Abstract

Structured illumination microscopy (SIM) improves resolution by down-modulating high-frequency information of an object to fit within the passband of the optical system. Generally, the reconstruction process requires prior knowledge of the illumination patterns, which implies a well-calibrated and aberration-free system. Here, we propose a new algorithmic self-calibration strategy for SIM that does not need to know the exact patterns a priori, but only their covariance. The algorithm, termed PE-SIMS, includes a pattern-estimation (PE) step requiring the uniformity of the sum of the illumination patterns and a SIM reconstruction procedure using a statistical prior (SIMS). Additionally, we perform a pixel reassignment process (SIMS-PR) to enhance the reconstruction quality. We achieve 2× better resolution than a conventional widefield microscope, while remaining insensitive to aberration-induced pattern distortion and robust against parameter tuning.

© 2017 Optical Society of America

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2016 (1)

2015 (5)

F. Ströhl and C. F. Kaminski, “A joint Richardson-Lucy deconvolution algorithm for the reconstruction of multifocal structured illumination microscopy data,” Methods Appl. Fluoresc. 3, 014002 (2015).
[Crossref]

M. Kim, C. Park, C. Rodriguez, Y. Park, and Y.-H. Cho, “Superresolution imaging with optical fluctuation using speckle patterns illumination,” Scientific Reports 5, 16525 (2015).
[Crossref] [PubMed]

A. Jost, E. Tolstik, P. Feldmann, K. Wicker, A. Sentenac, and R. Heintzmann, “Optical sectioning and high resolution in single-slice structured illumination microscopy by thick slice blind-SIM reconstruction,” PLoS ONE 10, e0132174 (2015).
[Crossref] [PubMed]

H. Yilmaz, E. G. V. Putten, J. Bertolotti, A. Lagendijk, W. L. Vos, and A. P. Mosk, “Speckle correlation resolution enhancement of wide-field fluorescence imaging,” Optica 2, 424–429 (2015).
[Crossref]

N. Chakrova, R. Heintzmann, B. Rieger, and S. Stallinga, “Studying different illumination patterns for resolution improvement in fluorescence microscopy,” Opt. Express 23, 31367–31383 (2015).
[Crossref] [PubMed]

2014 (1)

2013 (8)

K. Wicker, O. Mandula, G. Best, R. Fiolka, and R. Heintzmann, “Phase optimisation for structured illumination microscopy,” Opt. Express 21, 2032–2049 (2013).
[Crossref] [PubMed]

C. J. R. Sheppard, S. B. Mehta, and R. Heintzmann, “Superresolution by image scanning microscopy using pixel reassignment,” Opt. Lett. 38, 2889–2992 (2013).
[Crossref] [PubMed]

K. Wicker, “Non-iterative determination of pattern phase in structured illumination microscopy using auto-correlations in Fourier space,” Opt. Express 21, 24692–24701 (2013).
[Crossref] [PubMed]

R. Ayuk, H. Giovannini, A. Jost, E. Mudry, J. Girard, T. Mangeat, N. Sandeau, R. Heintzmann, K. Wicker, K. Belkebir, and A. Sentenac, “Structured illumination fluorescence microscopy with distorted excitations using a filtered blind-SIM algorithm,” Opt. Lett. 38, 4723–4726 (2013).
[Crossref] [PubMed]

J. Min, J. Jang, D. Keum, S.-W. Ryu, C. Choi, K.-H. Jeong, and J. C. Ye, “Fluorescent microscopy beyond diffraction limits using speckle illumination and joint support recovery,” Sci. Rep. 3, 2075 (2013).
[Crossref] [PubMed]

S. Roth, C. J. R. Sheppard, K. Wicker, and R. Heintzmann, “Optical photon reassignment microscopy (OPRA),” Optical Nanoscopy 2, 1–62013.

N. Parikh and S. Boyd, “Proximal algorithms,” Found. Trends Optim. 1, 123–231 (2013).

A. G. York, P. Chandris, D. D. Nogare, J. Head, P. Wawrzusin, R. S. Fischer, A. Chitnis, and H. Shroff, “Instant super-resolution imaging in live cells and embryos via analog image processing,” Nat. Methods 10, 1122–1126 (2013).
[Crossref] [PubMed]

2012 (2)

E. Mudry, K. Belkebir, J. Girard, J. Savatier, E. L. Moal, C. Nicoletti, M. Allain, and A. Sentenac, “Structured illumination microscopy using unknown speckle patterns,” Nat. Photon. 6, 312–315 (2012).
[Crossref]

A. G. York, S. H. Parekh, D. D. Nogare, R. S. Fischer, K. Temprine, M. Mione, A. B. Chitnis, C. A. Combs, and H. Shroff, “Resolution doubling in live, multicellular organisms via multifocal structured illumination microscopy,” Nat. Methods 9, 749–754 (2012).
[Crossref] [PubMed]

2010 (3)

2009 (3)

A. Beck and M. Teboulle, “A fast iterative shrinkage-thresholding algorithm for linear inverse problems,” SIAM J. Imaging Science 2, 183–202 (2009).
[Crossref]

S. A. Shroff, J. R. Fienup, and D. R. Williams, “Phase-shift estimation in sinusoidally illuminated images for lateral superresolution,” JOSA A 26, 413–424 (2009).
[Crossref] [PubMed]

T. Dertinger, R. Colyer, G. Iyer, S. Weiss, and J. Enderlein, “Fast, background-free, 3D super-resolution optical fluctuation imaging (SOFI),” PNAS 106, 22287–22292 (2009).
[Crossref] [PubMed]

2006 (3)

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, 1642–1645 (2006).
[Crossref] [PubMed]

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

R. Heintzmann and P. A. Benedetti, “High-resolution image reconstruction in fluorescence microscopy with patterned excitation,” Appl. Opt. 45, 5037–5045 (2006).
[Crossref] [PubMed]

2005 (1)

