Abstract

The virtual detection method provides us with a more simplified system setup and a considerable resolution enhancement compared with microscopies that apply physical modulation. However, the imaging speed of virtual modulation microscopy is much slower compared with wide-field imaging techniques such as structured illumination microscopy (SIM). In this study, we propose a multiplex virtual modulation microscopy that performs similarly to the former virtual modulation microscopy whose data are acquired using the same system as laser scanning microscopy (LSM) does, yet with a much higher theoretical speed. Under saturated conditions, the virtual modulation microscopy exhibits a high robustness against the aberrations and estimation errors of saturation level. We also show that by applying both virtual temporal modulation (VTM) and virtual spatial modulation (VSM), information can be extracted beyond the typical extended resolution support of linear superresolution methods such as confocal and SIM. However, it is demonstrated that VTM is more efficient in extracting high-spatial-frequency information than VSM. Biospecimen and fluorescent beads are imaged in order to verify the proposed method using the data acquired under the LSM system.

© 2017 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2017 (1)

2016 (2)

C. Kuang, Y. Ma, R. Zhou, G. Zheng, Y. Fang, Y. Xu, X. Liu, and P. T. So, “Virtual k-Space Modulation Optical Microscopy,” Phys. Rev. Lett. 117(2), 028102 (2016).
[Crossref] [PubMed]

F. Ströhl and C. F. Kaminski, “Frontiers in structured illumination microscopy,” Optica 3(6), 667–677 (2016).
[Crossref]

2015 (3)

J. Huff, “The Airyscan detector from ZEISS: confocal imaging with improved signal-to-noise ratio and super-resolution,” Nat. Methods 12, 388 (2015).

D. Li, L. Shao, B.-C. Chen, X. Zhang, M. Zhang, B. Moses, D. E. Milkie, J. R. Beach, J. A. Hammer, M. Pasham, T. Kirchhausen, M. A. Baird, M. W. Davidson, P. Xu, and E. Betzig, “Extended-resolution structured illumination imaging of endocytic and cytoskeletal dynamics,” Science 349(6251), aab3500 (2015).
[Crossref] [PubMed]

Y. Zhi, R. Lu, B. Wang, Q. Zhang, and X. Yao, “Rapid super-resolution line-scanning microscopy through virtually structured detection,” Opt. Lett. 40(8), 1683–1686 (2015).
[Crossref] [PubMed]

2014 (3)

2013 (3)

R.-W. Lu, B.-Q. Wang, Q.-X. Zhang, and X.-C. Yao, “Super-resolution scanning laser microscopy through virtually structured detection,” Biomed. Opt. Express 4(9), 1673–1682 (2013).
[Crossref] [PubMed]

G. Zheng, R. Horstmeyer, and C. Yang, “Wide-field, high-resolution Fourier ptychographic microscopy,” Nat. Photonics 7(9), 739–745 (2013).
[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(11), 1122–1126 (2013).
[Crossref] [PubMed]

2012 (3)

A. G. York, S. H. Parekh, D. Dalle 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(7), 749–754 (2012).
[Crossref] [PubMed]

E. Mudry, K. Belkebir, J. Girard, J. Savatier, E. Le Moal, C. Nicoletti, M. Allain, and A. Sentenac, “Structured illumination microscopy using unknown speckle patterns,” Nat. Photonics 6(5), 312–315 (2012).
[Crossref]

E. H. Rego, L. Shao, J. J. Macklin, L. Winoto, G. A. Johansson, N. Kamps-Hughes, M. W. Davidson, and M. G. Gustafsson, “Nonlinear structured-illumination microscopy with a photoswitchable protein reveals cellular structures at 50-nm resolution,” Proc. Natl. Acad. Sci. U.S.A. 109(3), E135–E143 (2012).
[Crossref] [PubMed]

2010 (1)

C. B. Müller and J. Enderlein, “Image scanning microscopy,” Phys. Rev. Lett. 104(19), 198101 (2010).
[Crossref] [PubMed]

2009 (1)

