Abstract

The stochastic transfer function (STF) has been introduced in previous publications [J. Opt. Soc. Am. A 26, 1622 (2009)]. This encompasses the conventional transfer function as well as a measure of the noise at each spatial frequency. We use the STF as a metric to characterize the noise performance of structured illumination microscopy where the final image is synthesized from several constituent images. In particular, we examine the effect of different processing strategies on the signal to noise at different spatial frequencies. We extend the so-called weighted average approach to account for different grating periods, where the noise in different image contributions is correlated. Finally, we demonstrate by simulation that a superior STF does lead to better imaging of a two-point object.

© 2011 Optical Society of America

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References

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  1. S. W. Hell, M. Schrader, and H. T. M. van der Voort, “Far-field fluorescence microscopy with three-dimensional resolution in the 100 nm range,” J. Microsc. 187, 1–7 (1997).
    [CrossRef] [PubMed]
  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. R. Heintzmann, “Saturated patterned excitation microscopy with two-dimensional excitation patterns,” Micron 34, 283–291(2003).
    [CrossRef] [PubMed]
  4. K. Hsu, M. G. Somekh, and M. C. Pitter, “Stochastic transfer function: application to fluorescence microscopy,” J. Opt. Soc. Am. A 26, 1622–1629 (2009).
    [CrossRef]
  5. M. G. Somekh, K. Hsu, and M. C. Pitter, “Stochastic transfer function for structured illumination microscopy,” J. Opt. Soc. Am. A 26, 1630–1637 (2009).
    [CrossRef]
  6. R. Heintzmann and C. Cremer, “Laterally modulated excitation microscopy: improvement of resolution by using a diffraction grating,” Proc. SPIE 3568, 185–196 (1999).
    [CrossRef]
  7. K. Hsu, “Stochastic analysis of lateral resolution and signal-to-noise ratio in fluorescence microscopy: application to structured illumination microscopy,” Ph.D. thesis (University of Nottingham, 2010).
  8. E. Chung, D. Kim, Y. Cui, Y.-H. Kim, and P. T. C. So, “Two-dimensional standing wave total internal reflection fluorescence microscopy: superresolution imaging of single molecular and biological specimens,” Biophys. J. 93, 1747–1757(2007).
    [CrossRef] [PubMed]
  9. M. G. Somekh, K. Hsu, and M. C. Pitter, “Resolution in structured illumination microscopy: an analytical probabilistic approach,” J. Opt. Soc. Am. A 25, 1319–1329 (2008).
    [CrossRef]
  10. A. Papoulis, Probability Random Variables and Stochastic Processes (McGraw Hill, 1965), Chap. 7, p. 226.
  11. C. W. See, C. J. Chuang, S. G. Liu, and M. G. Somekh, “Proximity projection grating structured light illumination microscopy,” Appl. Opt. 49, 6570–6576 (2010).
    [CrossRef] [PubMed]

2010 (1)

2009 (2)

2008 (1)

2007 (1)

E. Chung, D. Kim, Y. Cui, Y.-H. Kim, and P. T. C. So, “Two-dimensional standing wave total internal reflection fluorescence microscopy: superresolution imaging of single molecular and biological specimens,” Biophys. J. 93, 1747–1757(2007).
[CrossRef] [PubMed]

2003 (1)

R. Heintzmann, “Saturated patterned excitation microscopy with two-dimensional excitation patterns,” Micron 34, 283–291(2003).
[CrossRef] [PubMed]

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]

1997 (1)

S. W. Hell, M. Schrader, and H. T. M. van der Voort, “Far-field fluorescence microscopy with three-dimensional resolution in the 100 nm range,” J. Microsc. 187, 1–7 (1997).
[CrossRef] [PubMed]

Chuang, C. J.

Chung, E.

E. Chung, D. Kim, Y. Cui, Y.-H. Kim, and P. T. C. So, “Two-dimensional standing wave total internal reflection fluorescence microscopy: superresolution imaging of single molecular and biological specimens,” Biophys. J. 93, 1747–1757(2007).
[CrossRef] [PubMed]

Cremer, C.

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

Cui, Y.

E. Chung, D. Kim, Y. Cui, Y.-H. Kim, and P. T. C. So, “Two-dimensional standing wave total internal reflection fluorescence microscopy: superresolution imaging of single molecular and biological specimens,” Biophys. J. 93, 1747–1757(2007).
[CrossRef] [PubMed]

Gustafsson, M. G. L.