J. García, Z. Zalevsky, and D. Fixler, “Synthetic aperture superresolution by speckle pattern projection,” Opt. Express 13, 6075–6078 (2005).
[Crossref]

2004 (1)

S.-H. Jiang and J. G. Walker, “Experimental confirmation of non-scanning fluorescence confocal microscopy using speckle illumination,” Opt. Commun. 238, 1–12 (2004).
[Crossref]

2003 (1)

C. J. R. Sheppard, “Scanning confocal microscopy,” Encyclopedia of Opt. Engineering 20132525–2544 (2003).

2001 (1)

J. G. Walker, “Non-scanning confocal fluorescence microscopy using speckle illumination,” Opt. Commun. 189, 221–226 (2001).
[Crossref]

2000 (1)

M. G. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microscopy 198, 82–87 (2000).
[Crossref]

1997 (1)

1996 (1)

1994 (1)

1988 (1)

C. J. R. Sheppard, “Super-resolution in confocal imaging,” Optik 80, 53–54 (1988).

1983 (1)

Y. Nesterov, “A method for solving the convex programming problem with convergence rate O(1/k2),” Dokl. Akad. Nauk SSSR 269, 543–547 (1983).

1982 (1)

1967 (1)

Allain, M.

A. Negash, S. Labouesse, N. Sandeau, M. Allain, H. Giovannini, J. Idier, R. Heintzmann, P. C. Chaumet, K. Belkebir, and A. Sentenac, “Improving the axial and lateral resolution of three-dimensional fluorescence microscopy using random speckle illuminations,” J. Opt. Soc. Am. A 33, 1089–1094 (2016).
[Crossref]

E. Mudry, K. Belkebir, J. Girard, J. Savatier, E. L. Moal, C. Nicoletti, M. Allain, and A. Sentenac, “Structured illumination microscopy using unknown speckle patterns,” Nat. Photon. 6, 312–315 (2012).
[Crossref]

S. Labouesse, M. Allain, J. Idier, S. Bourguignon, A. Negash, P. Liu, and A. Sentenac, “Joint reconstruction strategy for structured illumination microscopy with unknown illuminations,” ArXiv: 1607.01980 (2016).

Ayuk, R.

Bates, M.

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

Beck, A.

A. Beck and M. Teboulle, “A fast iterative shrinkage-thresholding algorithm for linear inverse problems,” SIAM J. Imaging Science 2, 183–202 (2009).
[Crossref]

Belkebir, K.

Benedetti, P. A.

Bertolotti, J.

Best, G.

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, 1642–1645 (2006).
[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, 1642–1645 (2006).
[Crossref] [PubMed]

Bourguignon, S.

S. Labouesse, M. Allain, J. Idier, S. Bourguignon, A. Negash, P. Liu, and A. Sentenac, “Joint reconstruction strategy for structured illumination microscopy with unknown illuminations,” ArXiv: 1607.01980 (2016).

Boyd, S.

N. Parikh and S. Boyd, “Proximal algorithms,” Found. Trends Optim. 1, 123–231 (2013).

Chakrova, N.

Chandris, P.

A. G. York, P. Chandris, D. D. Nogare, J. Head, P. Wawrzusin, R. S. Fischer, A. Chitnis, and H. Shroff, “Instant super-resolution imaging in live cells and embryos via analog image processing,” Nat. Methods 10, 1122–1126 (2013).
[Crossref] [PubMed]

Chaumet, P. C.

Chitnis, A.

A. G. York, P. Chandris, D. D. Nogare, J. Head, P. Wawrzusin, R. S. Fischer, A. Chitnis, and H. Shroff, “Instant super-resolution imaging in live cells and embryos via analog image processing,” Nat. Methods 10, 1122–1126 (2013).
[Crossref] [PubMed]

Chitnis, A. B.

A. G. York, S. H. Parekh, D. D. Nogare, R. S. Fischer, K. Temprine, M. Mione, A. B. Chitnis, C. A. Combs, and H. Shroff, “Resolution doubling in live, multicellular organisms via multifocal structured illumination microscopy,” Nat. Methods 9, 749–754 (2012).
[Crossref] [PubMed]

Cho, Y.-H.

M. Kim, C. Park, C. Rodriguez, Y. Park, and Y.-H. Cho, “Superresolution imaging with optical fluctuation using speckle patterns illumination,” Scientific Reports 5, 16525 (2015).
[Crossref] [PubMed]

Choi, C.

J. Min, J. Jang, D. Keum, S.-W. Ryu, C. Choi, K.-H. Jeong, and J. C. Ye, “Fluorescent microscopy beyond diffraction limits using speckle illumination and joint support recovery,” Sci. Rep. 3, 2075 (2013).
[Crossref] [PubMed]

Colyer, R.

T. Dertinger, R. Colyer, G. Iyer, S. Weiss, and J. Enderlein, “Fast, background-free, 3D super-resolution optical fluctuation imaging (SOFI),” PNAS 106, 22287–22292 (2009).
[Crossref] [PubMed]

Combs, C. A.

A. G. York, S. H. Parekh, D. D. Nogare, R. S. Fischer, K. Temprine, M. Mione, A. B. Chitnis, C. A. Combs, and H. Shroff, “Resolution doubling in live, multicellular organisms via multifocal structured illumination microscopy,” Nat. Methods 9, 749–754 (2012).
[Crossref] [PubMed]

Cox, I. J.

Cremer, C.

R. Heintzmann and C. Cremer, “Laterally modulated excitation microscopy: improvement of resolution by using a diffraction grating,” Proc. SPIE3568, 185–196 (1999).
[Crossref]

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, 1642–1645 (2006).
[Crossref] [PubMed]

Dertinger, T.

T. Dertinger, R. Colyer, G. Iyer, S. Weiss, and J. Enderlein, “Fast, background-free, 3D super-resolution optical fluctuation imaging (SOFI),” PNAS 106, 22287–22292 (2009).
[Crossref] [PubMed]

Dong, S.

