메뉴 건너뛰기




Volumn 1, Issue 1, 2008, Pages 143-168

Bregman iterative algorithms for ℓ1-minimization with applications to compressed sensing

Author keywords

Basis pursuit; Bregman distances; Compressed sensing; Iterative regularization; 1 minimization

Indexed keywords

COMPRESSED SENSING; MATRIX ALGEBRA; NUMERICAL METHODS; SIGNAL RECONSTRUCTION;

EID: 84977895355     PISSN: None     EISSN: 19364954     Source Type: Journal    
DOI: 10.1137/070703983     Document Type: Article
Times cited : (1336)

References (102)
  • 6
    • 33645136947 scopus 로고    scopus 로고
    • Bayesian wavelet-based image deconvolution: A GEM algorithm exploiting a class of heavy-tailed priors
    • J. Bioucas-Dias, Bayesian wavelet-based image deconvolution: A GEM algorithm exploiting a class of heavy-tailed priors, IEEE Trans. Image Process., 15 (2006), pp. 937-951.
    • (2006) IEEE Trans. Image Process. , vol.15 , pp. 937-951
    • Bioucas-Dias, J.1
  • 8
  • 9
    • 49949144765 scopus 로고
    • The relaxation method of finding the common points of convex sets and its application to the solution of problems in convex programming
    • L. Bregman, The relaxation method of finding the common points of convex sets and its application to the solution of problems in convex programming, Comput. Math. Math. Phys., 7 (1967), pp. 200-217.
    • (1967) Comput. Math. Math. Phys. , vol.7 , pp. 200-217
    • Bregman, L.1
  • 11
    • 33646554522 scopus 로고    scopus 로고
    • Nonlinear inverse scale space methods for image restoration
    • Lecture Notes in Comput. Sci. 3752, Springer-Verlag, Berlin
    • M. Burger, S. Osher, J. Xu, and G. Gilboa, Nonlinear inverse scale space methods for image restoration, in Variational Geometric and Level Set Methods in Computer Vision, Lecture Notes in Comput. Sci. 3752, Springer-Verlag, Berlin, 2005, pp. 25-36.
    • (2005) Variational Geometric and Level Set Methods in Computer Vision , pp. 25-36
    • Burger, M.1    Osher, S.2    Xu, J.3    Gilboa, G.4
  • 12
    • 33744726904 scopus 로고    scopus 로고
    • Quantitative robust uncertainty principles and optimally sparse decompositions
    • E. Candès and J. Romberg, Quantitative robust uncertainty principles and optimally sparse decompositions, Found. Comput. Math., 6 (2006), pp. 227-254.
    • (2006) Found. Comput. Math. , vol.6 , pp. 227-254
    • Candès, E.1    Romberg, J.2
  • 13
    • 34249687049 scopus 로고    scopus 로고
    • Sparsity and incoherence in compressive sampling
    • E. Candès and J. Romberg, Sparsity and incoherence in compressive sampling, Inverse Problems, 23 (2007), pp. 969-985.
    • (2007) Inverse Problems , vol.23 , pp. 969-985
    • Candès, E.1    Romberg, J.2
  • 14
    • 31744440684 scopus 로고    scopus 로고
    • Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information
    • E. Candès, J. Romberg, and T. Tao, Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information, IEEE Trans. Inform. Theory, 52 (2006), pp. 489-509.
    • (2006) IEEE Trans. Inform. Theory , vol.52 , pp. 489-509
    • Candès, E.1    Romberg, J.2    Tao, T.3
  • 15
    • 29144439194 scopus 로고    scopus 로고
    • Decoding by linear programming
    • E. Candès and T. Tao, Decoding by linear programming, IEEE Trans. Inform. Theory, 51 (2005), pp. 4203-4215.
    • (2005) IEEE Trans. Inform. Theory , vol.51 , pp. 4203-4215
    • Candès, E.1    Tao, T.2
  • 16
    • 33947416035 scopus 로고    scopus 로고
    • Near optimal signal recovery from random projections: Universal encoding strategies
    • E. Candès and T. Tao, Near optimal signal recovery from random projections: Universal encoding strategies, IEEE Trans. Inform. Theory, 52 (2006), pp. 5406-5425.
