基于压缩传感原理的图像重建方法研究
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摘要
传统的Nyquist采样定理要求采样频率必须大于等于信号最高频率的两倍,但很多情况下信号带宽较大,采样频率达不到最高频率的两倍。最近Donoho和Candès提出了压缩传感CS(Compressed Sensing)理论。该理论利用原始信号或图像的稀疏性先验知识,通过合适的优化算法,可由少量的采样值或观测值来进行重建。目前该理论的研究尚处于初级阶段,大多是基于压缩传感基础理论的研究和一维信号的重建。本文将压缩传感理论应用于图像重建中,针对其重建速度慢和重建质量不高的缺点,在深入研究现有算法的基础上,从以下几方面进行研究:
(1)基于压缩传感和代数重建法的CT(Computed Tomography)重建结合压缩传感理论提出了一种基于代数重建法ART(Algebraic Reconstruction Technique)的高质量CT图像重建算法。该算法将CT图像的梯度稀疏性结合到ART图像重建中,在每次迭代中的投影操作结束后用梯度下降法调整全变差,减小图像梯度的1范数。实验结果表明了该算法的有效性。
(2)基于全变差多种范数的核磁共振图像重建利用核磁共振图像具有梯度和边缘稀疏性的先验知识来加快其成像速度,提出了一种基于全变差的核磁共振图像重建算法,并对1、p (0 < p< 1)与log和惩罚函数三种范数进行了实验和比较分析。
(3)基于线性Bregman和混合基稀疏表示的压缩传感图像重建提出了一种基于离散余弦变换和双树复数小波两种基混合的图像稀疏表示,利用线性Bregman迭代来进行重建的压缩传感系统。该算法在每一次迭代更新后用梯度下降法进行全变差调整,再分别在两种基上执行软阈值处理来减小图像的1范数。实验结果表明该算法有效提高了重建图像的质量。
The convertional Nyquist sampling theory requires the sampling frequency must at least twice the highest frequency of signal, but in many cases it doesn’t achieve the requirement, because of the large bandwidth. Recently Donoho and Candès have proposed compressed sensing theory, which applies the sparsity prior of the signals or images and can accurately reconstruct original signals or images from a small quantity of measurements, provided an appropriate optimizd procedure. At present the study of CS is still in the first stage, most of studies are about the basic theory and the recovery of one dimensional signal. Aimed at the disadvantage of the slow rcovered speed and the bad recovered quality, we apply the compressed sensing to the image reconstruction, in the foundation of the deeper research on the existing methods, and mainly research on the following aspects in this paper:
(1) Image reconstruction for CT based on compressed sensing and ART. We make use of the compressed sensing theory, and propose a method based on ART to improve the quality of the recovered image and the speed of reconstruction. The method, which combines the gradient sparsity of CT images and ART, reduces the 1 norm of the image gradient by regulating the total variation with the gradient descend method after completing the projection on the corresponding hyperplane in each iteration. The experimental results show the effectivity of the method.
(2) Sparse MRI reconstruction via multiple norms based on total variation. The compressed sensing makes use of the gradient and edge sparsities which are implicit in MR images to quicken the imaging speed. We propose a new MRI reconstruction method based on total variation, and then perform the experiments on 1, p (0 < p< 1) and log-sum penalty function and compare their performance.
(3) Compressed sensing image reconstruction based on linearized Bregman and mixed bases sparse representation. We propose a compressed sensing system based on the sparse representation of images on discrete cosine transform and the dual tree complex wavelet transform, making use of the linearized Bregman iteration to reconstruct the original image. The method regulates the total variation with the gradient descend method after updating in each iteration, and then performs the soft-thresholding on these two bases respectively to reduce the 1 norm of the image. The results of experiments show that our method effectively improves the quality of the recovered image.
