Abstract

High-density molecules localization algorithm is crucial to obtain sufficient temporal resolution in super-resolution fluorescence microscopy, particularly in view of the challenges associated with live-cell imaging. In this work, an algorithm based on augmented Lagrangian method (ALM) is proposed for reconstructing high-density molecules. The problem is firstly converted to an equivalent optimization problem with constraints using variable splitting, and then the alternating minimization method is applied to implement it straightforwardly. We also take advantage of quasi-Newton method to tackle the sub-problems for acceleration, and total variation regularization to reduce noise. Numerical results on both simulated and real data demonstrate that the algorithm can achieve using fewer frames of raw images to reconstruct high-resolution image with favorable performance in terms of detection rate and image quality.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2018 (1)

2017 (4)

W. Hao, R. Feng, Z.-F. Han, and C.-S. Leung, “ADMM-based algorithm for training fault tolerant RBF networks and selecting centers,” IEEE Trans. Neural Netw. Learn. Syst. 29(8), 3870–3878 (2017).
[PubMed]

T. Cheng, D. Chen, B. Yu, and H. Niu, “Reconstruction of super-resolution STORM images using compressed sensing based on low-resolution raw images and interpolation,” Biomed. Opt. Express 8(5), 2445–2457 (2017).
[Crossref] [PubMed]

A. Lee, K. Tsekouras, C. Calderon, C. Bustamante, and S. Pressé, “Unraveling the thousand word picture: an introduction to super-resolution data analysis,” Chem. Rev. 117(11), 7276–7330 (2017).
[Crossref] [PubMed]

J. Huang, M. Sun, J. Ma, and Y. Chi, “Super-resolution image reconstruction for high-density 3D single-molecule microscopy,” IEEE Trans. Comput. Imag. 3(4), 763–773 (2017).
[Crossref]

2016 (2)

K. Agarwal and R. Macháň, “Multiple signal classification algorithm for super-resolution fluorescence microscopy,” Nat. Commun. 7, 13752 (2016).
[Crossref] [PubMed]

N. Pavillon and N. I. Smith, “Compressed sensing laser scanning microscopy,” Opt. Express 24(26), 30038–30052 (2016).
[Crossref] [PubMed]

2015 (1)

D. Sage, H. Kirshner, T. Pengo, N. Stuurman, J. Min, S. Manley, and M. Unser, “Quantitative evaluation of software packages for single-molecule localization microscopy,” Nat. Methods 12(8), 717–724 (2015).
[Crossref] [PubMed]

2014 (6)

H. Bingsheng, H. Liu, Z. Wang, and X. Yuan, “A strictly contractive Peaceman-Rachford splitting method for convex programming,” SIAM J. Optim. 24(3), 1011–1040 (2014).
[Crossref] [PubMed]

M. Ovesný, P. Křížek, J. Borkovec, Z. Švindrych, and G. M. Hagen, “ThunderSTORM: a comprehensive ImageJ plug-in for PALM and STORM data analysis and super-resolution imaging,” Bioinformatics 30(16), 2389–2390 (2014).
[Crossref] [PubMed]

A. Small and S. Stahlheber, “Fluorophore localization algorithms for super-resolution microscopy,” Nat. Methods 11(3), 267–279 (2014).
[Crossref] [PubMed]

U. Köthe, F. Herrmannsdörfer, I. Kats, and F. A. Hamprecht, “SimpleSTORM: a fast, self-calibrating reconstruction algorithm for localization microscopy,” Histochem. Cell Biol. 141(6), 613–627 (2014).
[Crossref] [PubMed]

J. Min, C. Vonesch, H. Kirshner, L. Carlini, N. Olivier, S. Holden, S. Manley, J. C. Ye, and M. Unser, “FALCON: fast and unbiased reconstruction of high-density super-resolution microscopy data,” Sci. Rep. 4(1), 4577 (2014).
[Crossref] [PubMed]

M. Ovesný, P. Křížek, Z. Švindrych, and G. M. Hagen, “High density 3D localization microscopy using sparse support recovery,” Opt. Express 22(25), 31263–31276 (2014).
[Crossref] [PubMed]

2013 (5)

H. P. Babcock, J. R. Moffitt, Y. Cao, and X. Zhuang, “Fast compressed sensing analysis for super-resolution imaging using L1-homotopy,” Opt. Express 21(23), 28583–28596 (2013).
[Crossref] [PubMed]

M. J. Allison, S. Ramani, and J. A. Fessler, “Accelerated regularized estimation of MR coil sensitivities using augmented Lagrangian methods,” IEEE Trans. Med. Imaging 32(3), 556–564 (2013).
[Crossref] [PubMed]

R. P. J. Nieuwenhuizen, K. A. Lidke, M. Bates, D. L. Puig, D. Grünwald, S. Stallinga, and B. Rieger, “Measuring image resolution in optical nanoscopy,” Nat. Methods 10(6), 557–562 (2013).
[Crossref] [PubMed]

C. Li, W. Yin, H. Jiang, and Y. Zhang, “An efficient augmented Lagrangian method with applications to total variation minimization,” Comput. Optim. Appl. 56(3), 507–530 (2013).
[Crossref]

J. Li, S. Liu, and E. Y. Lam, “Efficient source and mask optimization with augmented Lagrangian methods in optical lithography,” Opt. Express 21(7), 8076–8090 (2013).
[Crossref] [PubMed]

2012 (3)

J. Li, Y. Shen, and E. Y. Lam, “Hotspot-aware fast source and mask optimization,” Opt. Express 20(19), 21792–21804 (2012).
[Crossref] [PubMed]

E. A. Mukamel, H. Babcock, and X. Zhuang, “Statistical deconvolution for superresolution fluorescence microscopy,” Biophys. J. 102(10), 2391–2400 (2012).
[Crossref] [PubMed]

L. Zhu, W. Zhang, D. Elnatan, and B. Huang, “Faster STORM using compressed sensing,” Nat. Methods 9(7), 721–723 (2012).
[Crossref] [PubMed]