    • (2006) IEEE Trans. Inform. Theory , vol.52 , pp. 5406-5425
    • Candès, E.1    Tao, T.2
  • 17
    • 1242352408 scopus 로고    scopus 로고
    • An algorithm for total variation minimization and applications
    • A. Chambolle, An algorithm for total variation minimization and applications, J. Math. Imaging Vision, 20 (2004), pp. 89-97.
    • (2004) J. Math. Imaging Vision , vol.20 , pp. 89-97
    • Chambolle, A.1
  • 19
    • 0032022704 scopus 로고    scopus 로고
    • Nonlinear wavelet image processing: Variational problems, compression, and noise removal through wavelet shrinkage
    • A. Chambolle, R. A. DeVore, N.-Y. Lee, and B. J. Lucier, Nonlinear wavelet image processing: Variational problems, compression, and noise removal through wavelet shrinkage, IEEE Trans. Image Process., 7 (1998), pp. 319-335.
    • (1998) IEEE Trans. Image Process. , vol.7 , pp. 319-335
    • Chambolle, A.1    DeVore, R.A.2    Lee, N.-Y.3    Lucier, B.J.4
  • 20
    • 34548724437 scopus 로고    scopus 로고
    • Exact reconstructions of sparse signals via nonconvex minimization
    • R. Chartrand, Exact reconstructions of sparse signals via nonconvex minimization, IEEE Signal Process. Lett., 14 (2007), pp. 707-710.
    • (2007) IEEE Signal Process. Lett. , vol.14 , pp. 707-710
    • Chartrand, R.1
  • 21
    • 34547496533 scopus 로고    scopus 로고
    • Nonconvex compressed sensing and error correction
    • Speech, and Signal Processing (ICASSP)
    • R. Chartrand, Nonconvex compressed sensing and error correction, in Proceedings of the 32nd International Conference on Acoustics, Speech, and Signal Processing (ICASSP), 2007, pp. 889-892.
    • (2007) Proceedings of the 32nd International Conference on Acoustics , pp. 889-892
    • Chartrand, R.1
  • 22
    • 51449111119 scopus 로고    scopus 로고
    • Iteratively reweighted algorithms for compressive sensing
    • submitted
    • R. Chartrand and W. Yin, Iteratively reweighted algorithms for compressive sensing, Proceedings of ICASSP’08, submitted.
    • Proceedings of ICASSP’08
    • Chartrand, R.1    Yin, W.2
  • 24
    • 33646697494 scopus 로고    scopus 로고
    • A new coarse-to-fine framework for 3D brain MR image registration
    • Lecture Notes in Comput. Sci. 3765, Springer-Verlag, Berlin
    • T. Chen, T. Huang, W. Yin, and X. S. Zhou, A new coarse-to-fine framework for 3D brain MR image registration, in Computer Vision for Biomedical Image Applications, Lecture Notes in Comput. Sci. 3765, Springer-Verlag, Berlin, 2005, pp. 114-124.
    • (2005) Computer Vision for Biomedical Image Applications , pp. 114-124
    • Chen, T.1    Huang, T.2    Yin, W.3    Zhou, X.S.4
  • 26
    • 34548677063 scopus 로고    scopus 로고
    • Proximal thresholding algorithm for minimization over orthonormal bases
    • P. L. Combettes and J.-C. Pesquet, Proximal thresholding algorithm for minimization over orthonormal bases, SIAM J. Optim., 18 (2007), pp. 1351-1376.
    • (2007) SIAM J. Optim. , vol.18 , pp. 1351-1376
    • Combettes, P.L.1    Pesquet, J.-C.2
  • 28
    • 33845482689 scopus 로고    scopus 로고
    • Image restoration with discrete constrained total variation, Part I: Fast and exact optimization
    • J. Darbon and M. Sigelle, Image restoration with discrete constrained total variation, Part I: Fast and exact optimization, J. Math. Imaging Vision, 26 (2006), pp. 261-276.