(1)基于压缩传感和代数重建法的CT(Computed Tomography)重建结合压缩传感理论提出了一种基于代数重建法ART(Algebraic Reconstruction Technique)的高质量CT图像重建算法。该算法将CT图像的梯度稀疏性结合到ART图像重建中,在每次迭代中的投影操作结束后用梯度下降法调整全变差,减小图像梯度的1范数。实验结果表明了该算法的有效性。
(2)基于全变差多种范数的核磁共振图像重建利用核磁共振图像具有梯度和边缘稀疏性的先验知识来加快其成像速度,提出了一种基于全变差的核磁共振图像重建算法,并对1、p (0 < p< 1)与log和惩罚函数三种范数进行了实验和比较分析。
(3)基于线性Bregman和混合基稀疏表示的压缩传感图像重建提出了一种基于离散余弦变换和双树复数小波两种基混合的图像稀疏表示,利用线性Bregman迭代来进行重建的压缩传感系统。该算法在每一次迭代更新后用梯度下降法进行全变差调整,再分别在两种基上执行软阈值处理来减小图像的1范数。实验结果表明该算法有效提高了重建图像的质量。
The convertional Nyquist sampling theory requires the sampling frequency must at least twice the highest frequency of signal, but in many cases it doesn’t achieve the requirement, because of the large bandwidth. Recently Donoho and Candès have proposed compressed sensing theory, which applies the sparsity prior of the signals or images and can accurately reconstruct original signals or images from a small quantity of measurements, provided an appropriate optimizd procedure. At present the study of CS is still in the first stage, most of studies are about the basic theory and the recovery of one dimensional signal. Aimed at the disadvantage of the slow rcovered speed and the bad recovered quality, we apply the compressed sensing to the image reconstruction, in the foundation of the deeper research on the existing methods, and mainly research on the following aspects in this paper:
(1) Image reconstruction for CT based on compressed sensing and ART. We make use of the compressed sensing theory, and propose a method based on ART to improve the quality of the recovered image and the speed of reconstruction. The method, which combines the gradient sparsity of CT images and ART, reduces the 1 norm of the image gradient by regulating the total variation with the gradient descend method after completing the projection on the corresponding hyperplane in each iteration. The experimental results show the effectivity of the method.
(2) Sparse MRI reconstruction via multiple norms based on total variation. The compressed sensing makes use of the gradient and edge sparsities which are implicit in MR images to quicken the imaging speed. We propose a new MRI reconstruction method based on total variation, and then perform the experiments on 1, p (0 < p< 1) and log-sum penalty function and compare their performance.
(3) Compressed sensing image reconstruction based on linearized Bregman and mixed bases sparse representation. We propose a compressed sensing system based on the sparse representation of images on discrete cosine transform and the dual tree complex wavelet transform, making use of the linearized Bregman iteration to reconstruct the original image. The method regulates the total variation with the gradient descend method after updating in each iteration, and then performs the soft-thresholding on these two bases respectively to reduce the 1 norm of the image. The results of experiments show that our method effectively improves the quality of the recovered image.
引文
1 D. Donoho. Compressed sensing. IEEE Transactions on Information Theory, 2006, 52(4):1289-1306
2 R. Baraniuk. Compressive Sensing. IEEE Signal Processing Magazine, 2007, 24(4): 118-121
3 E. Candès, J. Romberg, T. Tao. Robust Uncertainty Principles: Exact Signal Reconstruction from Highly Incomplete Frequency Information. IEEE Transactions on Information Theory, 2006, 52(2):489-509
4 E. J. Candès, M. Wakin. An Introduction to Compressive Sampling. IEEE Signal Processing Magazine, 2008, 25(2):21-30
