Abstract

The stochastic transfer function extends the concept of the conventional transfer function by incorporating noise statistics, thus giving a measure of the signal-to-noise ratio at each spatial frequency. This provides a convenient and standardized metric to assess the trade-off in terms of spatial frequency bandwidth and signal-to-noise ratio for a diverse range of resolution enhancement techniques. We apply this concept to structured illumination microscopy and compare its noise performance as a function of frequency with the conventional wide-field fluorescent microscope. The result suggests how a hybrid algorithm further improves the photon efficiency of the structured illumination technique.

© 2009 Optical Society of America

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  1. 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]
  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. B. Littleton, K. Lai, D. Longstaff, V. Sarafis, P. Munroe, N. Heckenberg, and H. Rubinsztein-Dunlop, “Coherent super-resolution microscopy via laterally structured illumination,” Micron 38, 150-157 (2007).
    [CrossRef]
  5. 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]
  6. S. G. Lipson, “Why is super-resolution so inefficient?” Micron 34, 309-312 (2003).
    [CrossRef] [PubMed]
  7. 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, 1642-1645 (2006).
    [CrossRef] [PubMed]
  8. 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]
  9. G. Donnert, J. Keller, R. Medda, M. A. Andrei, S. O. Rizzoli, R. Luhrmann, R. Jahn, C. Eggeling, and S. W. Hell, “Macromolecular-scale resolution in biological fluorescence microscopy,” Proc. Natl. Acad. Sci. U.S.A. 103, 11440-11445 (2006).
    [CrossRef] [PubMed]

2009

2007

B. Littleton, K. Lai, D. Longstaff, V. Sarafis, P. Munroe, N. Heckenberg, and H. Rubinsztein-Dunlop, “Coherent super-resolution microscopy via laterally structured illumination,” Micron 38, 150-157 (2007).
[CrossRef]

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]

2006

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, 1642-1645 (2006).
[CrossRef] [PubMed]

G. Donnert, J. Keller, R. Medda, M. A. Andrei, S. O. Rizzoli, R. Luhrmann, R. Jahn, C. Eggeling, and S. W. Hell, “Macromolecular-scale resolution in biological fluorescence microscopy,” Proc. Natl. Acad. Sci. U.S.A. 103, 11440-11445 (2006).
[CrossRef] [PubMed]

2003

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

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]

1994

Andrei, M. A.

G. Donnert, J. Keller, R. Medda, M. A. Andrei, S. O. Rizzoli, R. Luhrmann, R. Jahn, C. Eggeling, and S. W. Hell, “Macromolecular-scale resolution in biological fluorescence microscopy,” Proc. Natl. Acad. Sci. U.S.A. 103, 11440-11445 (2006).
[CrossRef] [PubMed]

Betzig, E.

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, 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, 1642-1645 (2006).
[CrossRef] [PubMed]

Davidson, M. W.

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, 1642-1645 (2006).
[CrossRef] [PubMed]

Donnert, G.

G. Donnert, J. Keller, R. Medda, M. A. Andrei, S. O. Rizzoli, R. Luhrmann, R. Jahn, C. Eggeling, and S. W. Hell, “Macromolecular-scale resolution in biological fluorescence microscopy,” Proc. Natl. Acad. Sci. U.S.A. 103, 11440-11445 (2006).
[CrossRef] [PubMed]

Eggeling, C.

G. Donnert, J. Keller, R. Medda, M. A. Andrei, S. O. Rizzoli, R. Luhrmann, R. Jahn, C. Eggeling, and S. W. Hell, “Macromolecular-scale resolution in biological fluorescence microscopy,” Proc. Natl. Acad. Sci. U.S.A. 103, 11440-11445 (2006).
[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]

Heckenberg, N.

B. Littleton, K. Lai, D. Longstaff, V. Sarafis, P. Munroe, N. Heckenberg, and H. Rubinsztein-Dunlop, “Coherent super-resolution microscopy via laterally structured illumination,” Micron 38, 150-157 (2007).
[CrossRef]

Heintzmann, R.

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

Hell, S. W.

G. Donnert, J. Keller, R. Medda, M. A. Andrei, S. O. Rizzoli, R. Luhrmann, R. Jahn, C. Eggeling, and S. W. Hell, “Macromolecular-scale resolution in biological fluorescence microscopy,” Proc. Natl. Acad. Sci. U.S.A. 103, 11440-11445 (2006).
[CrossRef] [PubMed]

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]

Hess, H. F.

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, 1642-1645 (2006).
[CrossRef] [PubMed]

Hsu, K.

Jahn, R.

G. Donnert, J. Keller, R. Medda, M. A. Andrei, S. O. Rizzoli, R. Luhrmann, R. Jahn, C. Eggeling, and S. W. Hell, “Macromolecular-scale resolution in biological fluorescence microscopy,” Proc. Natl. Acad. Sci. U.S.A. 103, 11440-11445 (2006).
[CrossRef] [PubMed]

Keller, J.