Enderlein, J.

C. B. Müller and J. Enderlein, “Image scanning microscopy,” Phys. Rev. Lett. 104, 198101 (2010).
[Crossref] [PubMed]

T. Dertinger, R. Colyer, G. Iyer, S. Weiss, and J. Enderlein, “Fast, background-free, 3D super-resolution optical fluctuation imaging (SOFI),” PNAS 106, 22287–22292 (2009).
[Crossref] [PubMed]

Feldmann, P.

A. Jost, E. Tolstik, P. Feldmann, K. Wicker, A. Sentenac, and R. Heintzmann, “Optical sectioning and high resolution in single-slice structured illumination microscopy by thick slice blind-SIM reconstruction,” PLoS ONE 10, e0132174 (2015).
[Crossref] [PubMed]

Fienup, J. R.

S. A. Shroff, J. R. Fienup, and D. R. Williams, “Lateral superresolution using a posteriori phase shift estimation for a moving object: experimental results,” J. Opt. Soc. Am. A 27, 1770–1782 (2010).
[Crossref]

S. A. Shroff, J. R. Fienup, and D. R. Williams, “Phase-shift estimation in sinusoidally illuminated images for lateral superresolution,” JOSA A 26, 413–424 (2009).
[Crossref] [PubMed]

Fiolka, R.

Fischer, R. S.

A. G. York, P. Chandris, D. D. Nogare, J. Head, P. Wawrzusin, R. S. Fischer, A. Chitnis, and H. Shroff, “Instant super-resolution imaging in live cells and embryos via analog image processing,” Nat. Methods 10, 1122–1126 (2013).
[Crossref] [PubMed]

A. G. York, S. H. Parekh, D. D. Nogare, R. S. Fischer, K. Temprine, M. Mione, A. B. Chitnis, C. A. Combs, and H. Shroff, “Resolution doubling in live, multicellular organisms via multifocal structured illumination microscopy,” Nat. Methods 9, 749–754 (2012).
[Crossref] [PubMed]

Fixler, D.

J. García, Z. Zalevsky, and D. Fixler, “Synthetic aperture superresolution by speckle pattern projection,” Opt. Express 13, 6075–6078 (2005).
[Crossref]

Garcá, J.

García, J.

J. García, Z. Zalevsky, and D. Fixler, “Synthetic aperture superresolution by speckle pattern projection,” Opt. Express 13, 6075–6078 (2005).
[Crossref]

Giovannini, H.

Girard, J.

Guo, K.

Gustafsson, M. G.

M. G. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microscopy 198, 82–87 (2000).
[Crossref]

Head, J.

A. G. York, P. Chandris, D. D. Nogare, J. Head, P. Wawrzusin, R. S. Fischer, A. Chitnis, and H. Shroff, “Instant super-resolution imaging in live cells and embryos via analog image processing,” Nat. Methods 10, 1122–1126 (2013).
[Crossref] [PubMed]

Heintzmann, R.

A. Negash, S. Labouesse, N. Sandeau, M. Allain, H. Giovannini, J. Idier, R. Heintzmann, P. C. Chaumet, K. Belkebir, and A. Sentenac, “Improving the axial and lateral resolution of three-dimensional fluorescence microscopy using random speckle illuminations,” J. Opt. Soc. Am. A 33, 1089–1094 (2016).
[Crossref]

N. Chakrova, R. Heintzmann, B. Rieger, and S. Stallinga, “Studying different illumination patterns for resolution improvement in fluorescence microscopy,” Opt. Express 23, 31367–31383 (2015).
[Crossref] [PubMed]

A. Jost, E. Tolstik, P. Feldmann, K. Wicker, A. Sentenac, and R. Heintzmann, “Optical sectioning and high resolution in single-slice structured illumination microscopy by thick slice blind-SIM reconstruction,” PLoS ONE 10, e0132174 (2015).
[Crossref] [PubMed]

R. Ayuk, H. Giovannini, A. Jost, E. Mudry, J. Girard, T. Mangeat, N. Sandeau, R. Heintzmann, K. Wicker, K. Belkebir, and A. Sentenac, “Structured illumination fluorescence microscopy with distorted excitations using a filtered blind-SIM algorithm,” Opt. Lett. 38, 4723–4726 (2013).
[Crossref] [PubMed]

C. J. R. Sheppard, S. B. Mehta, and R. Heintzmann, “Superresolution by image scanning microscopy using pixel reassignment,” Opt. Lett. 38, 2889–2992 (2013).
[Crossref] [PubMed]

K. Wicker, O. Mandula, G. Best, R. Fiolka, and R. Heintzmann, “Phase optimisation for structured illumination microscopy,” Opt. Express 21, 2032–2049 (2013).
[Crossref] [PubMed]

S. Roth, C. J. R. Sheppard, K. Wicker, and R. Heintzmann, “Optical photon reassignment microscopy (OPRA),” Optical Nanoscopy 2, 1–62013.

R. Heintzmann and P. A. Benedetti, “High-resolution image reconstruction in fluorescence microscopy with patterned excitation,” Appl. Opt. 45, 5037–5045 (2006).
[Crossref] [PubMed]

R. Heintzmann and C. Cremer, “Laterally modulated excitation microscopy: improvement of resolution by using a diffraction grating,” Proc. SPIE3568, 185–196 (1999).
[Crossref]

Hell, S. W.

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, 1642–1645 (2006).
[Crossref] [PubMed]

Idier, J.

A. Negash, S. Labouesse, N. Sandeau, M. Allain, H. Giovannini, J. Idier, R. Heintzmann, P. C. Chaumet, K. Belkebir, and A. Sentenac, “Improving the axial and lateral resolution of three-dimensional fluorescence microscopy using random speckle illuminations,” J. Opt. Soc. Am. A 33, 1089–1094 (2016).
[Crossref]

S. Labouesse, M. Allain, J. Idier, S. Bourguignon, A. Negash, P. Liu, and A. Sentenac, “Joint reconstruction strategy for structured illumination microscopy with unknown illuminations,” ArXiv: 1607.01980 (2016).

