, Tao T. Stable signal recovery from incomplete and inaccurate measurements. Comm. Pure Appl. Math., 2006, 59:1207-1223.
[2] 王忠梅, 许克明, 张焕水. 不确定量测联合信号重构的稳定性. 系统科学与数学, 2014, 34(10):1244-1251. (Wang Z M, Xu K M, Zhang H S. Stability of jointly sparse signal recovery with uncertain measurements. Journal of Systems Science and Mathematical Sciences, 2014, 34(10):1244-1251.)
[3] Cai T, Zhang A. Spares representation of a polytope and recovery of sparse signals and low-rank matrices. IEEE Trans. Inform. Theory, 2014, 60(1):122-132.
[4] 张万宏, 张妍, 周维琴. 一种基于野值剔除方法的稀疏重构算法. 系统科学与数学, 2015, 35(9):1018-1027. (Zhang W H, Zhang Y, Zhou W Q. A sparse reconstruction algorithm based on outlier deletion method. Journal of Systems Science and Mathematical Sciences, 2015, 35(9):1018-1027.)
[5] 王建军, 袁建军, 王尧. 基于混合$l_2/l_1$范数极小化方法的块稀疏信号重构条件. 数学学报, 2017, 60(4):619-630. (Wang J J, Yuan J J, Wang R. Improved condition of block-sparse signals recovery via mixed $l_2/l_1$ norm minimization. Acta Mathematica Sinica (Chinese Series), 2017, 60(4):619-630.)
[6] Zhang R, Li S. A proof of conjecture on restricted isometry property constants $\delta_{tk}$ ($0<t<\frac{4}{3}$). IEEE Trans. Inform. Theory, 2018, 64(3):1699-1705.
[7] 聂栋栋, 弓耀玲. 基于近似$l_0$范数的稀疏信号重构. 计算机研究与发展, 2018, 55(5):1090-1096. (Nie D D, Gong Y L. A sparse signal reconstruction algorithm based on approximate $l_0$ norm. Journal of Computer Researcher and Development, 2018, 55(5):1090-1096.)
[8] 周雪芹, 冯象初, 景明利. 一种自适应的拟牛顿投影稀疏信号恢复算法. 西安电子科技大学学报, 2019, 46(3):14-19. (Zhou X Q, Feng X C, Jing M L. Adaptive quasi-newton projection algorithm for sparse recover. Journal of Xidian University, 2019, 46(3):14-19.)
[9] Candès E J, Eldar Y C, Needell D, et al. Compressed sensing with coherent and redundant dictionaries. Appl. Comput. Harmon Anal., 2011, 31(1):59-73.
[10] Liu Y, Mi T, Li S. Compressed sensing with general frames via optimal-dual-based $l_1$-analysis. IEEE Trans. Inform. Theory, 2012, 58(7):4201-4214.
[11] Lin J, Li S, Shen Y. New bounds for resticted isometry constants with coherent tight frames. IEEE Trans. Signal Process, 2013, 61(3):611-621.
[12] Li S, Lin J. Compressed sensing with coherent tight frame via l_q-minimination for 0< q ≤ 1. Inverse Probl. Imaging, 2014, 8(3):761-777.
[13] Zhang R, Li S. Optimal D-RIP bounds in compressed sensing. Acta Math. Sin., 2015, 31(5):755-766.
[14] Chen W G, Li Y L. Stable recovery of signals with the high order D-RIP condition. Acta Math. Sci., 2016, 36B(6):1721-1730.
[15] Lin J H, Li S. Sparse recovery with coherent tight frame via analysis dantzig selector and analysis lasso. Appl. Comput. Harmon Anal., 2014, 37(1):126-139.
[16] Zhao T, Eldar Y C, Beck A, et al. Smoothing and decomposition for analysis sparse recovery. IEEE Trans. Signal Process, 2014, 62(7):1762-1774.
[17] Shen Y, Han B, Braverman E. Stable recovery of analysis based approaches. Appl. Comput. Harmon. Anal., 2015, 39(1):161-172.
[18] Ge H M, Wen J M, Chen W G, et al. Stable sparse recovery with three unconstrained analysis based approaches. http://alpha.math.uga.edu/mjlai/papers/20180126.pdf, 2018.
[19] Wang Y, Wang J J, Xu Z B. A note on block-sparse signals recovery with coherent tight frames. Discrete Dynamics in Nature and Society, 2013, 1-8, ID 905027.
[20] 张枫, 王建军. 基于冗余紧框架的$l_2/l_1$极小化块稀疏压缩感知. 纯粹数学与应用数学, 2019, 35(2):138-150. (Zhang F, Wang J J. Block-sparse compressed densing with redundant tight frames via $l_2/l_1$-minimization. Pure and Applied Mathematics, 2019, 35(2):138-150.)
[21] Chen W G, Li Y L. The high order block RIP condition for signal recovery. Journal of Computational Mathematics, 2019, 37(1):61-75.

[1]	王忠梅，许克明，张焕水. 不确定量测联合信号重构的稳定性[J]. 系统科学与数学, 2014, 34(10): 1244-1251.

阅读次数

全文

摘要