5 S. G. Mallat, Z. Zhang. Matching Pursuits with Time-frequency Dictionaries. IEEE Transactions on Signal Processing, 1993:3397-3415
6 J. Tropp, A. Gilbert. Signal Recovery from Random Measurements via Orthogonal Matching Pursuit. IEEE Trans. on Information Theory, 2007, 53(12):4655-4666
7 D. L. Donoho, Y. Tsaig, I. Drori, at el. Sparse Solution of Underdetermined Linear Equations by Stagewise Orthogonal Matching Pursuit. Tech. Report, Department of statistics, Standford university, 2006:5-10
8 T. Blumensath, M. E. Davies. Gradient Pursuits. IEEE Trans. on Signal Processing, 2008, 56(6):2370-2382
9 S. S. Chen, D. L. Donoho, M. A. Saunders. Atomic Decomposition by Basis Pursuit. SIAMJ. Sci. Comput., 1998, 20(1):36-61
10 A. Forsgren, P. E. Gill, M. H. Wright. Interior Methods for Nonlinear Optimization. SIAM Review, 2002, 44(4):525-597
11 E. J. Candes, J. Romberg. Practical Signal Recovery from Random Projections. In: Proceedings of SPIE Computational Imaging III, San Jose, 2005:76-86
12 I. Daubechies, M. Defrise, C. D. Mol. An Iterative Thresholding Algorithm for LinearInverse Problems with A Sparsity Constraint. Communications on Pure and Applied Mathematics, 2004, 57(11):1413-1457
13 J. M. Bioucas-Dias, Mario A. T. Figueiredo. A New TwIST: Two-step Iterative Shrinkage/Thresholding Algorithms for Image Restoration. IEEE Transactions on Image processing, 2007, 16(12):2992-3004
14 J. M. Bioucas-Dias, Mario A. T. Figueiredo. Two-step Algorithms for Linear Inverse Problems with Non-quadratic Regularization. Proceeding of ICIP, IEEE International Conference on Image Processing, 2007, 1:I-105-I-108
15 M. A. T. Figueiredo, R. D. Nowak, S. J. Wright. Gradient Projection for Sparse Reconstruction: Application to Compressed Sensing and Other Inverse Problems. IEEE Selected Topics in Signal Processing, 2007, 1(4):586-597
16 S. Kim, K. Koh, M. Lustig, at el. A Method for Largescale L1-regularized Least Squares Problems with Applications in Signal Processing and Statistics. Tech. Report, Dept. of Electrical Engineering, Stanford University, 2007:2-14
17 E. Berg, M. P. Friedlander. Probing the Pareto Frontier for Basis Pursuit Solutions. Tech. Reprot TR-2008-01, Department of Computer Science, University of British Columbia, 2008:6-18
18 W. Yin, S. Osher, D. Goldfarb, at el. Bregman Iterative Algorithms for L1-minimization With Applications to Compressed Sensing. SIAM J. Imaging Science, 2008, 1(1):143-168
19 S. Osher, Y. Mao, B. Dong, at el. Fast Linearized Bregman Iteration for Compressive Sensing and Sparse Denoising, Tech. Report, Department of Computational and Applied Mathematices, Rice University, 2008:3-21
20 J. Cai, S. Osher, Z. Shen. Linear Bregman Iterations for Compressed Sensing. UCLA CAM Report, Department of Mathematics, University of California, 2008:1-18
21 T. Goldstein, S. Osher. The Split Bregman Algorithm for L1 Regularized Problems. UCLA CAM Report, Department of Mathematics, University of California, 2008:1-25
22 R. Chartrand. Exact Reconstructions of Sparse Signals via Nonconvex Minimization. IEEE Signal Proc. Lett., 2007, 14(10):707-710
23 R. Chartrand. Nonconvex Compressed Sensing and Error Correction. IEEE Int. Conf. on Acoustics, Speech, and Signal Processing (ICASSP), Honolulu, Hawaii, 2007, 3:III-889-III-892
24 J. Trzasko, A. Manduca, E. Borisch. Highly Undersampled Magnetic Resonance Image Reconstruction via Homotopic Ell-0-minimization. IEEE Transactions on Medical Imaging, 2007, (99):1-20