J. Lu, W. Min, J.-A. Conchello, X. S. Xie, and J. W. Lichtman, “Super-Resolution Laser Scanning Microscopy through Spatiotemporal Modulation,” Nano Lett. 9(11), 3883–3889 (2009).
[Crossref] [PubMed]

2008 (2)

M. G. L. Gustafsson, L. Shao, P. M. Carlton, C. J. R. Wang, I. N. Golubovskaya, W. Z. Cande, D. A. Agard, and J. W. Sedat, “Three-Dimensional Resolution Doubling in Wide-Field Fluorescence Microscopy by Structured Illumination,” Biophys. J. 94(12), 4957–4970 (2008).
[Crossref] [PubMed]

L. Schermelleh, P. M. Carlton, S. Haase, L. Shao, L. Winoto, P. Kner, B. Burke, M. C. Cardoso, D. A. Agard, M. G. L. Gustafsson, H. Leonhardt, and J. W. Sedat, “Subdiffraction Multicolor Imaging of the Nuclear Periphery with 3D Structured Illumination Microscopy,” Science 320(5881), 1332–1336 (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), 793–796 (2006).
[Crossref] [PubMed]

2005 (1)

M. G. Gustafsson, “Nonlinear structured-illumination microscopy: wide-field fluorescence imaging with theoretically unlimited resolution,” Proc. Natl. Acad. Sci. U.S.A. 102(37), 13081–13086 (2005).
[Crossref] [PubMed]

2002 (1)

2000 (2)

M. G. L. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microsc. 198(2), 82–87 (2000).
[Crossref] [PubMed]

T. A. Klar, S. Jakobs, M. Dyba, A. Egner, and S. W. Hell, “Fluorescence microscopy with diffraction resolution barrier broken by stimulated emission,” Proc. Natl. Acad. Sci. U.S.A. 97(15), 8206–8210 (2000).
[Crossref] [PubMed]

1994 (1)

1988 (1)

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

1873 (1)

E. Abbe, “Beiträge zur Theorie des Mikroskops und der mikroskopischen Wahrnehmung,” Archiv für mikroskopische Anatomie 9, 413– 418 (1873).

Abbe, E.

E. Abbe, “Beiträge zur Theorie des Mikroskops und der mikroskopischen Wahrnehmung,” Archiv für mikroskopische Anatomie 9, 413– 418 (1873).

Agard, D. A.

M. G. L. Gustafsson, L. Shao, P. M. Carlton, C. J. R. Wang, I. N. Golubovskaya, W. Z. Cande, D. A. Agard, and J. W. Sedat, “Three-Dimensional Resolution Doubling in Wide-Field Fluorescence Microscopy by Structured Illumination,” Biophys. J. 94(12), 4957–4970 (2008).
[Crossref] [PubMed]

L. Schermelleh, P. M. Carlton, S. Haase, L. Shao, L. Winoto, P. Kner, B. Burke, M. C. Cardoso, D. A. Agard, M. G. L. Gustafsson, H. Leonhardt, and J. W. Sedat, “Subdiffraction Multicolor Imaging of the Nuclear Periphery with 3D Structured Illumination Microscopy,” Science 320(5881), 1332–1336 (2008).
[Crossref] [PubMed]

Allain, M.

E. Mudry, K. Belkebir, J. Girard, J. Savatier, E. Le Moal, C. Nicoletti, M. Allain, and A. Sentenac, “Structured illumination microscopy using unknown speckle patterns,” Nat. Photonics 6(5), 312–315 (2012).
[Crossref]

Baird, M. A.

D. Li, L. Shao, B.-C. Chen, X. Zhang, M. Zhang, B. Moses, D. E. Milkie, J. R. Beach, J. A. Hammer, M. Pasham, T. Kirchhausen, M. A. Baird, M. W. Davidson, P. Xu, and E. Betzig, “Extended-resolution structured illumination imaging of endocytic and cytoskeletal dynamics,” Science 349(6251), aab3500 (2015).
[Crossref] [PubMed]

Bates, M.

M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods 3(10), 793–796 (2006).
[Crossref] [PubMed]

Beach, J. R.