2011 (6)

S. J. Holden, S. Uphoff, and A. N. Kapanidis, “DAOSTORM: an algorithm for high- density super-resolution microscopy,” Nat. Methods 8(4), 279–280 (2011).
[Crossref] [PubMed]

F. Huang, S. L. Schwartz, J. M. Byars, and K. A. Lidke, “Simultaneous multiple-emitter fitting for single molecule super-resolution imaging,” Biomed. Opt. Express 2(5), 1377–1393 (2011).
[Crossref] [PubMed]

S. A. Jones, S. H. Shim, J. He, and X. Zhuang, “Fast, three-dimensional super-resolution imaging of live cells,” Nat. Methods 8(6), 499–508 (2011).
[Crossref] [PubMed]

S. Cox, E. Rosten, J. Monypenny, T. Jovanovic-Talisman, D. T. Burnette, J. Lippincott-Schwartz, G. E. Jones, and R. Heintzmann, “Bayesian localization microscopy reveals nanoscale podosome dynamics,” Nat. Methods 9(2), 195–200 (2011).
[Crossref] [PubMed]

S. H. Chan, R. Khoshabeh, K. B. Gibson, P. E. Gill, and T. Q. Nguyen, “An augmented Lagrangian method for total variation video restoration,” IEEE Trans. Image Process. 20(11), 3097–3111 (2011).
[Crossref] [PubMed]

J. L. Morales and J. Nocedal, “Remark on ‘algorithm 778: L-BFGS-B: Fortran subroutines for large-scale bound constrained optimization’,” ACM Trans. Math. Softw. 38(1), 1–4 (2011).
[Crossref]

2010 (1)

M. V. Afonso, J. M. Bioucas-Dias, and M. A. T. Figueiredo, “Fast image recovery using variable splitting and constrained optimization,” IEEE Trans. Image Process. 19(9), 2345–2356 (2010).
[Crossref] [PubMed]

2009 (2)

A. Beck and M. Teboulle, “Fast gradient-based algorithms for constrained total variation image denoising and deblurring problems,” IEEE Trans. Image Process. 18(11), 2419–2434 (2009).
[Crossref] [PubMed]

T. Goldstein and S. Osher, “The split Bregman algorithm forl1,” SIAM J. Imaging Sci. 2(2), 323–343 (2009).
[Crossref]

2008 (1)

G.-H. Chen, J. Tang, and S. Leng, “Prior image constrained compressed sensing (PICCS): a method to accurately reconstruct dynamic CT images from highly undersampled projection data sets,” Med. Phys. 35(2), 660–663 (2008).
[Crossref] [PubMed]

2006 (2)

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313(5793), 1642–1645 (2006).
[Crossref] [PubMed]

M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods 3(10), 793–795 (2006).
[Crossref] [PubMed]

1994 (1)

Afonso, M. V.

M. V. Afonso, J. M. Bioucas-Dias, and M. A. T. Figueiredo, “Fast image recovery using variable splitting and constrained optimization,” IEEE Trans. Image Process. 19(9), 2345–2356 (2010).
[Crossref] [PubMed]

Agarwal, K.

K. Agarwal and R. Macháň, “Multiple signal classification algorithm for super-resolution fluorescence microscopy,” Nat. Commun. 7, 13752 (2016).
[Crossref] [PubMed]

Allison, M. J.

M. J. Allison, S. Ramani, and J. A. Fessler, “Accelerated regularized estimation of MR coil sensitivities using augmented Lagrangian methods,” IEEE Trans. Med. Imaging 32(3), 556–564 (2013).
[Crossref] [PubMed]

Babcock, H.

E. A. Mukamel, H. Babcock, and X. Zhuang, “Statistical deconvolution for superresolution fluorescence microscopy,” Biophys. J. 102(10), 2391–2400 (2012).
[Crossref] [PubMed]

Babcock, H. P.

Bates, M.

R. P. J. Nieuwenhuizen, K. A. Lidke, M. Bates, D. L. Puig, D. Grünwald, S. Stallinga, and B. Rieger, “Measuring image resolution in optical nanoscopy,” Nat. Methods 10(6), 557–562 (2013).
[Crossref] [PubMed]

M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods 3(10), 793–795 (2006).
[Crossref] [PubMed]

Beck, A.

A. Beck and M. Teboulle, “Fast gradient-based algorithms for constrained total variation image denoising and deblurring problems,” IEEE Trans. Image Process. 18(11), 2419–2434 (2009).
[Crossref] [PubMed]

Betzig, E.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313(5793), 1642–1645 (2006).
[Crossref] [PubMed]

Bingsheng, H.

H. Bingsheng, H. Liu, Z. Wang, and X. Yuan, “A strictly contractive Peaceman-Rachford splitting method for convex programming,” SIAM J. Optim. 24(3), 1011–1040 (2014).
[Crossref] [PubMed]

Bioucas-Dias, J. M.

M. V. Afonso, J. M. Bioucas-Dias, and M. A. T. Figueiredo, “Fast image recovery using variable splitting and constrained optimization,” IEEE Trans. Image Process. 19(9), 2345–2356 (2010).
[Crossref] [PubMed]

Bonifacino, J. S.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313(5793), 1642–1645 (2006).
[Crossref] [PubMed]

Borkovec, J.

M. Ovesný, P. Křížek, J. Borkovec, Z. Švindrych, and G. M. Hagen, “ThunderSTORM: a comprehensive ImageJ plug-in for PALM and STORM data analysis and super-resolution imaging,” Bioinformatics 30(16), 2389–2390 (2014).
[Crossref] [PubMed]

Burnette, D. T.

S. Cox, E. Rosten, J. Monypenny, T. Jovanovic-Talisman, D. T. Burnette, J. Lippincott-Schwartz, G. E. Jones, and R. Heintzmann, “Bayesian localization microscopy reveals nanoscale podosome dynamics,” Nat. Methods 9(2), 195–200 (2011).
[Crossref] [PubMed]

Bustamante, C.