25 J. Trzasko, A. Manduca, E. Borisch. Sparse MRI Reconstruction via Multiscale L0-Continuation. IEEE Statistical Signal Processing, Madison, USA, 2007:176-180
26 R. Baraniuk, P. Steeghs. Compressive Radar Imaging. IEEE Radar Conference, Waltham, Massachusetts, 2007:128-133
27 L. Potter, P. Schniter, J. Ziniel. Sparse Reconstruction for Radar. SPIE Algorithms for Synthetic Aperture Radar Imagery XV, 2008, 6970:0-15
28 S. F. Cotter, B. D. Rao. Sparse Channel Estimation via Matching Pursuit with Application to Equalization. IEEE Trans. on Communications, 2002, 50(3):374-377
29 W. U. Bajwa, J. Haupt, G. Raz, at el. Compressed Channel Sensing. Conf. on Info. Sciences and Systems (CISS), Princeton, New Jersey, 2008:5-10
30 J. Ma, F. Le Dimet. Deblurring from Highly Incomplete Measurements for Remote Sensing. IEEE Trans. Geoscience and Remote Sensing, 2008, 46(12):51-69
31 M. Wakin, J. Laska, M. Duarte, at el. An Architecture for Compressive Imaging. In Proceedings of the IEEE International Conference on Image Processing (ICIP), Atlanta, GA, 2006:1273-1276
32 D. Takhar, J. Laska, M. Wakin, at el. A New Compressive Imaging Camera Architecture Using Optical-domain Compression. In Proceedings of Computational Imaging IV at the SPIE Symposium on Electronic Imaging, San Jose, USA, 2006, 6065: 1-10
33 D. Takhar, V. Bansal, M. Wakin, at el. A Compressed Sensing Camera: New Theory and An Implementation Using Digital Micromirrors. Tech. Reprot, Proc. Comp. Imaging IV at SPIE Electronic Imaging, San Jose, California, 2006:3-7
34 M. Duarte, M. Davenport, D. Takhar, at el. Single-pixel Imaging via CompressiveSampling. IEEE Signal Processing Magazine, 2008, 25(2):83-91
35 W. Wang, M. Garofalakis, K. Ramchandran. Distributed Sparse Random Projections for Refinable Approximation. In Proceedings of the International Conference on Information Processing in Sensor Networks (IPSN), Cambridge, USA, 2007:331-339
36 D. Baron, M. Wakin, M. Duarte, at el. Distributed Compressed Sensing of Jointly Sparse Signals. The Thirty-Ninth Asilomar Conference on Signals, Systems and Computers, 2005:1537-1541
37 M. Lustig, D. Donoho, J. M. Pauly. Sparse MRI: The Application of Compressed Sensing for Rapid MR Imaging. Magnetic Resonance in Medicine, 2007, 58(6): 1182-1195
38 S. Hu, M. Lustig, Albert P. Chen, at el. Compressed Sensing for Resolution Enhancement of Hyperpolarized 13C Flyback 3D-MRSI. Journal of Magnetic Resonance, 2008, 192(2):258-264
39 J. Laska, S. Kirolos, M. Duarte, at el. Theory and Implementation of An Analog-to-information Converter Using Random Demodulation. IEEE Int. Symp. on Circuits and Systems (ISCAS), New Orleans, Louisiana, 2007:1959-1962
40 D. S. Taubman, M. W. Marcellin. JPEG 2000: Image Compression Fundamentals, Standards and Practice. Kluwer International Series in Engineering and Computer Science, 2002, 11:20-63
41 H. Rauhut, K. Schass, P. Vandergheynst. Compressed Sensing and Redundant Dictionaries. IEEE Trans. on Information Theory, 2008, 54(5):2210-2219
42 M. Lustig, J. M. Santos, D. L. Donoho, at el. K-t SPARSE: High Frame Rate Dynamic MRI Exploiting Spatio-temporal Sparsity. ISMRM, Seattle, Washington, 2006:2420-2431
43 E. J. Candes, T. Tao. Decoding by Linear Programming. IEEE Trans. Inform. Theory, 2005, 51(12): 4203-4215
44方红,章权兵,韦穗.基于非常稀疏随机投影的图像重建方法.计算机工程与应用, 2007, 43(22):25-27
45 E. Candes, B. Wakin. "People Hearing Without Listening:" An Introduction toCompressive Sampling. Tech. Report, California Institute of technology, 2007:1-28
