Abstract

We comprehensively study the least-squares Gaussian approximations of the diffraction-limited 2D–3D paraxial–nonparaxial point-spread functions (PSFs) of the wide field fluorescence microscope (WFFM), the laser scanning confocal microscope (LSCM), and the disk scanning confocal microscope (DSCM). The PSFs are expressed using the Debye integral. Under an L constraint imposing peak matching, optimal and near-optimal Gaussian parameters are derived for the PSFs. With an L1 constraint imposing energy conservation, an optimal Gaussian parameter is derived for the 2D paraxial WFFM PSF. We found that (1) the 2D approximations are all very accurate; (2) no accurate Gaussian approximation exists for 3D WFFM PSFs; and (3) with typical pinhole sizes, the 3D approximations are accurate for the DSCM and nearly perfect for the LSCM. All the Gaussian parameters derived in this study are in explicit analytical form, allowing their direct use in practical applications.

© 2007 Optical Society of America

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References

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  1. M. Gu, Advanced Optical Imaging Theory (Springer-Verlag, 2000).
  2. C. M. Anderson, G. N. Georgiou, I. E. G. Morrison, G. V. W. Stevenson, and R. J. Cherry, "Tracking of cell surface receptors by fluorescence digital imaging microscopy using a charge-coupled device camera," J. Cell Sci. 101, 415-425 (1992).
    [PubMed]
  3. G. J. Schütz, H. Schindler, and T. Schmidt, "Single-molecule microscopy on model membranes reveals anomalous diffusion," Biophys. J. 73, 1073-1080 (1997).
    [CrossRef] [PubMed]
  4. M. K. Cheezum, W. F. Walker, and W. H. Guilford, "Quantitative comparison of algorithms for tracking single fluorescent particles," Biophys. J. 81, 2378-2388 (2001).
    [CrossRef] [PubMed]
  5. A. Santos and I. T. Young, "Model-based resolution: applying the theory in quantitative microscopy," Appl. Opt. 39, 2948-2958 (2000).
    [CrossRef]
  6. D. Thomann, D. R. Rines, P. K. Sorger, and G. Danuser, "Automatic fluorescent tag detection in 3D with super-resolution: application to the analysis of chromosome movement," J. Microsc. 208, 49-64 (2002).
    [CrossRef] [PubMed]
  7. F. Rooms, W. Philips, and D. S. Lidke, "Simultaneous degradation estimation and restoration of confocal images and performance evaluation by colocalization analysis," J. Microsc. 218, 22-36 (2005).
    [CrossRef] [PubMed]
  8. J. C. G. Blonk, A. Don, H. van Aalst, and J. J. Birmingham, "Fluorescence photobleaching recovery in the confocal scanning light microscope," J. Microsc. 169, 363-374 (1993).
    [CrossRef]
  9. K. Braeckmans, L. Peeters, N. N. Sanders, S. C. D. Smedt, and J. Demeester, "Three-dimensional fluorescence recovery after photobleaching with the confocal scanning laser microscope," Biophys. J. 85, 2240-2252 (2003).
    [CrossRef] [PubMed]
  10. G. J. Streekstra and J. van Pelt, "Analysis of tubular structures in three-dimensional confocal images," Network Comput. Neural Syst. 13, 381-395 (2002).
    [CrossRef]
  11. J.-A. Conchello, "Superresolution and convergence properties of the expectation-maximization algorithm for maximum-likelihood deconvolution of incoherent images," J. Opt. Soc. Am. A 15, 2609-2619 (1998).
    [CrossRef]
  12. L. J. van Vliet, "Grey-scale measurements in multi-dimensional digitized images," Ph.D. dissertation (Delft University, The Netherlands, 1993).
  13. B. Zhang, J. Zerubia, and J.-C. Olivo-Marin, "A study of Gaussian approximations of fluorescence microscopy PSF models," in Three-Dimensional and Multidimensional Microscopy: Image Acquisition and Processing XIII, J.-A. Conchello, C. J. Cogswell, and T. Wilson, eds., Proc. SPIE 6090, 60900K (2006).
  14. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 2002).
  15. L. Tao and C. Nicholson, "The three-dimensional point spread functions of a microscope objective in image and object space," J. Microsc. 178, 267-271 (1995).
    [CrossRef] [PubMed]
  16. L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media (Pergamon, 1984).
  17. D. R. Sandison, R. M. Williams, K. S. Wells, J. Strickler, and W. W. Webb, "Quantitative fluorescence confocal laser scanning microscopy (CLSM)," in Handbook of Biological Confocal Microscopy, J.B.Pawley, ed., 2nd ed. (Plenum, 1995), pp. 39-53.
  18. M. Petrán, M. Hadravský, J. Benes, R. Kucera, and A. Boyde, "The tandem scanning reflected light microscope. Part 1-The principle and its design," in Proceedings of the Royal Microscopical Society (Blackwell, 1985), Vol. 20, pp. 125-129.
  19. J.-A. Conchello and J. W. Lichtman, "Theoretical analysis of a rotating-disk partially confocal scanning microscope," Appl. Opt. 33, 585-596 (1994).
    [CrossRef] [PubMed]
  20. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 1967).