A. Lee, K. Tsekouras, C. Calderon, C. Bustamante, and S. Pressé, “Unraveling the thousand word picture: an introduction to super-resolution data analysis,” Chem. Rev. 117(11), 7276–7330 (2017).
[Crossref] [PubMed]

Byars, J. M.

Calderon, C.

A. Lee, K. Tsekouras, C. Calderon, C. Bustamante, and S. Pressé, “Unraveling the thousand word picture: an introduction to super-resolution data analysis,” Chem. Rev. 117(11), 7276–7330 (2017).
[Crossref] [PubMed]

Cao, Y.

Carlini, L.

J. Min, C. Vonesch, H. Kirshner, L. Carlini, N. Olivier, S. Holden, S. Manley, J. C. Ye, and M. Unser, “FALCON: fast and unbiased reconstruction of high-density super-resolution microscopy data,” Sci. Rep. 4(1), 4577 (2014).
[Crossref] [PubMed]

Chan, S. H.

S. H. Chan, R. Khoshabeh, K. B. Gibson, P. E. Gill, and T. Q. Nguyen, “An augmented Lagrangian method for total variation video restoration,” IEEE Trans. Image Process. 20(11), 3097–3111 (2011).
[Crossref] [PubMed]

Chen, D.

Chen, G.-H.

G.-H. Chen, J. Tang, and S. Leng, “Prior image constrained compressed sensing (PICCS): a method to accurately reconstruct dynamic CT images from highly undersampled projection data sets,” Med. Phys. 35(2), 660–663 (2008).
[Crossref] [PubMed]

Cheng, T.

Chi, Y.

J. Huang, M. Sun, J. Ma, and Y. Chi, “Super-resolution image reconstruction for high-density 3D single-molecule microscopy,” IEEE Trans. Comput. Imag. 3(4), 763–773 (2017).
[Crossref]

Cox, S.

S. Cox, E. Rosten, J. Monypenny, T. Jovanovic-Talisman, D. T. Burnette, J. Lippincott-Schwartz, G. E. Jones, and R. Heintzmann, “Bayesian localization microscopy reveals nanoscale podosome dynamics,” Nat. Methods 9(2), 195–200 (2011).
[Crossref] [PubMed]

Davidson, M. W.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313(5793), 1642–1645 (2006).
[Crossref] [PubMed]

Eldar, Y. C.

Elnatan, D.

L. Zhu, W. Zhang, D. Elnatan, and B. Huang, “Faster STORM using compressed sensing,” Nat. Methods 9(7), 721–723 (2012).
[Crossref] [PubMed]

Feng, R.

W. Hao, R. Feng, Z.-F. Han, and C.-S. Leung, “ADMM-based algorithm for training fault tolerant RBF networks and selecting centers,” IEEE Trans. Neural Netw. Learn. Syst. 29(8), 3870–3878 (2017).
[PubMed]

Fessler, J. A.

M. J. Allison, S. Ramani, and J. A. Fessler, “Accelerated regularized estimation of MR coil sensitivities using augmented Lagrangian methods,” IEEE Trans. Med. Imaging 32(3), 556–564 (2013).
[Crossref] [PubMed]

Figueiredo, M. A. T.

M. V. Afonso, J. M. Bioucas-Dias, and M. A. T. Figueiredo, “Fast image recovery using variable splitting and constrained optimization,” IEEE Trans. Image Process. 19(9), 2345–2356 (2010).
[Crossref] [PubMed]

Gibson, K. B.

S. H. Chan, R. Khoshabeh, K. B. Gibson, P. E. Gill, and T. Q. Nguyen, “An augmented Lagrangian method for total variation video restoration,” IEEE Trans. Image Process. 20(11), 3097–3111 (2011).
[Crossref] [PubMed]

Gill, P. E.

S. H. Chan, R. Khoshabeh, K. B. Gibson, P. E. Gill, and T. Q. Nguyen, “An augmented Lagrangian method for total variation video restoration,” IEEE Trans. Image Process. 20(11), 3097–3111 (2011).
[Crossref] [PubMed]

Goldstein, T.

T. Goldstein and S. Osher, “The split Bregman algorithm forl1,” SIAM J. Imaging Sci. 2(2), 323–343 (2009).
[Crossref]

Grünwald, D.

R. P. J. Nieuwenhuizen, K. A. Lidke, M. Bates, D. L. Puig, D. Grünwald, S. Stallinga, and B. Rieger, “Measuring image resolution in optical nanoscopy,” Nat. Methods 10(6), 557–562 (2013).
[Crossref] [PubMed]

Hagen, G. M.

M. Ovesný, P. Křížek, J. Borkovec, Z. Švindrych, and G. M. Hagen, “ThunderSTORM: a comprehensive ImageJ plug-in for PALM and STORM data analysis and super-resolution imaging,” Bioinformatics 30(16), 2389–2390 (2014).
[Crossref] [PubMed]

M. Ovesný, P. Křížek, Z. Švindrych, and G. M. Hagen, “High density 3D localization microscopy using sparse support recovery,” Opt. Express 22(25), 31263–31276 (2014).
[Crossref] [PubMed]

Hamprecht, F. A.

U. Köthe, F. Herrmannsdörfer, I. Kats, and F. A. Hamprecht, “SimpleSTORM: a fast, self-calibrating reconstruction algorithm for localization microscopy,” Histochem. Cell Biol. 141(6), 613–627 (2014).
[Crossref] [PubMed]

Han, Z.-F.

W. Hao, R. Feng, Z.-F. Han, and C.-S. Leung, “ADMM-based algorithm for training fault tolerant RBF networks and selecting centers,” IEEE Trans. Neural Netw. Learn. Syst. 29(8), 3870–3878 (2017).
[PubMed]

Hao, W.

W. Hao, R. Feng, Z.-F. Han, and C.-S. Leung, “ADMM-based algorithm for training fault tolerant RBF networks and selecting centers,” IEEE Trans. Neural Netw. Learn. Syst. 29(8), 3870–3878 (2017).
[PubMed]

He, J.