46 D. L. Donoho, X. Huo. Uncertainty Principles and Ideal Atomic Decomposition. IEEE Trans. Inform. Theory, 2001, 47(7):2845-2862
47 R. Coifman, F. Geshwind, Y. Meyer. Noiselets. Appl. Comput. Harmon. Anal., 2001, 10:27-44
48庄天戈. CT原理与算法.上海:上海交通大学出版社, 1992:256-345
49 K. Muller. Fast and Accurate Three-dimensional Reconstruction from Cone-beam Projection Data Using Algebraic Methods. The Ohio State University, 1998:32-43
50王宏钧,路宏年,傅健.代数重建技术中投影序列选择次序的研究.光学技术, 2006, 32(3):389-391
51 H. Guan, R. A. Gordon. Projection Access Order for Speedy Convergence of ART: A Multilevel Scheme for Computed Tomography. Physics in Medicine and Biology, 1994, 39(11):2005-2022
52 G. T. Herman, L. B. Meyer. Algebraic Reconstruction can be Made Computationally Efficient. IEEE Transactions on Medical Imaging, 1993, 12(3):600-609
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54王旭,陈志强,熊华,等.联合代数重建算法中基于像素的投影计算方法.核电子学与探测技术, 2005, 25(6):784-788
55 R. Gordon, R. Bender, G. T. Herman. Algebraic Reconstruction Techniques (ART) for Three-dimensional Electron Microscopy and X-ray Photography. Theoretical Biology, 1970, 29(3):471-481
56 K. Mueller, R. Yagel, J. F. Cornhill. The Weighted Distance Scheme: A Globally Optimizing Projection Ordering Method for ART. IEEE Transaction on Medical Image, 1997, 16(2):223-230
57张顺利,张定华,王凯,等.一种基于ART算法的快速图像重建技术.核电子学与探测技术, 2007, 27(3):479-483
58 M. Lustig, D. L. Donoho, J. M. Santos, at el. Compressed Sensing MRI. IEEE Signal Processing Magazine, 2008, 25(2):72-82
59 J. C. Ye, S. Tak, Y. Han, at el. Projection Reconstruction MR Imaging Using FOCUSS. Magn. Reson. Med., 2007, 57:764-775
60 M. Lustig, J. Lee, D. Donoho, at el. Faster Imaging with Randomly Perturbed, Undersampled Spirals and L1 Reconstruction. In: Proceedings of the 13th Annual Meeting of ISMRM, Miami Beach, 2005:685-701
61 E. J. Candes, M. B. Wakin, S. P. Boyd. Enhancing Sparsity by Reweighting L1 Minimization. Tech. Report, California Institute of Technology, 2007:12-25
62 I. W. Selesnick, R. G. Baraniuk, N. G. Kingsbury. The Dual Tree Complex Wavelet Transform. IEEE Signal Processing Magazine, 2005, 22(6):123-151
63 S. Osher, M. Burger, D. Goldfarb, at el. An Iterated Regularization Method for Total Variation-based Image Restoration. Multiscale Model. Simul., 2005, 4(2):460-489
64 L. He, T.-C. Chang, S. Osher, at el. MR Image Reconstruction by Using the Iterative Refinement Method and Nonlinear Inverse Scale Space Methods. UCLA CAM Report, Department of Mathematics, University of California, 2006:6-35
65 E. Hale, W. Yin, Y. Zhang. Fixed-Point Continuation for L1-Minimization: Methodology and Convergence. Technical Report TR07-07, Rice CAAM, 2007:1-28
2 R. Baraniuk. Compressive Sensing. IEEE Signal Processing Magazine, 2007, 24(4): 118-121
3 E. Candès, J. Romberg, T. Tao. Robust Uncertainty Principles: Exact Signal Reconstruction from Highly Incomplete Frequency Information. IEEE Transactions on Information Theory, 2006, 52(2):489-509
4 E. J. Candès, M. Wakin. An Introduction to Compressive Sampling. IEEE Signal Processing Magazine, 2008, 25(2):21-30
5 S. G. Mallat, Z. Zhang. Matching Pursuits with Time-frequency Dictionaries. IEEE Transactions on Signal Processing, 1993:3397-3415
6 J. Tropp, A. Gilbert. Signal Recovery from Random Measurements via Orthogonal Matching Pursuit. IEEE Trans. on Information Theory, 2007, 53(12):4655-4666