Ishida, H.

Iyer, G.

T. Dertinger, R. Colyer, G. Iyer, S. Weiss, and J. Enderlein, “Fast, background-free, 3D super-resolution optical fluctuation imaging (SOFI),” PNAS 106, 22287–22292 (2009).
[Crossref] [PubMed]

Jang, J.

J. Min, J. Jang, D. Keum, S.-W. Ryu, C. Choi, K.-H. Jeong, and J. C. Ye, “Fluorescent microscopy beyond diffraction limits using speckle illumination and joint support recovery,” Sci. Rep. 3, 2075 (2013).
[Crossref] [PubMed]

Jeong, K.-H.

J. Min, J. Jang, D. Keum, S.-W. Ryu, C. Choi, K.-H. Jeong, and J. C. Ye, “Fluorescent microscopy beyond diffraction limits using speckle illumination and joint support recovery,” Sci. Rep. 3, 2075 (2013).
[Crossref] [PubMed]

Jiang, S.-H.

S.-H. Jiang and J. G. Walker, “Experimental confirmation of non-scanning fluorescence confocal microscopy using speckle illumination,” Opt. Commun. 238, 1–12 (2004).
[Crossref]

Jost, A.

A. Jost, E. Tolstik, P. Feldmann, K. Wicker, A. Sentenac, and R. Heintzmann, “Optical sectioning and high resolution in single-slice structured illumination microscopy by thick slice blind-SIM reconstruction,” PLoS ONE 10, e0132174 (2015).
[Crossref] [PubMed]

R. Ayuk, H. Giovannini, A. Jost, E. Mudry, J. Girard, T. Mangeat, N. Sandeau, R. Heintzmann, K. Wicker, K. Belkebir, and A. Sentenac, “Structured illumination fluorescence microscopy with distorted excitations using a filtered blind-SIM algorithm,” Opt. Lett. 38, 4723–4726 (2013).
[Crossref] [PubMed]

Juškaitis, R.

Kaminski, C. F.

F. Ströhl and C. F. Kaminski, “A joint Richardson-Lucy deconvolution algorithm for the reconstruction of multifocal structured illumination microscopy data,” Methods Appl. Fluoresc. 3, 014002 (2015).
[Crossref]

Keum, D.

J. Min, J. Jang, D. Keum, S.-W. Ryu, C. Choi, K.-H. Jeong, and J. C. Ye, “Fluorescent microscopy beyond diffraction limits using speckle illumination and joint support recovery,” Sci. Rep. 3, 2075 (2013).
[Crossref] [PubMed]

Kim, M.

M. Kim, C. Park, C. Rodriguez, Y. Park, and Y.-H. Cho, “Superresolution imaging with optical fluctuation using speckle patterns illumination,” Scientific Reports 5, 16525 (2015).
[Crossref] [PubMed]

Kosugi, Y.

Labouesse, S.

A. Negash, S. Labouesse, N. Sandeau, M. Allain, H. Giovannini, J. Idier, R. Heintzmann, P. C. Chaumet, K. Belkebir, and A. Sentenac, “Improving the axial and lateral resolution of three-dimensional fluorescence microscopy using random speckle illuminations,” J. Opt. Soc. Am. A 33, 1089–1094 (2016).
[Crossref]

S. Labouesse, M. Allain, J. Idier, S. Bourguignon, A. Negash, P. Liu, and A. Sentenac, “Joint reconstruction strategy for structured illumination microscopy with unknown illuminations,” ArXiv: 1607.01980 (2016).

Lagendijk, A.

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, 1642–1645 (2006).
[Crossref] [PubMed]

Lippincott-Schwartz, J.

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, 1642–1645 (2006).
[Crossref] [PubMed]

Liu, P.

S. Labouesse, M. Allain, J. Idier, S. Bourguignon, A. Negash, P. Liu, and A. Sentenac, “Joint reconstruction strategy for structured illumination microscopy with unknown illuminations,” ArXiv: 1607.01980 (2016).

Lukosz, W.

Mandula, O.

Mangeat, T.

Mehta, S. B.

Micó, V.

Min, J.

J. Min, J. Jang, D. Keum, S.-W. Ryu, C. Choi, K.-H. Jeong, and J. C. Ye, “Fluorescent microscopy beyond diffraction limits using speckle illumination and joint support recovery,” Sci. Rep. 3, 2075 (2013).
[Crossref] [PubMed]

Mione, M.

A. G. York, S. H. Parekh, D. D. Nogare, R. S. Fischer, K. Temprine, M. Mione, A. B. Chitnis, C. A. Combs, and H. Shroff, “Resolution doubling in live, multicellular organisms via multifocal structured illumination microscopy,” Nat. Methods 9, 749–754 (2012).
[Crossref] [PubMed]

Moal, E. L.

E. Mudry, K. Belkebir, J. Girard, J. Savatier, E. L. Moal, C. Nicoletti, M. Allain, and A. Sentenac, “Structured illumination microscopy using unknown speckle patterns,” Nat. Photon. 6, 312–315 (2012).
[Crossref]

Mosk, A. P.

Mudry, E.

Müller, C. B.

C. B. Müller and J. Enderlein, “Image scanning microscopy,” Phys. Rev. Lett. 104, 198101 (2010).
[Crossref] [PubMed]

Nanda, P.

Negash, A.

A. Negash, S. Labouesse, N. Sandeau, M. Allain, H. Giovannini, J. Idier, R. Heintzmann, P. C. Chaumet, K. Belkebir, and A. Sentenac, “Improving the axial and lateral resolution of three-dimensional fluorescence microscopy using random speckle illuminations,” J. Opt. Soc. Am. A 33, 1089–1094 (2016).
[Crossref]

S. Labouesse, M. Allain, J. Idier, S. Bourguignon, A. Negash, P. Liu, and A. Sentenac, “Joint reconstruction strategy for structured illumination microscopy with unknown illuminations,” ArXiv: 1607.01980 (2016).

