压缩感知算法与应用研究
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摘要
本文在深入研究压缩感知基本理论的同时,将压缩感知理论应用于盲源信号分离和光声成像。
测量矩阵和恢复算法的设计是压缩感知的两大核心问题,首先从希尔伯特空间的基本性质和框架的定义入手深入分析了一个序列组可作为压缩感知观测向量组所需满足的条件,并据此引出了设计压缩感知测量矩阵的两个基本准则,即RIP条件和列相关性条件。然后在介绍了一些常见测量矩阵的同时,分析了包括随机测量矩阵和确定性测量矩阵在内的测量矩阵的优缺点。最后列举了两类算法中四种有代表性的恢复算法之后,通过大量的仿真实验全面比较这些算法的性能,并分析了这些算法各自的优势。
现有的大多数盲源分离方法都需要过量测量和很多先验知识,针对这一现状将压缩感知应用于盲源信号分离,该方法所需的信息量是欠定的。通过对欠定的盲源分离的一般模型中信息的重组建立了与压缩感知之间的联系,并提出了盲源分离中线性算子的设计需求。最后通过对稀疏随机信号和真实语音信号进行仿真实验来验证所提出的方法。
本文受到单像素相机设计原理的启发,用压缩感知原理设计了光声数据的采集模式,较已有的方法而言,其优势在于只需要较少的观测角度和观测数据就能达到传统方法的重建效果。根据光声数据在采集过程中会被噪声污染这一事实,将贝叶斯压缩感知算法应用于光声图像重建,实验结果表明这种方法有很好的重建效果。最后采用多角度融合的方法提高了成像分辨率。
Based on the in-depth study of the basic theory of Compressed Sensing, we apply the theory to blind source separation and photoacoustic imaging reconstruction.
The design of measurement matrix and recovery algorithm is the two core issues of Compressed Sensing. We first analyze what kind of sequences can be used as observation vectors by the basic properties of Hilbert space and tight frame, and thus leads to the two basic criteria of measurement matrix design: restricted isometry property (RIP) and the coherence of measurement matrix. Then we introduce some random matrices and deterministic matrices and analyze their advantages and disadvantages. After we list several algorithms of Compressed Sensing, a large number of simulation experiments are carried out to compare the performance of these algorithms.
Most of the existing methods for blind source separation (BSS) require much prior knowledge and many measurements, aiming at this actuality, Compressed Sensing is proposed to separate blind sources which requires undersampled data. We analyze some similarities between CS and BSS, furthermore, the relationship between them is built by equivalent transformation. And then we analyze how to design the momentous operator of BSS and give some valuable conclusions. At last, sparse random signals and real sound signals are applied to verify the effectiveness of the whole framework.
The photoacoustic data acquisition mode we design based on Compressed Sensing is prompted by the single-pixel camera. To have the same performance of traditional methods, the new mode requires only a few number of angles and observations. According to the fact that photoacoustic data is always polluted by noise data, we use Bayesian Compressed Sensing to reconstruct the photoacoustic imaging, the results of simulation experiments show that this algorithm has a good performance on photoacoustic imaging reconstruction. At last, by the multi-angle observation, it can reduce the number of measurements to improve the time resolution for a needed high-quality reconstruction image.