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.

R. Heintzmann, “Saturated patterned excitation microscopy with two-dimensional excitation patterns,” Micron 34, 283–291(2003).
[CrossRef] [PubMed]

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

Hell, S. W.

S. W. Hell, M. Schrader, and H. T. M. van der Voort, “Far-field fluorescence microscopy with three-dimensional resolution in the 100 nm range,” J. Microsc. 187, 1–7 (1997).
[CrossRef] [PubMed]

Hsu, K.

Kim, D.

E. Chung, D. Kim, Y. Cui, Y.-H. Kim, and P. T. C. So, “Two-dimensional standing wave total internal reflection fluorescence microscopy: superresolution imaging of single molecular and biological specimens,” Biophys. J. 93, 1747–1757(2007).
[CrossRef] [PubMed]

Kim, Y.-H.

E. Chung, D. Kim, Y. Cui, Y.-H. Kim, and P. T. C. So, “Two-dimensional standing wave total internal reflection fluorescence microscopy: superresolution imaging of single molecular and biological specimens,” Biophys. J. 93, 1747–1757(2007).
[CrossRef] [PubMed]

Liu, S. G.

Papoulis, A.

A. Papoulis, Probability Random Variables and Stochastic Processes (McGraw Hill, 1965), Chap. 7, p. 226.

Pitter, M. C.

Schrader, M.

S. W. Hell, M. Schrader, and H. T. M. van der Voort, “Far-field fluorescence microscopy with three-dimensional resolution in the 100 nm range,” J. Microsc. 187, 1–7 (1997).
[CrossRef] [PubMed]

See, C. W.

So, P. T. C.

E. Chung, D. Kim, Y. Cui, Y.-H. Kim, and P. T. C. So, “Two-dimensional standing wave total internal reflection fluorescence microscopy: superresolution imaging of single molecular and biological specimens,” Biophys. J. 93, 1747–1757(2007).
[CrossRef] [PubMed]

Somekh, M. G.

van der Voort, H. T. M.

S. W. Hell, M. Schrader, and H. T. M. van der Voort, “Far-field fluorescence microscopy with three-dimensional resolution in the 100 nm range,” J. Microsc. 187, 1–7 (1997).
[CrossRef] [PubMed]

Appl. Opt. (1)

Biophys. J. (1)

E. Chung, D. Kim, Y. Cui, Y.-H. Kim, and P. T. C. So, “Two-dimensional standing wave total internal reflection fluorescence microscopy: superresolution imaging of single molecular and biological specimens,” Biophys. J. 93, 1747–1757(2007).
[CrossRef] [PubMed]

J. Microsc. (2)

S. W. Hell, M. Schrader, and H. T. M. van der Voort, “Far-field fluorescence microscopy with three-dimensional resolution in the 100 nm range,” J. Microsc. 187, 1–7 (1997).
[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]

J. Opt. Soc. Am. A (3)

Micron (1)

R. Heintzmann, “Saturated patterned excitation microscopy with two-dimensional excitation patterns,” Micron 34, 283–291(2003).
[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 (2)

K. Hsu, “Stochastic analysis of lateral resolution and signal-to-noise ratio in fluorescence microscopy: application to structured illumination microscopy,” Ph.D. thesis (University of Nottingham, 2010).

A. Papoulis, Probability Random Variables and Stochastic Processes (McGraw Hill, 1965), Chap. 7, p. 226.

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Figures (8)

Fig. 1
Fig. 1

Relative position of spectral orders in SIM. (a) Position of spectral orders in cylindrical SIM, for maximum grating frequency, m g = 2 . (b) Position of spectral orders in cylindrical SIM, for grating frequency, m g < 2 . (c) Arrangement of spectral orders in spherical SIM with three azimuthal projections for m g = 2 . The angles of the azimuthal projections are 0 ° , 60 ° , and 120 ° , respectively, and the shaded region represents the spatial frequency coverage. The solid black circle represents the aperture of a conventional fluorescent microscope with radius of two spatial frequency units, whereas the dashed circle has radius of four spatial frequency units. (d) Summation of spectral orders to recover extended transfer function for m g = 2 . Sideband weighting is half of carrier weighting.