Neil, M. A. A.

Nesterov, Y.

Y. Nesterov, “A method for solving the convex programming problem with convergence rate O(1/k2),” Dokl. Akad. Nauk SSSR 269, 543–547 (1983).

Nicoletti, C.

E. Mudry, K. Belkebir, J. Girard, J. Savatier, E. L. Moal, C. Nicoletti, M. Allain, and A. Sentenac, “Structured illumination microscopy using unknown speckle patterns,” Nat. Photon. 6, 312–315 (2012).
[Crossref]

Nogare, D. D.

A. G. York, P. Chandris, D. D. Nogare, J. Head, P. Wawrzusin, R. S. Fischer, A. Chitnis, and H. Shroff, “Instant super-resolution imaging in live cells and embryos via analog image processing,” Nat. Methods 10, 1122–1126 (2013).
[Crossref] [PubMed]

A. G. York, S. H. Parekh, D. D. Nogare, R. S. Fischer, K. Temprine, M. Mione, A. B. Chitnis, C. A. Combs, and H. Shroff, “Resolution doubling in live, multicellular organisms via multifocal structured illumination microscopy,” Nat. Methods 9, 749–754 (2012).
[Crossref] [PubMed]

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, 1642–1645 (2006).
[Crossref] [PubMed]

Otsuki, S.

Parekh, S. H.

A. G. York, S. H. Parekh, D. D. Nogare, R. S. Fischer, K. Temprine, M. Mione, A. B. Chitnis, C. A. Combs, and H. Shroff, “Resolution doubling in live, multicellular organisms via multifocal structured illumination microscopy,” Nat. Methods 9, 749–754 (2012).
[Crossref] [PubMed]

Parikh, N.

N. Parikh and S. Boyd, “Proximal algorithms,” Found. Trends Optim. 1, 123–231 (2013).

Park, C.

M. Kim, C. Park, C. Rodriguez, Y. Park, and Y.-H. Cho, “Superresolution imaging with optical fluctuation using speckle patterns illumination,” Scientific Reports 5, 16525 (2015).
[Crossref] [PubMed]

Park, Y.

M. Kim, C. Park, C. Rodriguez, Y. Park, and Y.-H. Cho, “Superresolution imaging with optical fluctuation using speckle patterns illumination,” Scientific Reports 5, 16525 (2015).
[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, 1642–1645 (2006).
[Crossref] [PubMed]

Putten, E. G. V.

Rieger, B.

Rodriguez, C.

M. Kim, C. Park, C. Rodriguez, Y. Park, and Y.-H. Cho, “Superresolution imaging with optical fluctuation using speckle patterns illumination,” Scientific Reports 5, 16525 (2015).
[Crossref] [PubMed]

Roth, S.

S. Roth, C. J. R. Sheppard, K. Wicker, and R. Heintzmann, “Optical photon reassignment microscopy (OPRA),” Optical Nanoscopy 2, 1–62013.

Rust, M. J.

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

Ryu, S.-W.

J. Min, J. Jang, D. Keum, S.-W. Ryu, C. Choi, K.-H. Jeong, and J. C. Ye, “Fluorescent microscopy beyond diffraction limits using speckle illumination and joint support recovery,” Sci. Rep. 3, 2075 (2013).
[Crossref] [PubMed]

Sandeau, N.

Savatier, J.

E. Mudry, K. Belkebir, J. Girard, J. Savatier, E. L. Moal, C. Nicoletti, M. Allain, and A. Sentenac, “Structured illumination microscopy using unknown speckle patterns,” Nat. Photon. 6, 312–315 (2012).
[Crossref]

Sentenac, A.

A. Negash, S. Labouesse, N. Sandeau, M. Allain, H. Giovannini, J. Idier, R. Heintzmann, P. C. Chaumet, K. Belkebir, and A. Sentenac, “Improving the axial and lateral resolution of three-dimensional fluorescence microscopy using random speckle illuminations,” J. Opt. Soc. Am. A 33, 1089–1094 (2016).
[Crossref]

A. Jost, E. Tolstik, P. Feldmann, K. Wicker, A. Sentenac, and R. Heintzmann, “Optical sectioning and high resolution in single-slice structured illumination microscopy by thick slice blind-SIM reconstruction,” PLoS ONE 10, e0132174 (2015).
[Crossref] [PubMed]

R. Ayuk, H. Giovannini, A. Jost, E. Mudry, J. Girard, T. Mangeat, N. Sandeau, R. Heintzmann, K. Wicker, K. Belkebir, and A. Sentenac, “Structured illumination fluorescence microscopy with distorted excitations using a filtered blind-SIM algorithm,” Opt. Lett. 38, 4723–4726 (2013).
[Crossref] [PubMed]

E. Mudry, K. Belkebir, J. Girard, J. Savatier, E. L. Moal, C. Nicoletti, M. Allain, and A. Sentenac, “Structured illumination microscopy using unknown speckle patterns,” Nat. Photon. 6, 312–315 (2012).
[Crossref]

S. Labouesse, M. Allain, J. Idier, S. Bourguignon, A. Negash, P. Liu, and A. Sentenac, “Joint reconstruction strategy for structured illumination microscopy with unknown illuminations,” ArXiv: 1607.01980 (2016).

Sheppard, C. J. R.

C. J. R. Sheppard, S. B. Mehta, and R. Heintzmann, “Superresolution by image scanning microscopy using pixel reassignment,” Opt. Lett. 38, 2889–2992 (2013).
[Crossref] [PubMed]

S. Roth, C. J. R. Sheppard, K. Wicker, and R. Heintzmann, “Optical photon reassignment microscopy (OPRA),” Optical Nanoscopy 2, 1–62013.

