Measurement of the modulation transfer function of an X-ray microscope based on multiple Fourier orders analysis of a Siemens star

Joaquín Otón; Carlos Oscar S. Sorzano; Roberto Marabini; Eva Pereiro; Jose M. Carazo

doi:10.1364/OE.23.009567

Measurement of the modulation transfer function of an X-ray microscope based on multiple Fourier orders analysis of a Siemens star

Joaquín Otón, Carlos Oscar S. Sorzano, Roberto Marabini, Eva Pereiro, and Jose M. Carazo

Optics Express
Vol. 23,
Issue 8,
pp. 9567-9572
(2015)
•https://doi.org/10.1364/OE.23.009567

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Joaquín Otón, Carlos Oscar S. Sorzano, Roberto Marabini, Eva Pereiro, and Jose M. Carazo, "Measurement of the modulation transfer function of an X-ray microscope based on multiple Fourier orders analysis of a Siemens star," Opt. Express 23, 9567-9572 (2015)

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Related Topics
Table of Contents Category
- Microscopy
Optics & Photonics Topics
?

The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Modulation transfer function (110.4100)
- Optical transfer functions (110.4850)
- Soft x-rays (260.6048)
- Synchrotron radiation (340.6720)
- X-ray imaging (340.7440)
- X-ray microscopy (340.7460)

About this Article
History
- Original Manuscript: November 25, 2014
- Revised Manuscript: January 23, 2015
- Manuscript Accepted: January 27, 2015
- Published: April 6, 2015
Virtual Issues
Virtual Journal for Biomedical Optics Vol. 10, Iss. 5

Accessible

Open Access

Abstract

Soft X-ray tomography (SXT) is becoming a powerful imaging technique to analyze eukaryotic whole cells close to their native state. Central to the analysis of the quality of SXT 3D reconstruction is the estimation of the spatial resolution and Depth of Field of the X-ray microscope. In turn, the characterization of the Modulation Transfer Function (MTF) of the optical system is key to calculate both parameters. Consequently, in this work we introduce a fully automated technique to accurately estimate the transfer function of such an optical system. Our proposal is based on the preprocessing of the experimental images to obtain an estimate of the input pattern, followed by the analysis in Fourier space of multiple orders of a Siemens Star test sample, extending in this way its measured frequency range.

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References

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Author
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Publication

R. Burge, X.-C. Yuan, G. Morrison, P. Charalambous, M. Browne, and Z. An, “Incoherent imaging with the soft X-ray microscope,” Ultramicroscopy 83, 75–92 (2000).
[Crossref] [PubMed]
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[Crossref]
H. Takano, S. Konishi, T. Koyama, Y. Tsusaka, S. Ichimaru, T. Ohchi, H. Takenaka, and Y. Kagoshima, “Point spread function measurement of an X-ray beam focused by a multilayer zone plate with narrow annular aperture,” J. Synchrotron Radiat. 21, 446–448 (2014).
[Crossref] [PubMed]
S. Reichenbach, S. K. Park, and R. Narayanswamy, “Characterizing digital image acquisition devices,” Opt. Eng. 30, 170–177 (1991).
[Crossref]
C. D. Claxton and R. C. Staunton, “Measurement of the point-spread function of a noisy imaging system,” J. Opt. Soc. Am. A 25, 159–170 (2008).
[Crossref]
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[Crossref] [PubMed]
W. V. D. Broek, S. V. Aert, and D. V. Dyck, “Fully automated measurement of the modulation transfer function of charge-coupled devices above the Nyquist frequency,” Microsc. Microanal. 18, 336–342 (2012).
[Crossref] [PubMed]
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[Crossref] [PubMed]
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[Crossref] [PubMed]
M. C. Bertilson, O. von Hofsten, M. Lindblom, T. Wilhein, H. M. Hertz, and U. Vogt, “Compact high-resolution differential interference contrast soft X-ray microscopy,” Appl. Phys. Lett. 92, 064104 (2008).
[Crossref]
S. Rehbein, P. Guttmann, S. Werner, and G. Schneider, “Characterization of the resolving power and contrast transfer function of a transmission X-ray microscope with partially coherent illumination,” Opt. Express 20, 1–3 (2012).
[Crossref]
D. B. Carlson, J. Gelb, V. Palshin, and J. E. Evans, “Laboratory-based cryogenic soft X-ray tomography with correlative cryo-light and electron microscopy,” Microsc. Microanal. 19, 22–29 (2013).
[Crossref] [PubMed]
R. Reulke, S. Becker, N. Haala, and U. Tempelmann, “Determination and improvement of spatial resolution of the CCD-line-scanner system ADS40,” ISPRS J. Photogramm. Remote Sens. 60, 81–90 (2006).
[Crossref]
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[Crossref]
C. Chang and T. Nakamura, “Partially coherent image formation theory for X-ray microscopy,” in “Microscopy: Science, Technology, Applications and Education,”, A. Mendez-Vilas and J. Diaz, eds. (Formatex Research Center, 2010), 3, pp. 1897–1904, 4.
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[Crossref] [PubMed]