测量矩阵和恢复算法的设计是压缩感知的两大核心问题,首先从希尔伯特空间的基本性质和框架的定义入手深入分析了一个序列组可作为压缩感知观测向量组所需满足的条件,并据此引出了设计压缩感知测量矩阵的两个基本准则,即RIP条件和列相关性条件。然后在介绍了一些常见测量矩阵的同时,分析了包括随机测量矩阵和确定性测量矩阵在内的测量矩阵的优缺点。最后列举了两类算法中四种有代表性的恢复算法之后,通过大量的仿真实验全面比较这些算法的性能,并分析了这些算法各自的优势。
现有的大多数盲源分离方法都需要过量测量和很多先验知识,针对这一现状将压缩感知应用于盲源信号分离,该方法所需的信息量是欠定的。通过对欠定的盲源分离的一般模型中信息的重组建立了与压缩感知之间的联系,并提出了盲源分离中线性算子的设计需求。最后通过对稀疏随机信号和真实语音信号进行仿真实验来验证所提出的方法。
本文受到单像素相机设计原理的启发,用压缩感知原理设计了光声数据的采集模式,较已有的方法而言,其优势在于只需要较少的观测角度和观测数据就能达到传统方法的重建效果。根据光声数据在采集过程中会被噪声污染这一事实,将贝叶斯压缩感知算法应用于光声图像重建,实验结果表明这种方法有很好的重建效果。最后采用多角度融合的方法提高了成像分辨率。
Based on the in-depth study of the basic theory of Compressed Sensing, we apply the theory to blind source separation and photoacoustic imaging reconstruction.
The design of measurement matrix and recovery algorithm is the two core issues of Compressed Sensing. We first analyze what kind of sequences can be used as observation vectors by the basic properties of Hilbert space and tight frame, and thus leads to the two basic criteria of measurement matrix design: restricted isometry property (RIP) and the coherence of measurement matrix. Then we introduce some random matrices and deterministic matrices and analyze their advantages and disadvantages. After we list several algorithms of Compressed Sensing, a large number of simulation experiments are carried out to compare the performance of these algorithms.
Most of the existing methods for blind source separation (BSS) require much prior knowledge and many measurements, aiming at this actuality, Compressed Sensing is proposed to separate blind sources which requires undersampled data. We analyze some similarities between CS and BSS, furthermore, the relationship between them is built by equivalent transformation. And then we analyze how to design the momentous operator of BSS and give some valuable conclusions. At last, sparse random signals and real sound signals are applied to verify the effectiveness of the whole framework.
The photoacoustic data acquisition mode we design based on Compressed Sensing is prompted by the single-pixel camera. To have the same performance of traditional methods, the new mode requires only a few number of angles and observations. According to the fact that photoacoustic data is always polluted by noise data, we use Bayesian Compressed Sensing to reconstruct the photoacoustic imaging, the results of simulation experiments show that this algorithm has a good performance on photoacoustic imaging reconstruction. At last, by the multi-angle observation, it can reduce the number of measurements to improve the time resolution for a needed high-quality reconstruction image.
引文
[1] Donoho D L. Compressed Sensing [J]. IEEE Transactions on Information Theory, 2006 52(4):1289-1306.
[2] Candès E, Romberg J, and Tao T. Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information [J]. IEEE Transactions on Information Theory, 2006, 52(2): 489-509.
[3] Candès E. Compressive sampling[R]. In: Proceedings of International Congress of Mathematical Society Publishing House, 2006. 1433-1452.
[4] Duarte M F, Davenport M A, Takhar D, et al. Single-pixel imaging via compressive sampling [J]. IEEE Signal Processing Magazine, 2008, 25(2): 83-91.
[5] Rauhut H, Schnass K, Vandergheynst P. Compressed sensing and redundant dictionaries [J]. IEEE Transactions on Information Theory, 2008, 54(5): 2210-2219.
[6]孙玉宝,肖亮,韦志辉,邵文泽.基于Gabor感知多成分字典的图像稀疏表示算法研究[J].自动化学报, 2008, 34(11): 1379-1387.
[7] Mallat S. A Wavelet Tour of Signal Processing [M]. San Diego: Academic Press, 1996.
[8] E. Candès, Donoho D L. Curvelets-A Surprisingly Effective Nonadaptive Representation for Objects with Edges [R], Technical Report 1999-28, Department of Statistics Stanford University, USA, 1999.
[9] Mallat S, Zhang Z. Matching pursuit with time frequency dictionaries [J]. IEEE Tans. Signal Processing Magazine, 1993, 41(12): 3397-3415.