C. J. R. Sheppard, “Scanning confocal microscopy,” Encyclopedia of Opt. Engineering 20132525–2544 (2003).

C. J. R. Sheppard, “Super-resolution in confocal imaging,” Optik 80, 53–54 (1988).

I. J. Cox, C. J. R. Sheppard, and T. Wilson, “Improvement in resolution by nearly confocal microscopy,” Appl. Opt. 21, 778–781 (1982).
[Crossref] [PubMed]

T. Wilson and C. J. R. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic Press, 1984).

Shimizu, M.

Shiradkar, R.

Shroff, H.

A. G. York, P. Chandris, D. D. Nogare, J. Head, P. Wawrzusin, R. S. Fischer, A. Chitnis, and H. Shroff, “Instant super-resolution imaging in live cells and embryos via analog image processing,” Nat. Methods 10, 1122–1126 (2013).
[Crossref] [PubMed]

A. G. York, S. H. Parekh, D. D. Nogare, R. S. Fischer, K. Temprine, M. Mione, A. B. Chitnis, C. A. Combs, and H. Shroff, “Resolution doubling in live, multicellular organisms via multifocal structured illumination microscopy,” Nat. Methods 9, 749–754 (2012).
[Crossref] [PubMed]

Shroff, S. A.

S. A. Shroff, J. R. Fienup, and D. R. Williams, “Lateral superresolution using a posteriori phase shift estimation for a moving object: experimental results,” J. Opt. Soc. Am. A 27, 1770–1782 (2010).
[Crossref]

S. A. Shroff, J. R. Fienup, and D. R. Williams, “Phase-shift estimation in sinusoidally illuminated images for lateral superresolution,” JOSA A 26, 413–424 (2009).
[Crossref] [PubMed]

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, 1642–1645 (2006).
[Crossref] [PubMed]

Stallinga, S.

Ströhl, F.

F. Ströhl and C. F. Kaminski, “A joint Richardson-Lucy deconvolution algorithm for the reconstruction of multifocal structured illumination microscopy data,” Methods Appl. Fluoresc. 3, 014002 (2015).
[Crossref]

Sylman, D.

Tanaami, T.

Teboulle, M.

A. Beck and M. Teboulle, “A fast iterative shrinkage-thresholding algorithm for linear inverse problems,” SIAM J. Imaging Science 2, 183–202 (2009).
[Crossref]

Temprine, K.

A. G. York, S. H. Parekh, D. D. Nogare, R. S. Fischer, K. Temprine, M. Mione, A. B. Chitnis, C. A. Combs, and H. Shroff, “Resolution doubling in live, multicellular organisms via multifocal structured illumination microscopy,” Nat. Methods 9, 749–754 (2012).
[Crossref] [PubMed]

Tolstik, E.

A. Jost, E. Tolstik, P. Feldmann, K. Wicker, A. Sentenac, and R. Heintzmann, “Optical sectioning and high resolution in single-slice structured illumination microscopy by thick slice blind-SIM reconstruction,” PLoS ONE 10, e0132174 (2015).
[Crossref] [PubMed]

Tomosada, N.

Vos, W. L.

Walker, J. G.

S.-H. Jiang and J. G. Walker, “Experimental confirmation of non-scanning fluorescence confocal microscopy using speckle illumination,” Opt. Commun. 238, 1–12 (2004).
[Crossref]

J. G. Walker, “Non-scanning confocal fluorescence microscopy using speckle illumination,” Opt. Commun. 189, 221–226 (2001).
[Crossref]

Wawrzusin, P.

A. G. York, P. Chandris, D. D. Nogare, J. Head, P. Wawrzusin, R. S. Fischer, A. Chitnis, and H. Shroff, “Instant super-resolution imaging in live cells and embryos via analog image processing,” Nat. Methods 10, 1122–1126 (2013).
[Crossref] [PubMed]

Weiss, S.

T. Dertinger, R. Colyer, G. Iyer, S. Weiss, and J. Enderlein, “Fast, background-free, 3D super-resolution optical fluctuation imaging (SOFI),” PNAS 106, 22287–22292 (2009).
[Crossref] [PubMed]

Wichmann, J.

Wicker, K.

Williams, D. R.

S. A. Shroff, J. R. Fienup, and D. R. Williams, “Lateral superresolution using a posteriori phase shift estimation for a moving object: experimental results,” J. Opt. Soc. Am. A 27, 1770–1782 (2010).
[Crossref]

S. A. Shroff, J. R. Fienup, and D. R. Williams, “Phase-shift estimation in sinusoidally illuminated images for lateral superresolution,” JOSA A 26, 413–424 (2009).
[Crossref] [PubMed]

Wilson, T.

Ye, J. C.

J. Min, J. Jang, D. Keum, S.-W. Ryu, C. Choi, K.-H. Jeong, and J. C. Ye, “Fluorescent microscopy beyond diffraction limits using speckle illumination and joint support recovery,” Sci. Rep. 3, 2075 (2013).
[Crossref] [PubMed]

Yilmaz, H.

York, A. G.

A. G. York, P. Chandris, D. D. Nogare, J. Head, P. Wawrzusin, R. S. Fischer, A. Chitnis, and H. Shroff, “Instant super-resolution imaging in live cells and embryos via analog image processing,” Nat. Methods 10, 1122–1126 (2013).
[Crossref] [PubMed]

A. G. York, S. H. Parekh, D. D. Nogare, R. S. Fischer, K. Temprine, M. Mione, A. B. Chitnis, C. A. Combs, and H. Shroff, “Resolution doubling in live, multicellular organisms via multifocal structured illumination microscopy,” Nat. Methods 9, 749–754 (2012).
[Crossref] [PubMed]

Zalevsky, Z.

D. Sylman, V. Micó, J. Garcá, and Z. Zalevsky, “Random angular coding for superresolved imaging,” Appl. Opt. 49, 4874–4882 (2010).
[Crossref] [PubMed]

J. García, Z. Zalevsky, and D. Fixler, “Synthetic aperture superresolution by speckle pattern projection,” Opt. Express 13, 6075–6078 (2005).
[Crossref]

Zheng, G.

