Abstract

A three-dimensional (3D) particle tracking algorithm based on microscope off-focus images is presented in this paper. Subnanometer resolution in all three axes at 400Hz sampling rate is achieved using a complementary metal-oxide-semiconductor (CMOS) camera. At each sampling, the lateral position of the spherical particle is first estimated by the centroid method. The axial position is then estimated by comparing the radius vector, which is converted from the off-focus two-dimensional image of the particle with no information loss, with an object-specific model, calibrated automatically prior to each experiment. Estimation bias and variance of the 3D tracking algorithm are characterized through analytical analysis. It leads to an analytical model, enabling prediction of the measurement performance based on calibration data. Finally, experimental results are presented to illustrate the performance of the measurement method in terms of precision and range. The validity of the theoretical analysis is also experimentally confirmed.

© 2008 Optical Society of America

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  1. G. V. Soni, B. M. Jaffar Ali, Y. Hatwalne, and G. V. Shivashankar, “Single particle tracking of correlated bacterial dynamics,” Biophys. J. 84, 2634-2637 (2003).
    [CrossRef] [PubMed]
  2. R. Thar, N. Blackburn, and M. Kuhl, “A new system for three-dimensional tracking of motile microorganisms,” Appl. Environ. Microbiol. 66, 2238-2242 (2000).
    [CrossRef] [PubMed]
  3. M. Wu, J. W. Roberts, and M. Buckley, “Three-dimensional fluorescent particle tracking at micro-scale using a single camera,” Exp. Fluids 38, 461-465 (2005).
    [CrossRef]
  4. R. Luo, X. Y. Yang, X. F. Peng, and Y. F. Sun, “Three-dimensional tracking of fluorescent particles applied to micro-fluidic measurements,” J. Micromech. Microeng. 16, 1689-1699 (2006).
    [CrossRef]
  5. S. Devasenathipathy, J. G. Santiago, S. T. Wereley, C. D. Meinhart, and K. Takehara, “ Particle imaging techniques for microfabricated fluidic systems,” Exp. Fluids 34, 504-514 (2003).
  6. A. D. Dinsmore, Eric R. Weeks, V. Prasad, A. C. Levitt, and D. A. Weitz, “Three-dimensional confocal microscopy of colloids,” Appl. Opt. 40, 4152-4159 (2001).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  9. A. Rohrbach, C. Tischer, D. Neumayer, E.-L. Florin, and E. H. K Stelzer, “Trapping and tracking a local probe with a photonic force microscope,” Rev. Sci. Instrum. 75, 2197-2210 (2004).
    [CrossRef]
  10. V. Levi, Q. Ruan, and E. Gratton, “3-D particle tracking in a two-photon microscope: application to the study of molecular dynamics in cells,” Biophys. J. 88, 2919-2928 (2005).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  12. D. Li, J. Xiong, A. Qu, and T. Xu, “Three-dimensional tracking of single secretory granules in live pc12 cells,” Biophys. J. 87, 1991-2001 (2004).
    [CrossRef] [PubMed]
  13. D. Thomann, D. R. Rines, P. K. Sorger, and G. Danuster, “Automatic fluorescent tag detection in 3d with super-resolution application to the analysis of chromosome movement,” J. Microsc. 208, 49-64 (2002).
    [CrossRef] [PubMed]
  14. K. D. Kihm, A. Banerjee, C. K. Choi, and T. Takagi, “Near-wall hindered Brownian diffusion of nanoparticles examined by three-dimensional ratiometric total internal reflection fluorescence microscopy (3-D R-TIRFM),” Exp. Fluids 37, 811-824 (2004).
    [CrossRef]
  15. M. Speidel, A. Jonas, and E.-L. Florin, “Three-dimensional tracking of fluorescent nanoparticles with subnanometer precision by use of off-focus imaging,” Opt. Lett. 28, 69-71 (2003).
    [CrossRef] [PubMed]
  16. H. P. Kao and A. S. Verkman, “Tracking of single fluorescent particles in three dimensions: use of cylindrical optics to encode particle position,” Biophys. J. 67, 1291-1300 (1994).
    [CrossRef] [PubMed]
  17. T. Ragan, H. Huang, P. So, and E. Gratton, “3D particle tracking on a two-photon microscope,” J. Fluoresc. 16, 325-336(2006).
    [CrossRef] [PubMed]
  18. G. Cao and X. Yu, “Accuracy analysis of a Hartmann-Shack wavefront sensor operated with a faint object,” Opt. Eng. 33, 2331-2335 (1994).
    [CrossRef]
  19. B. F. Alexander and K. C. Ng, “Elimination of systematic error in subpixel accuracy centroid estimation,” Opt. Eng. 30, 1320-1331 (1991).
    [CrossRef]
  20. J. Arines and J. Ares, “Minimum variance centroid thresholding,” Opt. Lett. 27, 497-499 (2002).
    [CrossRef]
  21. J. Ares and J. Arines, “Influence of thresholding on centroid statistics: full analytical description,” Appl. Opt. 43, 5796-5805 (2004).
    [CrossRef] [PubMed]
  22. D. Robinson and P. Milanfar, “Fundamental performance limits in image registration,” IEEE Trans. Image Process. 13, 1185-1199 (2004).
    [CrossRef]

