Abstract

Structured illumination can be employed to extend the lateral resolution of wide-field fluorescence microscopy. Since a structured illumination microscopy image is reconstructed from a series of several acquired images, we develop a modified formulation of the imaging response of the microscope and a probabilistic analysis to assess the resolution performance. We use this model to compare the fluorescence imaging performance of structured illumination techniques to confocal microscopy. Specifically, we examine the trade-off between achievable lateral resolution and signal-to-noise ratio when photon shot noise is dominant. We conclude that for a given photon budget, structured illumination invariably achieves better lateral resolution than confocal microscopy.

© 2008 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. M. G. L. Gustafsson, D. A. Agard, and J. W. Sedat, “Doubling the lateral resolution of wide-field fluorescence microscopy using structured illumination,” Proc. SPIE 3919, 141-150 (2000).
    [CrossRef]
  3. H. H. Hopkins and P. M. Barham, “The influence of the condenser on microscopic resolution,” Proc. Phys. Soc. London, Sect. B 63, 737-744 (1950).
    [CrossRef]
  4. S. G. Lipson, “Why is super-resolution so inefficient?” Micron 34, 309-312 (2003).
    [CrossRef] [PubMed]
  5. G. Toraldo di Francia, “Super-gain antennas and optical resolving power,” Nuovo Cimento, Suppl. 9, 426-438 (1952).
    [CrossRef]
  6. 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]
  7. S. O. Rice, “Mathematical analysis of random noise,” Bell Syst. Tech. J. 23, 282-332 (1944).
  8. R. N. Bracewell, “Applications in statistics,” in The Fourier Transform and Its Applications, 3rd ed. (McGraw-Hill, 2000), pp. 428-438.
  9. A. Papoulis and S. U. Pillai, “Characteristic functions,” in Probability, Random Variables and Stochastic Processes (McGraw-Hill, 2002), Chap. 5, pp. 152-168.
  10. 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]
  11. R. Heintzmann, “Saturated patterned excitation microscopy with two-dimensional excitation patterns,” Micron 34, 283-291 (2003).
    [CrossRef] [PubMed]
  12. M. A. A. Neil, R. Juskaitis, and T. Wilson, “Method of obtaining optical sectioning by using structured light in a conventional microscope,” Opt. Lett. 22, 1905-1907 (1997).
    [CrossRef]

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 (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]

2000 (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]

M. G. L. Gustafsson, D. A. Agard, and J. W. Sedat, “Doubling the lateral resolution of wide-field fluorescence microscopy using structured illumination,” Proc. SPIE 3919, 141-150 (2000).
[CrossRef]

1997 (1)

1952 (1)

G. Toraldo di Francia, “Super-gain antennas and optical resolving power,” Nuovo Cimento, Suppl. 9, 426-438 (1952).
[CrossRef]

1950 (1)

H. H. Hopkins and P. M. Barham, “The influence of the condenser on microscopic resolution,” Proc. Phys. Soc. London, Sect. B 63, 737-744 (1950).
[CrossRef]

1944 (1)

S. O. Rice, “Mathematical analysis of random noise,” Bell Syst. Tech. J. 23, 282-332 (1944).

Agard, D. A.

M. G. L. Gustafsson, D. A. Agard, and J. W. Sedat, “Doubling the lateral resolution of wide-field fluorescence microscopy using structured illumination,” Proc. SPIE 3919, 141-150 (2000).
[CrossRef]

Barham, P. M.

H. H. Hopkins and P. M. Barham, “The influence of the condenser on microscopic resolution,” Proc. Phys. Soc. London, Sect. B 63, 737-744 (1950).
[CrossRef]

Bracewell, R. N.

R. N. Bracewell, “Applications in statistics,” in The Fourier Transform and Its Applications, 3rd ed. (McGraw-Hill, 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]

M. G. L. Gustafsson, D. A. Agard, and J. W. Sedat, “Doubling the lateral resolution of wide-field fluorescence microscopy using structured illumination,” Proc. SPIE 3919, 141-150 (2000).
[CrossRef]

Heintzmann, R.