    • (2006) J. Math. Imaging Vision , vol.26 , pp. 261-276
    • Darbon, J.1    Sigelle, M.2
  • 29
    • 7044231546 scopus 로고    scopus 로고
    • An iterative thresholding algorithm for linear inverse problems with a sparsity constraint
    • I. Daubechies, M. Defrise, and C. De Mol, An iterative thresholding algorithm for linear inverse problems with a sparsity constraint, Comm. Pure Appl. Math., 57 (2004), pp. 1413-1457.
    • (2004) Comm. Pure Appl. Math. , vol.57 , pp. 1413-1457
    • Daubechies, I.1    Defrise, M.2    De Mol, C.3
  • 31
    • 34548674463 scopus 로고    scopus 로고
    • A note on wavelet-based inversion algorithms
    • C. De Mol and M. Defrise, A note on wavelet-based inversion algorithms, Contemp. Math., 313 (2002), pp. 85-96.
    • (2002) Contemp. Math. , vol.313 , pp. 85-96
    • De Mol, C.1    Defrise, M.2
  • 32
    • 36248929865 scopus 로고    scopus 로고
    • Deterministic constructions of compressed sensing matrices
    • R. A. DeVore, Deterministic constructions of compressed sensing matrices, J. Complexity, 23 (2007), pp. 918-925.
    • (2007) J. Complexity , vol.23 , pp. 918-925
    • DeVore, R.A.1
  • 33
    • 85024093789 scopus 로고
    • Wavelets
    • Acta Numer., Cambridge University Press, Cambridge, UK
    • R. A. DeVore and B. J. Lucier, Wavelets, in Acta Numerica 1992, Acta Numer., Cambridge University Press, Cambridge, UK, 1992, pp. 54-81.
    • (1992) Acta Numerica 1992 , pp. 54-81
    • DeVore, R.A.1    Lucier, B.J.2
  • 35
  • 36
    • 22144471532 scopus 로고    scopus 로고
    • Neighborliness of randomly projected simplices in high dimensions
    • D. Donoho and J. Tanner, Neighborliness of randomly projected simplices in high dimensions, Proc. Natl. Acad. Sci. USA, 102 (2005), pp. 9452-9457.
    • (2005) Proc. Natl. Acad. Sci. USA , vol.102 , pp. 9452-9457
    • Donoho, D.1    Tanner, J.2
  • 37
    • 35548934118 scopus 로고    scopus 로고
    • Sparse solution of underdetermined linear equations by stagewise orthogonal matching pursuit
    • submitted
    • D. Donoho, Y. Tsaig, I. Drori, and J.-C. Starck, Sparse solution of underdetermined linear equations by stagewise orthogonal matching pursuit, IEEE Trans. Inform. Theory, submitted.
    • IEEE Trans. Inform. Theory
    • Donoho, D.1    Tsaig, Y.2    Drori, I.3    Starck, J.-C.4
  • 38
    • 33947361050 scopus 로고    scopus 로고
    • Why simple shrinkage is still relevant for redundant representations?
    • M. Elad, Why simple shrinkage is still relevant for redundant representations?, IEEE Trans. Inform. Theory, 52 (2006), pp. 5559-5569.
    • (2006) IEEE Trans. Inform. Theory , vol.52 , pp. 5559-5569
    • Elad, M.1
  • 39
    • 0036714214 scopus 로고    scopus 로고
    • A generalized uncertainty principle and sparse representations in pairs of bases
    • M. Elad and A. Bruckstein, A generalized uncertainty principle and sparse representations in pairs of bases, IEEE Trans. Inform. Theory, 48 (2002), pp. 2558-2567.
    • (2002) IEEE Trans. Inform. Theory , vol.48 , pp. 2558-2567
    • Elad, M.1    Bruckstein, A.2
  • 41
    • 35148813504 scopus 로고    scopus 로고
    • Coordinate and subspace optimization methods for linear least squares with non-quadratic regularization
    • M. Elad, B. Matalon, and M. Zibulevsky, Coordinate and subspace optimization methods for linear least squares with non-quadratic regularization, Appl. Comput. Harmon. Anal., 23 (2007), pp. 346-367.