2006 (1)

B. Zhang, J. Zerubia, and J.-C. Olivo-Marin, "A study of Gaussian approximations of fluorescence microscopy PSF models," in Three-Dimensional and Multidimensional Microscopy: Image Acquisition and Processing XIII, J.-A. Conchello, C. J. Cogswell, and T. Wilson, eds., Proc. SPIE 6090, 60900K (2006).

2005 (1)

F. Rooms, W. Philips, and D. S. Lidke, "Simultaneous degradation estimation and restoration of confocal images and performance evaluation by colocalization analysis," J. Microsc. 218, 22-36 (2005).
[CrossRef] [PubMed]

2003 (1)

K. Braeckmans, L. Peeters, N. N. Sanders, S. C. D. Smedt, and J. Demeester, "Three-dimensional fluorescence recovery after photobleaching with the confocal scanning laser microscope," Biophys. J. 85, 2240-2252 (2003).
[CrossRef] [PubMed]

2002 (2)

G. J. Streekstra and J. van Pelt, "Analysis of tubular structures in three-dimensional confocal images," Network Comput. Neural Syst. 13, 381-395 (2002).
[CrossRef]

D. Thomann, D. R. Rines, P. K. Sorger, and G. Danuser, "Automatic fluorescent tag detection in 3D with super-resolution: application to the analysis of chromosome movement," J. Microsc. 208, 49-64 (2002).
[CrossRef] [PubMed]

2001 (1)

M. K. Cheezum, W. F. Walker, and W. H. Guilford, "Quantitative comparison of algorithms for tracking single fluorescent particles," Biophys. J. 81, 2378-2388 (2001).
[CrossRef] [PubMed]

2000 (1)

1998 (1)

1997 (1)

G. J. Schütz, H. Schindler, and T. Schmidt, "Single-molecule microscopy on model membranes reveals anomalous diffusion," Biophys. J. 73, 1073-1080 (1997).
[CrossRef] [PubMed]

1995 (1)

L. Tao and C. Nicholson, "The three-dimensional point spread functions of a microscope objective in image and object space," J. Microsc. 178, 267-271 (1995).
[CrossRef] [PubMed]

1994 (1)

1993 (1)

J. C. G. Blonk, A. Don, H. van Aalst, and J. J. Birmingham, "Fluorescence photobleaching recovery in the confocal scanning light microscope," J. Microsc. 169, 363-374 (1993).
[CrossRef]

1992 (1)

C. M. Anderson, G. N. Georgiou, I. E. G. Morrison, G. V. W. Stevenson, and R. J. Cherry, "Tracking of cell surface receptors by fluorescence digital imaging microscopy using a charge-coupled device camera," J. Cell Sci. 101, 415-425 (1992).
[PubMed]

Anderson, C. M.

C. M. Anderson, G. N. Georgiou, I. E. G. Morrison, G. V. W. Stevenson, and R. J. Cherry, "Tracking of cell surface receptors by fluorescence digital imaging microscopy using a charge-coupled device camera," J. Cell Sci. 101, 415-425 (1992).
[PubMed]

Benes, J.

M. Petrán, M. Hadravský, J. Benes, R. Kucera, and A. Boyde, "The tandem scanning reflected light microscope. Part 1-The principle and its design," in Proceedings of the Royal Microscopical Society (Blackwell, 1985), Vol. 20, pp. 125-129.

Birmingham, J. J.

J. C. G. Blonk, A. Don, H. van Aalst, and J. J. Birmingham, "Fluorescence photobleaching recovery in the confocal scanning light microscope," J. Microsc. 169, 363-374 (1993).
[CrossRef]

Blonk, J. C. G.

J. C. G. Blonk, A. Don, H. van Aalst, and J. J. Birmingham, "Fluorescence photobleaching recovery in the confocal scanning light microscope," J. Microsc. 169, 363-374 (1993).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 2002).

Boyde, A.