[10] Aharon M, Elad M, Bruckstein A M. The K-SVD: an algorithm for designing of overcomplete dictionaries for sparse representations [J]. IEEE Transactions on Image Processing, 2006, 54(11): 4311-4322.
[11] Rauhut H, Schnass K, Vandergheynst P. Compressed sensing and redundant dictionaries [J]. IEEE Transactions on Information Theory, 2008, 54(5): 2210-2219.
[12] Lustig M, Donoho D L, Pauly L M. Sparse MRI: the application of compressed sensing for rapid MR imaging [J]. Magnetic Resonance in Medicine, 2007, 58(6): 1182-1195.
[13] M. Lustig, D. L. Donoho, J. M. Santos, et al. Compressed sensing MRI [J]. IEEE Signal Processing Magazine, 25(2): 72-82, 2008.
[14] Baraniuk R G. Compressive sensing [J]. IEEE Signal Processing Magazine, 2007,24(4): 118-124.
[15] Hengyong Yu and Ge Wang. Compressed sensing based interior tomography [J]. Physics in Medicine and Biology. 2009, 54(2009): 2791-2805.
[16] R. M. Willett, M. E. Gehm, D. J. Brady. Multiscale reconstruction for computational spectral imaging [C]. preprint, in Computational Imaging V ,C. A. Bouman, E. L. Miller, and I. Pollak, Eds. San Jose, CA: SPIE, Jan.2007 vol.6498, p.64980L
[17] Haupt J, Nowak R. Compressive sampling for signal detection [R]. In: Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing. Washington D.C., USA: IEEE, 2007. 1509-1512.
[18] Haupt J, Castro R, Nowak R, et al. Compressive sampling for signal classification [R]. In: Proceedings of the Fortieth Asilomar Conference on the Signals, Systems and Computers. Washington D.C.,USA: IEEE,2006.1430-1434
[19] Bajwa W U, Haupt J, Raz G, et al. Compressed channel sensing [C]. In: Proceedings of the 42nd Annual Conference on Information Sciences and Systems. Washington D.C., USA: IEEE, 2008. 5-10.
[20] Hennenfent G, Herrmann F J. Simply denoise: wave field reconstruction via jittered undersampling [J]. Geophysics, 2008, 73(3): 19-28.
[21] Bajwa W, Haupt J, Sayeed A, et al. Compressive wireless sensing [C]. In: Proceedings of the International Conference on Information Processing in Sensor Networks. Nashville, USA: ACM, 2006. 134-142.
[22] D. J. Brady, K. Choi, D. L. Marks, et al, Compressive Holography [J], Opt. Express, Vol.17, pp. 13040-13049, 2009.
[23] Dror Baron, Marco F. Duarte, Michael B. et al. Distributed Compressive Sensing [J]. Preprint, Jan, 2009, arXiv: 0901.3403v1 [cs. IT] 22 Jan 2009.
[24] P. Boufounos and R.G. Baraniuk. 1-Bit Compressive sensing [C], in Conf. on Info. Sciences and Systems (CISS), (Princeton, NJ), pp.16-21, March 2008.
[25] Tibshirani R. Regression shrinkage and selection via the Lasso [J]. Journal of the Royal Statistical Society, Series B, 1996, 58(1): 267-288.
[26] T. Strohmer and R. W. Heath. Grassmannian frames with applications to coding and communication [J], Appl. Compute. Harmon. Anal., vol. 14, no. 3, pp. 257-275, May 2003.
[27] Karin Schnass and Pierre Vandergheynst. Dictionary Preconditioning for Greedy Algorithms [J]. IEEE Transactions on Signal Processing, vol. 56, no. 5, May 2008.
[28] Richard Baraniuk, Mark Davenport, Ronald DeVore, et al. A simple proof of the restricted isometry property for random matrices [J]. Constructive Approximation, Dec.2008, 28(3), 256-263.
[29] E. Candès, T. Tao. Decoding by linear Programming [J]. IEEE Transactions on Information Theory, 2005, 51(12), 4203-4215.
