Abstract

We develop the concept of the stochastic transfer function (STF) and its application to high-resolution fluorescence microscopy. The STF is directly related to the conventional optical transfer functions but incorporates a probability density function at each spatial frequency. The mean of the STF yields the conventional transfer function; the variance of the STF gives a measure of the noise associated at each spatial frequency. The STF thus provides a more complete picture of the microscope performance in comparison with conventional transfer functions. Here we present the STF for a wide-field fluorescent microscope using both Monte Carlo simulation and probability theory.

© 2009 Optical Society of America

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References

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  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]
  2. R. Heintzmann, “Saturated patterned excitation microscopy with two-dimensional excitation patterns,” Micron 34, 283-291 (2003).
    [CrossRef] [PubMed]
  3. M. G. L. Gustafsson, “Nonlinear structured-illumination microscopy: Wide-field fluorescence imaging with theoretically unlimited resolution,” Proc. Natl. Acad. Sci. U.S.A. 102, 13081-13086 (2005).
    [CrossRef] [PubMed]
  4. S. W. Hell and J. Wichmann, “Breaking the diffraction resolution limit by stimulated emission: stimulated-emission-depletion fluorescence microscopy,” Opt. Lett. 19, 780-782 (1994).
    [CrossRef] [PubMed]
  5. V. Westphal, L. Kastrup, and S. W. Hell, “Lateral resolution of 28 nm(λ/25) in far-field fluorescence microscopy,” Appl. Phys. B: Lasers Opt. 77, 377-380 (2003).
    [CrossRef]
  6. S. G. Lipson, “Why is super-resolution so inefficient?” Micron 34, 309-312 (2003).
    [CrossRef] [PubMed]
  7. E. Stelzer, “Contrast, resolution, pixelation, dynamic range and signal-to-noise ratio: fundamental limits to resolution in fluorescence light microscopy,” J. Microsc. 189, 15-24 (1998).
    [CrossRef]
  8. S. Ram, E. S. Ward, and R. J. Ober, “Beyond Rayleigh's criterion: A resolution measure with application to single-molecule microscopy,” Proc. Natl. Acad. Sci. U.S.A. 103, 4457-4462 (2006).
    [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 (2007).
    [CrossRef]
  10. 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]
  11. R. N. Bracewell, “Applications in statistics,” in The Fourier Transform and Its Applications, 3rd ed. (McGraw-Hill Higher Education, 2000), pp. 428-438.

2009 (1)

2007 (1)

2006 (1)

S. Ram, E. S. Ward, and R. J. Ober, “Beyond Rayleigh's criterion: A resolution measure with application to single-molecule microscopy,” Proc. Natl. Acad. Sci. U.S.A. 103, 4457-4462 (2006).
[CrossRef] [PubMed]

2005 (1)

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

2003 (3)

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

V. Westphal, L. Kastrup, and S. W. Hell, “Lateral resolution of 28 nm(λ/25) in far-field fluorescence microscopy,” Appl. Phys. B: Lasers Opt. 77, 377-380 (2003).
[CrossRef]

S. G. Lipson, “Why is super-resolution so inefficient?” Micron 34, 309-312 (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]

1998 (1)

E. Stelzer, “Contrast, resolution, pixelation, dynamic range and signal-to-noise ratio: fundamental limits to resolution in fluorescence light microscopy,” J. Microsc. 189, 15-24 (1998).
[CrossRef]

1994 (1)

Bracewell, R. N.

R. N. Bracewell, “Applications in statistics,” in The Fourier Transform and Its Applications, 3rd ed. (McGraw-Hill Higher Education, 2000), pp. 428-438.

Gustafsson, M. G. L.

M. G. L. Gustafsson, “Nonlinear structured-illumination microscopy: Wide-field fluorescence imaging with theoretically unlimited resolution,” Proc. Natl. Acad. Sci. U.S.A. 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.

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

Hell, S. W.

V. Westphal, L. Kastrup, and S. W. Hell, “Lateral resolution of 28 nm(λ/25) in far-field fluorescence microscopy,” Appl. Phys. B: Lasers Opt. 77, 377-380 (2003).
[CrossRef]

S. W. Hell and J. Wichmann, “Breaking the diffraction resolution limit by stimulated emission: stimulated-emission-depletion fluorescence microscopy,” Opt. Lett. 19, 780-782 (1994).
[CrossRef] [PubMed]

Hsu, K.

