面向压缩感知的稀疏信号重构算法研究
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摘要
压缩感知(Compressed Sensing, CS)是近年来信号处理领域最热门的研究方向之一,由于其特殊的采样方式可以突破传统奈奎斯特(Nyquist)定理的限制,因此在雷达成像、无线传感器网络、射频通信、医学图像处理、图像设备采集等方面有非常广阔的应用前景。压缩感知的一个重要任务就是对压缩采样后的信号进行重构,目前引起了众多学者的关注和研究。
本文主要从压缩感知基本理论出发,对压缩感知重构算法目前存在的一些问题进行深入研究。从提高信号重构概率、降低复杂度等方面入手,首先对压缩感知标准稀疏信号常用算法进行了总结,尤其针对匹配追踪(Matching Pursuit, MP)类算法进行了详细阐述,然后研究了基于块稀疏信号模型的重构算法,最后研究了面向模拟信息转换(Analog to Information Converter, AIC)的重构算法,并通过仿真实验验证了算法的有效性。
本文的主要研究内容和取得的成果如下:
1.总结部分常用标准稀疏信号重构算法,并进行对比实验,尤其深入研究匹配追踪类算法,重点针对目前正交匹配追踪(Orthogonal Matching Pursuit, OMP)算法中匹配操作采用内积不准确的缺点,提出了一种基于相关系数的修正OMP算法。该算法利用相关系数代替内积进行原子匹配操作,提高了寻找信号支撑集的概率,从而提高了最后信号的重构概率。仿真实验表明该算法在一维信号的重构概率以及二维图像信号的重构信噪比等方面均优于标准OMP算法,具有较好的适用性。
2.研究基于块稀疏模型的信号重构算法,针对大多数块稀疏信号重构算法重构概率低、复杂度高以及所需先验知识多等缺点,提出了三种改进算法。首先引入子空间及回溯思想,提出了一种块稀疏子空间匹配追踪算法。该算法每次迭代对整个信号支撑块进行估计,且利用回溯对上一次估计的信号支撑集进行修正,该算法在复杂度和重构概率方面较多数块稀疏信号重构算法都有提高。然后,本文针对实际中块稀疏度未知的问题,提出了一种块稀疏度自适应迭代重构算法,该算法不需块稀疏度作为先验知识,只需初始化块稀疏度进行迭代,直到估计出块稀疏度和源信号为止。该算法在复杂度方面和原有多数重构算法具有相同的数量级,但重构概率有了提高。最后,本文针对实际中块稀疏度和块大小都未知的情况,提出了分块大小未知的自适应匹配追踪算法,该算法不需要块大小以及块稀疏度的先验知识,只需初始化块大小和块稀疏度,迭代过程中可以交替地估计块大小、块稀疏度和源信号,最后通过残差和估计信号的块稀疏度水平作为算法的终止条件。该算法在复杂度方面比多数算法略有提高,但所需先验知识少,重构概率高,在对实时性要求不太严格的情况下有较好的适用性。本文通过仿真实验验证了三种改进算法在块稀疏信号重构时的有效性。
3.研究面向模拟信息转换的信号重构算法,重点针对多频带信号调制宽带转换器(Modulated Wideband Converter, MWC)采样系统的重构算法进行深入研究。目前对MWC采样系统的重构算法多数采用同步正交匹配追踪(Simultaneous Orthogonal Matching Pursuit, SOMP),针对目前SOMP算法效率低、重构概率不高等缺点,本文提出了一种修正信号支撑频带的同步子空间追踪算法,该算法每次迭代过程中对整个信号支撑频带进行同步估计,并在下一次迭代过程中利用最小均方准则进行估计信号支撑频带的修正,最终确定信号支撑频带,从而重构出源信号。对MWC采样系统重构的仿真实验表明,本文算法在复杂度和重构概率上较SOMP算法都有一定的优势,且本文算法的抗噪性能也较好,具有很好的适用性。
Compressed Sensing (CS) is one emerging hotspot in signal processing. This technology employs a special samlping method which can capture and represent compressible signals at a rate significantly below the Nyquist rate, so there are wide application prospects in the areas of radar image, wireless sensor network (WSN), radio frequency communication, medical image processing, image device collecting and so on. One of the important tasks in CS is how to recover the signals more accurately and effectively, which is concerned by many researchers.
With the basic theory of CS, the recovery algorithms are discussed in this dissertation. Aimed to improve the recovery probability and reducing the complexity, firstly, this dissertation summarizes several recovery algorithms, especially the matching pursuit (MP) type algorithms in detailed. And then, the recovery algorithms for block-sparse signals are studied. Finally, the dissertation researches the recovery algorithms for analog to information converter (AIC). Simulation results demonstrate the effectiveness of our algorithms. The main contents and research contributions of this dissertation are listed as follows:
1.The MP type algorithms especially orthogonal matching pursuit (OMP) are studied. To solve the problem that the support set couldn’t be estimated accurately in standard OMP, a modified OMP using correlation coefficient is proposed. The basic idea of the algorithm is that the support set is searched by calculating correlation coefficients between the sensing matrix and the measurement vector instead of inner product. The correlation coefficient can describe the matching level better than inner product, so the proposed algorithm can determine the support set more accurately, which leads to high recovery probability. Simulation results show the proposed algorithm outperforms the standard OMP both on 1-D and 2-D signals.
2.Block CS recovery algorithms are studied. Focous on the problems that the most existing block CS recovery algorithms are of low accuracy, high complexity and require some prior, this dissertation proposes three improved algorithms. Firstly, based on the subspace and backtracking idea, a block-sparse subspace matching pursuit algorithm has been proposed. The algorithm determines an estimate of the correct support set during each iteration, and the estimate support set will be refined at next iteration using the backtracking. Compared with the most existing algorithms, our algorithm has high recovery probability and low computational complexity. Subsequently, to solve the problem that the most existing recovery algorithms require block sparsity as prior knowledge, a block sparsity adaptive iteration algorithm has been proposed when the block sparsity is unknown. The proposed algorithm initializes a block sparsity which will increase by steps, until the exact support set and original signal are acquired. The complexity of this algorithm equals to some existing block CS recovery algorithms, but this algorithm doesn’t require block sparsity as a prior and has high recovery probability. Finally, for the shortcoming that the most block CS recovery algorithms require the block size and block-sparsity as a prior, this dissertation proposes a block-sparse adaptive matching pursuit algorithm. The most innovation of the proposed algorithm lies in the idea initializing the block size and block sparsity which can alternatively estimate the block size, block sparsity and the target signal. Compared with some existing algorithms, the complexity of this algorithm is a little high, but the proposed algorithm require less prior knowledge, and has high recovery probability, which can be applied in some non-real time problems. Simulation results show that the three presented algorithm is valid in recovery target source.
3 . The recovery algorithms for AIC, especially modulated wideband converter (MWC) of multiband signals are studied. Most conventional MWC recovery algorithms employ simultaneous orthogonal matching pursuit (SOMP), which is ineffient and of low recovery probability. To solve the problem, a recovery algorithm which can refine the frequency support occupies is proposed in this dissertation, termed the simultaneous subspace pursuit. The proposed algorithm can estimate the whole frequency support occupy during each iteration, moreover, the frequency support occupies can be refined during next iteration using least mean square criterion. The recovery signal can be determined when the correct support band will be found. Compared with the SOMP, the proposed algorithm has low computational complexity, high recovery probability and good anti-noise performance. Simulation results for the practice multiband signals demonstrate its good performance.