[30] E. Candès, T. Tao. Near optimal signal recovery from random projections: Universal encoding strategies [J]? IEEE Transactions on Information Theory, 2006, 52(12), 5406-5425.
[31] T Do, D Trany, L Gan. Fast compressive sampling with structurally random matrices [J]. Proc. IEEE ASSP, 2008, 3369-3372.
[32] Ronald A. DeVore. Deterministic constructions of compressed sensing matrices [J]. Journal of Complexity, 2007, 23(4-6), 918-925.
[33] W. Xu and B. Hassibi. Further Results on Performance Analysis for Compressive Sensing Using Expander Graphs [C]. Conference Record of the Forty-First Asilomar Conference on Signal, Systems and Computers. ACSSC 2007. 4-7 Nov.2007 Page(s): 621-625, 2008.
[34] A. Gilbert and P. Indyk. Sparse recovery using sparse matrices [J]. Proc. IEEE, vol. 98, no. 6, pp. 959-971, 2010.
[35] V. Guruswami, C. Umans, and S. P. Vadhan. Unbalanced expanders and randomness extractors from Parvaresh-Vardy codes [C]. In IEEE Conference on Computational Complexity (CCC 2007), Pages 96-108, 2007.
[36] Donoho D L, Elad M, Temlyakov V N. Stable recovery of sparse overcomplete representations in the presence of noise [J]. IEEE Transactions on Information Theory, 2006, 52(1): 6-18.
[37] Chen S B, Donoho D L, Saunders M A. Atomic decomposition by basis pursuit [J]. SIAM Journal on Scientific Computing, 1998, 20(1): 33-61.
[38] Donoho D L, Tsaig Y. Fast Solution of l1-Norm Minimization Problems When the Solution May Be Sparse [R]. Technical Report, Department of Statistics, Stanford University, USA, 2008.
[39] B. Efron, T. Hastie, I. Johnstone, et al. Least angle regression [J]. Ann. Statist., vol. 32, no. 2, pp. 407-499, 2004.
[40] Tropp J A, Gilbert A C. Signal recovery from random measurements via orthogonal matching pursuit [J]. IEEE Transactions on Information Theory, 2007, 53(12): 4655- 4666.
[41] E. Liu and V. N. Temlyakov. Orthogonal super greedy algorithm and applications in compressed sensing. Preprint, 2010.
[42] Laura Rebollo Neira and David Lowe. Optimized Orthogonal Matching Pursuit Approach [J]. IEEE Signal Processing Letters, vol.9, no.4, pp.137-140, Apr.2002.
[43] Needell D, Vershynin R. Greedy signal recovery and uncertainty principles [C]. In: Proceedings of the Conference on Computational Imaging. San Jose, USA: SPIE, 2008. 1-12.
[44] Needell D, Tropp J A. CoSaMP: Iterative signal recovery from incomplete and inaccurate samples [J]. Applied and Computational Harmonic Analysis, 2008, 26(3): 301-321.
[45] W. Dai, O. Milenkovic. Subspace pursuit for compressive sensing signal reconstruction [J]. IEEE Trans. Inform. Theory, 2009, 55(5): 2230-2249.
[46] Qi Dai and Wei E.I. Sha. The physics of compressive sensing and the gradient based recovery algorithm. Research Report, http://arxiv.org/abs/0906.1487.
[47] J. L. Starck, Y. Moudden, J. Bobin, et al. Morphological component analysis [C].in proceedings of SPIE Conference on Wavelets, vol.5914, July, 2005.
[48] T. Blumensath and M. Davies. Compressed sensing and source separation [C]. In International Conference on Independent Component Analysis and Blind Source Separation, pp. 341-348, Sept. 2007.
[49] L. V. Wang. Prospects of PA tomography [J]. Medical Physics, vol.35, no.12, pp. 5758-5767, 2008.
[50] Provost J. and Lesage F. The Application of Compressed Sensing for Photo-Acoustic Tomography [J]. IEEE Transactions on Medical Imaging, vol. 28, pp.585-594, 2009.