Kastrup, L.

V. Westphal, L. Kastrup, and S. W. Hell, “Lateral resolution of 28 nm(λ/25) in far-field fluorescence microscopy,” Appl. Phys. B: Lasers Opt. 77, 377-380 (2003).
[CrossRef]

Lipson, S. G.

S. G. Lipson, “Why is super-resolution so inefficient?” Micron 34, 309-312 (2003).
[CrossRef] [PubMed]

Ober, R. J.

S. Ram, E. S. Ward, and R. J. Ober, “Beyond Rayleigh's criterion: A resolution measure with application to single-molecule microscopy,” Proc. Natl. Acad. Sci. U.S.A. 103, 4457-4462 (2006).
[CrossRef] [PubMed]

Pitter, M. C.

Ram, S.

S. Ram, E. S. Ward, and R. J. Ober, “Beyond Rayleigh's criterion: A resolution measure with application to single-molecule microscopy,” Proc. Natl. Acad. Sci. U.S.A. 103, 4457-4462 (2006).
[CrossRef] [PubMed]

Somekh, M. G.

Stelzer, E.

E. Stelzer, “Contrast, resolution, pixelation, dynamic range and signal-to-noise ratio: fundamental limits to resolution in fluorescence light microscopy,” J. Microsc. 189, 15-24 (1998).
[CrossRef]

Ward, E. S.

S. Ram, E. S. Ward, and R. J. Ober, “Beyond Rayleigh's criterion: A resolution measure with application to single-molecule microscopy,” Proc. Natl. Acad. Sci. U.S.A. 103, 4457-4462 (2006).
[CrossRef] [PubMed]

Westphal, V.

V. Westphal, L. Kastrup, and S. W. Hell, “Lateral resolution of 28 nm(λ/25) in far-field fluorescence microscopy,” Appl. Phys. B: Lasers Opt. 77, 377-380 (2003).
[CrossRef]

Wichmann, J.

Appl. Phys. B: Lasers Opt. (1)

V. Westphal, L. Kastrup, and S. W. Hell, “Lateral resolution of 28 nm(λ/25) in far-field fluorescence microscopy,” Appl. Phys. B: Lasers Opt. 77, 377-380 (2003).
[CrossRef]

J. Microsc. (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]

E. Stelzer, “Contrast, resolution, pixelation, dynamic range and signal-to-noise ratio: fundamental limits to resolution in fluorescence light microscopy,” J. Microsc. 189, 15-24 (1998).
[CrossRef]

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

Micron (2)

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

S. G. Lipson, “Why is super-resolution so inefficient?” Micron 34, 309-312 (2003).
[CrossRef] [PubMed]

Opt. Lett. (1)

Proc. Natl. Acad. Sci. U.S.A. (2)

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

S. Ram, E. S. Ward, and R. J. Ober, “Beyond Rayleigh's criterion: A resolution measure with application to single-molecule microscopy,” Proc. Natl. Acad. Sci. U.S.A. 103, 4457-4462 (2006).
[CrossRef] [PubMed]

Other (1)

R. N. Bracewell, “Applications in statistics,” in The Fourier Transform and Its Applications, 3rd ed. (McGraw-Hill Higher Education, 2000), pp. 428-438.

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Figures (6)

Fig. 1
Fig. 1

Transfer functions plotted against normalized spatial frequency for a fluorescent microscope for a 1D cylindrical imaging system (solid curve) and a 2D spherical imaging system (dashed curve).

Fig. 2
Fig. 2

Stochastic transfer functions calculated using Monte Carlo type simulations plotted against normalized spatial frequency, with 15 photons in the image: (a) Mean of the even component. (b) Mean of the odd component. (c) Variance of the even component. (d) Variance of the odd component.

Fig. 3
Fig. 3

Joint PDF corresponding to the odd and the even signals for different spatial frequencies with 15 photons in the image: (a) Normalized spatial frequency, m = 0.4 . (b) Normalized spatial frequency, m = 0.8 . (c) Normalized spatial frequency, m = 1.2 .

Fig. 4
Fig. 4

Mean and variance of the STF as a function of normalized spatial frequency m. The solid curve represents the mean of the STF (corresponding to the conventional transfer function), the dashed curve is the variance of the even component, and the dotted curve represents the variance of the odd component. The variances have been displaced vertically by one unit to separate them from the mean value. Note after m = 1 both the even and the odd components of the variance are equal.