本文主要从压缩感知基本理论出发,对压缩感知重构算法目前存在的一些问题进行深入研究。从提高信号重构概率、降低复杂度等方面入手,首先对压缩感知标准稀疏信号常用算法进行了总结,尤其针对匹配追踪(Matching Pursuit, MP)类算法进行了详细阐述,然后研究了基于块稀疏信号模型的重构算法,最后研究了面向模拟信息转换(Analog to Information Converter, AIC)的重构算法,并通过仿真实验验证了算法的有效性。
本文的主要研究内容和取得的成果如下:
1.总结部分常用标准稀疏信号重构算法,并进行对比实验,尤其深入研究匹配追踪类算法,重点针对目前正交匹配追踪(Orthogonal Matching Pursuit, OMP)算法中匹配操作采用内积不准确的缺点,提出了一种基于相关系数的修正OMP算法。该算法利用相关系数代替内积进行原子匹配操作,提高了寻找信号支撑集的概率,从而提高了最后信号的重构概率。仿真实验表明该算法在一维信号的重构概率以及二维图像信号的重构信噪比等方面均优于标准OMP算法,具有较好的适用性。
2.研究基于块稀疏模型的信号重构算法,针对大多数块稀疏信号重构算法重构概率低、复杂度高以及所需先验知识多等缺点,提出了三种改进算法。首先引入子空间及回溯思想,提出了一种块稀疏子空间匹配追踪算法。该算法每次迭代对整个信号支撑块进行估计,且利用回溯对上一次估计的信号支撑集进行修正,该算法在复杂度和重构概率方面较多数块稀疏信号重构算法都有提高。然后,本文针对实际中块稀疏度未知的问题,提出了一种块稀疏度自适应迭代重构算法,该算法不需块稀疏度作为先验知识,只需初始化块稀疏度进行迭代,直到估计出块稀疏度和源信号为止。该算法在复杂度方面和原有多数重构算法具有相同的数量级,但重构概率有了提高。最后,本文针对实际中块稀疏度和块大小都未知的情况,提出了分块大小未知的自适应匹配追踪算法,该算法不需要块大小以及块稀疏度的先验知识,只需初始化块大小和块稀疏度,迭代过程中可以交替地估计块大小、块稀疏度和源信号,最后通过残差和估计信号的块稀疏度水平作为算法的终止条件。该算法在复杂度方面比多数算法略有提高,但所需先验知识少,重构概率高,在对实时性要求不太严格的情况下有较好的适用性。本文通过仿真实验验证了三种改进算法在块稀疏信号重构时的有效性。
3.研究面向模拟信息转换的信号重构算法,重点针对多频带信号调制宽带转换器(Modulated Wideband Converter, MWC)采样系统的重构算法进行深入研究。目前对MWC采样系统的重构算法多数采用同步正交匹配追踪(Simultaneous Orthogonal Matching Pursuit, SOMP),针对目前SOMP算法效率低、重构概率不高等缺点,本文提出了一种修正信号支撑频带的同步子空间追踪算法,该算法每次迭代过程中对整个信号支撑频带进行同步估计,并在下一次迭代过程中利用最小均方准则进行估计信号支撑频带的修正,最终确定信号支撑频带,从而重构出源信号。对MWC采样系统重构的仿真实验表明,本文算法在复杂度和重构概率上较SOMP算法都有一定的优势,且本文算法的抗噪性能也较好,具有很好的适用性。
Compressed Sensing (CS) is one emerging hotspot in signal processing. This technology employs a special samlping method which can capture and represent compressible signals at a rate significantly below the Nyquist rate, so there are wide application prospects in the areas of radar image, wireless sensor network (WSN), radio frequency communication, medical image processing, image device collecting and so on. One of the important tasks in CS is how to recover the signals more accurately and effectively, which is concerned by many researchers.
With the basic theory of CS, the recovery algorithms are discussed in this dissertation. Aimed to improve the recovery probability and reducing the complexity, firstly, this dissertation summarizes several recovery algorithms, especially the matching pursuit (MP) type algorithms in detailed. And then, the recovery algorithms for block-sparse signals are studied. Finally, the dissertation researches the recovery algorithms for analog to information converter (AIC). Simulation results demonstrate the effectiveness of our algorithms. The main contents and research contributions of this dissertation are listed as follows:
1.The MP type algorithms especially orthogonal matching pursuit (OMP) are studied. To solve the problem that the support set couldn’t be estimated accurately in standard OMP, a modified OMP using correlation coefficient is proposed. The basic idea of the algorithm is that the support set is searched by calculating correlation coefficients between the sensing matrix and the measurement vector instead of inner product. The correlation coefficient can describe the matching level better than inner product, so the proposed algorithm can determine the support set more accurately, which leads to high recovery probability. Simulation results show the proposed algorithm outperforms the standard OMP both on 1-D and 2-D signals.
2.Block CS recovery algorithms are studied. Focous on the problems that the most existing block CS recovery algorithms are of low accuracy, high complexity and require some prior, this dissertation proposes three improved algorithms. Firstly, based on the subspace and backtracking idea, a block-sparse subspace matching pursuit algorithm has been proposed. The algorithm determines an estimate of the correct support set during each iteration, and the estimate support set will be refined at next iteration using the backtracking. Compared with the most existing algorithms, our algorithm has high recovery probability and low computational complexity. Subsequently, to solve the problem that the most existing recovery algorithms require block sparsity as prior knowledge, a block sparsity adaptive iteration algorithm has been proposed when the block sparsity is unknown. The proposed algorithm initializes a block sparsity which will increase by steps, until the exact support set and original signal are acquired. The complexity of this algorithm equals to some existing block CS recovery algorithms, but this algorithm doesn’t require block sparsity as a prior and has high recovery probability. Finally, for the shortcoming that the most block CS recovery algorithms require the block size and block-sparsity as a prior, this dissertation proposes a block-sparse adaptive matching pursuit algorithm. The most innovation of the proposed algorithm lies in the idea initializing the block size and block sparsity which can alternatively estimate the block size, block sparsity and the target signal. Compared with some existing algorithms, the complexity of this algorithm is a little high, but the proposed algorithm require less prior knowledge, and has high recovery probability, which can be applied in some non-real time problems. Simulation results show that the three presented algorithm is valid in recovery target source.
3 . The recovery algorithms for AIC, especially modulated wideband converter (MWC) of multiband signals are studied. Most conventional MWC recovery algorithms employ simultaneous orthogonal matching pursuit (SOMP), which is ineffient and of low recovery probability. To solve the problem, a recovery algorithm which can refine the frequency support occupies is proposed in this dissertation, termed the simultaneous subspace pursuit. The proposed algorithm can estimate the whole frequency support occupy during each iteration, moreover, the frequency support occupies can be refined during next iteration using least mean square criterion. The recovery signal can be determined when the correct support band will be found. Compared with the SOMP, the proposed algorithm has low computational complexity, high recovery probability and good anti-noise performance. Simulation results for the practice multiband signals demonstrate its good performance.