[51] Dong Liang, Hao F. Zhang and Leslie Ying. Compressed Sensing Photoacoustic Imaging based on Random Optical Illumination [J]. Int. J. Functional Informatics and Personalised Medicine, vol. 2, no. 4, pp. 394-405, 2009.
[52] Shihao Ji, Ya Xue and Lawrence Carin. Bayesian Compressive Sensing [J]. IEEE Transaction on Signal Proc., vol. 86, no. 6, pp. 2346-2356, June, 2008.
[53] Duarte M F, Davenport M A, Takhar D, et al. Single pixel imaging via compressive sampling [J]. IEEE Signal Processing Magazine, 2008, 25(2): 83-91.
[54] D. L. Donoho, Y. Tsaig, I. Drori, et al. Sparse solution of underdetermined linear equations by stage wise orthogonal matching pursuit [C]. Stanford Statistics Dept., Stanford Univ., Stanford, CA, TR-2006-2, Mar.2006, Preprint.
[55] M. Iwen. Simple deterministically constructible RIP matrices with sublinear Fourier sampling requirements [C]. Information Sciences and Systems, 2009. CISS 2009. 43rd Annual Conference on, pages 870-875, 2009.
[2] Candès E, Romberg J, and Tao T. Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information [J]. IEEE Transactions on Information Theory, 2006, 52(2): 489-509.
[3] Candès E. Compressive sampling[R]. In: Proceedings of International Congress of Mathematical Society Publishing House, 2006. 1433-1452.
[4] Duarte M F, Davenport M A, Takhar D, et al. Single-pixel imaging via compressive sampling [J]. IEEE Signal Processing Magazine, 2008, 25(2): 83-91.
[5] Rauhut H, Schnass K, Vandergheynst P. Compressed sensing and redundant dictionaries [J]. IEEE Transactions on Information Theory, 2008, 54(5): 2210-2219.
[6]孙玉宝,肖亮,韦志辉,邵文泽.基于Gabor感知多成分字典的图像稀疏表示算法研究[J].自动化学报, 2008, 34(11): 1379-1387.
[7] Mallat S. A Wavelet Tour of Signal Processing [M]. San Diego: Academic Press, 1996.
[8] E. Candès, Donoho D L. Curvelets-A Surprisingly Effective Nonadaptive Representation for Objects with Edges [R], Technical Report 1999-28, Department of Statistics Stanford University, USA, 1999.
[9] Mallat S, Zhang Z. Matching pursuit with time frequency dictionaries [J]. IEEE Tans. Signal Processing Magazine, 1993, 41(12): 3397-3415.
[10] Aharon M, Elad M, Bruckstein A M. The K-SVD: an algorithm for designing of overcomplete dictionaries for sparse representations [J]. IEEE Transactions on Image Processing, 2006, 54(11): 4311-4322.
[11] Rauhut H, Schnass K, Vandergheynst P. Compressed sensing and redundant dictionaries [J]. IEEE Transactions on Information Theory, 2008, 54(5): 2210-2219.
[12] Lustig M, Donoho D L, Pauly L M. Sparse MRI: the application of compressed sensing for rapid MR imaging [J]. Magnetic Resonance in Medicine, 2007, 58(6): 1182-1195.
[13] M. Lustig, D. L. Donoho, J. M. Santos, et al. Compressed sensing MRI [J]. IEEE Signal Processing Magazine, 25(2): 72-82, 2008.
[14] Baraniuk R G. Compressive sensing [J]. IEEE Signal Processing Magazine, 2007,24(4): 118-124.
[15] Hengyong Yu and Ge Wang. Compressed sensing based interior tomography [J]. Physics in Medicine and Biology. 2009, 54(2009): 2791-2805.