S. A. Jones, S. H. Shim, J. He, and X. Zhuang, “Fast, three-dimensional super-resolution imaging of live cells,” Nat. Methods 8(6), 499–508 (2011).
[Crossref] [PubMed]

Heintzmann, R.

S. Cox, E. Rosten, J. Monypenny, T. Jovanovic-Talisman, D. T. Burnette, J. Lippincott-Schwartz, G. E. Jones, and R. Heintzmann, “Bayesian localization microscopy reveals nanoscale podosome dynamics,” Nat. Methods 9(2), 195–200 (2011).
[Crossref] [PubMed]

Hell, S. W.

Herrmannsdörfer, F.

U. Köthe, F. Herrmannsdörfer, I. Kats, and F. A. Hamprecht, “SimpleSTORM: a fast, self-calibrating reconstruction algorithm for localization microscopy,” Histochem. Cell Biol. 141(6), 613–627 (2014).
[Crossref] [PubMed]

Hess, H. F.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313(5793), 1642–1645 (2006).
[Crossref] [PubMed]

Holden, S.

J. Min, C. Vonesch, H. Kirshner, L. Carlini, N. Olivier, S. Holden, S. Manley, J. C. Ye, and M. Unser, “FALCON: fast and unbiased reconstruction of high-density super-resolution microscopy data,” Sci. Rep. 4(1), 4577 (2014).
[Crossref] [PubMed]

Holden, S. J.

S. J. Holden, S. Uphoff, and A. N. Kapanidis, “DAOSTORM: an algorithm for high- density super-resolution microscopy,” Nat. Methods 8(4), 279–280 (2011).
[Crossref] [PubMed]

Huang, B.

L. Zhu, W. Zhang, D. Elnatan, and B. Huang, “Faster STORM using compressed sensing,” Nat. Methods 9(7), 721–723 (2012).
[Crossref] [PubMed]

Huang, F.

Huang, J.

J. Huang, M. Sun, J. Ma, and Y. Chi, “Super-resolution image reconstruction for high-density 3D single-molecule microscopy,” IEEE Trans. Comput. Imag. 3(4), 763–773 (2017).
[Crossref]

Jiang, H.

C. Li, W. Yin, H. Jiang, and Y. Zhang, “An efficient augmented Lagrangian method with applications to total variation minimization,” Comput. Optim. Appl. 56(3), 507–530 (2013).
[Crossref]

Jones, G. E.

S. Cox, E. Rosten, J. Monypenny, T. Jovanovic-Talisman, D. T. Burnette, J. Lippincott-Schwartz, G. E. Jones, and R. Heintzmann, “Bayesian localization microscopy reveals nanoscale podosome dynamics,” Nat. Methods 9(2), 195–200 (2011).
[Crossref] [PubMed]

Jones, S. A.

S. A. Jones, S. H. Shim, J. He, and X. Zhuang, “Fast, three-dimensional super-resolution imaging of live cells,” Nat. Methods 8(6), 499–508 (2011).
[Crossref] [PubMed]

Jovanovic-Talisman, T.

S. Cox, E. Rosten, J. Monypenny, T. Jovanovic-Talisman, D. T. Burnette, J. Lippincott-Schwartz, G. E. Jones, and R. Heintzmann, “Bayesian localization microscopy reveals nanoscale podosome dynamics,” Nat. Methods 9(2), 195–200 (2011).
[Crossref] [PubMed]

Kapanidis, A. N.

S. J. Holden, S. Uphoff, and A. N. Kapanidis, “DAOSTORM: an algorithm for high- density super-resolution microscopy,” Nat. Methods 8(4), 279–280 (2011).
[Crossref] [PubMed]

Kats, I.

U. Köthe, F. Herrmannsdörfer, I. Kats, and F. A. Hamprecht, “SimpleSTORM: a fast, self-calibrating reconstruction algorithm for localization microscopy,” Histochem. Cell Biol. 141(6), 613–627 (2014).
[Crossref] [PubMed]

Khoshabeh, R.

S. H. Chan, R. Khoshabeh, K. B. Gibson, P. E. Gill, and T. Q. Nguyen, “An augmented Lagrangian method for total variation video restoration,” IEEE Trans. Image Process. 20(11), 3097–3111 (2011).
[Crossref] [PubMed]

Kirshner, H.

D. Sage, H. Kirshner, T. Pengo, N. Stuurman, J. Min, S. Manley, and M. Unser, “Quantitative evaluation of software packages for single-molecule localization microscopy,” Nat. Methods 12(8), 717–724 (2015).
[Crossref] [PubMed]

J. Min, C. Vonesch, H. Kirshner, L. Carlini, N. Olivier, S. Holden, S. Manley, J. C. Ye, and M. Unser, “FALCON: fast and unbiased reconstruction of high-density super-resolution microscopy data,” Sci. Rep. 4(1), 4577 (2014).
[Crossref] [PubMed]

Köthe, U.

U. Köthe, F. Herrmannsdörfer, I. Kats, and F. A. Hamprecht, “SimpleSTORM: a fast, self-calibrating reconstruction algorithm for localization microscopy,” Histochem. Cell Biol. 141(6), 613–627 (2014).
[Crossref] [PubMed]

Krížek, P.

M. Ovesný, P. Křížek, J. Borkovec, Z. Švindrych, and G. M. Hagen, “ThunderSTORM: a comprehensive ImageJ plug-in for PALM and STORM data analysis and super-resolution imaging,” Bioinformatics 30(16), 2389–2390 (2014).
[Crossref] [PubMed]

M. Ovesný, P. Křížek, Z. Švindrych, and G. M. Hagen, “High density 3D localization microscopy using sparse support recovery,” Opt. Express 22(25), 31263–31276 (2014).
[Crossref] [PubMed]

Lam, E. Y.

Lee, A.