引文
1 H. Nyquist. Certain topics in telegraph transmission theory. Transactions of the A.I.E.E., 1928: 617~644
2 C. E. Shannon. Communication in the presence of noise. Proc. Institute of Radio Engineers, 1949, 37(1): 10~21
3 E. Candès. Compressive sampling. Proceedings of the International Congress of Mathematicians. Madrid, Spain, 2006, 3: 1433~1452
4 E. Candès, J. Romberg, Terence Tao. Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information. IEEE Transactions on Information Theory, 2006, 52(2): 489~509
5 E. Candès and J. Romberg. Quantitative robust uncertainty principles and optimally sparse decompositions. Foundations of Comput Math, 2006, 6(2): 227~254
6 D. L. Donoho. Compressed sensing. IEEE Transactions on Information Theory, 2006, 52(4): 1289~1306
7石光明,刘丹华,高大华等.压缩感知理论及其研究进展.电子学报, 2009, 37(5): 1070~1081
8 S. G. Mallat, Z. Zhang. Matching pursuits with time-frequency dictionaries. IEEE Transactions on Signal Processing, 1993, 41(12): 3397~3415
9 S. S. Chen, D. L. Donoho, M. A. Saunders. Atomic decomposition by basis pursuit. SIAM Journal of Science Computation, 1998, 20(1): 33~61
10 E. Candès and T. Tao. Near optimal signal recovery from random projections: Universal encoding strategies? IEEE Transactions on Information Theory, 2006, 52(12): 5406~5425
11 E. Candès. Ridgelets: theory and applications. Stanford: Stanford University, 1998
12 E. Candès, D. L. Donoho. Curvelets. USA: Department of Statistics, Stanford University, 1999
13 E. L. Pennec, S. Mallat. Image compression with geometrical wavelets. IEEE International Conference on Image Processing, ICIP’2000. Vancouver, BC: IEEE Computer Society, 2000, 1: 661~664
14 D. L. Donoho, M. Vetterli. Contourlets: A new directional multiresolutionimage representation. Conference Record of the Asilomar Conference on Signals, Systems and Computers. Pacific Groove, CA, United States: IEEE Compute Society, 2002. 1: 497~501
15 J. L. Starck, M. Elad, D. L. Donoho. Image decomposition via the combination of sparse representations and a variational approach. IEEE Transactions on Image Processing, 2005, 14 (10): 1570~1582
16 J. L. Starck, M. Elad, D. Donoho. Redundant multiscale transforms and their application for morphological component analysis. Advances in Imaging and Electron Physics, 2004, 132(82): 287~348
17练秋生,陈书贞.基于混合基稀疏图像表示的压缩传感图像重构.自动化学报, 2010, 36(2): 385~391
18练秋生,陈书贞.基于解析轮廓波的图像稀疏表示及其在压缩传感中的应用.电子学报, 2010, 38(6): 1293~1298
19 X. Qu, X. Cao, D. Guo, C. Hu and Z. Chen. Combined sparsifying transforms for compressed sensing MRI. Electronics Letters, 2010, 46(2): 121~123
20 E. Candès and T. Tao. Decoding by linear programming. IEEE Transactions on Information Theory, 2005, 51(12): 4203~4215
21 R. Baraniuk. A lecture on compressive sensing. IEEE Signal Processing Magazine, 2007, 24 (4): 118~121
22 E. Candès, J. Romberg, Terence Tao. Stable signal recovery from incomplete and inaccurate measurements. Communications on Pure and Applied Mathematics, 2006, 59(8): 1207~1223
23 Y. Tsaig, D. L. Donoho. Extensions of compressed sensing. Signal Processing, 2006, 86(3): 549~571
24 F. Sebert, L. Ying and Y. M. Zou. Toeplitz block matrices in compressed sensing and their applications in imaging. In: Proceedings of International Conference on Technology and Applications in Biomedicine. Washington D. C., USA: IEEE, 2008: 47~50
25 T. T. Do, T. D. Trany, L. Gan. Fast compressive sampling with structurally random matrices. IEEE International Conference on Acoustics, Speech and Signal Processing. Washington D. C., USA: IEEE, 2008: 3369~3372
26 D. L. Donoho. For most large underdetermined systems of linear equations, the minimal l1-norm solution is also the sparsest solution. Communications on Pure and Applied Mathematics, 2006, 59(6): 797~829
27 S. Bhattacharya, T. Blumensath, B. Mulgrew, et al. Fast encoding of synthetic aperture radar raw data using compressed sensing. IEEE Workshop on Statistical Signal Processing. Madison, Wisconsin, 2007: 448~452
28刘记红,徐少坤,高勋章,黎湘,庄钊文.压缩感知雷达成像技术综述.信号处理, 2011, 27(2): 251~260
29 W. Bajwa, J. Haupt, A. Sayeed, et al. Compressive wireless sensing. Proceedings of the fifth International Conference on Information Processing in Sensor Networks, IPSN’06. New York: Association for Computing Machinery. 2006: 134~142
30唐亮,周正,石磊,姚海鹏,张静.基于LEACH和压缩感知的无线传感器网络目标探测.北京邮电大学学报, 2011 (优先出版)
31胡海峰,杨震.无线传感器网络中基于空间相关性的分布式压缩感知.南京邮电大学学报. 2009, 29(6): 12~16
32 D. Takhar, J. Laska, M. Wakin, et al. A new compressive imaging camera architecture using optical2domain compression. Proceedings of SPIE. Bellingham WA: International Society for Optical Engineering. 2006: 60~65
33吴敏,韦志辉,汤黎明,孙玉宝,肖亮.基于稀疏逼近的EEG信号的压缩感知重建研究.中国医疗器械杂志, 2010, 34(4): 241~245
34 M. Lustig, D. L. Donoho, J. M. Pauly. Rapid MR imaging with compressed sensing and randomly under-sampled 3DFT trajectories. Proceedings of the 14th. Annual Meeting of ISMRM. Seattle, WA. 2006
35 F. Parvaresh, H. Vikalo, S. Misra, and B. Hassibi. Recovering sparse signals using sparse measurement matrices in compressed DNA microarrays. IEEE Journal of Selected Topics in Signal Processing, 2008, 2(3): 275~285
36 P. Borgnat, P. Flandrin. Time-frequency localization from sparsity constraints. IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP). Piscataway: Institute of Electrical and Electronics Engineers Inc., 2008: 3785~3788
37 R. Willett, M. Gehm, D. Brady. Multiscale reconstruction for computational spectral imaging. Proceedings of SPIE-The International Society for Optical Engineering, 2007
38 J. W. Ma, F. X. L. Dimet. Deblurring from highly incomplete measurements for remote sensing. IEEE Transactions on Geoscience and Remote Sensing, 2009, 47(3): 792~802
39 F. J. Herrmann, G. Hennenfent. Non-parametric seismic data recovery with curvelet frames. UBC Earth & Ocean Sciences Department Technical Report TR2200721, 2007
40 F. J. Herrmann, D. Wang, G. Hennenfent et al. Curvelet-based seismic data processing: a multiscale and nonlinear approach. Geophysic. 2008, 73(1): A1~A5
41 D. Takhar, V. Bansal, M. Wakin, et al. A compressed sensing camera: New theory and an implementation using digital micromirrors. SPIE Electronic Imaging: Computational Imaging. San Jose. 2006
42 D. Healy. Analog-to-information. 2005, BAA #05-35. Available from http://www.darpa.mil/mto/solicitations/baa05-35/s/index.html.
43 J. A. Tropp, M. B. Wakin, M. F. Duarte, et al. Random filters for compressive sampling and reconstruction. IEEE International Conference on Acoustics, Speech and Signal Processing, Toulouse, France, 2006: 872~875
44 J. Laska, S. Kirolos, Y. Massoud, R.d Baraniuk, A. Gilbert, M. Iwen, and M. Strauss. Random sampling for analog-to-information conversion of wideband signals. IEEE Dallas Circuits and Systems Workshop (DCAS), Richardson, TX, United states, 2006: 119~122
45 S. Kirolos, J. Laska, M. Wakin, M. Duarte, D. Baron, T. Ragheb, Y. Mas- soud, and R. Baraniuk. Analog-to-information conversion via random demodulation. IEEE Dallas Circuits and Systems Workshop (DCAS), Richardson, TX, United states, 2006: 71~74
46 O. Taheri and S. A. Vorobyov. Segmented compressed sampling for analog-to-information conversion. IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP), Aruba, Netherlands, 2009: 113~116
47 O. Taheri and S. A. Vorobyov. Segmented compressed sampling for analog-to-information conversion: Method and performance analysis. IEEE Transactions on Signal Processing, 2011, 59(2): 554~572
48 M. Mishali, Y. C. Eldar, A. Elron. Xampling: Signal Acquisition and Processing in Union of Subspaces. CCIT Report #747 Oct-09, EE Pub No. 1704, EE Dept., Technion; [Online] arXiv 0911.0519, Oct. 2009.