    • (2007) Appl. Comput. Harmon. Anal. , vol.23 , pp. 346-367
    • Elad, M.1    Matalon, B.2    Zibulevsky, M.3
  • 42
    • 27844460570 scopus 로고    scopus 로고
    • Simultaneous cartoon and texture image inpainting using morphological component analysis (MCA)
    • M. Elad, J.-C. Starck, P. Querre, and D. Donoho, Simultaneous cartoon and texture image inpainting using morphological component analysis (MCA), Appl. Comput. Harmon. Anal., 19 (2005), pp. 340-358.
    • (2005) Appl. Comput. Harmon. Anal. , vol.19 , pp. 340-358
    • Elad, M.1    Starck, J.-C.2    Querre, P.3    Donoho, D.4
  • 44
    • 0038718564 scopus 로고    scopus 로고
    • On sparse representation in pairs of bases
    • A. Feuer and A. Nemirovski, On sparse representation in pairs of bases, IEEE Trans. Inform. Theory, 49 (2003), pp. 1579-1581.
    • (2003) IEEE Trans. Inform. Theory , vol.49 , pp. 1579-1581
    • Feuer, A.1    Nemirovski, A.2
  • 45
    • 36749072615 scopus 로고    scopus 로고
    • Majorization-minimization algorithms for waveletbased image restoration
    • M. Figueiredo, J. Bioucas-Dias, and R. Nowak, Majorization-minimization algorithms for waveletbased image restoration, IEEE Trans. Image Process., 16 (2007), pp. 2980-2991.
    • (2007) IEEE Trans. Image Process. , vol.16 , pp. 2980-2991
    • Figueiredo, M.1    Bioucas-Dias, J.2    Nowak, R.3
  • 46
    • 0142200827 scopus 로고    scopus 로고
    • An EM algorithm for wavelet-based image restoration
    • M. Figueiredo and R. Nowak, An EM algorithm for wavelet-based image restoration, IEEE Trans. Image Process., 12 (2003), pp. 906-916.
    • (2003) IEEE Trans. Image Process. , vol.12 , pp. 906-916
    • Figueiredo, M.1    Nowak, R.2
  • 49
    • 84926080848 scopus 로고    scopus 로고
    • Gradient projection for sparse reconstruction: Application to compressed sensing and other inverse problems
    • to appear
    • M. Figueiredo, R. Nowak, and S. J. Wright, Gradient projection for sparse reconstruction: Application to compressed sensing and other inverse problems, IEEE J. Selected Topics in Signal Processing, to appear.
    • IEEE J. Selected Topics in Signal Processing
    • Figueiredo, M.1    Nowak, R.2    Wright, S.J.3
  • 52
    • 0031312521 scopus 로고    scopus 로고
    • WaveShrink with firm shrinkage
    • H.-Y. Gao and A. G. Bruce, WaveShrink with firm shrinkage, Statist. Sinica, 7 (1997), pp. 855-874.
    • (1997) Statist. Sinica , vol.7 , pp. 855-874
    • Gao, H.-Y.1    Bruce, A.G.2
  • 57
    • 85024107401 scopus 로고    scopus 로고
    • A numerical study of fixed-point continuation applied to compressed sensing
    • submitted
    • E. Hale, W. Yin, and Y. Zhang, A numerical study of fixed-point continuation applied to compressed sensing, IEEE Trans. Inform. Theory, submitted.
    • IEEE Trans. Inform. Theory
    • Hale, E.1    Yin, W.2    Zhang, Y.3
  • 60
    • 0014604308 scopus 로고
    • Multiplier and gradient methods
    • M. R. Hestenes, Multiplier and gradient methods, J. Optim. Theory Appl., 4 (1969), pp. 303-320.
    • (1969) J. Optim. Theory Appl. , vol.4 , pp. 303-320
    • Hestenes, M.R.1
  • 61
    • 34248999153 scopus 로고    scopus 로고
    • Improved k-t BLAST and k-t SENSE using FOCUSS
    • H. Jung, J. Ye, and E. Kim, Improved k-t BLAST and k-t SENSE using FOCUSS, Phys. Med. Biol., 52 (2007), pp. 3201-3226.