A. Lee, K. Tsekouras, C. Calderon, C. Bustamante, and S. Pressé, “Unraveling the thousand word picture: an introduction to super-resolution data analysis,” Chem. Rev. 117(11), 7276–7330 (2017).
[Crossref] [PubMed]

Leng, S.

G.-H. Chen, J. Tang, and S. Leng, “Prior image constrained compressed sensing (PICCS): a method to accurately reconstruct dynamic CT images from highly undersampled projection data sets,” Med. Phys. 35(2), 660–663 (2008).
[Crossref] [PubMed]

Leung, C.-S.

W. Hao, R. Feng, Z.-F. Han, and C.-S. Leung, “ADMM-based algorithm for training fault tolerant RBF networks and selecting centers,” IEEE Trans. Neural Netw. Learn. Syst. 29(8), 3870–3878 (2017).
[PubMed]

Li, C.

C. Li, W. Yin, H. Jiang, and Y. Zhang, “An efficient augmented Lagrangian method with applications to total variation minimization,” Comput. Optim. Appl. 56(3), 507–530 (2013).
[Crossref]

Li, J.

Lidke, K. A.

R. P. J. Nieuwenhuizen, K. A. Lidke, M. Bates, D. L. Puig, D. Grünwald, S. Stallinga, and B. Rieger, “Measuring image resolution in optical nanoscopy,” Nat. Methods 10(6), 557–562 (2013).
[Crossref] [PubMed]

F. Huang, S. L. Schwartz, J. M. Byars, and K. A. Lidke, “Simultaneous multiple-emitter fitting for single molecule super-resolution imaging,” Biomed. Opt. Express 2(5), 1377–1393 (2011).
[Crossref] [PubMed]

Lindwasser, O. W.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313(5793), 1642–1645 (2006).
[Crossref] [PubMed]

Lippincott-Schwartz, J.

S. Cox, E. Rosten, J. Monypenny, T. Jovanovic-Talisman, D. T. Burnette, J. Lippincott-Schwartz, G. E. Jones, and R. Heintzmann, “Bayesian localization microscopy reveals nanoscale podosome dynamics,” Nat. Methods 9(2), 195–200 (2011).
[Crossref] [PubMed]

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313(5793), 1642–1645 (2006).
[Crossref] [PubMed]

Liu, H.

H. Bingsheng, H. Liu, Z. Wang, and X. Yuan, “A strictly contractive Peaceman-Rachford splitting method for convex programming,” SIAM J. Optim. 24(3), 1011–1040 (2014).
[Crossref] [PubMed]

Liu, S.

Ma, J.

J. Huang, M. Sun, J. Ma, and Y. Chi, “Super-resolution image reconstruction for high-density 3D single-molecule microscopy,” IEEE Trans. Comput. Imag. 3(4), 763–773 (2017).
[Crossref]

Machán, R.

K. Agarwal and R. Macháň, “Multiple signal classification algorithm for super-resolution fluorescence microscopy,” Nat. Commun. 7, 13752 (2016).
[Crossref] [PubMed]

Manley, S.

D. Sage, H. Kirshner, T. Pengo, N. Stuurman, J. Min, S. Manley, and M. Unser, “Quantitative evaluation of software packages for single-molecule localization microscopy,” Nat. Methods 12(8), 717–724 (2015).
[Crossref] [PubMed]

J. Min, C. Vonesch, H. Kirshner, L. Carlini, N. Olivier, S. Holden, S. Manley, J. C. Ye, and M. Unser, “FALCON: fast and unbiased reconstruction of high-density super-resolution microscopy data,” Sci. Rep. 4(1), 4577 (2014).
[Crossref] [PubMed]

Min, J.

D. Sage, H. Kirshner, T. Pengo, N. Stuurman, J. Min, S. Manley, and M. Unser, “Quantitative evaluation of software packages for single-molecule localization microscopy,” Nat. Methods 12(8), 717–724 (2015).
[Crossref] [PubMed]

J. Min, C. Vonesch, H. Kirshner, L. Carlini, N. Olivier, S. Holden, S. Manley, J. C. Ye, and M. Unser, “FALCON: fast and unbiased reconstruction of high-density super-resolution microscopy data,” Sci. Rep. 4(1), 4577 (2014).
[Crossref] [PubMed]

Moffitt, J. R.

Monypenny, J.

S. Cox, E. Rosten, J. Monypenny, T. Jovanovic-Talisman, D. T. Burnette, J. Lippincott-Schwartz, G. E. Jones, and R. Heintzmann, “Bayesian localization microscopy reveals nanoscale podosome dynamics,” Nat. Methods 9(2), 195–200 (2011).
[Crossref] [PubMed]

Morales, J. L.

J. L. Morales and J. Nocedal, “Remark on ‘algorithm 778: L-BFGS-B: Fortran subroutines for large-scale bound constrained optimization’,” ACM Trans. Math. Softw. 38(1), 1–4 (2011).
[Crossref]

Mukamel, E. A.

E. A. Mukamel, H. Babcock, and X. Zhuang, “Statistical deconvolution for superresolution fluorescence microscopy,” Biophys. J. 102(10), 2391–2400 (2012).
[Crossref] [PubMed]

Mutzafi, M.

Nguyen, T. Q.

S. H. Chan, R. Khoshabeh, K. B. Gibson, P. E. Gill, and T. Q. Nguyen, “An augmented Lagrangian method for total variation video restoration,” IEEE Trans. Image Process. 20(11), 3097–3111 (2011).
[Crossref] [PubMed]

Nieuwenhuizen, R. P. J.

R. P. J. Nieuwenhuizen, K. A. Lidke, M. Bates, D. L. Puig, D. Grünwald, S. Stallinga, and B. Rieger, “Measuring image resolution in optical nanoscopy,” Nat. Methods 10(6), 557–562 (2013).
[Crossref] [PubMed]

Niu, H.

Nocedal, J.

J. L. Morales and J. Nocedal, “Remark on ‘algorithm 778: L-BFGS-B: Fortran subroutines for large-scale bound constrained optimization’,” ACM Trans. Math. Softw. 38(1), 1–4 (2011).
[Crossref]

Olenych, S.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313(5793), 1642–1645 (2006).
[Crossref] [PubMed]

Olivier, N.