[16] R. M. Willett, M. E. Gehm, D. J. Brady. Multiscale reconstruction for computational spectral imaging [C]. preprint, in Computational Imaging V ,C. A. Bouman, E. L. Miller, and I. Pollak, Eds. San Jose, CA: SPIE, Jan.2007 vol.6498, p.64980L
[17] Haupt J, Nowak R. Compressive sampling for signal detection [R]. In: Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing. Washington D.C., USA: IEEE, 2007. 1509-1512.
[18] Haupt J, Castro R, Nowak R, et al. Compressive sampling for signal classification [R]. In: Proceedings of the Fortieth Asilomar Conference on the Signals, Systems and Computers. Washington D.C.,USA: IEEE,2006.1430-1434
[19] Bajwa W U, Haupt J, Raz G, et al. Compressed channel sensing [C]. In: Proceedings of the 42nd Annual Conference on Information Sciences and Systems. Washington D.C., USA: IEEE, 2008. 5-10.
[20] Hennenfent G, Herrmann F J. Simply denoise: wave field reconstruction via jittered undersampling [J]. Geophysics, 2008, 73(3): 19-28.
[21] Bajwa W, Haupt J, Sayeed A, et al. Compressive wireless sensing [C]. In: Proceedings of the International Conference on Information Processing in Sensor Networks. Nashville, USA: ACM, 2006. 134-142.
[22] D. J. Brady, K. Choi, D. L. Marks, et al, Compressive Holography [J], Opt. Express, Vol.17, pp. 13040-13049, 2009.
[23] Dror Baron, Marco F. Duarte, Michael B. et al. Distributed Compressive Sensing [J]. Preprint, Jan, 2009, arXiv: 0901.3403v1 [cs. IT] 22 Jan 2009.
[24] P. Boufounos and R.G. Baraniuk. 1-Bit Compressive sensing [C], in Conf. on Info. Sciences and Systems (CISS), (Princeton, NJ), pp.16-21, March 2008.
[25] Tibshirani R. Regression shrinkage and selection via the Lasso [J]. Journal of the Royal Statistical Society, Series B, 1996, 58(1): 267-288.
[26] T. Strohmer and R. W. Heath. Grassmannian frames with applications to coding and communication [J], Appl. Compute. Harmon. Anal., vol. 14, no. 3, pp. 257-275, May 2003.
[27] Karin Schnass and Pierre Vandergheynst. Dictionary Preconditioning for Greedy Algorithms [J]. IEEE Transactions on Signal Processing, vol. 56, no. 5, May 2008.
[28] Richard Baraniuk, Mark Davenport, Ronald DeVore, et al. A simple proof of the restricted isometry property for random matrices [J]. Constructive Approximation, Dec.2008, 28(3), 256-263.
[29] E. Candès, T. Tao. Decoding by linear Programming [J]. IEEE Transactions on Information Theory, 2005, 51(12), 4203-4215.
[30] E. Candès, T. Tao. Near optimal signal recovery from random projections: Universal encoding strategies [J]? IEEE Transactions on Information Theory, 2006, 52(12), 5406-5425.
[31] T Do, D Trany, L Gan. Fast compressive sampling with structurally random matrices [J]. Proc. IEEE ASSP, 2008, 3369-3372.
[32] Ronald A. DeVore. Deterministic constructions of compressed sensing matrices [J]. Journal of Complexity, 2007, 23(4-6), 918-925.
[33] W. Xu and B. Hassibi. Further Results on Performance Analysis for Compressive Sensing Using Expander Graphs [C]. Conference Record of the Forty-First Asilomar Conference on Signal, Systems and Computers. ACSSC 2007. 4-7 Nov.2007 Page(s): 621-625, 2008.
[34] A. Gilbert and P. Indyk. Sparse recovery using sparse matrices [J]. Proc. IEEE, vol. 98, no. 6, pp. 959-971, 2010.
[35] V. Guruswami, C. Umans, and S. P. Vadhan. Unbalanced expanders and randomness extractors from Parvaresh-Vardy codes [C]. In IEEE Conference on Computational Complexity (CCC 2007), Pages 96-108, 2007.