D. Li, L. Shao, B.-C. Chen, X. Zhang, M. Zhang, B. Moses, D. E. Milkie, J. R. Beach, J. A. Hammer, M. Pasham, T. Kirchhausen, M. A. Baird, M. W. Davidson, P. Xu, and E. Betzig, “Extended-resolution structured illumination imaging of endocytic and cytoskeletal dynamics,” Science 349(6251), aab3500 (2015).
[Crossref] [PubMed]

Belkebir, K.

E. Mudry, K. Belkebir, J. Girard, J. Savatier, E. Le Moal, C. Nicoletti, M. Allain, and A. Sentenac, “Structured illumination microscopy using unknown speckle patterns,” Nat. Photonics 6(5), 312–315 (2012).
[Crossref]

Betzig, E.

D. Li, L. Shao, B.-C. Chen, X. Zhang, M. Zhang, B. Moses, D. E. Milkie, J. R. Beach, J. A. Hammer, M. Pasham, T. Kirchhausen, M. A. Baird, M. W. Davidson, P. Xu, and E. Betzig, “Extended-resolution structured illumination imaging of endocytic and cytoskeletal dynamics,” Science 349(6251), aab3500 (2015).
[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]

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]

Burke, B.

L. Schermelleh, P. M. Carlton, S. Haase, L. Shao, L. Winoto, P. Kner, B. Burke, M. C. Cardoso, D. A. Agard, M. G. L. Gustafsson, H. Leonhardt, and J. W. Sedat, “Subdiffraction Multicolor Imaging of the Nuclear Periphery with 3D Structured Illumination Microscopy,” Science 320(5881), 1332–1336 (2008).
[Crossref] [PubMed]

Cande, W. Z.

M. G. L. Gustafsson, L. Shao, P. M. Carlton, C. J. R. Wang, I. N. Golubovskaya, W. Z. Cande, D. A. Agard, and J. W. Sedat, “Three-Dimensional Resolution Doubling in Wide-Field Fluorescence Microscopy by Structured Illumination,” Biophys. J. 94(12), 4957–4970 (2008).
[Crossref] [PubMed]

Cardoso, M. C.

L. Schermelleh, P. M. Carlton, S. Haase, L. Shao, L. Winoto, P. Kner, B. Burke, M. C. Cardoso, D. A. Agard, M. G. L. Gustafsson, H. Leonhardt, and J. W. Sedat, “Subdiffraction Multicolor Imaging of the Nuclear Periphery with 3D Structured Illumination Microscopy,” Science 320(5881), 1332–1336 (2008).
[Crossref] [PubMed]

Carlton, P. M.

M. G. L. Gustafsson, L. Shao, P. M. Carlton, C. J. R. Wang, I. N. Golubovskaya, W. Z. Cande, D. A. Agard, and J. W. Sedat, “Three-Dimensional Resolution Doubling in Wide-Field Fluorescence Microscopy by Structured Illumination,” Biophys. J. 94(12), 4957–4970 (2008).
[Crossref] [PubMed]

L. Schermelleh, P. M. Carlton, S. Haase, L. Shao, L. Winoto, P. Kner, B. Burke, M. C. Cardoso, D. A. Agard, M. G. L. Gustafsson, H. Leonhardt, and J. W. Sedat, “Subdiffraction Multicolor Imaging of the Nuclear Periphery with 3D Structured Illumination Microscopy,” Science 320(5881), 1332–1336 (2008).
[Crossref] [PubMed]

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(11), 1122–1126 (2013).
[Crossref] [PubMed]

Chen, B.-C.

D. Li, L. Shao, B.-C. Chen, X. Zhang, M. Zhang, B. Moses, D. E. Milkie, J. R. Beach, J. A. Hammer, M. Pasham, T. Kirchhausen, M. A. Baird, M. W. Davidson, P. Xu, and E. Betzig, “Extended-resolution structured illumination imaging of endocytic and cytoskeletal dynamics,” Science 349(6251), aab3500 (2015).
[Crossref] [PubMed]

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(11), 1122–1126 (2013).
[Crossref] [PubMed]

Chitnis, A. B.