2014 (1)

H. Takano, S. Konishi, T. Koyama, Y. Tsusaka, S. Ichimaru, T. Ohchi, H. Takenaka, and Y. Kagoshima, “Point spread function measurement of an X-ray beam focused by a multilayer zone plate with narrow annular aperture,” J. Synchrotron Radiat. 21, 446–448 (2014).
[Crossref] [PubMed]

2013 (2)

R. S. Ruskin, Z. Yu, and N. Grigorieff, “Quantitative characterization of electron detectors for transmission electron microscopy,” J. Struct. Biol. 184, 385–393 (2013).
[Crossref] [PubMed]

D. B. Carlson, J. Gelb, V. Palshin, and J. E. Evans, “Laboratory-based cryogenic soft X-ray tomography with correlative cryo-light and electron microscopy,” Microsc. Microanal. 19, 22–29 (2013).
[Crossref] [PubMed]

2012 (3)

S. Rehbein, P. Guttmann, S. Werner, and G. Schneider, “Characterization of the resolving power and contrast transfer function of a transmission X-ray microscope with partially coherent illumination,” Opt. Express 20, 1–3 (2012).
[Crossref]

L. Chen, J. McGinty, H. B. Taylor, L. Bugeon, J. R. Lamb, M. J. Dallman, and P. M. W. French, “Incorporation of an experimentally determined MTF for spatial frequency filtering and deconvolution during optical projection tomography reconstruction,” Opt. Express 20, 7323–7337 (2012).
[Crossref] [PubMed]

W. V. D. Broek, S. V. Aert, and D. V. Dyck, “Fully automated measurement of the modulation transfer function of charge-coupled devices above the Nyquist frequency,” Microsc. Microanal. 18, 336–342 (2012).
[Crossref] [PubMed]

2009 (1)

E. Pereiro, J. Nicolás, S. Ferrer, and M. R. Howells, “A soft X-ray beamline for transmission X-ray microscopy at ALBA,” J. Synchrotron Radiat. 16, 505–512 (2009).
[Crossref] [PubMed]

2008 (2)

C. D. Claxton and R. C. Staunton, “Measurement of the point-spread function of a noisy imaging system,” J. Opt. Soc. Am. A 25, 159–170 (2008).
[Crossref]

M. C. Bertilson, O. von Hofsten, M. Lindblom, T. Wilhein, H. M. Hertz, and U. Vogt, “Compact high-resolution differential interference contrast soft X-ray microscopy,” Appl. Phys. Lett. 92, 064104 (2008).
[Crossref]

2007 (1)

P. a. C. Takman, H. Stollberg, G. A. Johansson, A. Holmberg, M. Lindblom, and H. M. Hertz, “High-resolution compact X-ray microscopy,” J. Microsc. 226, 175–181 (2007).
[Crossref] [PubMed]

2006 (1)

R. Reulke, S. Becker, N. Haala, and U. Tempelmann, “Determination and improvement of spatial resolution of the CCD-line-scanner system ADS40,” ISPRS J. Photogramm. Remote Sens. 60, 81–90 (2006).
[Crossref]

2000 (2)

R. Burge, X.-C. Yuan, G. Morrison, P. Charalambous, M. Browne, and Z. An, “Incoherent imaging with the soft X-ray microscope,” Ultramicroscopy 83, 75–92 (2000).
[Crossref] [PubMed]

R. R. Meyer and A. I. Kirkland, “Characterisation of the signal and noise transfer of CCD cameras for electron detection,” Microsc. Res. Techniq. 49, 269–280 (2000).
[Crossref]

1991 (1)

S. Reichenbach, S. K. Park, and R. Narayanswamy, “Characterizing digital image acquisition devices,” Opt. Eng. 30, 170–177 (1991).
[Crossref]

Aert, S. V.