[36] Donoho D L, Elad M, Temlyakov V N. Stable recovery of sparse overcomplete representations in the presence of noise [J]. IEEE Transactions on Information Theory, 2006, 52(1): 6-18.
[37] Chen S B, Donoho D L, Saunders M A. Atomic decomposition by basis pursuit [J]. SIAM Journal on Scientific Computing, 1998, 20(1): 33-61.
[38] Donoho D L, Tsaig Y. Fast Solution of l1-Norm Minimization Problems When the Solution May Be Sparse [R]. Technical Report, Department of Statistics, Stanford University, USA, 2008.
[39] B. Efron, T. Hastie, I. Johnstone, et al. Least angle regression [J]. Ann. Statist., vol. 32, no. 2, pp. 407-499, 2004.
[40] Tropp J A, Gilbert A C. Signal recovery from random measurements via orthogonal matching pursuit [J]. IEEE Transactions on Information Theory, 2007, 53(12): 4655- 4666.
[41] E. Liu and V. N. Temlyakov. Orthogonal super greedy algorithm and applications in compressed sensing. Preprint, 2010.
[42] Laura Rebollo Neira and David Lowe. Optimized Orthogonal Matching Pursuit Approach [J]. IEEE Signal Processing Letters, vol.9, no.4, pp.137-140, Apr.2002.
[43] Needell D, Vershynin R. Greedy signal recovery and uncertainty principles [C]. In: Proceedings of the Conference on Computational Imaging. San Jose, USA: SPIE, 2008. 1-12.
[44] Needell D, Tropp J A. CoSaMP: Iterative signal recovery from incomplete and inaccurate samples [J]. Applied and Computational Harmonic Analysis, 2008, 26(3): 301-321.
[45] W. Dai, O. Milenkovic. Subspace pursuit for compressive sensing signal reconstruction [J]. IEEE Trans. Inform. Theory, 2009, 55(5): 2230-2249.
[46] Qi Dai and Wei E.I. Sha. The physics of compressive sensing and the gradient based recovery algorithm. Research Report, http://arxiv.org/abs/0906.1487.
[47] J. L. Starck, Y. Moudden, J. Bobin, et al. Morphological component analysis [C].in proceedings of SPIE Conference on Wavelets, vol.5914, July, 2005.
[48] T. Blumensath and M. Davies. Compressed sensing and source separation [C]. In International Conference on Independent Component Analysis and Blind Source Separation, pp. 341-348, Sept. 2007.
[49] L. V. Wang. Prospects of PA tomography [J]. Medical Physics, vol.35, no.12, pp. 5758-5767, 2008.
[50] Provost J. and Lesage F. The Application of Compressed Sensing for Photo-Acoustic Tomography [J]. IEEE Transactions on Medical Imaging, vol. 28, pp.585-594, 2009.
[51] Dong Liang, Hao F. Zhang and Leslie Ying. Compressed Sensing Photoacoustic Imaging based on Random Optical Illumination [J]. Int. J. Functional Informatics and Personalised Medicine, vol. 2, no. 4, pp. 394-405, 2009.
[52] Shihao Ji, Ya Xue and Lawrence Carin. Bayesian Compressive Sensing [J]. IEEE Transaction on Signal Proc., vol. 86, no. 6, pp. 2346-2356, June, 2008.
[53] Duarte M F, Davenport M A, Takhar D, et al. Single pixel imaging via compressive sampling [J]. IEEE Signal Processing Magazine, 2008, 25(2): 83-91.
[54] D. L. Donoho, Y. Tsaig, I. Drori, et al. Sparse solution of underdetermined linear equations by stage wise orthogonal matching pursuit [C]. Stanford Statistics Dept., Stanford Univ., Stanford, CA, TR-2006-2, Mar.2006, Preprint.
[55] M. Iwen. Simple deterministically constructible RIP matrices with sublinear Fourier sampling requirements [C]. Information Sciences and Systems, 2009. CISS 2009. 43rd Annual Conference on, pages 870-875, 2009.
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