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

Hopkins, H. H.

H. H. Hopkins and P. M. Barham, “The influence of the condenser on microscopic resolution,” Proc. Phys. Soc. London, Sect. B 63, 737-744 (1950).
[CrossRef]

Juskaitis, R.

Lipson, S. G.

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

Neil, M. A. A.

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]

Papoulis, A.

A. Papoulis and S. U. Pillai, “Characteristic functions,” in Probability, Random Variables and Stochastic Processes (McGraw-Hill, 2002), Chap. 5, pp. 152-168.

Pillai, S. U.

A. Papoulis and S. U. Pillai, “Characteristic functions,” in Probability, Random Variables and Stochastic Processes (McGraw-Hill, 2002), Chap. 5, pp. 152-168.

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]

Rice, S. O.

S. O. Rice, “Mathematical analysis of random noise,” Bell Syst. Tech. J. 23, 282-332 (1944).

Sedat, J. W.

M. G. L. Gustafsson, D. A. Agard, and J. W. Sedat, “Doubling the lateral resolution of wide-field fluorescence microscopy using structured illumination,” Proc. SPIE 3919, 141-150 (2000).
[CrossRef]

Toraldo di Francia, G.

G. Toraldo di Francia, “Super-gain antennas and optical resolving power,” Nuovo Cimento, Suppl. 9, 426-438 (1952).
[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]

Wilson, T.

Bell Syst. Tech. J. (1)

S. O. Rice, “Mathematical analysis of random noise,” Bell Syst. Tech. J. 23, 282-332 (1944).

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]

Micron (2)

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

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

Nuovo Cimento, Suppl. (1)

G. Toraldo di Francia, “Super-gain antennas and optical resolving power,” Nuovo Cimento, Suppl. 9, 426-438 (1952).
[CrossRef]

Opt. Lett. (1)

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

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]

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]

Proc. Phys. Soc. London, Sect. B (1)

H. H. Hopkins and P. M. Barham, “The influence of the condenser on microscopic resolution,” Proc. Phys. Soc. London, Sect. B 63, 737-744 (1950).
[CrossRef]

Proc. SPIE (1)

M. G. L. Gustafsson, D. A. Agard, and J. W. Sedat, “Doubling the lateral resolution of wide-field fluorescence microscopy using structured illumination,” Proc. SPIE 3919, 141-150 (2000).
[CrossRef]

Other (2)

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

A. Papoulis and S. U. Pillai, “Characteristic functions,” in Probability, Random Variables and Stochastic Processes (McGraw-Hill, 2002), Chap. 5, pp. 152-168.

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Figures (14)

Fig. 1
Fig. 1

Schematic diagrams of the microscope systems analyzed in this paper. (a) Confocal fluorescence microscope (CFM). (b) Structured illumination microscope (SIM).

Fig. 2
Fig. 2

One-dimensional transfer functions of wide-field fluorescence microscope or CFM with large pinhole (solid, bottom curve), CFM with small pinhole (dashed, middle curve), and wide-field fluorescence microscope with doubled bandwidth (dotted, top curve).

Fig. 3
Fig. 3

Synthesis of transfer function with doubled bandwidth (dotted curve) by adding three transfer functions: conventional fluorescence microscope transfer function (solid curve), downconverted (dashed curve) and upconverted (dashed curve).

Fig. 4
Fig. 4

Definition of resolution criterion used. (a) PSF with x 25 and x 75 marked. (b) Cumulative values of the PSF showing how x 25 and x 75 are obtained.

Fig. 5
Fig. 5

CFM response for a two-point object with a separation of 0.315 NW with a confocal pinhole width of 0.04 NW .

Fig. 6
Fig. 6

CFM response for a two-point object with a separation of 0.315 NW with a confocal pinhole width of 1.0 NW .

Fig. 7
Fig. 7

Probability that the detected number of photons in the outer region exceeds the number detected in the inner region versus separation of the two object points; 50 detected photons from each point corresponding to 100 detected photons in the image.