49 M. Mishali, Y. C. Eldar, O. Dounaevsky and E. Shoshan. Xampling: Analog to Digital at Sub-Nyquist Rates. To appear in IET, Circuits, Devices & Systems;CCIT Report #751 Dec-09, EE Pub No. 1708, EE Dept., Technion, Dec. 2009
50 M. Mishali, Y. C. Eldar. Xampling: Analog data compression. Data Compression Conference Proceedings, Snowbird, UT, United states, 2010: 366~375
51 Y. C. Eldar. Compressed sensing of analog signals in shift-invariant spaces. IEEE Transactions on Signal Processing, 2009, 57(8): 2986~2997
52 Y. C. Eldar. Uncertainty relations for shift-invariant analog signals. IEEE Transactions on Information Theory, 2009, 55(12): 5742~5757
53 D. L. Donoho, M. Elad, V. N. Temlyakov. Stable recovery of sparse overcomplete representations in the presence of noise. IEEE Transactions on Information Theory, 2006, 52(1): 6~18
54 E. Candès, J. Romberg. Signal recovery from random projections. Proceedings of SPIE-The International Society for Optical Engineering, 2005: 76~86
55 S. J. Kim, K. Koh, M. Lustig, S. Boyd, D. Gorinevsky. An interior-point method for large-scale l1 regularized least squares. IEEE Journal of Selected Topics in Signal Processing, 2007, 1(4): 606~617
56 D. L. Donoho, Y. Tsaig. Fast Solution of l1-norm minimization problems when the solution may be sparse. Technical Report, Department of Statistics, Stanford University, USA, 2008: 1~45
57 E. Candès, N. Braun, M. B. Wakin. Sparse signal and image recovery from compressive samples. IEEE International Symposium on Biomedical Imaging: From Nano to Macro. Washington D. C., USA: IEEE, 2007: 976~979
58 J. A. Tropp, A. C. Gilbert. Signal recovery from random measurements via orthogonal matching pursuit. IEEE Transactions on Information Theory, 2007, 53(12): 4655~4666
59 D. Needell, R. Vershynin. Greedy signal recovery and uncertainty principles. In: Proceedings of the Conference on Computational Imaging. San Jose, USA: SPIE, 2008: 1~12
60 E. Livshitz. On efficiency of orthogonal matching pursuit. Preprint 2010.
61 Z. Q. Xu. A remark about orthogonal matching pursuit algorithm. 2011 (arXiv:1005.3093)
62 S. S. Huang, J. B. Zhu. Recovery of sparse signals using OMP and its variants: convergence analysis based on RIP. Inverse Problems, 2011, 27(3)
63 C. La, M. N. Do. Signal reconstruction using sparse tree representation.Proceedings of SPIE. San Diego, CA, United States: International Society for Optical Engineering. 2005, 5914: 1~11
64 D. Needell and R. Vershynin. Uniform uncertainty principle and signal recovery via regularized orthogonal matching pursuit. Foundations of Computational Mathematics. 2009, 9: 317~334
65 N. H. Nguyen and T. D. Tran. The stability of regularized orthogonal mathcing pursuit. http://dsp.rice.edu/files/cs/Stability_of_ROMP.pdf
66 D. L. Donoho, Y. Tsaig, I. Drori et al. Sparse solution of underdetermined linear equations by stagewise orthogonal matching pursuit. Technical Report, 2006
67 D. Needell, J. A. Tropp. CoSaMP: Iterative signal recovery from incomplete and inaccurate samples. Applied and Computational Harmonic Analysis, 2009, 26(3): 301~321.
68 W. Dai, O. Milenkovic. Subspace pursuit for compressive sensing signal reconstruction. IEEE Transactions on Information Theory, 2009, 55(5): 2230~2249
69 B. Varadarajan, S. Khudanpur, T. D. Tran. Stepwise optimal subspace pursuit for improving sparse recovery. IEEE Signal Processing Letters, 2011, 18(1): 27~30
70 T. T. Do, L. Gan, N. Nguyen, and T. D. Tran. Sparsity adaptive matching pursuit algorithm for practical compressed sensing. In Proceedings of the 42th Asilomar Conference on Signals, Systems, and Computers, Pacific Grove, California, 2008: 581~587
71高睿,赵瑞珍,胡绍海.基于压缩感知的变步长自适应匹配追踪重建算法.光学学报, 2010, 30(6): 1639~1644
72刘亚新,赵瑞珍,胡绍海,姜春晖.用于压缩感知信号重建的正则化自适应匹配追踪算法.电子与信息学报, 2010, 32(11): 2713~2717
73方红,杨海蓉.基于压缩感知的后退型自适应匹配追踪算法.计算机工程与应用, 2011 (优先出版)
74 M. Fornasier and H. Rauhut. Iterative thresholding algorithms. Applied and Computational Harmonic Analysis, 2008, 25(2): 187~208
75 T. Blumensath and M. E. Davies. Iterative thresholding for sparse approximations. Applied and Computational Harmonic Analysis, 2009, 27(3): 265~274
76 T. Blumensath. Accelerated iterative hard thresholding. IEEE Signal ProcessingLetters. Prepint 2011
77 K. K. Herrity, A. C. Gilbert, J. A. Tropp. Sparse approximation via iterative thresholding. In: Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing. Washington D. C., USA: IEEE, 2006: 624~627
78 R. Chartrand and W. Yin. Iteratively reweighted algorithms for compressive sensing. IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Las Vegas, NV, United states, 2008: 3869~3872
79 I. Daubechies, M. Defrise and C. De Mol. An iterative thresholding algorithm for linear inverse problems with a sparsity constraint, Comm. Pure Appl. Math, 2004, 57(11): 1413~1457
80 H. Mohimani, M. Babaie-Zadeh, C. Jutten. A fast approach for overcomplete sparse decomposition based on smoothed l0 norm. IEEE Transactions on Signal Processing, 2009, 57(1): 289~301
81 E. Candes and J. Romberg. l1-Magic: Recovery of sparse signals via convex programming, 2005
82 L. Rudin, S. Osher, and E. Fatemi. Nonlinear total variation based noise removal algorithms. Physica D: Nonlinear Phenomena, 1992, 60(1): 259~268
83 M. S. Lobo, L. Vandenberghe, S. Boyd, and H. Lebret. Applications of second-order cone programming. Linear Algebra and Its Applications, 1998, 284(1~3): 193~228
84 M. S. Lobo, L. Vandenberghe, S. Boyd. SOCP: Software for second-order cone programming. Information Systems Laboratory, Stanford University, 1997
85 http://cvxr.com/cvx/
86 M. A. T. Fiqueiredo, R. D. Nowak, S. J. Wright. Gradient projection for sparse reconstruction: application to compressed sensing and other inverse problems. IEEE Journal of Selected Topics in Signal Processing, 2007, 1(4): 586~597
87梁瑞宇,邹采容,王青云,张学武.基于自适应次梯度投影算法的压缩感知信号重构.信号处理, 2010, 26(12): 1883~1889
88 T. Blumensath and M. E. Davies. Gradient pursuits. IEEE Transactions on Signal Processing, 2008, 56(6): 2370~2382
89 I. Daubechies, M. Fornasier, I. Loris. Accelerated projected gradient method for linear inverse problems with sparsity constraints. Journal of Fourier Analysis and Applications, 2008, 14(5~6):746~792
90 G. Teschke, C. Borries. Accelerated projected steepest descent method for nonlinear inverse problems with sparsity constraints. Inverse Problem, 2010, 26(2)
91甘伟,许录平,苏哲.一种压缩感知重构算法.电子与信息学报, 2010, 32(9): 2151~2155
92 A. C. Gilbert, M. J. Strauss, J. A. Tropp, et al. Algorithmic linear dimension reduction in the l1 norm for sparse vectors. The 44th Annual Allerton Conference on Communication, Control, and Computing. 2006
93 S. H. Ji, Y. Xue, and L. Carin. Bayesian compressive sensing. IEEE Transactions on Signal Processing, 2008, 56(6): 2346~2356
94 A. C. Gilbert, M. J. Strauss, J. A. Tropp, et al. One sketch for all: Fast algorithms for compressed sensing. Proceedings of the 39th Annual ACM Symposium on Theory of Computing. New York: Association for Computing Machiner. 2007: 237~246
95 S. Gleichman and Y. C. Eldar. Blind compressed sensing: Theory. Latent Variable Analysis and Signal Separation-9th International Conference, LVA/ICA, St. Malo, France, 2010: 386~393