    • (2007) Phys. Med. Biol. , vol.52 , pp. 3201-3226
    • Jung, H.1    Ye, J.2    Kim, E.3
  • 67
    • 0026839499 scopus 로고
    • On the linear convergence of descent methods for convex essentially smooth minimization
    • Z.-Q. Luo and P. Tseng, On the linear convergence of descent methods for convex essentially smooth minimization, SIAM J. Control Optim., 30 (1992), pp. 408-425.
    • (1992) SIAM J. Control Optim. , vol.30 , pp. 408-425
    • Luo, Z.-Q.1    Tseng, P.2
  • 68
    • 36849088522 scopus 로고    scopus 로고
    • Sparse MRI: The application of compressed sensing for rapid MR imaging
    • M. Lustig, D. Donoho, and J. M. Pauly, Sparse MRI: The application of compressed sensing for rapid MR imaging, Magnetic Resonance in Medicine, 58 (2007), pp. 1182-1195.
    • (2007) Magnetic Resonance in Medicine , vol.58 , pp. 1182-1195
    • Lustig, M.1    Donoho, D.2    Pauly, J.M.3
  • 72
    • 0035573098 scopus 로고    scopus 로고
    • Fast wavelet-based image deconvolution using the EM algorithm
    • Systems, and Computers, Monterey, CA
    • R. Nowak and M. Figueiredo, Fast wavelet-based image deconvolution using the EM algorithm,, in Proceedings of the 35th Asilomar Conference on Signals, Systems, and Computers, Monterey, CA, 2001.
    • (2001) Proceedings of the 35th Asilomar Conference on Signals
    • Nowak, R.1    Figueiredo, M.2
  • 73
    • 19844370110 scopus 로고    scopus 로고
    • An iterated regularization method for total variation-based image restoration
    • S. Osher, M. Burger, D. Goldfarb, J. Xu, and W. Yin, An iterated regularization method for total variation-based image restoration, Multiscale Model. Simul., 4 (2005), pp. 460-489.
    • (2005) Multiscale Model. Simul. , vol.4 , pp. 460-489
    • Osher, S.1    Burger, M.2    Goldfarb, D.3    Xu, J.4    Yin, W.5
  • 74
    • 0001170028 scopus 로고
    • A posteriori error bounds for the linearly-constrained variational inequality problem
    • J.-S. Pang, A posteriori error bounds for the linearly-constrained variational inequality problem, Math. Oper. Res., 12 (1987), pp. 474-484.
    • (1987) Math. Oper. Res. , vol.12 , pp. 474-484
    • Pang, J.-S.1
  • 75
    • 0002380692 scopus 로고
    • A method for nonlinear constraints in minimization problems
    • R. Fletcher, ed., Academic Press, New York
    • M. J. D. Powell, A method for nonlinear constraints in minimization problems, in Optimization, R. Fletcher, ed., Academic Press, New York, 1972, pp. 283-298.
    • (1972) Optimization , pp. 283-298
    • Powell, M.J.D.1
  • 79
    • 0015727623 scopus 로고
    • A dual approach to solving nonlinear programming problems by unconstrained optimization
    • R. T. Rockafellar, A dual approach to solving nonlinear programming problems by unconstrained optimization, Math. Programming, 5 (1973), pp. 354-373.
    • (1973) Math. Programming , vol.5 , pp. 354-373
    • Rockafellar, R.T.1
  • 80
    • 33746801607 scopus 로고    scopus 로고
    • Geometric approach to error correcting codes and reconstruction of signals
    • M. Rudelson and R. Vershynin, Geometric approach to error correcting codes and reconstruction of signals, Int. Math. Res. Not., 64 (2005), pp. 4019-4041.