M. Petrán, M. Hadravský, J. Benes, R. Kucera, and A. Boyde, "The tandem scanning reflected light microscope. Part 1-The principle and its design," in Proceedings of the Royal Microscopical Society (Blackwell, 1985), Vol. 20, pp. 125-129.

Braeckmans, K.

K. Braeckmans, L. Peeters, N. N. Sanders, S. C. D. Smedt, and J. Demeester, "Three-dimensional fluorescence recovery after photobleaching with the confocal scanning laser microscope," Biophys. J. 85, 2240-2252 (2003).
[CrossRef] [PubMed]

Cheezum, M. K.

M. K. Cheezum, W. F. Walker, and W. H. Guilford, "Quantitative comparison of algorithms for tracking single fluorescent particles," Biophys. J. 81, 2378-2388 (2001).
[CrossRef] [PubMed]

Cherry, R. J.

C. M. Anderson, G. N. Georgiou, I. E. G. Morrison, G. V. W. Stevenson, and R. J. Cherry, "Tracking of cell surface receptors by fluorescence digital imaging microscopy using a charge-coupled device camera," J. Cell Sci. 101, 415-425 (1992).
[PubMed]

Conchello, J.-A.

Danuser, G.

D. Thomann, D. R. Rines, P. K. Sorger, and G. Danuser, "Automatic fluorescent tag detection in 3D with super-resolution: application to the analysis of chromosome movement," J. Microsc. 208, 49-64 (2002).
[CrossRef] [PubMed]

Demeester, J.

K. Braeckmans, L. Peeters, N. N. Sanders, S. C. D. Smedt, and J. Demeester, "Three-dimensional fluorescence recovery after photobleaching with the confocal scanning laser microscope," Biophys. J. 85, 2240-2252 (2003).
[CrossRef] [PubMed]

Don, A.

J. C. G. Blonk, A. Don, H. van Aalst, and J. J. Birmingham, "Fluorescence photobleaching recovery in the confocal scanning light microscope," J. Microsc. 169, 363-374 (1993).
[CrossRef]

Georgiou, G. N.

C. M. Anderson, G. N. Georgiou, I. E. G. Morrison, G. V. W. Stevenson, and R. J. Cherry, "Tracking of cell surface receptors by fluorescence digital imaging microscopy using a charge-coupled device camera," J. Cell Sci. 101, 415-425 (1992).
[PubMed]

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 1967).

Gu, M.

M. Gu, Advanced Optical Imaging Theory (Springer-Verlag, 2000).

Guilford, W. H.

M. K. Cheezum, W. F. Walker, and W. H. Guilford, "Quantitative comparison of algorithms for tracking single fluorescent particles," Biophys. J. 81, 2378-2388 (2001).
[CrossRef] [PubMed]

Hadravský, M.

M. Petrán, M. Hadravský, J. Benes, R. Kucera, and A. Boyde, "The tandem scanning reflected light microscope. Part 1-The principle and its design," in Proceedings of the Royal Microscopical Society (Blackwell, 1985), Vol. 20, pp. 125-129.

Kucera, R.

M. Petrán, M. Hadravský, J. Benes, R. Kucera, and A. Boyde, "The tandem scanning reflected light microscope. Part 1-The principle and its design," in Proceedings of the Royal Microscopical Society (Blackwell, 1985), Vol. 20, pp. 125-129.

Landau, L. D.

L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media (Pergamon, 1984).

Lichtman, J. W.

Lidke, D. S.

F. Rooms, W. Philips, and D. S. Lidke, "Simultaneous degradation estimation and restoration of confocal images and performance evaluation by colocalization analysis," J. Microsc. 218, 22-36 (2005).
[CrossRef] [PubMed]

Lifshitz, E. M.

L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media (Pergamon, 1984).

Morrison, I. E. G.

C. M. Anderson, G. N. Georgiou, I. E. G. Morrison, G. V. W. Stevenson, and R. J. Cherry, "Tracking of cell surface receptors by fluorescence digital imaging microscopy using a charge-coupled device camera," J. Cell Sci. 101, 415-425 (1992).
[PubMed]

Nicholson, C.

L. Tao and C. Nicholson, "The three-dimensional point spread functions of a microscope objective in image and object space," J. Microsc. 178, 267-271 (1995).
[CrossRef] [PubMed]

Olivo-Marin, J.-C.

B. Zhang, J. Zerubia, and J.-C. Olivo-Marin, "A study of Gaussian approximations of fluorescence microscopy PSF models," in Three-Dimensional and Multidimensional Microscopy: Image Acquisition and Processing XIII, J.-A. Conchello, C. J. Cogswell, and T. Wilson, eds., Proc. SPIE 6090, 60900K (2006).