96 R. G. Baraniuk, V. Cevher, M. F. Duarte, and C. Hegde. Model-based compressive sensing. IEEE Transactions on Information Theory, 2010, 56(4): 1982~2001
97 Y. C. Eldar and M. Mishali. Robust recovery of signals from a structured union of subspaces. IEEE Transactions on Information Theory, 2009, 55(11): 5302~5316
98 Y. C. Eldar and H. B?lcski. Block-sparsity: coherence and efficient recovery. IEEE International Conference on Acoustics, Speech, and Signal Processing-Proceedings (ICASSP), Taipei, Taiwan, 2009: 2885~2888
99 Y. C. Eldar, P. Kuppinger, and H. B?lcskei. Compressed sensing of block-sparse signals: uncertainty relations and efficient recovery. IEEE Transactions on Signal Processing, 2010, 58 (6): 3042~3054
100 M. Stojnic, F. Parvaresh and B. Hassibi. On the reconstruction of block-sparse signals with an optimal number of measurements. IEEE Transactions on Signal Processing, 2009, 57(8): 3075~3085
101 D. Baron, M. B. Wakin, M. Duarte, et al. Distributed compressed sensing. 2005 [Online]. Available: http://dsp.rice.edu/cs/DCS112005.pdf
102 T. Ragheb, S. Kirolos, J. Laska, A. Gilbert, M. Strauss, R. Baraniuk, and Y. Massoud. Implementation models for analog-to-information conversion via random sampling. Midwest Symposium on Circuits and Systems (MWSCAS), Montreal, QC, Canada, 2007: 325~328
103 Stephen Pfetsch, Tamer Ragheb, Jason Laska, et al. On the feasibility of hardware implementation of sub-Nyquist random-sampling based analog-to-information conversion. IEEE International Symposium on Circuits and Systems, Seattle, WA, United states, 2008: 1480~1483
104 J. Laska, S. Kirolos, M. Duarte, and T. Ragheb. Theory and implementation of an analog-to-information converter using random demodulation. IEEE International Symposium on Circuits and Systems (ISCAS), New Orleans, LA, United states, 2007: 1959~1962
105 J. A. Tropp, J. N. Laska, M. F. Duarte, J. K. Romberg, and R. G. Baraniuk. Beyond nyquist: Efficient sampling of sparse bandlimited signals. IEEE Transactions on Information Theory, 2010, 56(1): 520~544
106 D. Angelosante, G. B. Giannakis. RLS-weighted Lasso for adaptive estimation of sparse signals. IEEE International Conference on Acoustics, Speech and Signal Processing, Taipei, Taiwan, 2009: 3245~3248
107 M. Mishali, Y. C. Eldar, and J. A. Tropp. Efficient sampling of sparse wideband analog signals. IEEE Convention of Electrical and Electronics Engineers in Israel, Proceedings, Eilat, Israel, 2008: 290~294
108 M. Mishali, Y. C. Eldar. From theory to practice: Sub-Nyquist sampling of sparse wideband analog signals. IEEE Journal on Selected Topics in Signal Processing, 2010, 4(2): 375~391
109 M. Mishali and Y. C. Eldar. Reduce and boost: Recovering arbitrary sets of jointly sparse vectors. IEEE Transactions on Signal Processing, 2008, 56(10): 4692~4702
110 J. A. Tropp, A. C. Gilbert, and M. J. Strauss. Simultaneous sparse approximation via greedy pursuit. IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Philadelphia, PA, United states, 2005: 721~724
111 J. A. Tropp, A. C. Gilbert, and M. J. Strauss. Algorithms for simultaneous sparse approximation. Part I: Greedy pursuit. Signal Process, 2006, 86(3): 572~588
112 J. A. Tropp, A. C. Gilbert, and M. J. Strauss. Algorithms for simultaneous sparse approximation. Part II: Convex relaxation. Signal Process, 2006, 86(3): 589~602
113 Y. C. Eldar and T. Michaeli. Beyond bandlimited sampling: A review of nonlinearities, smoothness, and sparsity. IEEE Signal Processing Magazine, 2009, 26(3): 48~68
114 D. L. Donoho, Y. Tsaig. Extension of compressed sensing. Signal Processing, 2006, 86(3): 544~548
115 http://sparselab.stanford.edu/
116 Golub, G. H. and F. V. Loan. Matrix Computations (3rd ed.). John Hopkins University, 1996
117 R. A. DeVore and V. N. Temlyakov. Some remarks on greedy algorithms. Advances in Computational Mathematics, 1996: 173~187
118 E. Candès, R. Mark, T. Tao, and R. Vershynin. Error correction via linear programming. IEEE Symposium on Foundations of Computer Science (FOCS), 2005: 295~308
119 G. H. Qu, D. L. Zhang, and P. Yan. Information measure for performance of image fusion. Electronics Letters, 2002, 38(7): 313~315
120 M. Mishali, Y. C. Eldar. Blind multi-band signal reconstruction: Compressed sensing for analog signals. IEEE Transactions on Signal Processing, 2009, 57(3): 993~1009
121 H. J. Landau. Necessary density conditions for sampling and interpolation of certain entire functions. Acta Math, 1967, 117(1): 37~52
122 S. F. Cotter, B. D. Rao, K. Engan, and K. Kreutz-Delgado. Sparse solutions to linear inverse problems with multiple measurement vectors. IEEE Transactions on Signal Processing, 2005, 53(7): 2477~2488
123 Y. M. Lu and M. N. Do. A theory for sampling signals from a union of subspaces. IEEE Transactions on Signal Processing, 2008, 56(6): 2334~2345
124 J. Tropp. Greedy is good: Algorithmic results for sparse approximation. IEEE Transactions on Information Theory, 2004, 50(10): 2231~2242
125 D. L. Donodo and X. Huo. Uncertainty principles and ideal atomic decompositions. IEEE Transactions on Information Theory, 2001, 47(7): 2845~2862
126 Y. C. Eldar, M. Mishali. Block sparsity and sampling over a union of subspaces.DSP 2009: 16th International Conference on Digital Signal Processing, Proceedings, Santorini, Greece, 2009
127 Y. Han, C. Hartmann, and C.-C. Chen. Efficient priority-first search maximum-likelihood soft-decision decoding of linear block codes. IEEE Transactions on Information Theory, 1993, 39(5): 1514~1523
128 R. G. Vaughan, N. L. Scott, and D. R. White. The theory of bandpass sampling. IEEE Transactions on Signal Processing, 1991, 39(9): 1973~1984
129 Y.-P. Lin and P. P. Vaidyanathan. Periodically nonuniform sampling of bandpass signals. IEEE Transactions Circuits Syst. II, 1998, 45(3): 340~351
130 R. Venkataramani and Y. Bresler. Perfect reconstruction formulas and bounds on aliasing error in sub-Nyquist nonuniform sampling of multiband signals. IEEE Transactions on Information Theory, 2000, 46(6): 2173~2183
131 G. L. Fudge, R. E. Bland, M. A. Chivers, S. Ravindran, J. Haupt, and P. E. Pace. A Nyquist folding analog-to-information receiver. In Proc. 42nd Asilomar Conf. on Signals, Systems and Computers, Pacific Grove, CA, United states, 2008: 541~545
132 Z. Yu, S. Hoyos, and B. M. Sadler. Mixed-signal parallel compressed sensing and reception for cognitive radio. IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Las Vegas, NV, United states, 2008: 3861~3864
133 M. Mishali, Y. C. Eldar, O. Dounaevsky, E. Shoshan. Xampling: Analog to digital at sub-Nyquist rates. IET Circuits, Devices and Systems, 2011, 5(1): 8~20
134 M. Mishali, Y. C. Eldar. Expected RIP: Conditioning of The Modulated Wideband Converter. IEEE Information Theory Workshop (ITW), Taormina, Sicily, Italy, 2009: 343~347
2 C. E. Shannon. Communication in the presence of noise. Proc. Institute of Radio Engineers, 1949, 37(1): 10~21
3 E. Candès. Compressive sampling. Proceedings of the International Congress of Mathematicians. Madrid, Spain, 2006, 3: 1433~1452
4 E. Candès, J. Romberg, Terence Tao. Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information. IEEE Transactions on Information Theory, 2006, 52(2): 489~509