2007 (1)

2006 (2)

R. Luo, X. Y. Yang, X. F. Peng, and Y. F. Sun, “Three-dimensional tracking of fluorescent particles applied to micro-fluidic measurements,” J. Micromech. Microeng. 16, 1689-1699 (2006).
[CrossRef]

T. Ragan, H. Huang, P. So, and E. Gratton, “3D particle tracking on a two-photon microscope,” J. Fluoresc. 16, 325-336(2006).
[CrossRef] [PubMed]

2005 (3)

M. Wu, J. W. Roberts, and M. Buckley, “Three-dimensional fluorescent particle tracking at micro-scale using a single camera,” Exp. Fluids 38, 461-465 (2005).
[CrossRef]

V. Levi, Q. Ruan, and E. Gratton, “3-D particle tracking in a two-photon microscope: application to the study of molecular dynamics in cells,” Biophys. J. 88, 2919-2928 (2005).
[CrossRef] [PubMed]

F. Aguet, D. Van De Ville, and M. Unser, “A maximum-likelihood formalism for sub-resolution axial localization of fluorescent nanoparticles,” Opt. Express 13, 10503-10522(2005).
[CrossRef] [PubMed]

2004 (5)

J. Ares and J. Arines, “Influence of thresholding on centroid statistics: full analytical description,” Appl. Opt. 43, 5796-5805 (2004).
[CrossRef] [PubMed]

A. Rohrbach, C. Tischer, D. Neumayer, E.-L. Florin, and E. H. K Stelzer, “Trapping and tracking a local probe with a photonic force microscope,” Rev. Sci. Instrum. 75, 2197-2210 (2004).
[CrossRef]

D. Robinson and P. Milanfar, “Fundamental performance limits in image registration,” IEEE Trans. Image Process. 13, 1185-1199 (2004).
[CrossRef]

D. Li, J. Xiong, A. Qu, and T. Xu, “Three-dimensional tracking of single secretory granules in live pc12 cells,” Biophys. J. 87, 1991-2001 (2004).
[CrossRef] [PubMed]

K. D. Kihm, A. Banerjee, C. K. Choi, and T. Takagi, “Near-wall hindered Brownian diffusion of nanoparticles examined by three-dimensional ratiometric total internal reflection fluorescence microscopy (3-D R-TIRFM),” Exp. Fluids 37, 811-824 (2004).
[CrossRef]

2003 (3)

S. Devasenathipathy, J. G. Santiago, S. T. Wereley, C. D. Meinhart, and K. Takehara, “ Particle imaging techniques for microfabricated fluidic systems,” Exp. Fluids 34, 504-514 (2003).

G. V. Soni, B. M. Jaffar Ali, Y. Hatwalne, and G. V. Shivashankar, “Single particle tracking of correlated bacterial dynamics,” Biophys. J. 84, 2634-2637 (2003).
[CrossRef] [PubMed]

M. Speidel, A. Jonas, and E.-L. Florin, “Three-dimensional tracking of fluorescent nanoparticles with subnanometer precision by use of off-focus imaging,” Opt. Lett. 28, 69-71 (2003).
[CrossRef] [PubMed]

2002 (2)

J. Arines and J. Ares, “Minimum variance centroid thresholding,” Opt. Lett. 27, 497-499 (2002).
[CrossRef]

D. Thomann, D. R. Rines, P. K. Sorger, and G. Danuster, “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)

2000 (1)

R. Thar, N. Blackburn, and M. Kuhl, “A new system for three-dimensional tracking of motile microorganisms,” Appl. Environ. Microbiol. 66, 2238-2242 (2000).
[CrossRef] [PubMed]

1999 (1)

H. Bornfleth, P. Edelmann, D. Zink, T. Cremer, and C. Cremer, “Quantitative motion analysis of subchromosomal foci in living cells using four-dimensional microscopy,” Biophys. J. 77, 2871-2886 (1999).
[CrossRef] [PubMed]

1994 (2)

G. Cao and X. Yu, “Accuracy analysis of a Hartmann-Shack wavefront sensor operated with a faint object,” Opt. Eng. 33, 2331-2335 (1994).
[CrossRef]

H. P. Kao and A. S. Verkman, “Tracking of single fluorescent particles in three dimensions: use of cylindrical optics to encode particle position,” Biophys. J. 67, 1291-1300 (1994).
[CrossRef] [PubMed]

1991 (1)

B. F. Alexander and K. C. Ng, “Elimination of systematic error in subpixel accuracy centroid estimation,” Opt. Eng. 30, 1320-1331 (1991).
[CrossRef]

Aguet, F.