J. Min, C. Vonesch, H. Kirshner, L. Carlini, N. Olivier, S. Holden, S. Manley, J. C. Ye, and M. Unser, “FALCON: fast and unbiased reconstruction of high-density super-resolution microscopy data,” Sci. Rep. 4(1), 4577 (2014).
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T. Goldstein and S. Osher, “The split Bregman algorithm forl1,” SIAM J. Imaging Sci. 2(2), 323–343 (2009).
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Ovesný, M.

M. Ovesný, P. Křížek, J. Borkovec, Z. Švindrych, and G. M. Hagen, “ThunderSTORM: a comprehensive ImageJ plug-in for PALM and STORM data analysis and super-resolution imaging,” Bioinformatics 30(16), 2389–2390 (2014).
[Crossref] [PubMed]

M. Ovesný, P. Křížek, Z. Švindrych, and G. M. Hagen, “High density 3D localization microscopy using sparse support recovery,” Opt. Express 22(25), 31263–31276 (2014).
[Crossref] [PubMed]

Patterson, G. H.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313(5793), 1642–1645 (2006).
[Crossref] [PubMed]

Pavillon, N.

Pengo, T.

D. Sage, H. Kirshner, T. Pengo, N. Stuurman, J. Min, S. Manley, and M. Unser, “Quantitative evaluation of software packages for single-molecule localization microscopy,” Nat. Methods 12(8), 717–724 (2015).
[Crossref] [PubMed]

Pressé, S.

A. Lee, K. Tsekouras, C. Calderon, C. Bustamante, and S. Pressé, “Unraveling the thousand word picture: an introduction to super-resolution data analysis,” Chem. Rev. 117(11), 7276–7330 (2017).
[Crossref] [PubMed]

Puig, D. L.

R. P. J. Nieuwenhuizen, K. A. Lidke, M. Bates, D. L. Puig, D. Grünwald, S. Stallinga, and B. Rieger, “Measuring image resolution in optical nanoscopy,” Nat. Methods 10(6), 557–562 (2013).
[Crossref] [PubMed]

Ramani, S.

M. J. Allison, S. Ramani, and J. A. Fessler, “Accelerated regularized estimation of MR coil sensitivities using augmented Lagrangian methods,” IEEE Trans. Med. Imaging 32(3), 556–564 (2013).
[Crossref] [PubMed]

Rieger, B.

R. P. J. Nieuwenhuizen, K. A. Lidke, M. Bates, D. L. Puig, D. Grünwald, S. Stallinga, and B. Rieger, “Measuring image resolution in optical nanoscopy,” Nat. Methods 10(6), 557–562 (2013).
[Crossref] [PubMed]

Rosten, E.

S. Cox, E. Rosten, J. Monypenny, T. Jovanovic-Talisman, D. T. Burnette, J. Lippincott-Schwartz, G. E. Jones, and R. Heintzmann, “Bayesian localization microscopy reveals nanoscale podosome dynamics,” Nat. Methods 9(2), 195–200 (2011).
[Crossref] [PubMed]

Rust, M. J.

M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods 3(10), 793–795 (2006).
[Crossref] [PubMed]

Sage, D.

D. Sage, H. Kirshner, T. Pengo, N. Stuurman, J. Min, S. Manley, and M. Unser, “Quantitative evaluation of software packages for single-molecule localization microscopy,” Nat. Methods 12(8), 717–724 (2015).
[Crossref] [PubMed]

Schwartz, S. L.

Segev, M.

Shen, Y.

Shim, S. H.

S. A. Jones, S. H. Shim, J. He, and X. Zhuang, “Fast, three-dimensional super-resolution imaging of live cells,” Nat. Methods 8(6), 499–508 (2011).
[Crossref] [PubMed]

Small, A.

A. Small and S. Stahlheber, “Fluorophore localization algorithms for super-resolution microscopy,” Nat. Methods 11(3), 267–279 (2014).
[Crossref] [PubMed]

Smith, N. I.

Solomon, O.

Sougrat, R.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313(5793), 1642–1645 (2006).
[Crossref] [PubMed]

Stahlheber, S.

A. Small and S. Stahlheber, “Fluorophore localization algorithms for super-resolution microscopy,” Nat. Methods 11(3), 267–279 (2014).
[Crossref] [PubMed]

Stallinga, S.

R. P. J. Nieuwenhuizen, K. A. Lidke, M. Bates, D. L. Puig, D. Grünwald, S. Stallinga, and B. Rieger, “Measuring image resolution in optical nanoscopy,” Nat. Methods 10(6), 557–562 (2013).
[Crossref] [PubMed]

Stuurman, N.

D. Sage, H. Kirshner, T. Pengo, N. Stuurman, J. Min, S. Manley, and M. Unser, “Quantitative evaluation of software packages for single-molecule localization microscopy,” Nat. Methods 12(8), 717–724 (2015).
[Crossref] [PubMed]

Sun, M.

J. Huang, M. Sun, J. Ma, and Y. Chi, “Super-resolution image reconstruction for high-density 3D single-molecule microscopy,” IEEE Trans. Comput. Imag. 3(4), 763–773 (2017).
[Crossref]

Švindrych, Z.

M. Ovesný, P. Křížek, J. Borkovec, Z. Švindrych, and G. M. Hagen, “ThunderSTORM: a comprehensive ImageJ plug-in for PALM and STORM data analysis and super-resolution imaging,” Bioinformatics 30(16), 2389–2390 (2014).
[Crossref] [PubMed]

M. Ovesný, P. Křížek, Z. Švindrych, and G. M. Hagen, “High density 3D localization microscopy using sparse support recovery,” Opt. Express 22(25), 31263–31276 (2014).
[Crossref] [PubMed]

Tang, J.

