Abstract

The artefact-free reconstruction of structured illumination microscopy images requires precise knowledge of the pattern phases in the raw images. If this parameter cannot be controlled precisely enough in an experimental setup, the phases have to be determined a posteriori from the acquired data. While an iterative optimisation based on cross-correlations between individual Fourier images yields accurate results, it is rather time-consuming. Here I present a fast non-iterative technique which determines each pattern phase from an auto-correlation of the respective Fourier image. In addition to improving the speed of the reconstruction, simulations show that this method is also more robust, yielding errors of typically less than λ/500 under realistic signal-to-noise levels.

© 2013 OSA

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References

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  1. R. Heintzmann and C. Cremer, “Laterally modulated excitation microscopy: improvement of resolution by using a diffraction grating,” Proc. SPIE 3568, 185–196 (1999).
    [Crossref]
  2. M. G. L. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microsc. 198, 82–87 (2000).
    [Crossref] [PubMed]
  3. J. T. Frohn, “Super-resolution fluorescence microscopy by structured light illumination,” Ph.D. thesis, Eidgenössische Technische Hochschule Zürich, Switzerland (2000).
  4. 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]
  5. 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]
  6. 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, 4957–4970 (2008).
    [Crossref] [PubMed]
  7. R. Heintzmann, T. M. Jovin, and C. Cremer, “Saturated patterned excitation microscopy - a concept for optical resolution improvement,” JOSA A 19, 1599–1609 (2002).
    [Crossref]
  8. M. G. L. Gustafsson, “Nonlinear structured-illumination microscopy: wide-field fluorescence imaging with theoretically unlimited resolution,” PNAS 102, 13081–13086 (2005).
    [Crossref] [PubMed]
  9. K. Wicker and R. Heintzmann, “Single-shot optical sectioning using polarization-coded structured illumination,” J. Opt. 12, 084010 (2010).
    [Crossref]
  10. G. Best, R. Amberger, D. Baddeley, T. Ach, S. Dithmar, R. Heintzmann, and C. Cremer, “Structured illumination microscopy of autofluorescent aggregations in human tissue,” Micron 42, 330–335 (2011).
    [Crossref]

2013 (1)

2011 (1)

G. Best, R. Amberger, D. Baddeley, T. Ach, S. Dithmar, R. Heintzmann, and C. Cremer, “Structured illumination microscopy of autofluorescent aggregations in human tissue,” Micron 42, 330–335 (2011).
[Crossref]

2010 (1)

K. Wicker and R. Heintzmann, “Single-shot optical sectioning using polarization-coded structured illumination,” J. Opt. 12, 084010 (2010).
[Crossref]

2009 (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]

2008 (1)

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, 4957–4970 (2008).
[Crossref] [PubMed]

2005 (1)

M. G. L. Gustafsson, “Nonlinear structured-illumination microscopy: wide-field fluorescence imaging with theoretically unlimited resolution,” PNAS 102, 13081–13086 (2005).
[Crossref] [PubMed]

2002 (1)

R. Heintzmann, T. M. Jovin, and C. Cremer, “Saturated patterned excitation microscopy - a concept for optical resolution improvement,” JOSA A 19, 1599–1609 (2002).
[Crossref]

2000 (1)

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

1999 (1)

R. Heintzmann and C. Cremer, “Laterally modulated excitation microscopy: improvement of resolution by using a diffraction grating,” Proc. SPIE 3568, 185–196 (1999).
[Crossref]

Ach, T.

G. Best, R. Amberger, D. Baddeley, T. Ach, S. Dithmar, R. Heintzmann, and C. Cremer, “Structured illumination microscopy of autofluorescent aggregations in human tissue,” Micron 42, 330–335 (2011).
[Crossref]

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, 4957–4970 (2008).
[Crossref] [PubMed]

Amberger, R.

G. Best, R. Amberger, D. Baddeley, T. Ach, S. Dithmar, R. Heintzmann, and C. Cremer, “Structured illumination microscopy of autofluorescent aggregations in human tissue,” Micron 42, 330–335 (2011).
[Crossref]

Baddeley, D.

G. Best, R. Amberger, D. Baddeley, T. Ach, S. Dithmar, R. Heintzmann, and C. Cremer, “Structured illumination microscopy of autofluorescent aggregations in human tissue,” Micron 42, 330–335 (2011).
[Crossref]

Best, G.