A. G. York, S. H. Parekh, D. Dalle 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(7), 749–754 (2012).
[Crossref] [PubMed]

Combs, C. A.

A. G. York, S. H. Parekh, D. Dalle 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(7), 749–754 (2012).
[Crossref] [PubMed]

Conchello, J.-A.

J. Lu, W. Min, J.-A. Conchello, X. S. Xie, and J. W. Lichtman, “Super-Resolution Laser Scanning Microscopy through Spatiotemporal Modulation,” Nano Lett. 9(11), 3883–3889 (2009).
[Crossref] [PubMed]

Cremer, C.

Dalle Nogare, D.

A. G. York, S. H. Parekh, D. Dalle 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(7), 749–754 (2012).
[Crossref] [PubMed]

Davidson, M. W.

D. Li, L. Shao, B.-C. Chen, X. Zhang, M. Zhang, B. Moses, D. E. Milkie, J. R. Beach, J. A. Hammer, M. Pasham, T. Kirchhausen, M. A. Baird, M. W. Davidson, P. Xu, and E. Betzig, “Extended-resolution structured illumination imaging of endocytic and cytoskeletal dynamics,” Science 349(6251), aab3500 (2015).
[Crossref] [PubMed]

E. H. Rego, L. Shao, J. J. Macklin, L. Winoto, G. A. Johansson, N. Kamps-Hughes, M. W. Davidson, and M. G. Gustafsson, “Nonlinear structured-illumination microscopy with a photoswitchable protein reveals cellular structures at 50-nm resolution,” Proc. Natl. Acad. Sci. U.S.A. 109(3), E135–E143 (2012).
[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]

Dong, S.

Dyba, M.

T. A. Klar, S. Jakobs, M. Dyba, A. Egner, and S. W. Hell, “Fluorescence microscopy with diffraction resolution barrier broken by stimulated emission,” Proc. Natl. Acad. Sci. U.S.A. 97(15), 8206–8210 (2000).
[Crossref] [PubMed]

Egner, A.

T. A. Klar, S. Jakobs, M. Dyba, A. Egner, and S. W. Hell, “Fluorescence microscopy with diffraction resolution barrier broken by stimulated emission,” Proc. Natl. Acad. Sci. U.S.A. 97(15), 8206–8210 (2000).
[Crossref] [PubMed]

Enderlein, J.

C. B. Müller and J. Enderlein, “Image scanning microscopy,” Phys. Rev. Lett. 104(19), 198101 (2010).
[Crossref] [PubMed]

Fang, Y.

C. Kuang, Y. Ma, R. Zhou, G. Zheng, Y. Fang, Y. Xu, X. Liu, and P. T. So, “Virtual k-Space Modulation Optical Microscopy,” Phys. Rev. Lett. 117(2), 028102 (2016).
[Crossref] [PubMed]

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(11), 1122–1126 (2013).
[Crossref] [PubMed]

A. G. York, S. H. Parekh, D. Dalle 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(7), 749–754 (2012).
[Crossref] [PubMed]

Girard, J.

E. Mudry, K. Belkebir, J. Girard, J. Savatier, E. Le Moal, C. Nicoletti, M. Allain, and A. Sentenac, “Structured illumination microscopy using unknown speckle patterns,” Nat. Photonics 6(5), 312–315 (2012).
[Crossref]

Golubovskaya, I. N.

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Supplementary Material (1)

NameDescription
» Visualization 1       Procedure for virtual modulation

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Figures (14)