G.-H. Chen, J. Tang, and S. Leng, “Prior image constrained compressed sensing (PICCS): a method to accurately reconstruct dynamic CT images from highly undersampled projection data sets,” Med. Phys. 35(2), 660–663 (2008).
[Crossref] [PubMed]

Teboulle, M.

A. Beck and M. Teboulle, “Fast gradient-based algorithms for constrained total variation image denoising and deblurring problems,” IEEE Trans. Image Process. 18(11), 2419–2434 (2009).
[Crossref] [PubMed]

Tsekouras, K.

A. Lee, K. Tsekouras, C. Calderon, C. Bustamante, and S. Pressé, “Unraveling the thousand word picture: an introduction to super-resolution data analysis,” Chem. Rev. 117(11), 7276–7330 (2017).
[Crossref] [PubMed]

Unser, M.

D. Sage, H. Kirshner, T. Pengo, N. Stuurman, J. Min, S. Manley, and M. Unser, “Quantitative evaluation of software packages for single-molecule localization microscopy,” Nat. Methods 12(8), 717–724 (2015).
[Crossref] [PubMed]

J. Min, C. Vonesch, H. Kirshner, L. Carlini, N. Olivier, S. Holden, S. Manley, J. C. Ye, and M. Unser, “FALCON: fast and unbiased reconstruction of high-density super-resolution microscopy data,” Sci. Rep. 4(1), 4577 (2014).
[Crossref] [PubMed]

Uphoff, S.

S. J. Holden, S. Uphoff, and A. N. Kapanidis, “DAOSTORM: an algorithm for high- density super-resolution microscopy,” Nat. Methods 8(4), 279–280 (2011).
[Crossref] [PubMed]

Vonesch, C.

J. Min, C. Vonesch, H. Kirshner, L. Carlini, N. Olivier, S. Holden, S. Manley, J. C. Ye, and M. Unser, “FALCON: fast and unbiased reconstruction of high-density super-resolution microscopy data,” Sci. Rep. 4(1), 4577 (2014).
[Crossref] [PubMed]

Wang, Z.

H. Bingsheng, H. Liu, Z. Wang, and X. Yuan, “A strictly contractive Peaceman-Rachford splitting method for convex programming,” SIAM J. Optim. 24(3), 1011–1040 (2014).
[Crossref] [PubMed]

Wichmann, J.

Ye, J. C.

J. Min, C. Vonesch, H. Kirshner, L. Carlini, N. Olivier, S. Holden, S. Manley, J. C. Ye, and M. Unser, “FALCON: fast and unbiased reconstruction of high-density super-resolution microscopy data,” Sci. Rep. 4(1), 4577 (2014).
[Crossref] [PubMed]

Yin, W.

C. Li, W. Yin, H. Jiang, and Y. Zhang, “An efficient augmented Lagrangian method with applications to total variation minimization,” Comput. Optim. Appl. 56(3), 507–530 (2013).
[Crossref]

Yu, B.

Yuan, X.

H. Bingsheng, H. Liu, Z. Wang, and X. Yuan, “A strictly contractive Peaceman-Rachford splitting method for convex programming,” SIAM J. Optim. 24(3), 1011–1040 (2014).
[Crossref] [PubMed]

Zhang, W.

L. Zhu, W. Zhang, D. Elnatan, and B. Huang, “Faster STORM using compressed sensing,” Nat. Methods 9(7), 721–723 (2012).
[Crossref] [PubMed]

Zhang, Y.

C. Li, W. Yin, H. Jiang, and Y. Zhang, “An efficient augmented Lagrangian method with applications to total variation minimization,” Comput. Optim. Appl. 56(3), 507–530 (2013).
[Crossref]

Zhu, L.

L. Zhu, W. Zhang, D. Elnatan, and B. Huang, “Faster STORM using compressed sensing,” Nat. Methods 9(7), 721–723 (2012).
[Crossref] [PubMed]

Zhuang, X.

H. P. Babcock, J. R. Moffitt, Y. Cao, and X. Zhuang, “Fast compressed sensing analysis for super-resolution imaging using L1-homotopy,” Opt. Express 21(23), 28583–28596 (2013).
[Crossref] [PubMed]

E. A. Mukamel, H. Babcock, and X. Zhuang, “Statistical deconvolution for superresolution fluorescence microscopy,” Biophys. J. 102(10), 2391–2400 (2012).
[Crossref] [PubMed]

S. A. Jones, S. H. Shim, J. He, and X. Zhuang, “Fast, three-dimensional super-resolution imaging of live cells,” Nat. Methods 8(6), 499–508 (2011).
[Crossref] [PubMed]

M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods 3(10), 793–795 (2006).
[Crossref] [PubMed]

ACM Trans. Math. Softw. (1)

J. L. Morales and J. Nocedal, “Remark on ‘algorithm 778: L-BFGS-B: Fortran subroutines for large-scale bound constrained optimization’,” ACM Trans. Math. Softw. 38(1), 1–4 (2011).
[Crossref]

Bioinformatics (1)

M. Ovesný, P. Křížek, J. Borkovec, Z. Švindrych, and G. M. Hagen, “ThunderSTORM: a comprehensive ImageJ plug-in for PALM and STORM data analysis and super-resolution imaging,” Bioinformatics 30(16), 2389–2390 (2014).
[Crossref] [PubMed]

Biomed. Opt. Express (2)

Biophys. J. (1)

E. A. Mukamel, H. Babcock, and X. Zhuang, “Statistical deconvolution for superresolution fluorescence microscopy,” Biophys. J. 102(10), 2391–2400 (2012).
[Crossref] [PubMed]

Chem. Rev. (1)

A. Lee, K. Tsekouras, C. Calderon, C. Bustamante, and S. Pressé, “Unraveling the thousand word picture: an introduction to super-resolution data analysis,” Chem. Rev. 117(11), 7276–7330 (2017).
[Crossref] [PubMed]

Comput. Optim. Appl. (1)

C. Li, W. Yin, H. Jiang, and Y. Zhang, “An efficient augmented Lagrangian method with applications to total variation minimization,” Comput. Optim. Appl. 56(3), 507–530 (2013).
[Crossref]