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]

G. Best, R. Amberger, D. Baddeley, T. Ach, S. Dithmar, R. Heintzmann, and C. Cremer, “Structured illumination microscopy of autofluorescent aggregations in human tissue,” Micron 42, 330–335 (2011).
[Crossref]

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, 4957–4970 (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, 4957–4970 (2008).
[Crossref] [PubMed]

Cremer, C.

G. Best, R. Amberger, D. Baddeley, T. Ach, S. Dithmar, R. Heintzmann, and C. Cremer, “Structured illumination microscopy of autofluorescent aggregations in human tissue,” Micron 42, 330–335 (2011).
[Crossref]

R. Heintzmann, T. M. Jovin, and C. Cremer, “Saturated patterned excitation microscopy - a concept for optical resolution improvement,” JOSA A 19, 1599–1609 (2002).
[Crossref]

R. Heintzmann and C. Cremer, “Laterally modulated excitation microscopy: improvement of resolution by using a diffraction grating,” Proc. SPIE 3568, 185–196 (1999).
[Crossref]

Dithmar, S.

G. Best, R. Amberger, D. Baddeley, T. Ach, S. Dithmar, R. Heintzmann, and C. Cremer, “Structured illumination microscopy of autofluorescent aggregations in human tissue,” Micron 42, 330–335 (2011).
[Crossref]

Fienup, J. R.

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.

Frohn, J. T.

J. T. Frohn, “Super-resolution fluorescence microscopy by structured light illumination,” Ph.D. thesis, Eidgenössische Technische Hochschule Zürich, Switzerland (2000).

Golubovskaya, I. N.

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, 4957–4970 (2008).
[Crossref] [PubMed]

Gustafsson, M. G. L.

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, 4957–4970 (2008).
[Crossref] [PubMed]

M. G. L. Gustafsson, “Nonlinear structured-illumination microscopy: wide-field fluorescence imaging with theoretically unlimited resolution,” PNAS 102, 13081–13086 (2005).
[Crossref] [PubMed]

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

Heintzmann, R.

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]

G. Best, R. Amberger, D. Baddeley, T. Ach, S. Dithmar, R. Heintzmann, and C. Cremer, “Structured illumination microscopy of autofluorescent aggregations in human tissue,” Micron 42, 330–335 (2011).
[Crossref]

K. Wicker and R. Heintzmann, “Single-shot optical sectioning using polarization-coded structured illumination,” J. Opt. 12, 084010 (2010).
[Crossref]

R. Heintzmann, T. M. Jovin, and C. Cremer, “Saturated patterned excitation microscopy - a concept for optical resolution improvement,” JOSA A 19, 1599–1609 (2002).
[Crossref]

R. Heintzmann and C. Cremer, “Laterally modulated excitation microscopy: improvement of resolution by using a diffraction grating,” Proc. SPIE 3568, 185–196 (1999).
[Crossref]

Jovin, T. M.

R. Heintzmann, T. M. Jovin, and C. Cremer, “Saturated patterned excitation microscopy - a concept for optical resolution improvement,” JOSA A 19, 1599–1609 (2002).
[Crossref]

Mandula, O.

Sedat, J. W.

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, 4957–4970 (2008).
[Crossref] [PubMed]

Shao, L.

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, 4957–4970 (2008).
[Crossref] [PubMed]

Shroff, S. A.

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]

Wang, C. J. R.

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, 4957–4970 (2008).
[Crossref] [PubMed]

Wicker, K.

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]

K. Wicker and R. Heintzmann, “Single-shot optical sectioning using polarization-coded structured illumination,” J. Opt. 12, 084010 (2010).
[Crossref]

Williams, D. R.