G. Donnert, J. Keller, R. Medda, M. A. Andrei, S. O. Rizzoli, R. Luhrmann, R. Jahn, C. Eggeling, and S. W. Hell, “Macromolecular-scale resolution in biological fluorescence microscopy,” Proc. Natl. Acad. Sci. U.S.A. 103, 11440-11445 (2006).
[CrossRef] [PubMed]

Lai, K.

B. Littleton, K. Lai, D. Longstaff, V. Sarafis, P. Munroe, N. Heckenberg, and H. Rubinsztein-Dunlop, “Coherent super-resolution microscopy via laterally structured illumination,” Micron 38, 150-157 (2007).
[CrossRef]

Lindwasser, O. W.

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, 1642-1645 (2006).
[CrossRef] [PubMed]

Lippincott-Schwartz, J.

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, 1642-1645 (2006).
[CrossRef] [PubMed]

Lipson, S. G.

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

Littleton, B.

B. Littleton, K. Lai, D. Longstaff, V. Sarafis, P. Munroe, N. Heckenberg, and H. Rubinsztein-Dunlop, “Coherent super-resolution microscopy via laterally structured illumination,” Micron 38, 150-157 (2007).
[CrossRef]

Longstaff, D.

B. Littleton, K. Lai, D. Longstaff, V. Sarafis, P. Munroe, N. Heckenberg, and H. Rubinsztein-Dunlop, “Coherent super-resolution microscopy via laterally structured illumination,” Micron 38, 150-157 (2007).
[CrossRef]

Luhrmann, R.

G. Donnert, J. Keller, R. Medda, M. A. Andrei, S. O. Rizzoli, R. Luhrmann, R. Jahn, C. Eggeling, and S. W. Hell, “Macromolecular-scale resolution in biological fluorescence microscopy,” Proc. Natl. Acad. Sci. U.S.A. 103, 11440-11445 (2006).
[CrossRef] [PubMed]

Medda, R.

G. Donnert, J. Keller, R. Medda, M. A. Andrei, S. O. Rizzoli, R. Luhrmann, R. Jahn, C. Eggeling, and S. W. Hell, “Macromolecular-scale resolution in biological fluorescence microscopy,” Proc. Natl. Acad. Sci. U.S.A. 103, 11440-11445 (2006).
[CrossRef] [PubMed]

Munroe, P.

B. Littleton, K. Lai, D. Longstaff, V. Sarafis, P. Munroe, N. Heckenberg, and H. Rubinsztein-Dunlop, “Coherent super-resolution microscopy via laterally structured illumination,” Micron 38, 150-157 (2007).
[CrossRef]

Olenych, 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, 1642-1645 (2006).
[CrossRef] [PubMed]

Patterson, G. H.

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, 1642-1645 (2006).
[CrossRef] [PubMed]

Pitter, M. C.

Rizzoli, S. O.

G. Donnert, J. Keller, R. Medda, M. A. Andrei, S. O. Rizzoli, R. Luhrmann, R. Jahn, C. Eggeling, and S. W. Hell, “Macromolecular-scale resolution in biological fluorescence microscopy,” Proc. Natl. Acad. Sci. U.S.A. 103, 11440-11445 (2006).
[CrossRef] [PubMed]

Rubinsztein-Dunlop, H.

B. Littleton, K. Lai, D. Longstaff, V. Sarafis, P. Munroe, N. Heckenberg, and H. Rubinsztein-Dunlop, “Coherent super-resolution microscopy via laterally structured illumination,” Micron 38, 150-157 (2007).
[CrossRef]

Sarafis, V.

B. Littleton, K. Lai, D. Longstaff, V. Sarafis, P. Munroe, N. Heckenberg, and H. Rubinsztein-Dunlop, “Coherent super-resolution microscopy via laterally structured illumination,” Micron 38, 150-157 (2007).
[CrossRef]

Somekh, M. G.

Sougrat, R.

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, 1642-1645 (2006).
[CrossRef] [PubMed]

Wichmann, J.

J. Microsc.

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

Micron

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]

B. Littleton, K. Lai, D. Longstaff, V. Sarafis, P. Munroe, N. Heckenberg, and H. Rubinsztein-Dunlop, “Coherent super-resolution microscopy via laterally structured illumination,” Micron 38, 150-157 (2007).
[CrossRef]

Opt. Lett.

Proc. Natl. Acad. Sci. U.S.A.

G. Donnert, J. Keller, R. Medda, M. A. Andrei, S. O. Rizzoli, R. Luhrmann, R. Jahn, C. Eggeling, and S. W. Hell, “Macromolecular-scale resolution in biological fluorescence microscopy,” Proc. Natl. Acad. Sci. U.S.A. 103, 11440-11445 (2006).
[CrossRef] [PubMed]

Science

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, 1642-1645 (2006).
[CrossRef] [PubMed]

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Figures (7)

Fig. 1
Fig. 1

Transfer functions plotted against normalized spatial frequency ( N A λ ) . Only positive spatial frequencies are shown since these are symmetrical for the negative spatial frequencies. (a) Transfer functions corresponding to a cylindrical geometry. Conventional fluorescent microscope (light curve); SIM (heavy curve). (b) Transfer functions corresponding to a spherical geometry. Conventional fluorescent microscope (light curve); SIM with grating projected at large (infinite) number of axial directions (dotted curve). Heavy solid curve and dashed curves correspond to SIM with two orthogonal projections; the heavy solid curve shows a section of the transfer function parallel to one of the directions of projection and the dashed curve at 45° to this direction.