Zhuang, X.

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

Appl. Opt. (4)

Dokl. Akad. Nauk SSSR (1)

Y. Nesterov, “A method for solving the convex programming problem with convergence rate O(1/k2),” Dokl. Akad. Nauk SSSR 269, 543–547 (1983).

Encyclopedia of Opt. Engineering (1)

C. J. R. Sheppard, “Scanning confocal microscopy,” Encyclopedia of Opt. Engineering 20132525–2544 (2003).

Found. Trends Optim. (1)

N. Parikh and S. Boyd, “Proximal algorithms,” Found. Trends Optim. 1, 123–231 (2013).

J. Microscopy (1)

M. G. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microscopy 198, 82–87 (2000).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

JOSA A (1)

S. A. Shroff, J. R. Fienup, and D. R. Williams, “Phase-shift estimation in sinusoidally illuminated images for lateral superresolution,” JOSA A 26, 413–424 (2009).
[Crossref] [PubMed]

Methods Appl. Fluoresc. (1)

F. Ströhl and C. F. Kaminski, “A joint Richardson-Lucy deconvolution algorithm for the reconstruction of multifocal structured illumination microscopy data,” Methods Appl. Fluoresc. 3, 014002 (2015).
[Crossref]

Nat. Methods (2)

A. G. York, S. H. Parekh, D. D. Nogare, R. S. Fischer, K. Temprine, M. Mione, A. B. Chitnis, C. A. Combs, and H. Shroff, “Resolution doubling in live, multicellular organisms via multifocal structured illumination microscopy,” Nat. Methods 9, 749–754 (2012).
[Crossref] [PubMed]

A. G. York, P. Chandris, D. D. Nogare, J. Head, P. Wawrzusin, R. S. Fischer, A. Chitnis, and H. Shroff, “Instant super-resolution imaging in live cells and embryos via analog image processing,” Nat. Methods 10, 1122–1126 (2013).
[Crossref] [PubMed]

Nat. Photon. (1)

E. Mudry, K. Belkebir, J. Girard, J. Savatier, E. L. Moal, C. Nicoletti, M. Allain, and A. Sentenac, “Structured illumination microscopy using unknown speckle patterns,” Nat. Photon. 6, 312–315 (2012).
[Crossref]

Nature Methods (1)

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

Opt. Commun. (2)

J. G. Walker, “Non-scanning confocal fluorescence microscopy using speckle illumination,” Opt. Commun. 189, 221–226 (2001).
[Crossref]

S.-H. Jiang and J. G. Walker, “Experimental confirmation of non-scanning fluorescence confocal microscopy using speckle illumination,” Opt. Commun. 238, 1–12 (2004).
[Crossref]

Opt. Express (5)

Opt. Lett. (4)

Optica (1)

Optical Nanoscopy (1)

S. Roth, C. J. R. Sheppard, K. Wicker, and R. Heintzmann, “Optical photon reassignment microscopy (OPRA),” Optical Nanoscopy 2, 1–62013.

Optik (1)

C. J. R. Sheppard, “Super-resolution in confocal imaging,” Optik 80, 53–54 (1988).

Phys. Rev. Lett. (1)

C. B. Müller and J. Enderlein, “Image scanning microscopy,” Phys. Rev. Lett. 104, 198101 (2010).
[Crossref] [PubMed]

PLoS ONE (1)

A. Jost, E. Tolstik, P. Feldmann, K. Wicker, A. Sentenac, and R. Heintzmann, “Optical sectioning and high resolution in single-slice structured illumination microscopy by thick slice blind-SIM reconstruction,” PLoS ONE 10, e0132174 (2015).
[Crossref] [PubMed]

PNAS (1)

T. Dertinger, R. Colyer, G. Iyer, S. Weiss, and J. Enderlein, “Fast, background-free, 3D super-resolution optical fluctuation imaging (SOFI),” PNAS 106, 22287–22292 (2009).
[Crossref] [PubMed]

Sci. Rep. (1)

J. Min, J. Jang, D. Keum, S.-W. Ryu, C. Choi, K.-H. Jeong, and J. C. Ye, “Fluorescent microscopy beyond diffraction limits using speckle illumination and joint support recovery,” Sci. Rep. 3, 2075 (2013).
[Crossref] [PubMed]

Science (1)

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, 1642–1645 (2006).
[Crossref] [PubMed]

Scientific Reports (1)

M. Kim, C. Park, C. Rodriguez, Y. Park, and Y.-H. Cho, “Superresolution imaging with optical fluctuation using speckle patterns illumination,” Scientific Reports 5, 16525 (2015).
[Crossref] [PubMed]

SIAM J. Imaging Science (1)

A. Beck and M. Teboulle, “A fast iterative shrinkage-thresholding algorithm for linear inverse problems,” SIAM J. Imaging Science 2, 183–202 (2009).
[Crossref]

Other (3)

S. Labouesse, M. Allain, J. Idier, S. Bourguignon, A. Negash, P. Liu, and A. Sentenac, “Joint reconstruction strategy for structured illumination microscopy with unknown illuminations,” ArXiv: 1607.01980 (2016).

R. Heintzmann and C. Cremer, “Laterally modulated excitation microscopy: improvement of resolution by using a diffraction grating,” Proc. SPIE3568, 185–196 (1999).
[Crossref]

T. Wilson and C. J. R. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic Press, 1984).

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Figures (8)

Fig. 1
Fig. 1

Example experimental setup for structured illumination microscopy (SIM) using a deformable mirror device (DMD) to capture low-resolution images of the object modulated by different illumination patterns. Our IPE-SIMS algorithm reconstructs both the super-resoloved image and the unknown arbitrary illumination patterns.

Fig. 2
Fig. 2

The first part of our algorithm, Pattern Estimation (PE), iteratively estimates the illumination patterns from an approximated object given by the deconvolved widefield image.