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]

Biophys. J. (1)

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, 4957–4970 (2008).
[Crossref] [PubMed]

J. Microsc. (1)

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

J. Opt. (1)

K. Wicker and R. Heintzmann, “Single-shot optical sectioning using polarization-coded structured illumination,” J. Opt. 12, 084010 (2010).
[Crossref]

JOSA A (2)

R. Heintzmann, T. M. Jovin, and C. Cremer, “Saturated patterned excitation microscopy - a concept for optical resolution improvement,” JOSA A 19, 1599–1609 (2002).
[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]

Micron (1)

G. Best, R. Amberger, D. Baddeley, T. Ach, S. Dithmar, R. Heintzmann, and C. Cremer, “Structured illumination microscopy of autofluorescent aggregations in human tissue,” Micron 42, 330–335 (2011).
[Crossref]

Opt. Express (1)

PNAS (1)

M. G. L. Gustafsson, “Nonlinear structured-illumination microscopy: wide-field fluorescence imaging with theoretically unlimited resolution,” PNAS 102, 13081–13086 (2005).
[Crossref] [PubMed]

Proc. SPIE (1)

R. Heintzmann and C. Cremer, “Laterally modulated excitation microscopy: improvement of resolution by using a diffraction grating,” Proc. SPIE 3568, 185–196 (1999).
[Crossref]

Other (1)

J. T. Frohn, “Super-resolution fluorescence microscopy by structured light illumination,” Ph.D. thesis, Eidgenössische Technische Hochschule Zürich, Switzerland (2000).

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Figures (3)

Fig. 1
Fig. 1

The synthetic sample used for simulations.

Fig. 2
Fig. 2

Performance of phase optimisation. The red line shows the average phase error over the photon level (expectation value of the brightest pixel in a set of three raw images) of the non-iterative (single step) auto-correlation technique and as such is a measure for the accuracy for the technique. The red shaded area shows the standard deviation around this average phase error and is a measure for the robustness of the technique. The blue line and shaded area show the corresponding average phase errors for the iterative approach [5]. (a) shows the results for the coarser pattern with a period of 210 nm, (b) shows the finer pattern with a period of 185 nm. For realistic photon levels (> 100) the single step approach yields more accurate and more robust results than the iterative approach.

Fig. 3
Fig. 3

SIM reconstructions of retinal pigment epithelial (RPE). (a) shows the reconstruction with the (obviously) wrong assumption of equidistant phase stepping, leading to strong artefacts. In the corresponding Fourier image (b) residual pattern peaks can be observed, which stem from a bad separation of components. Both iterative phase optimisation (c) and single step phase optimisation (e) lead to an artefact-free reconstruction with no visibly discernible differences. Their respective Fourier images (d) and (f) are void of any unwanted residual pattern peaks. Images reconstructed from data provided by Gerrit Best from the group of Christoph Cremer.

Equations (13)

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D n ( r ) = m = M M [ S ( r ) a m exp { ι ( 2 π m p r + m ϕ n ) } ] h ( r ) ,
D ˜ n ( k ) = m = M M exp { ι m ϕ n } a m S ˜ ( k m p ) h ˜ ( k ) , = : C ˜ m ( k )
D ˜ ( k ) = M C ˜ ( k ) ,
M n m = exp { ι m ϕ n } .
C ˜ ( k ) = M 1 D ˜ ( k ) .
S ˜ ( k ) = q , m { a m h ˜ * ( k + m p q ) C ˜ q , m ( k + m p q ) } q , m { | a m h ˜ ( k + m p q ) | 2 } + w A ˜ ( k ) ,
𝒟 n = [ D ˜ n D ˜ n ] ( p ) = D ˜ n ( k ) D ˜ n * ( k + p ) d 3 k = m = M M m = M M { exp { ι ( m m ) ϕ n } a m a m × S ˜ ( k m p ) S ˜ * ( k ( m 1 ) p ) | h ˜ ( k ) | 2 | h ˜ ( k + p ) | 2 d 3 k } = : 𝒞 m , m .
𝒞 m , m + 1 = | S ˜ ( k m p ) | 2 | h ˜ ( k ) | 2 | h ˜ ( k + p ) | 2 d 3 k ,
𝒞 m , m m + 1 = S ˜ ( k m p ) S ˜ * ( k ( m 1 ) p ) | h ˜ ( k ) | 2 | h ˜ ( k + p ) | 2 d 3 k .
𝒟 n exp { ι ϕ n } m = M M 1 { a m a m + 1 𝒞 m , m + 1 } .
ϕ n = arg { 𝒟 n } .
𝒟 n exp { ι ϕ n } 2 a 0 a 1 𝒞 0 , 1 ,
a 1 ( n ) / a 1 ( n ) = | 𝒟 n | / | 𝒟 n | .

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