Peeters, L.

K. Braeckmans, L. Peeters, N. N. Sanders, S. C. D. Smedt, and J. Demeester, "Three-dimensional fluorescence recovery after photobleaching with the confocal scanning laser microscope," Biophys. J. 85, 2240-2252 (2003).
[CrossRef] [PubMed]

Petrán, M.

M. Petrán, M. Hadravský, J. Benes, R. Kucera, and A. Boyde, "The tandem scanning reflected light microscope. Part 1-The principle and its design," in Proceedings of the Royal Microscopical Society (Blackwell, 1985), Vol. 20, pp. 125-129.

Philips, W.

F. Rooms, W. Philips, and D. S. Lidke, "Simultaneous degradation estimation and restoration of confocal images and performance evaluation by colocalization analysis," J. Microsc. 218, 22-36 (2005).
[CrossRef] [PubMed]

Pitaevskii, L. P.

L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media (Pergamon, 1984).

Rines, D. R.

D. Thomann, D. R. Rines, P. K. Sorger, and G. Danuser, "Automatic fluorescent tag detection in 3D with super-resolution: application to the analysis of chromosome movement," J. Microsc. 208, 49-64 (2002).
[CrossRef] [PubMed]

Rooms, F.

F. Rooms, W. Philips, and D. S. Lidke, "Simultaneous degradation estimation and restoration of confocal images and performance evaluation by colocalization analysis," J. Microsc. 218, 22-36 (2005).
[CrossRef] [PubMed]

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 1967).

Sanders, N. N.

K. Braeckmans, L. Peeters, N. N. Sanders, S. C. D. Smedt, and J. Demeester, "Three-dimensional fluorescence recovery after photobleaching with the confocal scanning laser microscope," Biophys. J. 85, 2240-2252 (2003).
[CrossRef] [PubMed]

Sandison, D. R.

D. R. Sandison, R. M. Williams, K. S. Wells, J. Strickler, and W. W. Webb, "Quantitative fluorescence confocal laser scanning microscopy (CLSM)," in Handbook of Biological Confocal Microscopy, J.B.Pawley, ed., 2nd ed. (Plenum, 1995), pp. 39-53.

Santos, A.

Schindler, H.

G. J. Schütz, H. Schindler, and T. Schmidt, "Single-molecule microscopy on model membranes reveals anomalous diffusion," Biophys. J. 73, 1073-1080 (1997).
[CrossRef] [PubMed]

Schmidt, T.

G. J. Schütz, H. Schindler, and T. Schmidt, "Single-molecule microscopy on model membranes reveals anomalous diffusion," Biophys. J. 73, 1073-1080 (1997).
[CrossRef] [PubMed]

Schütz, G. J.

G. J. Schütz, H. Schindler, and T. Schmidt, "Single-molecule microscopy on model membranes reveals anomalous diffusion," Biophys. J. 73, 1073-1080 (1997).
[CrossRef] [PubMed]

Smedt, S. C. D.

K. Braeckmans, L. Peeters, N. N. Sanders, S. C. D. Smedt, and J. Demeester, "Three-dimensional fluorescence recovery after photobleaching with the confocal scanning laser microscope," Biophys. J. 85, 2240-2252 (2003).
[CrossRef] [PubMed]

Sorger, P. K.

D. Thomann, D. R. Rines, P. K. Sorger, and G. Danuser, "Automatic fluorescent tag detection in 3D with super-resolution: application to the analysis of chromosome movement," J. Microsc. 208, 49-64 (2002).
[CrossRef] [PubMed]

Stevenson, G. V. W.

C. M. Anderson, G. N. Georgiou, I. E. G. Morrison, G. V. W. Stevenson, and R. J. Cherry, "Tracking of cell surface receptors by fluorescence digital imaging microscopy using a charge-coupled device camera," J. Cell Sci. 101, 415-425 (1992).
[PubMed]

Streekstra, G. J.

G. J. Streekstra and J. van Pelt, "Analysis of tubular structures in three-dimensional confocal images," Network Comput. Neural Syst. 13, 381-395 (2002).
[CrossRef]

Strickler, J.

D. R. Sandison, R. M. Williams, K. S. Wells, J. Strickler, and W. W. Webb, "Quantitative fluorescence confocal laser scanning microscopy (CLSM)," in Handbook of Biological Confocal Microscopy, J.B.Pawley, ed., 2nd ed. (Plenum, 1995), pp. 39-53.

Tao, L.

L. Tao and C. Nicholson, "The three-dimensional point spread functions of a microscope objective in image and object space," J. Microsc. 178, 267-271 (1995).
[CrossRef] [PubMed]

Thomann, D.