Fig. 5
Fig. 5

(a) Real part of the characteristic function of the even component, with N = 15 , m = 0 . The characteristic function repeats periodically for every unit increment of ϕ (b) Real part of the characteristic function of the even component, with N = 15 , m = 0.2 .

Fig. 6
Fig. 6

The different regions plotted against N and m. Region A corresponds to the distribution approximating a Gaussian in all respects, within region B the skewness is greater than our threshold value, and in region C the distribution still has evidence of its origin from the discrete Poisson distribution. The circle, square, and triangle correspond to the data associated with Figs. 3a, 3b, 3c, respectively.

Equations (23)

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c ( m , n ) = H ( x , y ) exp [ 2 π j ( m x + n y ) ] d x d y ,
c ( m , n ) = H ( x , y ) cos [ 2 π ( m x + n y ) ] d x d y .
p ( 0 ) = exp [ i ( x ) Δ x ] 1 i ( x ) Δ x ,
p ( 1 ) i ( x ) Δ x .
p ( f 1 ( x ) , f 2 ( x ) ) ( 1 i ( x ) Δ x ) δ ( q , s ) + i ( x ) Δ x δ ( q f 1 ( x ) , s f 2 ( x ) )
A ( ϕ , ψ ) Lim Δ x 0 = exp [ i ( x ) Δ x ( exp { j 2 π [ f 1 ( x ) ϕ + f 2 ( x ) ψ ] } 1 ) ] .
A ( ϕ , ψ ) = exp [ R ( exp { j 2 π [ f 1 ( x ) ϕ + f 2 ( x ) ψ ] } 1 ) i ( x ) d x ] ,
A ( ϕ , ψ ) = exp [ R ( exp { j 2 π [ f 1 ( x , y ) ϕ + f 2 ( x , y ) ψ ] } 1 ) i ( x , y ) d x d y ] .
A ( ϕ , ψ ) = exp [ N R ( exp { 2 π j [ cos ( 2 π m x ) ϕ + sin ( 2 π m x ) ψ ] } 1 ) H ( x ) d x ] ,
p ( q , s ) = A ( ϕ , ψ ) exp [ 2 π j ( ϕ q + ψ s ) ] d ϕ d ψ .
A e ( ϕ ) = exp [ N ( exp { 2 π j ϕ [ cos ( 2 π m x ) ] } 1 ) H ( x ) d x ] ,
A 0 ( ψ ) = exp [ N ( exp { 2 π j ψ [ sin ( 2 π m x ) ] } 1 ) H ( x ) d x ] .
arg [ Re ( A e ( ϕ ) ) ] = N { cos [ 2 π ϕ cos ( 2 π m x ) ] 1 } H ( x ) d x ,
arg [ Im ( A e ( ϕ ) ) ] = N { sin [ 2 π ϕ cos ( 2 π m x ) ] } H ( x ) d x ,
arg [ Re ( A 0 ( ϕ ) ) ] = N { cos [ 2 π ϕ sin ( 2 π m x ) ] 1 } H ( x ) d x ,
arg [ Im ( A 0 ( ϕ ) ) ] = N { sin [ 2 π ϕ sin ( 2 π m x ) ] } H ( x ) d x .
arg [ Re ( A e ( ϕ ) ) ] N 2 π 2 ϕ 2 cos 2 ( 2 π m x ) H ( x ) d x = N π 2 ϕ 2 [ 1 + cos ( 4 π m x ) ] H ( x ) d x .
arg [ Re ( A e ( ϕ ) ) ] = N π 2 ϕ 2 [ 1 + c ( 2 m ) ] .
arg [ Re ( A 0 ( ϕ ) ) ] = N π 2 ϕ 2 [ 1 c ( 2 m ) ] .
arg [ Im ( A e ( ϕ ) ) ] = 2 π N ϕ c ( m ) , arg [ Im ( A 0 ( ϕ ) ) ] = 0 .
μ = N c ( m ) , σ e 2 = N 2 [ 1 + c ( 2 m ) ] , σ 0 2 = N 2 [ 1 c ( 2 m ) ] .
μ 2 σ T 2 = N [ c ( m ) ] 2 .
μ = A e ( 0 ) 2 π i A e ( 0 ) , σ e 2 = A e ( 0 ) 4 π 2 A e ( 0 ) + [ A e ( 0 ) ] 2 4 π 2 [ A e ( 0 ) ] 2 ,

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