    • (2005) Int. Math. Res. Not. , vol.64 , pp. 4019-4041
    • Rudelson, M.1    Vershynin, R.2
  • 81
    • 44049111982 scopus 로고
    • Nonlinear total variation based noise removal algorithms
    • L. Rudin, S. Osher, and E. Fatemi, Nonlinear total variation based noise removal algorithms, Phys. D, 60 (1992), pp. 259-268.
    • (1992) Phys. D , vol.60 , pp. 259-268
    • Rudin, L.1    Osher, S.2    Fatemi, E.3
  • 85
    • 0001287271 scopus 로고    scopus 로고
    • Regression shrinkage and selection via the lasso
    • R. Tibshirani, Regression shrinkage and selection via the lasso, J. Roy. Statist. Soc. Ser. B, 58 (1996), pp. 267-288.
    • (1996) J. Roy. Statist. Soc. Ser. B , vol.58 , pp. 267-288
    • Tibshirani, R.1
  • 86
    • 33645712308 scopus 로고    scopus 로고
    • Just relax: Convex programming methods for identifying sparse signals
    • J. Tropp, Just relax: Convex programming methods for identifying sparse signals, IEEE Trans. Inform. Theory, 51 (2006), pp. 1030-1051.
    • (2006) IEEE Trans. Inform. Theory , vol.51 , pp. 1030-1051
    • Tropp, J.1
  • 88
    • 30844433547 scopus 로고    scopus 로고
    • Extensions of compressed sensing
    • Y. Tsaig and D. L. Donoho, Extensions of compressed sensing, Signal Processing, 86 (2006), pp. 549-571.
    • (2006) Signal Processing , vol.86 , pp. 549-571
    • Tsaig, Y.1    Donoho, D.L.2
  • 89
    • 41249087776 scopus 로고    scopus 로고
    • UBC Computer Science Technical report TR-2007-16, University of British Columbia, Vancouver
    • E. Van den Berg and M. P. Friedlander, In Pursuit of a Root, UBC Computer Science Technical report TR-2007-16, University of British Columbia, Vancouver, 2007.
    • (2007) In Pursuit of a Root
    • Van Den Berg, E.1    Friedlander, M.P.2
  • 95
    • 33847715169 scopus 로고    scopus 로고
    • Iterative regularization and nonlinear inverse scale space applied to wavelet-based denoising
    • J. Xu and S. Osher, Iterative regularization and nonlinear inverse scale space applied to wavelet-based denoising, IEEE Trans. Image Process., 16 (2006), pp. 534-544.
    • (2006) IEEE Trans. Image Process. , vol.16 , pp. 534-544
    • Xu, J.1    Osher, S.2
  • 96
    • 34548781325 scopus 로고    scopus 로고
    • Compressed sensing shape estimation of star-shaped objects in Fourier imaging
    • J. Ye, Compressed sensing shape estimation of star-shaped objects in Fourier imaging, IEEE Signal Process. Lett., 14 (2007), pp. 750-753.
    • (2007) IEEE Signal Process. Lett. , vol.14 , pp. 750-753
    • Ye, J.1
  • 98
    • 34247608671 scopus 로고    scopus 로고
    • A comparison of three total variation-based texture extraction models
    • W. Yin, D. Goldfarb, and S. Osher, A comparison of three total variation-based texture extraction models, J. Visual Commun. Image Representation, 18 (2007), pp. 240-252.
    • (2007) J. Visual Commun. Image Representation , vol.18 , pp. 240-252
    • Yin, W.1    Goldfarb, D.2    Osher, S.3
  • 100
    • 67749135852 scopus 로고    scopus 로고
    • CAAM Technical report TR05-09, Rice University, Houston, TX
    • 1-Minimization: Go Over or Under?, CAAM Technical report TR05-09, Rice University, Houston, TX, 2005.
    • (2005) 1-Minimization: Go Over or Under?
    • Zhang, Y.1
  • 101
    • 67749091294 scopus 로고    scopus 로고
    • CAAM Technical report TR06-15, Rice University, Houston, TX
    • Y. Zhang, When Is Missing Data Recoverable?, CAAM Technical report TR06-15, Rice University, Houston, TX, 2006.
    • (2006) When Is Missing Data Recoverable?
    • Zhang, Y.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.