5 E. Candès and J. Romberg. Quantitative robust uncertainty principles and optimally sparse decompositions. Foundations of Comput Math, 2006, 6(2): 227~254
6 D. L. Donoho. Compressed sensing. IEEE Transactions on Information Theory, 2006, 52(4): 1289~1306
7石光明,刘丹华,高大华等.压缩感知理论及其研究进展.电子学报, 2009, 37(5): 1070~1081
8 S. G. Mallat, Z. Zhang. Matching pursuits with time-frequency dictionaries. IEEE Transactions on Signal Processing, 1993, 41(12): 3397~3415
9 S. S. Chen, D. L. Donoho, M. A. Saunders. Atomic decomposition by basis pursuit. SIAM Journal of Science Computation, 1998, 20(1): 33~61
10 E. Candès and T. Tao. Near optimal signal recovery from random projections: Universal encoding strategies? IEEE Transactions on Information Theory, 2006, 52(12): 5406~5425
11 E. Candès. Ridgelets: theory and applications. Stanford: Stanford University, 1998
12 E. Candès, D. L. Donoho. Curvelets. USA: Department of Statistics, Stanford University, 1999
13 E. L. Pennec, S. Mallat. Image compression with geometrical wavelets. IEEE International Conference on Image Processing, ICIP’2000. Vancouver, BC: IEEE Computer Society, 2000, 1: 661~664
14 D. L. Donoho, M. Vetterli. Contourlets: A new directional multiresolutionimage representation. Conference Record of the Asilomar Conference on Signals, Systems and Computers. Pacific Groove, CA, United States: IEEE Compute Society, 2002. 1: 497~501
15 J. L. Starck, M. Elad, D. L. Donoho. Image decomposition via the combination of sparse representations and a variational approach. IEEE Transactions on Image Processing, 2005, 14 (10): 1570~1582
16 J. L. Starck, M. Elad, D. Donoho. Redundant multiscale transforms and their application for morphological component analysis. Advances in Imaging and Electron Physics, 2004, 132(82): 287~348
17练秋生,陈书贞.基于混合基稀疏图像表示的压缩传感图像重构.自动化学报, 2010, 36(2): 385~391
18练秋生,陈书贞.基于解析轮廓波的图像稀疏表示及其在压缩传感中的应用.电子学报, 2010, 38(6): 1293~1298
19 X. Qu, X. Cao, D. Guo, C. Hu and Z. Chen. Combined sparsifying transforms for compressed sensing MRI. Electronics Letters, 2010, 46(2): 121~123
20 E. Candès and T. Tao. Decoding by linear programming. IEEE Transactions on Information Theory, 2005, 51(12): 4203~4215
21 R. Baraniuk. A lecture on compressive sensing. IEEE Signal Processing Magazine, 2007, 24 (4): 118~121
22 E. Candès, J. Romberg, Terence Tao. Stable signal recovery from incomplete and inaccurate measurements. Communications on Pure and Applied Mathematics, 2006, 59(8): 1207~1223
23 Y. Tsaig, D. L. Donoho. Extensions of compressed sensing. Signal Processing, 2006, 86(3): 549~571
24 F. Sebert, L. Ying and Y. M. Zou. Toeplitz block matrices in compressed sensing and their applications in imaging. In: Proceedings of International Conference on Technology and Applications in Biomedicine. Washington D. C., USA: IEEE, 2008: 47~50
25 T. T. Do, T. D. Trany, L. Gan. Fast compressive sampling with structurally random matrices. IEEE International Conference on Acoustics, Speech and Signal Processing. Washington D. C., USA: IEEE, 2008: 3369~3372
26 D. L. Donoho. For most large underdetermined systems of linear equations, the minimal l1-norm solution is also the sparsest solution. Communications on Pure and Applied Mathematics, 2006, 59(6): 797~829
27 S. Bhattacharya, T. Blumensath, B. Mulgrew, et al. Fast encoding of synthetic aperture radar raw data using compressed sensing. IEEE Workshop on Statistical Signal Processing. Madison, Wisconsin, 2007: 448~452
28刘记红,徐少坤,高勋章,黎湘,庄钊文.压缩感知雷达成像技术综述.信号处理, 2011, 27(2): 251~260
29 W. Bajwa, J. Haupt, A. Sayeed, et al. Compressive wireless sensing. Proceedings of the fifth International Conference on Information Processing in Sensor Networks, IPSN’06. New York: Association for Computing Machinery. 2006: 134~142
30唐亮,周正,石磊,姚海鹏,张静.基于LEACH和压缩感知的无线传感器网络目标探测.北京邮电大学学报, 2011 (优先出版)
31胡海峰,杨震.无线传感器网络中基于空间相关性的分布式压缩感知.南京邮电大学学报. 2009, 29(6): 12~16
32 D. Takhar, J. Laska, M. Wakin, et al. A new compressive imaging camera architecture using optical2domain compression. Proceedings of SPIE. Bellingham WA: International Society for Optical Engineering. 2006: 60~65
33吴敏,韦志辉,汤黎明,孙玉宝,肖亮.基于稀疏逼近的EEG信号的压缩感知重建研究.中国医疗器械杂志, 2010, 34(4): 241~245
34 M. Lustig, D. L. Donoho, J. M. Pauly. Rapid MR imaging with compressed sensing and randomly under-sampled 3DFT trajectories. Proceedings of the 14th. Annual Meeting of ISMRM. Seattle, WA. 2006
35 F. Parvaresh, H. Vikalo, S. Misra, and B. Hassibi. Recovering sparse signals using sparse measurement matrices in compressed DNA microarrays. IEEE Journal of Selected Topics in Signal Processing, 2008, 2(3): 275~285
36 P. Borgnat, P. Flandrin. Time-frequency localization from sparsity constraints. IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP). Piscataway: Institute of Electrical and Electronics Engineers Inc., 2008: 3785~3788
37 R. Willett, M. Gehm, D. Brady. Multiscale reconstruction for computational spectral imaging. Proceedings of SPIE-The International Society for Optical Engineering, 2007
38 J. W. Ma, F. X. L. Dimet. Deblurring from highly incomplete measurements for remote sensing. IEEE Transactions on Geoscience and Remote Sensing, 2009, 47(3): 792~802
39 F. J. Herrmann, G. Hennenfent. Non-parametric seismic data recovery with curvelet frames. UBC Earth & Ocean Sciences Department Technical Report TR2200721, 2007
40 F. J. Herrmann, D. Wang, G. Hennenfent et al. Curvelet-based seismic data processing: a multiscale and nonlinear approach. Geophysic. 2008, 73(1): A1~A5
41 D. Takhar, V. Bansal, M. Wakin, et al. A compressed sensing camera: New theory and an implementation using digital micromirrors. SPIE Electronic Imaging: Computational Imaging. San Jose. 2006
42 D. Healy. Analog-to-information. 2005, BAA #05-35. Available from http://www.darpa.mil/mto/solicitations/baa05-35/s/index.html.
43 J. A. Tropp, M. B. Wakin, M. F. Duarte, et al. Random filters for compressive sampling and reconstruction. IEEE International Conference on Acoustics, Speech and Signal Processing, Toulouse, France, 2006: 872~875
44 J. Laska, S. Kirolos, Y. Massoud, R.d Baraniuk, A. Gilbert, M. Iwen, and M. Strauss. Random sampling for analog-to-information conversion of wideband signals. IEEE Dallas Circuits and Systems Workshop (DCAS), Richardson, TX, United states, 2006: 119~122
45 S. Kirolos, J. Laska, M. Wakin, M. Duarte, D. Baron, T. Ragheb, Y. Mas- soud, and R. Baraniuk. Analog-to-information conversion via random demodulation. IEEE Dallas Circuits and Systems Workshop (DCAS), Richardson, TX, United states, 2006: 71~74
46 O. Taheri and S. A. Vorobyov. Segmented compressed sampling for analog-to-information conversion. IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP), Aruba, Netherlands, 2009: 113~116
47 O. Taheri and S. A. Vorobyov. Segmented compressed sampling for analog-to-information conversion: Method and performance analysis. IEEE Transactions on Signal Processing, 2011, 59(2): 554~572
48 M. Mishali, Y. C. Eldar, A. Elron. Xampling: Signal Acquisition and Processing in Union of Subspaces. CCIT Report #747 Oct-09, EE Pub No. 1704, EE Dept., Technion; [Online] arXiv 0911.0519, Oct. 2009.