Fig. 2
Fig. 2

Variance of the STF for the structured illumination microscope with cylindrical geometry in units of N, the total number of photons. Variance of the even part (solid curve); variance of the odd part (dashed curve). Note that above m = 2 the two curves overlay.

Fig. 3
Fig. 3

Joint PDFs for SIM in a cylindrical geometry where the expected number N of photons in the image is equal to 15. (a) Corresponds to the case where m = 0.8 . (b) Corresponds to the case where m = 1.6 . (c) Corresponds to the case where m = 2.4 .

Fig. 4
Fig. 4

Evaluation of the STF of the SIM system using Monte Carlo simulations (20 000 repeats) in a spherical geometry with eight grating directions (giving a virtually isotropic response). (a) Mean value of the even part of the STF against normalized spatial frequency m. (b) Mean value of the odd part of the STF against normalized spatial frequency m. (c) Variance of the even part of the STF against normalized spatial frequency m. (d) Variance of the odd part of the STF against normalized spatial frequency m.

Fig. 5
Fig. 5

SNR versus normalized spatial frequency m for WFM (solid curve) and SIM (dashed curve) obtained using analytical Gaussian approximation for a cylindrical geometry. The figure shows that SNR of the SIM system is better for spatial frequencies greater than 1.19.

Fig. 6
Fig. 6

SNR versus normalized spatial frequency plot using Monte Carlo evaluation for WFM (solid curve) and SIM (dashed curve) with eight grating directions in a spherical geometry. The results are plotted in two sections for clarity of the values at the high frequencies.

Fig. 7
Fig. 7

Comparison of WFM and SIM showing applicability of STF for periodic objects in a cylindrical geometry. (a) Diagram showing the basis of the simulation; we consider the probability that the measured signal in region B exceeds that in region A. Contrast of the object is 0.1, and the total number of photons per cycle is 500. (b) Probability that signal in region B exceeds signal in region A for WFM (solid curve) and SIM (dashed curve).

Tables (2)

Tables Icon

Table 1 Real Part of the Complex Argument of the Characteristic Function Expressed by Eqs. (4, 5) a

Tables Icon

Table 2 Imaginary Part of the Complex Argument of the Characteristic Function Expressed by Eqs. (4, 5) a

Equations (17)

Equations on this page are rendered with MathJax. Learn more.

i s ( x ) = { [ 1 + 2 cos ( 2 π m g x ) ] i 1 + [ 1 2 sin ( 2 π m g x ) ] i 2 + [ 1 2 cos ( 2 π m g x ) ] i 3 + [ 1 + 2 sin ( 2 π m g x ) ] i 4 } ,
ill n = N 4 [ 1 + cos ( 2 π m g x + ( n 1 ) π 2 ) ] .
i 1 ( x ) = 1 2 N H ( x ) , i 2 ( x ) = 1 4 N H ( x ) , i 3 ( x ) = 0 ,
i 4 ( x ) = 1 4 N H ( x ) .
A e , n ( ϕ ) = exp ( a n N { exp [ 2 π i ϕ f e , n ( x ) ] 1 } H ( x ) d x ) ,
A o , n ( ϕ ) = exp ( a n N { exp [ 2 π i ϕ f o , n ( x ) ] 1 } H ( x ) d x ) .
a 1 = 1 2 , a 2 = 1 4 , a 3 = 0 , a 4 = 1 4 ,
{ f e 1 ( x ) f o 1 ( x ) } = [ 1 + 2 cos ( 2 π m g x ) ] { cos ( 2 π m x ) sin ( 2 π m x ) } ,
{ f e 2 ( x ) f o 2 ( x ) } = [ 1 2 sin ( 2 π m g x ) ] { cos ( 2 π m x ) sin ( 2 π m x ) } ,
{ f e 4 ( x ) f o 4 ( x ) } = [ 1 + 2 sin ( 2 π m g x ) ] { cos ( 2 π m x ) sin ( 2 π m x ) } .
A e o ( ϕ ) = n A e o , n ( ϕ ) .
cos [ 2 π ( k m + l m g ) x ] H ( x ) d x = c ( k m + l m g ) ,
σ e 2 = N 2 [ 3 + 3 c ( 2 m ) + c ( 2 m m g ) + c ( 2 m + m g ) ] ,
σ o 2 = N 2 [ 3 3 c ( 2 m ) c ( 2 m m g ) c ( 2 m + m g ) ] ,
μ = N 2 [ 2 c ( m ) + c ( m + m g ) + c ( m m g ) ] .
μ 2 σ e 2 + σ o 2 = N [ c ( m 2 ) ] 2 3 .
A n ( ϕ , ψ ) = exp [ R ( exp { j 2 π [ f e , n ( x , y ) ϕ + f o , n ( x , y ) ψ ] } 1 ) i ( x , y ) d x d y ] .

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