D. Thomann, D. R. Rines, P. K. Sorger, and G. Danuser, "Automatic fluorescent tag detection in 3D with super-resolution: application to the analysis of chromosome movement," J. Microsc. 208, 49-64 (2002).
[CrossRef] [PubMed]

van Aalst, H.

J. C. G. Blonk, A. Don, H. van Aalst, and J. J. Birmingham, "Fluorescence photobleaching recovery in the confocal scanning light microscope," J. Microsc. 169, 363-374 (1993).
[CrossRef]

van Pelt, J.

G. J. Streekstra and J. van Pelt, "Analysis of tubular structures in three-dimensional confocal images," Network Comput. Neural Syst. 13, 381-395 (2002).
[CrossRef]

van Vliet, L. J.

L. J. van Vliet, "Grey-scale measurements in multi-dimensional digitized images," Ph.D. dissertation (Delft University, The Netherlands, 1993).

Walker, W. F.

M. K. Cheezum, W. F. Walker, and W. H. Guilford, "Quantitative comparison of algorithms for tracking single fluorescent particles," Biophys. J. 81, 2378-2388 (2001).
[CrossRef] [PubMed]

Webb, W. W.

D. R. Sandison, R. M. Williams, K. S. Wells, J. Strickler, and W. W. Webb, "Quantitative fluorescence confocal laser scanning microscopy (CLSM)," in Handbook of Biological Confocal Microscopy, J.B.Pawley, ed., 2nd ed. (Plenum, 1995), pp. 39-53.

Wells, K. S.

D. R. Sandison, R. M. Williams, K. S. Wells, J. Strickler, and W. W. Webb, "Quantitative fluorescence confocal laser scanning microscopy (CLSM)," in Handbook of Biological Confocal Microscopy, J.B.Pawley, ed., 2nd ed. (Plenum, 1995), pp. 39-53.

Williams, R. M.

D. R. Sandison, R. M. Williams, K. S. Wells, J. Strickler, and W. W. Webb, "Quantitative fluorescence confocal laser scanning microscopy (CLSM)," in Handbook of Biological Confocal Microscopy, J.B.Pawley, ed., 2nd ed. (Plenum, 1995), pp. 39-53.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 2002).

Young, I. T.

Zerubia, J.

B. Zhang, J. Zerubia, and J.-C. Olivo-Marin, "A study of Gaussian approximations of fluorescence microscopy PSF models," in Three-Dimensional and Multidimensional Microscopy: Image Acquisition and Processing XIII, J.-A. Conchello, C. J. Cogswell, and T. Wilson, eds., Proc. SPIE 6090, 60900K (2006).

Zhang, B.

B. Zhang, J. Zerubia, and J.-C. Olivo-Marin, "A study of Gaussian approximations of fluorescence microscopy PSF models," in Three-Dimensional and Multidimensional Microscopy: Image Acquisition and Processing XIII, J.-A. Conchello, C. J. Cogswell, and T. Wilson, eds., Proc. SPIE 6090, 60900K (2006).

Appl. Opt. (2)

Biophys. J. (3)

G. J. Schütz, H. Schindler, and T. Schmidt, "Single-molecule microscopy on model membranes reveals anomalous diffusion," Biophys. J. 73, 1073-1080 (1997).
[CrossRef] [PubMed]

M. K. Cheezum, W. F. Walker, and W. H. Guilford, "Quantitative comparison of algorithms for tracking single fluorescent particles," Biophys. J. 81, 2378-2388 (2001).
[CrossRef] [PubMed]

K. Braeckmans, L. Peeters, N. N. Sanders, S. C. D. Smedt, and J. Demeester, "Three-dimensional fluorescence recovery after photobleaching with the confocal scanning laser microscope," Biophys. J. 85, 2240-2252 (2003).
[CrossRef] [PubMed]

J. Cell Sci. (1)

C. M. Anderson, G. N. Georgiou, I. E. G. Morrison, G. V. W. Stevenson, and R. J. Cherry, "Tracking of cell surface receptors by fluorescence digital imaging microscopy using a charge-coupled device camera," J. Cell Sci. 101, 415-425 (1992).
[PubMed]

J. Microsc. (4)

D. Thomann, D. R. Rines, P. K. Sorger, and G. Danuser, "Automatic fluorescent tag detection in 3D with super-resolution: application to the analysis of chromosome movement," J. Microsc. 208, 49-64 (2002).
[CrossRef] [PubMed]