49 M. Mishali, Y. C. Eldar, O. Dounaevsky and E. Shoshan. Xampling: Analog to Digital at Sub-Nyquist Rates. To appear in IET, Circuits, Devices & Systems;CCIT Report #751 Dec-09, EE Pub No. 1708, EE Dept., Technion, Dec. 2009
50 M. Mishali, Y. C. Eldar. Xampling: Analog data compression. Data Compression Conference Proceedings, Snowbird, UT, United states, 2010: 366~375
51 Y. C. Eldar. Compressed sensing of analog signals in shift-invariant spaces. IEEE Transactions on Signal Processing, 2009, 57(8): 2986~2997
52 Y. C. Eldar. Uncertainty relations for shift-invariant analog signals. IEEE Transactions on Information Theory, 2009, 55(12): 5742~5757
53 D. L. Donoho, M. Elad, V. N. Temlyakov. Stable recovery of sparse overcomplete representations in the presence of noise. IEEE Transactions on Information Theory, 2006, 52(1): 6~18
54 E. Candès, J. Romberg. Signal recovery from random projections. Proceedings of SPIE-The International Society for Optical Engineering, 2005: 76~86
55 S. J. Kim, K. Koh, M. Lustig, S. Boyd, D. Gorinevsky. An interior-point method for large-scale l1 regularized least squares. IEEE Journal of Selected Topics in Signal Processing, 2007, 1(4): 606~617
56 D. L. Donoho, Y. Tsaig. Fast Solution of l1-norm minimization problems when the solution may be sparse. Technical Report, Department of Statistics, Stanford University, USA, 2008: 1~45
57 E. Candès, N. Braun, M. B. Wakin. Sparse signal and image recovery from compressive samples. IEEE International Symposium on Biomedical Imaging: From Nano to Macro. Washington D. C., USA: IEEE, 2007: 976~979
58 J. A. Tropp, A. C. Gilbert. Signal recovery from random measurements via orthogonal matching pursuit. IEEE Transactions on Information Theory, 2007, 53(12): 4655~4666
59 D. Needell, R. Vershynin. Greedy signal recovery and uncertainty principles. In: Proceedings of the Conference on Computational Imaging. San Jose, USA: SPIE, 2008: 1~12
60 E. Livshitz. On efficiency of orthogonal matching pursuit. Preprint 2010.
61 Z. Q. Xu. A remark about orthogonal matching pursuit algorithm. 2011 (arXiv:1005.3093)
62 S. S. Huang, J. B. Zhu. Recovery of sparse signals using OMP and its variants: convergence analysis based on RIP. Inverse Problems, 2011, 27(3)
63 C. La, M. N. Do. Signal reconstruction using sparse tree representation.Proceedings of SPIE. San Diego, CA, United States: International Society for Optical Engineering. 2005, 5914: 1~11
64 D. Needell and R. Vershynin. Uniform uncertainty principle and signal recovery via regularized orthogonal matching pursuit. Foundations of Computational Mathematics. 2009, 9: 317~334
65 N. H. Nguyen and T. D. Tran. The stability of regularized orthogonal mathcing pursuit. http://dsp.rice.edu/files/cs/Stability_of_ROMP.pdf
66 D. L. Donoho, Y. Tsaig, I. Drori et al. Sparse solution of underdetermined linear equations by stagewise orthogonal matching pursuit. Technical Report, 2006
67 D. Needell, J. A. Tropp. CoSaMP: Iterative signal recovery from incomplete and inaccurate samples. Applied and Computational Harmonic Analysis, 2009, 26(3): 301~321.
68 W. Dai, O. Milenkovic. Subspace pursuit for compressive sensing signal reconstruction. IEEE Transactions on Information Theory, 2009, 55(5): 2230~2249
69 B. Varadarajan, S. Khudanpur, T. D. Tran. Stepwise optimal subspace pursuit for improving sparse recovery. IEEE Signal Processing Letters, 2011, 18(1): 27~30
70 T. T. Do, L. Gan, N. Nguyen, and T. D. Tran. Sparsity adaptive matching pursuit algorithm for practical compressed sensing. In Proceedings of the 42th Asilomar Conference on Signals, Systems, and Computers, Pacific Grove, California, 2008: 581~587
71高睿,赵瑞珍,胡绍海.基于压缩感知的变步长自适应匹配追踪重建算法.光学学报, 2010, 30(6): 1639~1644
72刘亚新,赵瑞珍,胡绍海,姜春晖.用于压缩感知信号重建的正则化自适应匹配追踪算法.电子与信息学报, 2010, 32(11): 2713~2717
73方红,杨海蓉.基于压缩感知的后退型自适应匹配追踪算法.计算机工程与应用, 2011 (优先出版)
74 M. Fornasier and H. Rauhut. Iterative thresholding algorithms. Applied and Computational Harmonic Analysis, 2008, 25(2): 187~208
75 T. Blumensath and M. E. Davies. Iterative thresholding for sparse approximations. Applied and Computational Harmonic Analysis, 2009, 27(3): 265~274
76 T. Blumensath. Accelerated iterative hard thresholding. IEEE Signal ProcessingLetters. Prepint 2011
77 K. K. Herrity, A. C. Gilbert, J. A. Tropp. Sparse approximation via iterative thresholding. In: Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing. Washington D. C., USA: IEEE, 2006: 624~627
78 R. Chartrand and W. Yin. Iteratively reweighted algorithms for compressive sensing. IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Las Vegas, NV, United states, 2008: 3869~3872
79 I. Daubechies, M. Defrise and C. De Mol. An iterative thresholding algorithm for linear inverse problems with a sparsity constraint, Comm. Pure Appl. Math, 2004, 57(11): 1413~1457
80 H. Mohimani, M. Babaie-Zadeh, C. Jutten. A fast approach for overcomplete sparse decomposition based on smoothed l0 norm. IEEE Transactions on Signal Processing, 2009, 57(1): 289~301
81 E. Candes and J. Romberg. l1-Magic: Recovery of sparse signals via convex programming, 2005
82 L. Rudin, S. Osher, and E. Fatemi. Nonlinear total variation based noise removal algorithms. Physica D: Nonlinear Phenomena, 1992, 60(1): 259~268
83 M. S. Lobo, L. Vandenberghe, S. Boyd, and H. Lebret. Applications of second-order cone programming. Linear Algebra and Its Applications, 1998, 284(1~3): 193~228
84 M. S. Lobo, L. Vandenberghe, S. Boyd. SOCP: Software for second-order cone programming. Information Systems Laboratory, Stanford University, 1997
85 http://cvxr.com/cvx/
86 M. A. T. Fiqueiredo, R. D. Nowak, S. J. Wright. Gradient projection for sparse reconstruction: application to compressed sensing and other inverse problems. IEEE Journal of Selected Topics in Signal Processing, 2007, 1(4): 586~597
87梁瑞宇,邹采容,王青云,张学武.基于自适应次梯度投影算法的压缩感知信号重构.信号处理, 2010, 26(12): 1883~1889
88 T. Blumensath and M. E. Davies. Gradient pursuits. IEEE Transactions on Signal Processing, 2008, 56(6): 2370~2382
89 I. Daubechies, M. Fornasier, I. Loris. Accelerated projected gradient method for linear inverse problems with sparsity constraints. Journal of Fourier Analysis and Applications, 2008, 14(5~6):746~792
90 G. Teschke, C. Borries. Accelerated projected steepest descent method for nonlinear inverse problems with sparsity constraints. Inverse Problem, 2010, 26(2)
91甘伟,许录平,苏哲.一种压缩感知重构算法.电子与信息学报, 2010, 32(9): 2151~2155
92 A. C. Gilbert, M. J. Strauss, J. A. Tropp, et al. Algorithmic linear dimension reduction in the l1 norm for sparse vectors. The 44th Annual Allerton Conference on Communication, Control, and Computing. 2006
93 S. H. Ji, Y. Xue, and L. Carin. Bayesian compressive sensing. IEEE Transactions on Signal Processing, 2008, 56(6): 2346~2356
94 A. C. Gilbert, M. J. Strauss, J. A. Tropp, et al. One sketch for all: Fast algorithms for compressed sensing. Proceedings of the 39th Annual ACM Symposium on Theory of Computing. New York: Association for Computing Machiner. 2007: 237~246