Fig. 2
Fig. 2

Normalized SNR associated with the conventional fluorescent microscope, solid curve, and the structured light microscope (with no prefiltering), dashed curve. The values presented here should be multiplied by the number of photons in the image.

Fig. 3
Fig. 3

Processing approaches for SIM. (a) SIM without prefiltering of noise in each sideband. (b) SIM prefiltering with the objective bandwidth prior to image reconstruction (in-band filtering). The vertical line at spatial frequency, m, indicates the presence of duplicate information in carrier and sideband that can be used to further increase the SNR.

Fig. 4
Fig. 4

SNR obtained from STFs for SIM under different filtering conditions, for m g = 2 and no correlation between carrier and sidebands. (a) SNR obtained from STF for cylindrical microscope system; solid curve (red), no filtering; dashed curve (green), in-band filtering; dotted curve (blue), weighted average; NCM with doubled bandwidth, bold curve (black). (b) SNR obtained from STF for spherical microscope system; solid curve (red), no filtering; dashed curve (green), in-band filtering; dotted curve (blue), weighted average; NCM with doubled bandwidth, bold curve (black). The curves correspond to three azimuthal directions showing the response along the grating direction.

Fig. 5
Fig. 5

Monte Carlo calculation of correlation between carrier and a single sideband (solid curve) and between upper and lower sidebands (dashed curve).

Fig. 6
Fig. 6

SNRs obtained from cylindrical STFs for different grating vectors. Black (solid), m g = 2 no correlation; magenta (dashed), m g = 1.5 ; blue (dotted), m g 1 ; green (dotted–dashed), m g = 0.5 ; red (solid), m g = 0 .

Fig. 7
Fig. 7

Optimum weighting values for different grating frequencies. (a) Weighting for m g = 2 ; solid line (green), carrier weighting; dashed line (blue), weighting of upper sideband. (b) Weighting for m g = 1 ; solid line (green), carrier weighting; dashed line (blue), weighting of upper sideband; dotted line (red), weighting of lower sideband. (c) Weighting for m g = 1 , of lower sideband, rescaled.

Fig. 8
Fig. 8

Calculated probabilities of resolving two point objects for 100 photons in the image. (a) Diagram explain ing resolution criterion. (b) Calculated probability that the inner region > outer region . Bold solid (black), conventional microscope; dashed (green), raw SIM; fine solid (red), SIM in-band filtering; dotted (blue), weighted average.

Tables (4)

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Table 1 Equating Terms with Characteristic Function of Eq. (11) and Equating Imaginary Terms Corresponding to the Mean Values a

Tables Icon

Table 2 Equating Terms with Characteristic Function of Eq. (11) and Equating Terms in ϕ 2 a

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Table 3 Equating Terms with Characteristic Function of Eq. (11) and Equating Terms in ψ 2 a

Tables Icon

Table 4 Equating Terms with Characteristic Function of Eq. (11) and Equating Terms in ϕ ψ a

Equations (29)