An, Z.

R. Burge, X.-C. Yuan, G. Morrison, P. Charalambous, M. Browne, and Z. An, “Incoherent imaging with the soft X-ray microscope,” Ultramicroscopy 83, 75–92 (2000).
[Crossref] [PubMed]

Becker, S.

Bertilson, M. C.

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999), 7.
[Crossref]

Broek, W. V. D.

Browne, M.

R. Burge, X.-C. Yuan, G. Morrison, P. Charalambous, M. Browne, and Z. An, “Incoherent imaging with the soft X-ray microscope,” Ultramicroscopy 83, 75–92 (2000).
[Crossref] [PubMed]

Bugeon, L.

Burge, R.

R. Burge, X.-C. Yuan, G. Morrison, P. Charalambous, M. Browne, and Z. An, “Incoherent imaging with the soft X-ray microscope,” Ultramicroscopy 83, 75–92 (2000).
[Crossref] [PubMed]

Carlson, D. B.

Chang, C.

C. Chang and T. Nakamura, “Partially coherent image formation theory for X-ray microscopy,” in “Microscopy: Science, Technology, Applications and Education,”, A. Mendez-Vilas and J. Diaz, eds. (Formatex Research Center, 2010), 3, pp. 1897–1904, 4.

Charalambous, P.

R. Burge, X.-C. Yuan, G. Morrison, P. Charalambous, M. Browne, and Z. An, “Incoherent imaging with the soft X-ray microscope,” Ultramicroscopy 83, 75–92 (2000).
[Crossref] [PubMed]

Chen, L.

Claxton, C. D.

C. D. Claxton and R. C. Staunton, “Measurement of the point-spread function of a noisy imaging system,” J. Opt. Soc. Am. A 25, 159–170 (2008).
[Crossref]

Dallman, M. J.

Dyck, D. V.

Evans, J. E.

Ferrer, S.

E. Pereiro, J. Nicolás, S. Ferrer, and M. R. Howells, “A soft X-ray beamline for transmission X-ray microscopy at ALBA,” J. Synchrotron Radiat. 16, 505–512 (2009).
[Crossref] [PubMed]

French, P. M. W.

Gelb, J.

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, 2000).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

Grigorieff, N.

R. S. Ruskin, Z. Yu, and N. Grigorieff, “Quantitative characterization of electron detectors for transmission electron microscopy,” J. Struct. Biol. 184, 385–393 (2013).
[Crossref] [PubMed]

Guttmann, P.

Haala, N.

Hertz, H. M.

P. a. C. Takman, H. Stollberg, G. A. Johansson, A. Holmberg, M. Lindblom, and H. M. Hertz, “High-resolution compact X-ray microscopy,” J. Microsc. 226, 175–181 (2007).
[Crossref] [PubMed]

Holmberg, A.

P. a. C. Takman, H. Stollberg, G. A. Johansson, A. Holmberg, M. Lindblom, and H. M. Hertz, “High-resolution compact X-ray microscopy,” J. Microsc. 226, 175–181 (2007).
[Crossref] [PubMed]

Howells, M. R.

E. Pereiro, J. Nicolás, S. Ferrer, and M. R. Howells, “A soft X-ray beamline for transmission X-ray microscopy at ALBA,” J. Synchrotron Radiat. 16, 505–512 (2009).
[Crossref] [PubMed]

Ichimaru, S.

Johansson, G. A.

P. a. C. Takman, H. Stollberg, G. A. Johansson, A. Holmberg, M. Lindblom, and H. M. Hertz, “High-resolution compact X-ray microscopy,” J. Microsc. 226, 175–181 (2007).
[Crossref] [PubMed]

Kagoshima, Y.

Kirkland, A. I.

R. R. Meyer and A. I. Kirkland, “Characterisation of the signal and noise transfer of CCD cameras for electron detection,” Microsc. Res. Techniq. 49, 269–280 (2000).
[Crossref]

Konishi, S.