Fig. 8
Fig. 8

Probability that the detected number of photons in the outer region exceeds the number detected in the inner region versus separation of the two object points; 200 detected photons in the image.

Fig. 9
Fig. 9

Probability that the detected number of photons in the outer region exceeds the number detected in the inner region versus separation of the two object points; 1000 detected photons in the image.

Fig. 10
Fig. 10

Real part of the characteristic function of the SIM system with a total of 400 detected photons in the image for a two-point object with a separation of 0.15 NW . The solid curve corresponds to the inner region, and the dashed curve corresponds to the outer region.

Fig. 11
Fig. 11

Probability density distribution for the SIM system corresponding to an object separation of 0.15 NW and 400 detected photons in the image. The solid curve denotes the probability density distribution corresponding to the inner region, and the dashed curve corresponds to the outer region.

Fig. 12
Fig. 12

Probability density distribution for a conventional fluorescence microscope with the same conditions as in Fig. 10.

Fig. 13
Fig. 13

Probability of detection for a structured illumination microscope with 100 (black solid curve), 200 (dashed curve), and 1000 photons (light solid curve) used to form the image.

Fig. 14
Fig. 14

Total number of image photons required to ensure a probability of 0.99 that the detected number of photons in the outer region exceeds the detected number of photons in the inner region. (Top to bottom) CFM using a 0.04 NW pinhole width (dashed curve); CFM using a 4 NW pinhole width (dotted–dashed curve), which is essentially the case of a conventional wide-field fluorescence microscope; CFM using a 0.25 NW pinhole width (light gray curve). The thick solid curve corresponds to the structured light microscope. The thin solid curve corresponds to a conventional fluorescence microscope with doubled bandwidth.

Tables (1)

Tables Icon

Table 1 Comparison of the Number of Photons Required to Ensure 0.99 Probability That the Outer Region is Greater Than the Inner Region for an Object Separation of 0.15 NW

Equations (38)