95 S. Gleichman and Y. C. Eldar. Blind compressed sensing: Theory. Latent Variable Analysis and Signal Separation-9th International Conference, LVA/ICA, St. Malo, France, 2010: 386~393
96 R. G. Baraniuk, V. Cevher, M. F. Duarte, and C. Hegde. Model-based compressive sensing. IEEE Transactions on Information Theory, 2010, 56(4): 1982~2001
97 Y. C. Eldar and M. Mishali. Robust recovery of signals from a structured union of subspaces. IEEE Transactions on Information Theory, 2009, 55(11): 5302~5316
98 Y. C. Eldar and H. B?lcski. Block-sparsity: coherence and efficient recovery. IEEE International Conference on Acoustics, Speech, and Signal Processing-Proceedings (ICASSP), Taipei, Taiwan, 2009: 2885~2888
99 Y. C. Eldar, P. Kuppinger, and H. B?lcskei. Compressed sensing of block-sparse signals: uncertainty relations and efficient recovery. IEEE Transactions on Signal Processing, 2010, 58 (6): 3042~3054
100 M. Stojnic, F. Parvaresh and B. Hassibi. On the reconstruction of block-sparse signals with an optimal number of measurements. IEEE Transactions on Signal Processing, 2009, 57(8): 3075~3085
101 D. Baron, M. B. Wakin, M. Duarte, et al. Distributed compressed sensing. 2005 [Online]. Available: http://dsp.rice.edu/cs/DCS112005.pdf
102 T. Ragheb, S. Kirolos, J. Laska, A. Gilbert, M. Strauss, R. Baraniuk, and Y. Massoud. Implementation models for analog-to-information conversion via random sampling. Midwest Symposium on Circuits and Systems (MWSCAS), Montreal, QC, Canada, 2007: 325~328
103 Stephen Pfetsch, Tamer Ragheb, Jason Laska, et al. On the feasibility of hardware implementation of sub-Nyquist random-sampling based analog-to-information conversion. IEEE International Symposium on Circuits and Systems, Seattle, WA, United states, 2008: 1480~1483
104 J. Laska, S. Kirolos, M. Duarte, and T. Ragheb. Theory and implementation of an analog-to-information converter using random demodulation. IEEE International Symposium on Circuits and Systems (ISCAS), New Orleans, LA, United states, 2007: 1959~1962
105 J. A. Tropp, J. N. Laska, M. F. Duarte, J. K. Romberg, and R. G. Baraniuk. Beyond nyquist: Efficient sampling of sparse bandlimited signals. IEEE Transactions on Information Theory, 2010, 56(1): 520~544
106 D. Angelosante, G. B. Giannakis. RLS-weighted Lasso for adaptive estimation of sparse signals. IEEE International Conference on Acoustics, Speech and Signal Processing, Taipei, Taiwan, 2009: 3245~3248
107 M. Mishali, Y. C. Eldar, and J. A. Tropp. Efficient sampling of sparse wideband analog signals. IEEE Convention of Electrical and Electronics Engineers in Israel, Proceedings, Eilat, Israel, 2008: 290~294
108 M. Mishali, Y. C. Eldar. From theory to practice: Sub-Nyquist sampling of sparse wideband analog signals. IEEE Journal on Selected Topics in Signal Processing, 2010, 4(2): 375~391
109 M. Mishali and Y. C. Eldar. Reduce and boost: Recovering arbitrary sets of jointly sparse vectors. IEEE Transactions on Signal Processing, 2008, 56(10): 4692~4702
110 J. A. Tropp, A. C. Gilbert, and M. J. Strauss. Simultaneous sparse approximation via greedy pursuit. IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Philadelphia, PA, United states, 2005: 721~724
111 J. A. Tropp, A. C. Gilbert, and M. J. Strauss. Algorithms for simultaneous sparse approximation. Part I: Greedy pursuit. Signal Process, 2006, 86(3): 572~588
112 J. A. Tropp, A. C. Gilbert, and M. J. Strauss. Algorithms for simultaneous sparse approximation. Part II: Convex relaxation. Signal Process, 2006, 86(3): 589~602
113 Y. C. Eldar and T. Michaeli. Beyond bandlimited sampling: A review of nonlinearities, smoothness, and sparsity. IEEE Signal Processing Magazine, 2009, 26(3): 48~68
114 D. L. Donoho, Y. Tsaig. Extension of compressed sensing. Signal Processing, 2006, 86(3): 544~548
115 http://sparselab.stanford.edu/
116 Golub, G. H. and F. V. Loan. Matrix Computations (3rd ed.). John Hopkins University, 1996
117 R. A. DeVore and V. N. Temlyakov. Some remarks on greedy algorithms. Advances in Computational Mathematics, 1996: 173~187
118 E. Candès, R. Mark, T. Tao, and R. Vershynin. Error correction via linear programming. IEEE Symposium on Foundations of Computer Science (FOCS), 2005: 295~308
119 G. H. Qu, D. L. Zhang, and P. Yan. Information measure for performance of image fusion. Electronics Letters, 2002, 38(7): 313~315
120 M. Mishali, Y. C. Eldar. Blind multi-band signal reconstruction: Compressed sensing for analog signals. IEEE Transactions on Signal Processing, 2009, 57(3): 993~1009
121 H. J. Landau. Necessary density conditions for sampling and interpolation of certain entire functions. Acta Math, 1967, 117(1): 37~52
122 S. F. Cotter, B. D. Rao, K. Engan, and K. Kreutz-Delgado. Sparse solutions to linear inverse problems with multiple measurement vectors. IEEE Transactions on Signal Processing, 2005, 53(7): 2477~2488
123 Y. M. Lu and M. N. Do. A theory for sampling signals from a union of subspaces. IEEE Transactions on Signal Processing, 2008, 56(6): 2334~2345
124 J. Tropp. Greedy is good: Algorithmic results for sparse approximation. IEEE Transactions on Information Theory, 2004, 50(10): 2231~2242
125 D. L. Donodo and X. Huo. Uncertainty principles and ideal atomic decompositions. IEEE Transactions on Information Theory, 2001, 47(7): 2845~2862
126 Y. C. Eldar, M. Mishali. Block sparsity and sampling over a union of subspaces.DSP 2009: 16th International Conference on Digital Signal Processing, Proceedings, Santorini, Greece, 2009
127 Y. Han, C. Hartmann, and C.-C. Chen. Efficient priority-first search maximum-likelihood soft-decision decoding of linear block codes. IEEE Transactions on Information Theory, 1993, 39(5): 1514~1523
128 R. G. Vaughan, N. L. Scott, and D. R. White. The theory of bandpass sampling. IEEE Transactions on Signal Processing, 1991, 39(9): 1973~1984
129 Y.-P. Lin and P. P. Vaidyanathan. Periodically nonuniform sampling of bandpass signals. IEEE Transactions Circuits Syst. II, 1998, 45(3): 340~351
130 R. Venkataramani and Y. Bresler. Perfect reconstruction formulas and bounds on aliasing error in sub-Nyquist nonuniform sampling of multiband signals. IEEE Transactions on Information Theory, 2000, 46(6): 2173~2183
131 G. L. Fudge, R. E. Bland, M. A. Chivers, S. Ravindran, J. Haupt, and P. E. Pace. A Nyquist folding analog-to-information receiver. In Proc. 42nd Asilomar Conf. on Signals, Systems and Computers, Pacific Grove, CA, United states, 2008: 541~545
132 Z. Yu, S. Hoyos, and B. M. Sadler. Mixed-signal parallel compressed sensing and reception for cognitive radio. IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Las Vegas, NV, United states, 2008: 3861~3864
133 M. Mishali, Y. C. Eldar, O. Dounaevsky, E. Shoshan. Xampling: Analog to digital at sub-Nyquist rates. IET Circuits, Devices and Systems, 2011, 5(1): 8~20
134 M. Mishali, Y. C. Eldar. Expected RIP: Conditioning of The Modulated Wideband Converter. IEEE Information Theory Workshop (ITW), Taormina, Sicily, Italy, 2009: 343~347