7 D. L. Donoho, Y. Tsaig, I. Drori, at el. Sparse Solution of Underdetermined Linear Equations by Stagewise Orthogonal Matching Pursuit. Tech. Report, Department of statistics, Standford university, 2006:5-10
8 T. Blumensath, M. E. Davies. Gradient Pursuits. IEEE Trans. on Signal Processing, 2008, 56(6):2370-2382
9 S. S. Chen, D. L. Donoho, M. A. Saunders. Atomic Decomposition by Basis Pursuit. SIAMJ. Sci. Comput., 1998, 20(1):36-61
10 A. Forsgren, P. E. Gill, M. H. Wright. Interior Methods for Nonlinear Optimization. SIAM Review, 2002, 44(4):525-597
11 E. J. Candes, J. Romberg. Practical Signal Recovery from Random Projections. In: Proceedings of SPIE Computational Imaging III, San Jose, 2005:76-86
12 I. Daubechies, M. Defrise, C. D. Mol. An Iterative Thresholding Algorithm for LinearInverse Problems with A Sparsity Constraint. Communications on Pure and Applied Mathematics, 2004, 57(11):1413-1457
13 J. M. Bioucas-Dias, Mario A. T. Figueiredo. A New TwIST: Two-step Iterative Shrinkage/Thresholding Algorithms for Image Restoration. IEEE Transactions on Image processing, 2007, 16(12):2992-3004
14 J. M. Bioucas-Dias, Mario A. T. Figueiredo. Two-step Algorithms for Linear Inverse Problems with Non-quadratic Regularization. Proceeding of ICIP, IEEE International Conference on Image Processing, 2007, 1:I-105-I-108
15 M. A. T. Figueiredo, R. D. Nowak, S. J. Wright. Gradient Projection for Sparse Reconstruction: Application to Compressed Sensing and Other Inverse Problems. IEEE Selected Topics in Signal Processing, 2007, 1(4):586-597
16 S. Kim, K. Koh, M. Lustig, at el. A Method for Largescale L1-regularized Least Squares Problems with Applications in Signal Processing and Statistics. Tech. Report, Dept. of Electrical Engineering, Stanford University, 2007:2-14
17 E. Berg, M. P. Friedlander. Probing the Pareto Frontier for Basis Pursuit Solutions. Tech. Reprot TR-2008-01, Department of Computer Science, University of British Columbia, 2008:6-18
18 W. Yin, S. Osher, D. Goldfarb, at el. Bregman Iterative Algorithms for L1-minimization With Applications to Compressed Sensing. SIAM J. Imaging Science, 2008, 1(1):143-168
19 S. Osher, Y. Mao, B. Dong, at el. Fast Linearized Bregman Iteration for Compressive Sensing and Sparse Denoising, Tech. Report, Department of Computational and Applied Mathematices, Rice University, 2008:3-21
20 J. Cai, S. Osher, Z. Shen. Linear Bregman Iterations for Compressed Sensing. UCLA CAM Report, Department of Mathematics, University of California, 2008:1-18
21 T. Goldstein, S. Osher. The Split Bregman Algorithm for L1 Regularized Problems. UCLA CAM Report, Department of Mathematics, University of California, 2008:1-25
22 R. Chartrand. Exact Reconstructions of Sparse Signals via Nonconvex Minimization. IEEE Signal Proc. Lett., 2007, 14(10):707-710
23 R. Chartrand. Nonconvex Compressed Sensing and Error Correction. IEEE Int. Conf. on Acoustics, Speech, and Signal Processing (ICASSP), Honolulu, Hawaii, 2007, 3:III-889-III-892
24 J. Trzasko, A. Manduca, E. Borisch. Highly Undersampled Magnetic Resonance Image Reconstruction via Homotopic Ell-0-minimization. IEEE Transactions on Medical Imaging, 2007, (99):1-20
25 J. Trzasko, A. Manduca, E. Borisch. Sparse MRI Reconstruction via Multiscale L0-Continuation. IEEE Statistical Signal Processing, Madison, USA, 2007:176-180
26 R. Baraniuk, P. Steeghs. Compressive Radar Imaging. IEEE Radar Conference, Waltham, Massachusetts, 2007:128-133