Koyama, T.

Lamb, J. R.

Lindblom, M.

P. a. C. Takman, H. Stollberg, G. A. Johansson, A. Holmberg, M. Lindblom, and H. M. Hertz, “High-resolution compact X-ray microscopy,” J. Microsc. 226, 175–181 (2007).
[Crossref] [PubMed]

McGinty, J.

Meyer, R. R.

R. R. Meyer and A. I. Kirkland, “Characterisation of the signal and noise transfer of CCD cameras for electron detection,” Microsc. Res. Techniq. 49, 269–280 (2000).
[Crossref]

Morrison, G.

R. Burge, X.-C. Yuan, G. Morrison, P. Charalambous, M. Browne, and Z. An, “Incoherent imaging with the soft X-ray microscope,” Ultramicroscopy 83, 75–92 (2000).
[Crossref] [PubMed]

Nakamura, T.

Narayanswamy, R.

S. Reichenbach, S. K. Park, and R. Narayanswamy, “Characterizing digital image acquisition devices,” Opt. Eng. 30, 170–177 (1991).
[Crossref]

Nicolás, J.

E. Pereiro, J. Nicolás, S. Ferrer, and M. R. Howells, “A soft X-ray beamline for transmission X-ray microscopy at ALBA,” J. Synchrotron Radiat. 16, 505–512 (2009).
[Crossref] [PubMed]

Ohchi, T.

Palshin, V.

Park, S. K.

S. Reichenbach, S. K. Park, and R. Narayanswamy, “Characterizing digital image acquisition devices,” Opt. Eng. 30, 170–177 (1991).
[Crossref]

Pereiro, E.

E. Pereiro, J. Nicolás, S. Ferrer, and M. R. Howells, “A soft X-ray beamline for transmission X-ray microscopy at ALBA,” J. Synchrotron Radiat. 16, 505–512 (2009).
[Crossref] [PubMed]

Rehbein, S.

Reichenbach, S.

S. Reichenbach, S. K. Park, and R. Narayanswamy, “Characterizing digital image acquisition devices,” Opt. Eng. 30, 170–177 (1991).
[Crossref]

Reulke, R.

Ruskin, R. S.

R. S. Ruskin, Z. Yu, and N. Grigorieff, “Quantitative characterization of electron detectors for transmission electron microscopy,” J. Struct. Biol. 184, 385–393 (2013).
[Crossref] [PubMed]

Schneider, G.

Staunton, R. C.

C. D. Claxton and R. C. Staunton, “Measurement of the point-spread function of a noisy imaging system,” J. Opt. Soc. Am. A 25, 159–170 (2008).
[Crossref]

Stollberg, H.

P. a. C. Takman, H. Stollberg, G. A. Johansson, A. Holmberg, M. Lindblom, and H. M. Hertz, “High-resolution compact X-ray microscopy,” J. Microsc. 226, 175–181 (2007).
[Crossref] [PubMed]

Takano, H.

Takenaka, H.

Takman, P. a. C.

P. a. C. Takman, H. Stollberg, G. A. Johansson, A. Holmberg, M. Lindblom, and H. M. Hertz, “High-resolution compact X-ray microscopy,” J. Microsc. 226, 175–181 (2007).
[Crossref] [PubMed]

Taylor, H. B.

Tempelmann, U.

Tsusaka, Y.

Vogt, U.

von Hofsten, O.

Werner, S.

Wilhein, T.

Wolf, E.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999), 7.
[Crossref]

Yu, Z.

R. S. Ruskin, Z. Yu, and N. Grigorieff, “Quantitative characterization of electron detectors for transmission electron microscopy,” J. Struct. Biol. 184, 385–393 (2013).
[Crossref] [PubMed]

Yuan, X.-C.