Fig. 1
Fig. 1 (a) Scanning process of LSM and (b) multi-spot complex detection system. The green star in (a) and (b) denotes the position of the object, and the blue arrows denote the scanning route of the pattern. The gray grids represent the conjugated sensor plane with each square denoting a single pixel. The illumination pattern is represented by the yellow and red dot.
Fig. 2
Fig. 2 (a) Procedure before VIKMOM algorithm is applied when illuminated by the multi-spot pattern. The stacked images denote the data acquired by the detector and the Siemens star is the object in the simulation. After the sub-data set along the tx and ty dimensions is extracted (process ①), the data should be rearranged to their true position in the same way LSM acquire data along the tx and ty dimensions (process ②) using knowledge of the estimated initial position and the scanning step of the multi-spot pattern. (b) Crosstalk between pixel A and B when estimating the initial position of the multi-spot pattern. The excitation photon flux is represented by a dashed line and the fluorescent flux is represented by a solid line. Two different imaging processes are distinguished by different colors; the conjugated detection pixels are represented by gray parallelograms and the green star denotes the position of the fluorophore.
Fig. 3
Fig. 3 Procedure for the virtual modulation microscopy. The image captured by the ith detector can be digitally modulated in two ways. The center of the detector array is labeled as “1,” which corresponds to the center of the focusing spot. The blue images denote the modulation patterns that are applied in the virtual modulation step. And the orange and purple squares, which is plotted as the reference of the shifting directions, denote the final image size which depends on the original image size and the maximum value of | k i | . The vector k i describes how the image shifts relative to the image captured by the central detector and all the captured images should be placed at the same position referring to the square, which is termed the “initial position”. The value of | k i | is the distance between the center of the projecting detectors on the sample plane, which depends on the magnification power and the distance between the detector i and the central detector.
Fig. 4
Fig. 4 Simulation of saturated virtual modulation microscopy. (a) 2 × bandwidth-filtered image with NA = 1.49 × 2. (b) Reconstruction image under VTM and (c) VSM. (d) Multiplex detection method with NA = 1.49 and f s = 6 when the maximum intensity of PSF equals unity. The diameter of the widefield result is normalized to unity.
Fig. 5
Fig. 5 Simulation of reconstruction results of MVTM against noise. (a) Noise-free, (b) 5%, (c) 10%, and (d) 20% Gaussian noise added to the raw data. The diameter of the wide-field result, which is not shown here, is normalized to unity.
Fig. 6
Fig. 6 Reconstruction result with different numbers of modulation patterns. (a) Result employing patterns with two different modulation frequencies (1.5 × and 2.3 × f c ), (b) four frequencies (1.1 × , 1.5 × , 1.9 × , and 2.3 × f c ), (c) six frequencies (0.9 × , 1.1 × , 1.5 × , 1.7 × , 2.1 × , and 2.3 × f c ), and (d) eight frequencies (0.9 × , 1.1 × , 1.3 × , 1.5 × , 1.7 × , 1.9 × , 2.1 × , and 2.3 × f c ). 10% Gaussian-type noise is added to the raw data. (e) VTM frequency and its equivalent OTF under saturation level of 6.
Fig. 7
Fig. 7 Simulation results of MVTM under different excitation intensity. (a) Reconstructed result when the fluorescence signal is proportional to the excitation laser intensity. Results when the maximum intensity of the excitation spot equal (b) 0.1, (c) 0.5 and (d) 1 demonstrate resolution enhancement beyond a factor of two. The intensity profile of the corresponding effective PSFs are shown in (e). The saturation level is assumed to be 6 in this simulation.
Fig. 8
Fig. 8 Simulation results of MVTM when there are estimation errors in the saturated level. (a)–(e) Reconstruction results using different saturation factors of 2, 4, 6, 8, and 10, respectively. The true saturation factor is set to 6. (f) Normalized effective excitation point-spread-function (PSF) profile under different saturation conditions and a normal PSF (blue line) drawn for reference. (g) Corresponding effective OTF curve under different saturation levels.
Fig. 9
Fig. 9 Simulations of aberrations. The color bar at the upper right shows the phase shift of the pupil that will induce aberrations. The corresponding detection PSFs are presented in the second row. From the results shown in the last row, the MVTM exhibits a strong resistance to aberrations.
Fig. 10
Fig. 10 Estimation of initial position. (a) Ground truth image. (b) Widefield result and (c) 2 × bandwidth filtered result, which can be achieved by methods such as structured illumination microscopy [6, 13, 27] (SIM). (d)–(f) Results of using different sampling sizes, which are marked by different colors in (a), to estimate the initial position. (d) uses the data covered by the orange area, (e) uses the blue and orange area, and (f) uses the full sampling area.
Fig. 11
Fig. 11 Simulations of shift error. The maximum probable relative shift error is shown in the left column. As the iteration time that applied in the reconstruction algorithm goes, the shift error gradually reduces the image quality.
Fig. 12
Fig. 12 Experimental results of (a)–(c) microtubule (An effective PSF under saturation condition can be achieved when fs = 6.2 and maximum intensity of the actual excitation PSF equals unity) and (e)–(g) beads (fs = 9.1, maximum excitation intensity equals one under saturation condition). (a) Confocal results with 1 airy unit (AU) detector and (e) 0.2AU detector, (b)(f) airyscan results under unsaturated condition, and (c)(g) VTM reconstruction images under saturated condition. (d) Normalized intensity profiles of the cutline indicated by the arrows in (a)–(c) and (h) the profiles of the cutline in (e)–(g). The scale bar is 500 nm for (e)–(g) and the nominal bead diameter is 200 nm.
Fig. 13
Fig. 13 Experiment results of 100-nm beads. (a) Confocal result with 0.2AU detector and (b) 1AU detector, (c) airyscan result, (d) VSM result, and (e) VTM result (fs = 3.5, maximum excitation intensity is unity for the data of VSM and VTM). (f) Intensity profiles of the cut-lines shown in the lower right of (a)–(e).
Fig. 14
Fig. 14 Experimental results of nuclear pore complex. (a) Confocal result with 0.2AU detector and (b) 1 AU detector, (c) airyscan result, and (d) VTM result (fs = 5.7, maximum excitation intensity is unity).