27 L. Potter, P. Schniter, J. Ziniel. Sparse Reconstruction for Radar. SPIE Algorithms for Synthetic Aperture Radar Imagery XV, 2008, 6970:0-15
28 S. F. Cotter, B. D. Rao. Sparse Channel Estimation via Matching Pursuit with Application to Equalization. IEEE Trans. on Communications, 2002, 50(3):374-377
29 W. U. Bajwa, J. Haupt, G. Raz, at el. Compressed Channel Sensing. Conf. on Info. Sciences and Systems (CISS), Princeton, New Jersey, 2008:5-10
30 J. Ma, F. Le Dimet. Deblurring from Highly Incomplete Measurements for Remote Sensing. IEEE Trans. Geoscience and Remote Sensing, 2008, 46(12):51-69
31 M. Wakin, J. Laska, M. Duarte, at el. An Architecture for Compressive Imaging. In Proceedings of the IEEE International Conference on Image Processing (ICIP), Atlanta, GA, 2006:1273-1276
32 D. Takhar, J. Laska, M. Wakin, at el. A New Compressive Imaging Camera Architecture Using Optical-domain Compression. In Proceedings of Computational Imaging IV at the SPIE Symposium on Electronic Imaging, San Jose, USA, 2006, 6065: 1-10
33 D. Takhar, V. Bansal, M. Wakin, at el. A Compressed Sensing Camera: New Theory and An Implementation Using Digital Micromirrors. Tech. Reprot, Proc. Comp. Imaging IV at SPIE Electronic Imaging, San Jose, California, 2006:3-7
34 M. Duarte, M. Davenport, D. Takhar, at el. Single-pixel Imaging via CompressiveSampling. IEEE Signal Processing Magazine, 2008, 25(2):83-91
35 W. Wang, M. Garofalakis, K. Ramchandran. Distributed Sparse Random Projections for Refinable Approximation. In Proceedings of the International Conference on Information Processing in Sensor Networks (IPSN), Cambridge, USA, 2007:331-339
36 D. Baron, M. Wakin, M. Duarte, at el. Distributed Compressed Sensing of Jointly Sparse Signals. The Thirty-Ninth Asilomar Conference on Signals, Systems and Computers, 2005:1537-1541
37 M. Lustig, D. Donoho, J. M. Pauly. Sparse MRI: The Application of Compressed Sensing for Rapid MR Imaging. Magnetic Resonance in Medicine, 2007, 58(6): 1182-1195
38 S. Hu, M. Lustig, Albert P. Chen, at el. Compressed Sensing for Resolution Enhancement of Hyperpolarized 13C Flyback 3D-MRSI. Journal of Magnetic Resonance, 2008, 192(2):258-264
39 J. Laska, S. Kirolos, M. Duarte, at el. Theory and Implementation of An Analog-to-information Converter Using Random Demodulation. IEEE Int. Symp. on Circuits and Systems (ISCAS), New Orleans, Louisiana, 2007:1959-1962
40 D. S. Taubman, M. W. Marcellin. JPEG 2000: Image Compression Fundamentals, Standards and Practice. Kluwer International Series in Engineering and Computer Science, 2002, 11:20-63
41 H. Rauhut, K. Schass, P. Vandergheynst. Compressed Sensing and Redundant Dictionaries. IEEE Trans. on Information Theory, 2008, 54(5):2210-2219
42 M. Lustig, J. M. Santos, D. L. Donoho, at el. K-t SPARSE: High Frame Rate Dynamic MRI Exploiting Spatio-temporal Sparsity. ISMRM, Seattle, Washington, 2006:2420-2431
43 E. J. Candes, T. Tao. Decoding by Linear Programming. IEEE Trans. Inform. Theory, 2005, 51(12): 4203-4215
44方红,章权兵,韦穗.基于非常稀疏随机投影的图像重建方法.计算机工程与应用, 2007, 43(22):25-27
45 E. Candes, B. Wakin. "People Hearing Without Listening:" An Introduction toCompressive Sampling. Tech. Report, California Institute of technology, 2007:1-28
46 D. L. Donoho, X. Huo. Uncertainty Principles and Ideal Atomic Decomposition. IEEE Trans. Inform. Theory, 2001, 47(7):2845-2862
47 R. Coifman, F. Geshwind, Y. Meyer. Noiselets. Appl. Comput. Harmon. Anal., 2001, 10:27-44
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