Histochem. Cell Biol. (1)

U. Köthe, F. Herrmannsdörfer, I. Kats, and F. A. Hamprecht, “SimpleSTORM: a fast, self-calibrating reconstruction algorithm for localization microscopy,” Histochem. Cell Biol. 141(6), 613–627 (2014).
[Crossref] [PubMed]

IEEE Trans. Comput. Imag. (1)

J. Huang, M. Sun, J. Ma, and Y. Chi, “Super-resolution image reconstruction for high-density 3D single-molecule microscopy,” IEEE Trans. Comput. Imag. 3(4), 763–773 (2017).
[Crossref]

IEEE Trans. Image Process. (3)

M. V. Afonso, J. M. Bioucas-Dias, and M. A. T. Figueiredo, “Fast image recovery using variable splitting and constrained optimization,” IEEE Trans. Image Process. 19(9), 2345–2356 (2010).
[Crossref] [PubMed]

S. H. Chan, R. Khoshabeh, K. B. Gibson, P. E. Gill, and T. Q. Nguyen, “An augmented Lagrangian method for total variation video restoration,” IEEE Trans. Image Process. 20(11), 3097–3111 (2011).
[Crossref] [PubMed]

A. Beck and M. Teboulle, “Fast gradient-based algorithms for constrained total variation image denoising and deblurring problems,” IEEE Trans. Image Process. 18(11), 2419–2434 (2009).
[Crossref] [PubMed]

IEEE Trans. Med. Imaging (1)

M. J. Allison, S. Ramani, and J. A. Fessler, “Accelerated regularized estimation of MR coil sensitivities using augmented Lagrangian methods,” IEEE Trans. Med. Imaging 32(3), 556–564 (2013).
[Crossref] [PubMed]

IEEE Trans. Neural Netw. Learn. Syst. (1)

W. Hao, R. Feng, Z.-F. Han, and C.-S. Leung, “ADMM-based algorithm for training fault tolerant RBF networks and selecting centers,” IEEE Trans. Neural Netw. Learn. Syst. 29(8), 3870–3878 (2017).
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Med. Phys. (1)

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Figures (5)

Fig. 1
Fig. 1 Simulated STORM image analysis using the proposed algorithm. (a) True molecule positions with magnified reconstructed results for the red square region. (b) Low-resolution raw image.
Fig. 2
Fig. 2 Performance comparison of the proposed ALM and CS method on simulated STORM data. (a), (b) and (c) are comparisons of recall, JAC values and localization accuracy, respectively.
Fig. 3
Fig. 3 Comparison of ThunderSTORM, FACON and ALM on simulated STORM data. (a) True molecule positions. (b) Plots of intensity profiles along the cyan lines. (c) - (e) are reconstructed images using 120 frames of raw images with ThunderSTORM, FALCON and ALM respectively. (f) - (h) are reconstructed images using 30 frames of raw images with ThunderSTORM, FALCON and ALM, respectively. Yellow squares in lower left corners are correspondingly magnified reconstructed results.
Fig. 4
Fig. 4 Comparison of FALCON, MUSICAL and ALM on experimental STORM data from EPFL website. (a) Reconstructed image using 500 frames of raw images with ThunderSTORM. (b) A frame of raw STORM image. (c), (d) and (e) are reconstructed images respectively using FALCON, MUSICAL and ALM with 500 frames. (f), (g) and (h) are reconstructed images respectively using FALCON, MUSICAL and ALM with 50 frames. White squares in upper right corners are correspondingly magnified reconstructed results.
Fig. 5
Fig. 5 (a) and (b) are plots of intensity profiles of FALCON reconstructed images along the blue and cyan lines, respectively. (c) and (d) are plots of intensity profiles of MUSICAL reconstructed images along the blue and cyan lines, respectively. (e) and (f) are plots of intensity profiles of ALM reconstructed images along the blue and cyan lines, respectively.

Tables (2)

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Table 1 Pseudo-code of optimization procedure

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Table 2 Comparison of performance with different frames of raw images and methods.

Equations (17)

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b = A s + n ,
A = N × exp [ 4 ln 2 ( x 2 + y 2 σ 2 ) ] p 2 p 2 p 2 p 2 exp [ 4 ln 2 ( x 2 + y 2 σ 2 ) ] d x d y .
minimize ϕ ( s i ) subject to b i = A s i + n .
D x ( s ) = V ( S x S ) and D y ( s ) = V ( S y S ) ,
F ( s ) = A s b 2 2 .
R ( s ) = D ( s ) 1 .
C = F ( s ) + R ( s ) = μ 2 A s b 2 2 + D ( s ) 1 .
minimize s μ 2 A s b 2 2 + u 1 subject to u = D ( s ) , s 0 .
L ρ ( s , u , d ) = μ 2 A s b 2 2 + u 1 d T [ u D ( s ) ] + ρ 2 u D ( s ) 2 2 .
s k + 1 = arg min s μ 2 A s b 2 2 d k T [ u k D ( s ) ] + ρ 2 u k D ( s ) 2 2 subject to s 0 ,
u k + 1 = arg min u u 1 d k T [ u D ( s k + 1 ) ] + ρ 2 u D ( s k + 1 ) 2 2 ,
d k + 1 = d k ρ [ u k + 1 D ( s k + 1 ) ] ,
μ A T A s + ρ D T D ( s ) = μ A T b D T ( d k ) + ρ D T u k .
u k + 1 = max { | d k ρ + D ( s k + 1 ) | 1 ρ , 0 } sgn [ d k ρ + D ( s k + 1 ) ] ,
sgn ( x ) = { 1 x > 0 0 x = 0 1 x < 0 .
recall = TP FN+TP , JAC = TP FN+FP+TP , and RMSE = 1 TP i = 1 TP ( x i x 0 ) 2 + ( y i y 0 ) 2 .
SNR = 10 log 10 s 0 2 s 0 s 2 ,

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