Equations (16)

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I ( x , y , t x , t y ) = h i l l ( u t x , v t y ) S ( u , v ) h det ( x u , y v ) d u d v ,
V t ( x , y ) = I ( x , y , t x , t y ) M ( t x , t y ) d t x d t y = [ ( h i l l M ) S ] h det ,
I ( x , y , t x , t y ) = h i l l ( u t x , v t y ) S ( u , v ) h det ( x u , y v ) d u d v = h i l l ( t x u , t y v ) S ( u , v ) h det ( u x , v y ) d u d v .
V s ( t x , t y ) = I ( x , y , t x , t y ) M ( x , y ) d t x d t y = [ ( h det M ) S ] h i l l ,
I e m = η ψ e x c 1 σ τ + ψ e x c ,
I ( x , y , t x , t y ) = ( η f s h i l l 1 + f s h i l l S ) h det .
V t ( x , y ) = [ ( h s M ) S ] h det V s ( t x , t y ) = [ ( h det M ) S ] h s .
I ( x , y , t x , t y ) = p ( u t x , v t y ) S ( u , v ) h det ( x u , y v ) d u d v ,
minimize Δ f ( Δ ) = p { | m ( r ) m [ Δ a ( p p 0 ) ] | w ( p ) }
m ( v ) = m ( v 1 , v 2 ) = ( v 1 mod L x , v 2 mod L y )
I ˜ = c o n j ( O T F det ) I ˜ | O T F det | + α
M t i e f f = h s M i ; M s i e f f = h det M i .
S u p d a t e = S + M t i e f f max ( M t i e f f ) 2 [ V t ( M t i e f f S ) h det ] h det ; S u p d a t e = S + M s i e f f max ( M s i e f f ) 2 [ V s ( M s i e f f S ) h s ] h s .
O T F det u p d a t e = O T F det + n | S ˜ M ˜ t i e f f | c o n j ( S ˜ M ˜ t i e f f ) [ V ˜ t O T F det ( S ˜ M ˜ t i e f f ) ] max ( | S ˜ M ˜ t i e f f | ) [ | S ˜ M ˜ t i e f f | 2 + ε ] ;
V ˜ t = ( S ˜ ( k ) { O T F s [ 1 + a δ ( k k 0 ) + a ¯ δ ( k + k 0 ) ] } ) O T F det , V ˜ s = ( S ˜ ( k ) { O T F det [ 1 + a δ ( k k 0 ) + a ¯ δ ( k + k 0 ) ] } ) O T F s .
k 0 < support of O T F s for V ˜ t ; k 0 < support of O T F det for V ˜ s

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