Fig. 3
Fig. 3

The second part of our algorithm, termed structured illumination microscopy with a statistical prior (SIMS), estimates the high-resolution object from the measured images and the estimated illumination patterns obtained in Part 1.

Fig. 4
Fig. 4

(a) Simulated reconstructions of a Siemens star target under a widefield microscope, deconvolved widefield, confocal microscope, deconvolved confocal, blind SIM [16], S-SOFI [20], our PE-SIMS and PE-SIMS-PR algorithms. (b) The effective modulation transfer function (MTF) of each method, given by the contrast of the reconstructed Siemens star image at different radii.

Fig. 5
Fig. 5

Reconstructions of red fluorescent beads (Ex:580 nm/Em:605 nm) from the experiment using random pattern illumination with 20 × 20 scanning step.

Fig. 6
Fig. 6

Comparison of our algorithm on dataset from Multispot SIM (MSIM) which uses with Nimg = 224 scanned multi-spot patterns from [10]. We show the deconvolved widefield image and the reconstructions using MSIM with known patterns, as well as our blind PE-SIMS algorithm with and without pixel reassignment.

Fig. 7
Fig. 7

Results with simulated and experimental (fluorescent beads) datasets comparing random speckle and multi-spot illumination patterns. (middle row) Shading maps overlaid on the object. Decreasing the number of random patterns results in shading artifacts in the reconstruction. The random patterns are scanned in 20 × 20, 10 × 10, and 6 × 6 steps with the same step size of 0.6 FWHM of the PSF, while the multi-spot pattern is scanned with 6 × 6 steps.

Fig. 8
Fig. 8

(a) Comparison of the PSF and OTF for SIMS and SIMS with pixel reassignment (PR). (b) Comparisons of the deconvolved widefield image and the reconstructions of the 6 × 6 multi-spot scanned fluorescent beads with and without pixel reassignment.

Tables (2)

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Subroutine 1 Pattern Estimation

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Table 1 Achieved resolution for different algorithms

Equations (19)

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I ( r ) = [ o ( r ) p ( r ) ] h det ( r ) = o ( r ) p ( r ) h det ( r r ) d 2 r .
I avg ( r ) = I ( r ) = [ o ( r ) p ( r ) ] h det ( r ) p 0 o ( r ) h det ( r ) ,
o est ( r ) = 1 { I ˜ avg ( u ) h ˜ det ( u ) | h ˜ det ( u ) | 2 + β } ,
minimize p f ( p ) = f diff ( p ) + I C ( p ) = r | I ( r ) [ o est ( r ) p ( r ) ] h det ( r ) | 2 + I C ( p ) , where I C ( p ) = { 0 , p C + , p C , C = { p ( r ) | p ˜ ( u ) = 0 , u > 2 N A λ illu } ,
g ( k ) ( r ) = f diff ( p ( k ) ) p = 2 o est ( r ) [ h det ( r ) ( I ( r ) [ o est ( r ) p ( k ) ( r ) ] h det ( r ) ) ] ,
C ( y ) = 1 { { y } | h ˜ illu ( u ) | 2 | h ˜ illu ( u ) | 2 + δ } ,
I cov ( r ) = Δ p ( r ) Δ I ( r ) = o ( r ) Δ p ( r ) Δ p ( r ) h det ( r r ) d 2 r ,
p ( r ) = t ( r ) h illu ( r r ) d 2 r ,
Δ p ( r ) Δ p ( r ) = Δ t ( r 1 ) Δ t ( r 2 ) h illu ( r r 1 ) h illu ( r r 2 ) d 2 r 1 d 2 r 2 = γ t Δ t 2 ( r 1 ) δ ( r 1 r 2 ) h illu ( r r 1 ) h illu ( r r 2 ) d 2 r 1 d 2 r 2 α t h illu ( r r 1 ) h illu ( r r 1 ) d 2 r 1 = α t ( h illu h illu ) ( r r ) ,
Δ t ( r 1 ) Δ t ( r 2 ) = γ t Δ t 2 ( r 1 ) δ ( r 1 r 2 ) α t δ ( r 1 r 2 ) ,
I cov ( r ) = Δ p ( r ) Δ I ( r ) = α t o ( r ) [ ( h illu h illu ) h det ] ( r r ) d 2 r .
I cov , dec ( r ) = 1 { I ˜ cov ( u ) H ( u ) | H ( u ) | 2 + ξ } ,
α t ( r ) = γ t Δ t 2 ( r ) γ t Δ p 2 ( r ) .
I SIMS ( r ) = I cov , dec ( r ) α t ( r ) α t 2 ( r ) + ϵ ,
t ( r ) = Λ m , n δ ( r r m n r ) + t 0 Δ t ( r ) Λ 2 m , n δ ( r r m n r ) ,
Δ t ( r 1 ) Δ t ( r 2 ) = Δ t ( r 1 r ) Δ t ( r 2 r ) d 2 r = Λ 4 m , n δ ( r 1 r 2 r m n ) m , n δ ( r 1 r 2 r m n ) Λ 4 η m , n δ ( r 1 r 2 r m n ) ,
Δ p ( r ) Δ p ( r ) = ( h illu h illu ) ( r r ) Λ 4 η m , n δ ( r r r m n ) .
I cov s ( r , r s ) = Δ p ( r r s ) Δ I ( r ) = o ( r ) Δ p ( r r s ) Δ p ( r ) h ( r r ) d 2 r = α t o ( r ) ( h illu h illu ) ( r r s r ) h det ( r r ) d 2 r .
I PR ( r ) = I cov s ( r + r s 2 , r s ) d 2 r s = α t o ( r ) [ ( h illu h illu ) ( r r s 2 r ) h det ( r + r s 2 r ) d 2 r s ] d 2 r = α t o ( r ) [ ( h illu h illu ) h det ] ( 2 ( r r ) ) d 2 r

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