R. Burge, X.-C. Yuan, G. Morrison, P. Charalambous, M. Browne, and Z. An, “Incoherent imaging with the soft X-ray microscope,” Ultramicroscopy 83, 75–92 (2000).
[Crossref] [PubMed]

Appl. Phys. Lett. (1)

ISPRS J. Photogramm. Remote Sens. (1)

J. Microsc. (1)

P. a. C. Takman, H. Stollberg, G. A. Johansson, A. Holmberg, M. Lindblom, and H. M. Hertz, “High-resolution compact X-ray microscopy,” J. Microsc. 226, 175–181 (2007).
[Crossref] [PubMed]

J. Opt. Soc. Am. A (1)

C. D. Claxton and R. C. Staunton, “Measurement of the point-spread function of a noisy imaging system,” J. Opt. Soc. Am. A 25, 159–170 (2008).
[Crossref]

J. Struct. Biol. (1)

R. S. Ruskin, Z. Yu, and N. Grigorieff, “Quantitative characterization of electron detectors for transmission electron microscopy,” J. Struct. Biol. 184, 385–393 (2013).
[Crossref] [PubMed]

J. Synchrotron Radiat. (2)

E. Pereiro, J. Nicolás, S. Ferrer, and M. R. Howells, “A soft X-ray beamline for transmission X-ray microscopy at ALBA,” J. Synchrotron Radiat. 16, 505–512 (2009).
[Crossref] [PubMed]

Microsc. Microanal. (2)

Microsc. Res. Techniq. (1)

R. R. Meyer and A. I. Kirkland, “Characterisation of the signal and noise transfer of CCD cameras for electron detection,” Microsc. Res. Techniq. 49, 269–280 (2000).
[Crossref]

Opt. Eng. (1)

S. Reichenbach, S. K. Park, and R. Narayanswamy, “Characterizing digital image acquisition devices,” Opt. Eng. 30, 170–177 (1991).
[Crossref]

Opt. Express (2)

Ultramicroscopy (1)

R. Burge, X.-C. Yuan, G. Morrison, P. Charalambous, M. Browne, and Z. An, “Incoherent imaging with the soft X-ray microscope,” Ultramicroscopy 83, 75–92 (2000).
[Crossref] [PubMed]

Other (4)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999), 7.
[Crossref]

J. W. Goodman, Statistical Optics (Wiley, 2000).

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Fig. 1

(a) Siemens star test pattern; (b) Polar decomposition of (a); (c) $I_{S S}^{r}_{, r e f}$ profile calculated from the binarization of $I_{S S}^{r}$ by thresholding, remarked by red lines in (d) and (b), respectively; (d) $I_{S S}^{r}_{, r e f}$ image in polar coordinates; (e) 1D Fourier transform of profile at largest radius from (d).

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Fig. 2

(a) Plot of the Fourier coefficient values for 1^st and 3^rd diffraction orders for $I_{S S}^{r}$ and $I_{S S}^{r}_{, r e f}$ against the corresponding spatial frequency; (b) $| ℋ_{A} |$ curves obtained from plots in (a); (c) Final experimental $| ℋ_{A} |$ curve obtained by combining curves in (b); (d) Comparison of simulated and measured MTF profiles for ZP with dr_N = 25 nm at different defocus distances.

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Equations (6)

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

ℋ_{A} (f_{x}, f_{y}) = \frac{{\tilde{I}}_{o u t} (f_{x}, f_{y})}{{\tilde{I}}_{i n} (f_{x}, f_{y})},

f (θ) = \sum_{m = - \infty}^{\infty} C_{m} \cos (\frac{2 π m θ}{Λ}),

I_{S S}^{r} (θ) = \sum_{m = - \infty}^{\infty} C_{m} \cos (\frac{2 π m θ}{Λ}) = \sum_{m = - \infty}^{\infty} C_{m} \cos (\frac{2 π m l}{P (r)}) = I_{S S} (l, r),

{\tilde{I}}_{S S}^{r} (ω) = \sum_{m = - \infty}^{\infty} \frac{C_{m}}{2} [δ (ω - \frac{m}{Λ}) + δ (ω + \frac{m}{Λ})],

{\tilde{I}}_{S S} (u, r) = \frac{1}{| r |} \sum_{m = - \infty}^{\infty} \frac{C_{m}}{2} [δ (u - \frac{m}{Λ r}) + δ (u + \frac{m}{Λ r})],

ℋ_{A} (u = \frac{m}{Λ r}) = \frac{{\tilde{I}}_{S S}^{r} (ω = \frac{m}{Λ})}{{\tilde{I}}_{S S, r e f}^{r} (ω = \frac{m}{Λ})} .