F. Rooms, W. Philips, and D. S. Lidke, "Simultaneous degradation estimation and restoration of confocal images and performance evaluation by colocalization analysis," J. Microsc. 218, 22-36 (2005).
[CrossRef] [PubMed]

J. C. G. Blonk, A. Don, H. van Aalst, and J. J. Birmingham, "Fluorescence photobleaching recovery in the confocal scanning light microscope," J. Microsc. 169, 363-374 (1993).
[CrossRef]

L. Tao and C. Nicholson, "The three-dimensional point spread functions of a microscope objective in image and object space," J. Microsc. 178, 267-271 (1995).
[CrossRef] [PubMed]

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

Network Comput. Neural Syst. (1)

G. J. Streekstra and J. van Pelt, "Analysis of tubular structures in three-dimensional confocal images," Network Comput. Neural Syst. 13, 381-395 (2002).
[CrossRef]

Proc. SPIE (1)

B. Zhang, J. Zerubia, and J.-C. Olivo-Marin, "A study of Gaussian approximations of fluorescence microscopy PSF models," in Three-Dimensional and Multidimensional Microscopy: Image Acquisition and Processing XIII, J.-A. Conchello, C. J. Cogswell, and T. Wilson, eds., Proc. SPIE 6090, 60900K (2006).

Other (7)

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 2002).

L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media (Pergamon, 1984).

D. R. Sandison, R. M. Williams, K. S. Wells, J. Strickler, and W. W. Webb, "Quantitative fluorescence confocal laser scanning microscopy (CLSM)," in Handbook of Biological Confocal Microscopy, J.B.Pawley, ed., 2nd ed. (Plenum, 1995), pp. 39-53.

M. Petrán, M. Hadravský, J. Benes, R. Kucera, and A. Boyde, "The tandem scanning reflected light microscope. Part 1-The principle and its design," in Proceedings of the Royal Microscopical Society (Blackwell, 1985), Vol. 20, pp. 125-129.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 1967).

L. J. van Vliet, "Grey-scale measurements in multi-dimensional digitized images," Ph.D. dissertation (Delft University, The Netherlands, 1993).

M. Gu, Advanced Optical Imaging Theory (Springer-Verlag, 2000).

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Figures (3)

Fig. 1
Fig. 1

Focusing of an illumination wave by an objective lens.

Fig. 2
Fig. 2

Examples of the Gaussian approximations of WFFM, LSCM, and DSCM PSFs with an L constraint. Nonparaxial cases, λ e x = 488   nm , λ e m = 520   nm , n = 1.515 , NA = 1.0 , and the pinhole diameter D = 0.5   AU in the LSCM and the DSCM. (a) Lateral WFFM PSF, (b) lateral LSCM PSF, (c) lateral DSCM PSF ( d = d 0 ) , (d) axial WFFM PSF, (e) axial LSCM PSF, and (f) axial DSCM PSF ( d = d 0 ) .

Fig. 3
Fig. 3

Examples of the Gaussian approximations of 2D paraxial WFFM PSF. (a) Approximation with an L constraint, and (b) approximation with an L 1 constraint. λ e m = 520   nm , n = 1.515 , and NA = 0.3 .

Tables (7)

Tables Icon

Table 1 Gaussian Parameters for 2D PSFs

Tables Icon

Table 2 Lateral Gaussian Parameters for 3D PSFs ( L Constraint)

Tables Icon

Table 3 Axial Gaussian Parameters for 3D PSFs ( L Constraint)

Tables Icon

Table 4 Approximation Errors a on WFFM PSFs ( L Constraint)

Tables Icon

Table 5 Approximation Errors a on LSCM PSFs ( L Constraint)

Tables Icon

Table 6 Approximation Errors a on DSCM PSFs ( L Constraint)

Tables Icon

Table 7 Approximation Errors a on the 2D Paraxial WFFM PSF ( L 1 Constraint)

Equations (41)