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ill n ( x ) = N 4 [ 1 + cos ( 2 π m g x + ( n 1 ) π 2 ) ] ,
i ^ carrier ( m ) = i ^ 1 ( m ) + i ^ 2 ( m ) + i ^ 4 ( m ) ,
i ^ upper ( m m g ) = i ^ 1 ( m m g ) + j i ^ 2 ( m m g ) j i ^ 4 ( m m g ) ,
i ^ lower ( m + m g ) = i ^ 1 ( m + m g ) j i ^ 2 ( m + m g ) + j i ^ 4 ( m + m g ) ,
A ( ϕ , ψ ) = exp R i ( x ) { exp [ j 2 π ( f a ( x ) ϕ + f b ( x ) ψ ) ] 1 } d x ,
i 1 ( x ) = N 2 H ( x ) , i 2 ( x ) = i 4 ( x ) = N 4 H ( x ) .
i s ( x ) = [ 1 + 2 cos ( 2 π m g x ) ] i 1 ( x ) + [ 1 2 sin ( 2 π m g x ) ] i 2 ( x ) + [ 1 + 2 sin ( 2 π m g x ) ] i 4 ( x ) ,
A e ( ϕ , ψ ) = n A en ( ϕ , ψ ) ,
A o ( ϕ , ψ ) = n A on ( ϕ , ψ ) .
f a e 1 ( x ) f a o 1 ( x ) } = { cos 2 π m x sin 2 π m x , f a e 2 ( x ) f a o 2 ( x ) } = { cos 2 π m x sin 2 π m x , f a e 4 ( x ) f a o 4 ( x ) } = { cos 2 π m x sin 2 π m x ,
f b e 1 ( x ) f b o 1 ( x ) } = 2 cos 2 π m g x { cos 2 π m x sin 2 π m x , f b e 2 ( x ) f b o 2 ( x ) } = 2 sin 2 π m g x { cos 2 π m x sin 2 π m x , f b e 4 ( x ) f b o 4 ( x ) } = 2 sin 2 π m g x { cos 2 π m x sin 2 π m x .
A ( ϕ , ψ ) = exp { R N H ( x ) { 2 π j [ f a ( x ) ϕ + f b ( x ) ψ ] 2 π 2 [ f a 2 ( x ) ϕ 2 + f b 2 ( x ) ψ 2 + 2 f a ( x ) f b ( x ) ϕ ψ ] } d x } .
H ( x ) { cos ( k m + l m g ) x sin ( k m + l m g ) x d x = { c ( k m + l m g ) 0 ,
CF = exp 2 π [ j ( μ 1 ϕ + μ 2 ψ ) π ( σ 1 2 ϕ 2 + σ 2 2 ψ 2 + 2 ρ σ 1 σ 2 ϕ ψ ) ] ,
μ 1 = N c ( m ) ,
μ 2 = N 2 [ c ( m + m g ) + c ( m m g ) ] .
σ 1 2 = N , σ 2 2 = 2 N .
cov = N c ( m g ) ,
ρ = cov σ 1 2 σ 2 2 = 1 2 c ( m g ) .
ρ = N c ( m g ) 2 N N = c ( m g ) 2 .
Minimizing : W 1 2 V 1 + W 2 2 V 2 + W 3 2 V 3 + 2 ρ 12 W 1 W 2 V 1 V 2 + 2 ρ 13 W 1 W 3 V 1 V 3 + 2 ρ 23 W 2 W 3 V 2 V 3 ,
F = W 1 2 V 1 + W 2 2 V 2 + W 3 2 V 3 + 2 ρ 12 W 1 W 2 V 1 V 2 + 2 ρ 13 W 1 W 3 V 1 V 3 + 2 ρ 23 W 2 W 3 V 2 V 3 + λ ( W 1 + W 2 + W 3 1 ) .
F W 1 = 2 W 1 V 1 + 2 ρ 12 W 2 V 1 V 2 + 2 ρ 13 W 3 V 1 V 3 + λ = 0 , F W 2 = 2 W 2 V 2 + 2 ρ 12 W 1 V 1 V 2 + 2 ρ 23 W 3 V 2 V 3 + λ = 0 , F W 3 = 2 W 3 V 3 + 2 ρ 13 W 1 V 1 V 3 + 2 ρ 23 W 2 V 2 V 3 + λ = 0.
[ cov ] [ W 1 W 2 W 3 ] = λ 2 [ 1 1 1 ] ,
W 1 = α 1 α 1 + α 2 + α 3 ,
W 2 = α 2 α 1 + α 2 + α 3 ,
W 3 = α 3 α 1 + α 2 + α 3 ,
α 1 = V 2 V 3 ( 1 ρ 23 2 ) + V 3 V 1 V 2 ( ρ 13 ρ 23 ρ 12 ) + V 2 V 1 V 3 ( ρ 12 ρ 23 ρ 13 ) , α 2 = V 1 V 3 ( 1 ρ 13 2 ) + V 3 V 1 V 2 ( ρ 13 ρ 23 ρ 12 ) + V 1 V 2 V 3 ( ρ 12 ρ 13 ρ 23 ) , α 3 = V 1 V 2 ( 1 ρ 12 2 ) + V 1 V 2 V 3 ( ρ 12 ρ 13 ρ 23 ) + V 2 V 1 V 3 ( ρ 12 ρ 23 ρ 13 ) .
W 1 = V 2 ρ V 1 V 2 V 1 + V 2 2 ρ V 1 V 2 , W 2 = V 1 ρ V 1 V 2 V 1 + V 2 2 ρ V 1 V 2 ,

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