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i 1 ( x ) = 1 2 ( 1 + cos k g x ) s ( x ) H ( x ) ,
i 2 ( x ) = 1 2 ( 1 sin k g x ) s ( x ) H ( x ) ,
i 3 ( x ) = 1 2 ( 1 cos k g x ) s ( x ) H ( x ) ,
i 4 ( x ) = 1 2 ( 1 + sin k g x ) s ( x ) H ( x ) ,
s ( x ) H ( x ) 1 2 [ 1 + cos ( k g x ) ] ,
1 2 s ( x ) H ( x ) = 1 4 ( i 1 + i 2 + i 3 + i 4 ) .
2 i ̂ 1 ( k ) = H ̂ ( k ) s ̂ ( k ) + 1 2 H ̂ ( k ) s ̂ ( k k g ) + 1 2 H ̂ ( k ) s ̂ ( k + k g ) ,
2 i ̂ 2 ( k ) = H ̂ ( k ) s ̂ ( k ) + j 2 H ̂ ( k ) s ̂ ( k k g ) j 2 H ̂ ( k ) s ̂ ( k + k g ) ,
2 i ̂ 3 ( k ) = H ̂ ( k ) s ̂ ( k ) 1 2 H ̂ ( k ) s ̂ ( k k g ) 1 2 H ̂ ( k ) s ̂ ( k + k g ) ,
2 i ̂ 4 ( k ) = H ̂ ( k ) s ̂ ( k ) j 2 H ̂ ( k ) s ̂ ( k k g ) + j 2 H ̂ ( k ) s ̂ ( k + k g ) ,
H ̂ ( k ) s ̂ ( k k g ) = [ i ̂ 1 ( k ) i ̂ 3 ( k ) ] j [ i ̂ 2 ( k ) i ̂ 4 ( k ) ] ,
H ̂ ( k ) s ̂ ( k + k g ) = [ i ̂ 1 ( k ) i ̂ 3 ( k ) ] + j [ i ̂ 2 ( k ) i ̂ 4 ( k ) ] .
[ i ̂ 1 ( k + k g ) i ̂ 3 ( k + k g ) ] j [ i ̂ 2 ( k + k g ) i ̂ 4 ( k + k g ) ] = s ̂ ( k ) H ̂ ( k + k g ) ,
[ i ̂ 1 ( k k g ) i ̂ 3 ( k k g ) ] + j [ i ̂ 2 ( k k g ) i ̂ 4 ( k k g ) ] = s ̂ ( k ) H ̂ ( k k g ) .
s ( x ) H ( x ) exp ( j k g x ) = { [ i 1 ( x ) i 3 ( x ) ] j [ i 2 ( x ) i 4 ( x ) ] } exp ( j k g x ) ,
s ( x ) H ( x ) exp ( j k g x ) = { [ i 1 ( x ) i 3 ( x ) ] + j [ i 2 ( x ) i 4 ( x ) ] } exp ( j k g x ) .
s ( x ) H ( x ) 1 2 [ 1 + cos ( k g x ) ] = 1 4 [ ( 1 + 2 cos k g x ) i 1 + ( 1 2 sin k g x ) i 2 + ( 1 2 cos k g x ) i 3 + ( 1 + 2 sin k g x ) i 4 ] .
p ( 0 ) = exp [ i n ( x ) Δ x ] 1 i n ( x ) Δ x ,
p ( 1 ) = i n ( x ) Δ x exp [ i n ( x ) Δ x ] i n ( x ) Δ x .
C ( ϕ ) = p ( y ) exp ( j ϕ y ) d y ,
p ( 0 ) = exp [ i n ( x ) Δ x ] δ ( y ) [ 1 i n ( x ) Δ x ] δ ( y ) ,
p [ f ( x ) ] = i n ( x ) Δ x exp { i n ( x ) Δ x } δ [ y f ( x ) ] i n ( x ) Δ x δ [ y f ( x ) ] .
C n ( ϕ ) 1 + i n ( x ) Δ x { exp [ j ϕ f n ( x ) ] 1 } .
Lim Δ x 0 C n ( ϕ ) exp ( i n ( x ) Δ x { exp [ j ϕ f n ( x ) ] 1 } ) .
C n R ( ϕ ) = exp ( R { exp [ j ϕ f n ( x ) ] 1 } i n ( x ) d x ) ,
C R ( ϕ ) = 1 4 C n R ( ϕ , ) .
( σ 2 μ 2 ) u n p r o c e s s e d ( σ 2 μ 2 ) p r o c e s s e d = y 2 y 2 y 2 ,
C ( ϕ ) = exp ( R [ exp ( j ϕ f ( x ) ) 1 ] i ( x ) d x ) .
C ( ϕ ) ϕ = 0 = j y ; C ( ϕ ) ϕ = 0 = y 2 .
y 2 y 2 y 2 = C ( ϕ ) ϕ = 0 C ( ϕ ) ϕ = 0 2 C ( ϕ ) ϕ = 0 2 .
Lim Δ ϕ 0 = C ( ϕ + Δ ϕ ) C ( ϕ ) Δ ϕ .
C ( ϕ ) = j C ( ϕ ) R f ( x ) i ( x ) d x ,
C ( ϕ ) = j C ( ϕ ) R f ( x ) i ( x ) exp [ j ϕ f ( x ) ] d x C ( ϕ ) R [ f ( x ) ] 2 i ( x ) exp [ j ϕ f ( x ) ] d x .
C ( ϕ ) ϕ = 0 = j R f ( x ) i ( x ) d x ,
C ( ϕ ) ϕ = 0 = [ R f ( x ) i ( x ) d x ] 2 R [ f ( x ) ] 2 i ( x ) d x .
y 2 y 2 y 2 = R [ f ( x ) ] 2 i ( x ) d x [ R f ( x ) i ( x ) d x ] 2 .
R [ f ( x ) ] 2 i ( x ) d x [ R f ( x ) i ( x ) d x ] 2 R i ( x ) d x [ R i ( x ) d x ] 2 = 1 R i ( x ) d x .
R [ β ( x ) ] 2 d x [ R α ( x ) β ( x ) d x ] 2 1 R [ α ( x ) ] 2 d x .

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