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

h ( x , y , z ; λ ) = C 0 0 α cos θ J 0 ( k ρ sin θ ) e i k z cos θ sin θ  d θ ,
h ( x , y , z ; λ ) = C 1 e i k z 0 1 J 0 ( k ρ t sin α ) e ( i / 2 ) k z t 2 sin 2 α t d t ,
PSF W F F M ( x , y , z ) = | h ( x , y , z ; λ e m ) | 2 ,
PSF L S C M ( x , y , z ) = | h ( x , y , z ; λ e x ) | 2 × { x 1 2 + y 1 2 r 2 } | h ( x x 1 , y y 1 , z ; λ e m ) | 2 d x 1 d y 1 ,
PSF D S C M ( x , y , z ) = | ( n x , n y ) D h ( x d 2 ( n x + n y ) , y 3 2 d ( n x n y ) , z ; λ e x ) | 2 × { x 1 2 + y 1 2 r 2 } | h ( x x 1 , y y 1 , z ; λ e m ) | 2 d x 1 d y 1 ,
  g σ ρ ( x , y ) := A 1 exp ( x 2 + y 2 2 σ ρ 2 ) = A 1 exp ( ρ 2 2 σ ρ 2 ) ,
g σ ρ , σ z ( x , y , z ) := A 2 exp ( x 2 + y 2 2 σ ρ 2 z 2 2 σ z 2 ) = A 2 exp ( ρ 2 2 σ ρ 2 z 2 2 σ z 2 ) .
σ * = argmin σ > 0 PSF g σ 2 2 .
PSF W F F M ( ρ ) = [ 2 J 1 ( k e m NA ρ ) k e m NA ρ ] 2 ,
σ ρ * 0.21 λ e m NA .
         σ ^ ρ * = 1 n k e m [ 4 7 cos 3 / 2 α + 3 cos 7 / 2 α 7 ( 1 cos 3 / 2 α ) ] 1 / 2 .
lim α 0 + E q . ( 11 ) 2 / ( k e m NA ) = 1.
σ ^ ρ * = 2 [ c 1 2 r 2 + 4 c 2 J 0 ( c 2 ) J 1 ( c 2 ) 8 J 1 2 ( c 2 ) r 2 [ J 0 2 ( c 2 ) + J 1 2 ( c 2 ) 1 ] ] 1 / 2 ,
σ ^ ρ * = 2 [ 2 σ e m , ρ 4 [ exp ( r 2 / 2 σ e m , ρ 2 ) 1 ] + r 2 σ e x , ρ 2 σ e x , ρ 2 σ e m , ρ 4 [ exp ( r 2 / 2 σ e m , ρ 2 ) 1 ] ] 1 / 2 ,
lim α 0 + E q . ( 14 ) E q . ( 13 ) = 1.
σ ^ ρ * = 2 2 π NA λ e x λ e m ( λ e x 2 + λ e m 2 ) 1 / 2 ,
σ ^ ρ * = λ e x λ e m 2 π n ( λ e x 2 + λ e m 2 ) 1 / 2 × [ 7 cos 3 / 2 α 3 cos 7 / 2 α 4 7 ( cos 3 / 2 α 1 ) ] 1 / 2 .
lim r 0 + E q . ( 13 ) = E q . ( 16 ) , lim r 0 + E q . ( 14 ) = E q . ( 17 ) .
lim r + E q . ( 13 ) = 2 k e x NA 0.225 λ e x NA ,
lim r + E q . ( 14 ) = 1 n k e x [ 7 cos 3 / 2 α 3 cos 7 / 2 α 4 7 ( cos 3 / 2 α 1 ) ] 1 / 2 .
d d 0 := 1 2 ( λ e m λ e x + D + 1 ) AU,
σ ρ * 0.22 λ e m NA .
RSE   := PSF g σ ^ * 2 2 PSF 2 2 ,
g Σ ( x ) := ( 2 π ) ( 3 / 2 ) | Σ | ( 1 / 2 ) × exp ( 1 2 ( x μ ) T Σ 1 ( x μ ) ) ,
h ( ρ ) = [ 2 J 1 ( c ρ ) c ρ ] 2 , g σ ( ρ ) = exp ( ρ 2 2 σ 2 ) ,
E p ( σ ) := h h p g σ g σ p 2 2 .
σ * { 0.22 λ e m / NA p = 1 0.21 λ e m / NA p = .
E 1 σ ( σ ) = 3 e c 2 σ 2 4 I 0 ( c 2 σ 2 ) 8 I 1 ( c 2 σ 2 ) 2 π σ 3 e c 2 σ 2 .
f ( u ) = 1 + k = 1 ( k + 1 ) ( k + 2 ) c k u k ,
c k = { 3 ( 2 n ) ! 4 2 2 n ( n ! ) 2 k = 2 n 2 3 ( 2 n + 1 ) ! 4 2 2 n n ! ( n + 1 ) ! k = 2 n 1 n = 1 , 2 , . . .
c 2 h p ( p 2 π ) 1 / p σ 2 ( 2 / p ) ( 1 2 p ) = 8 [ p 1 ( e c 2 σ 2 I 0 ( c 2 σ 2 ) 1 ) + e c 2 σ 2 I 1 ( c 2 σ 2 ) ( p 1 + 1 ) ] .
d d σ h h p g σ g σ p 2 2 = 2 π 0 ρ d d σ ( h ( ρ ) h p g σ ( ρ ) g σ p ) 2 d ρ = 4 π ( p 2 π ) 1 / p ( T 0 + T 1 + T 2 ) ,
T 0 = ( p 2 π ) 1 / p σ ( 4 / p ) 1 0 exp ( ρ 2 σ 2 ) [ σ 2 ρ 2 2 p ] ρ d ρ = ( p 2 π ) 1 / p p 2 2 p σ 1 ( 4 / p ) , T 1 = 8 σ ( 2 / p ) 1 p c 2 h p 0 J 1 2 ( c ρ ) exp ( ρ 2 2 σ 2 ) ρ 1 d ρ = 4 σ ( 2 / p ) + 1 p h p 1 exp ( c 2 σ 2 ) [ I 0 ( c 2 σ 2 ) + I 1 ( c 2 σ 2 ) ] c 2 σ 2 , T 2 = 4 σ ( 2 / p ) 3 h p c 2 0 J 1 2 ( c ρ ) exp ( ρ 2 2 σ 2 ) ρ d ρ = 4 σ ( 2 / p ) 1 h p c 2 exp ( c 2 σ 2 ) I 1 ( c 2 σ 2 ) .
h p ( x , y , z ; λ ) := | 0 1 J 0 ( k t x 2 + y 2 sin α ) e ( i / 2 ) k z t 2 sin 2 α t d t | 2 ,
h n p ( x , y , z ; λ ) := | 0 α cos θ J 0 ( k x 2 + y 2 sin θ ) e i k z cos θ × sin θ  d θ | 2 ,
g σ ρ , σ z ( x , y , z ) := exp ( x 2 + y 2 2 σ ρ 2 z 2 2 σ z 2 ) ,
g σ ρ , σ z ( x , y , z ) = 1 1 2 σ ρ 2 ( x 2 + y 2 ) 1 2 σ z 2 z 2 + o ( | x | 2 ) .
4 h p ( x , y , z ; λ e m ) = 1 k e m 2 NA 2 4 ( x 2 + y 2 ) k e m 2 NA 2 sin 2 α 48 z 2 + o ( | x | 2 ) ,
9 4 ( 1 cos 3 / 2 α ) 2 h n p ( x , y , z ; λ e m ) = 1 n 2 k e m 2 ( 4 7 cos 3 / 2 α + 3 cos 3 / 2 α ) 14 ( 1 cos 3 / 2 α ) ( x 2 + y 2 ) , 3 n 2 k e m 2 ( 4 + 4 cos 5 α 25 cos 7 / 2 α + 42 cos 5 / 2 α 25 cos 3 / 2 α ) 175 ( 1 cos 3 / 2 α ) 2 z 2 + o ( | x | 2 ) .
4 k e m 2 NA 2 π [ 1 J 0 2 ( c 2 ) J 1 2 ( c 2 ) ] h p ( x , y , z ; λ e x ) × t 1 2 + t 2 2 r 2 h p ( x t 1 , y t 2 , z ; λ e m ) d t 1 d t 2 = 1 1 4 [ c 1 2 r 2 + 4 c 2 J 0 ( c 2 ) J 1 ( c 2 ) 8 J 1 2 ( c 2 ) r 2 [ J 0 2 ( c 2 ) + J 1 2 ( c 2 ) 1 ] ] ( x 2 + y 2 ) 1 48 [ c 1 2 NA 2 r 2 n 2 48 c 2 2 [ J 0 2 ( c 2 ) + J 1 2 ( c 2 ) ] 192 J 1 2 ( c 2 ) n 2 k e m 2 r 4 [ J 0 2 ( c 2 ) + J 1 2 ( c 2 ) 1 ] ] z 2 + o ( | x | 2 ) .
exp ( r 2 2 σ e m , ρ 2 ) 2 π σ e m , ρ 2 [ exp ( r 2 2 σ e m , ρ 2 ) 1 ] g σ e x , ρ , σ e x , z ( x , y , z ) t 1 2 + t 2 2 r 2 g σ e m , ρ , σ e m , z ( x t 1 , y t 2 , z ) d t 1 d t 2 = 1 1 4 2 σ e m , ρ 4 [ exp ( r 2 2 σ e m , ρ 2 ) 1 ] + r 2 σ e x , ρ 2 σ e x , ρ 2 σ e m , ρ 4 [ exp ( r 2 2 σ e m , ρ 2 ) 1 ] ( x 2 + y 2 ) 1 2 σ e x , z 2 + σ e m , z 2 σ e x , z 2 σ e m , z 2 z 2 + o ( | x | 2 ) .

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