Alexander, B. F.

B. F. Alexander and K. C. Ng, “Elimination of systematic error in subpixel accuracy centroid estimation,” Opt. Eng. 30, 1320-1331 (1991).
[CrossRef]

Ares, J.

Arines, J.

Banerjee, A.

K. D. Kihm, A. Banerjee, C. K. Choi, and T. Takagi, “Near-wall hindered Brownian diffusion of nanoparticles examined by three-dimensional ratiometric total internal reflection fluorescence microscopy (3-D R-TIRFM),” Exp. Fluids 37, 811-824 (2004).
[CrossRef]

Blackburn, N.

R. Thar, N. Blackburn, and M. Kuhl, “A new system for three-dimensional tracking of motile microorganisms,” Appl. Environ. Microbiol. 66, 2238-2242 (2000).
[CrossRef] [PubMed]

Bornfleth, H.

H. Bornfleth, P. Edelmann, D. Zink, T. Cremer, and C. Cremer, “Quantitative motion analysis of subchromosomal foci in living cells using four-dimensional microscopy,” Biophys. J. 77, 2871-2886 (1999).
[CrossRef] [PubMed]

Buckley, M.

M. Wu, J. W. Roberts, and M. Buckley, “Three-dimensional fluorescent particle tracking at micro-scale using a single camera,” Exp. Fluids 38, 461-465 (2005).
[CrossRef]

Cao, G.

G. Cao and X. Yu, “Accuracy analysis of a Hartmann-Shack wavefront sensor operated with a faint object,” Opt. Eng. 33, 2331-2335 (1994).
[CrossRef]

Choi, C. K.

K. D. Kihm, A. Banerjee, C. K. Choi, and T. Takagi, “Near-wall hindered Brownian diffusion of nanoparticles examined by three-dimensional ratiometric total internal reflection fluorescence microscopy (3-D R-TIRFM),” Exp. Fluids 37, 811-824 (2004).
[CrossRef]

Cizmar, T.

Cremer, C.

H. Bornfleth, P. Edelmann, D. Zink, T. Cremer, and C. Cremer, “Quantitative motion analysis of subchromosomal foci in living cells using four-dimensional microscopy,” Biophys. J. 77, 2871-2886 (1999).
[CrossRef] [PubMed]

Cremer, T.

H. Bornfleth, P. Edelmann, D. Zink, T. Cremer, and C. Cremer, “Quantitative motion analysis of subchromosomal foci in living cells using four-dimensional microscopy,” Biophys. J. 77, 2871-2886 (1999).
[CrossRef] [PubMed]

Danuster, G.

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

Devasenathipathy, S.

S. Devasenathipathy, J. G. Santiago, S. T. Wereley, C. D. Meinhart, and K. Takehara, “ Particle imaging techniques for microfabricated fluidic systems,” Exp. Fluids 34, 504-514 (2003).

Dinsmore, A. D.

Edelmann, P.

H. Bornfleth, P. Edelmann, D. Zink, T. Cremer, and C. Cremer, “Quantitative motion analysis of subchromosomal foci in living cells using four-dimensional microscopy,” Biophys. J. 77, 2871-2886 (1999).
[CrossRef] [PubMed]

Florin, E.-L.

A. Rohrbach, C. Tischer, D. Neumayer, E.-L. Florin, and E. H. K Stelzer, “Trapping and tracking a local probe with a photonic force microscope,” Rev. Sci. Instrum. 75, 2197-2210 (2004).
[CrossRef]

M. Speidel, A. Jonas, and E.-L. Florin, “Three-dimensional tracking of fluorescent nanoparticles with subnanometer precision by use of off-focus imaging,” Opt. Lett. 28, 69-71 (2003).
[CrossRef] [PubMed]

Gratton, E.

T. Ragan, H. Huang, P. So, and E. Gratton, “3D particle tracking on a two-photon microscope,” J. Fluoresc. 16, 325-336(2006).
[CrossRef] [PubMed]

V. Levi, Q. Ruan, and E. Gratton, “3-D particle tracking in a two-photon microscope: application to the study of molecular dynamics in cells,” Biophys. J. 88, 2919-2928 (2005).
[CrossRef] [PubMed]

Hatwalne, Y.

G. V. Soni, B. M. Jaffar Ali, Y. Hatwalne, and G. V. Shivashankar, “Single particle tracking of correlated bacterial dynamics,” Biophys. J. 84, 2634-2637 (2003).
[CrossRef] [PubMed]

Huang, H.

T. Ragan, H. Huang, P. So, and E. Gratton, “3D particle tracking on a two-photon microscope,” J. Fluoresc. 16, 325-336(2006).
[CrossRef] [PubMed]

Jaffar Ali, B. M.

G. V. Soni, B. M. Jaffar Ali, Y. Hatwalne, and G. V. Shivashankar, “Single particle tracking of correlated bacterial dynamics,” Biophys. J. 84, 2634-2637 (2003).
[CrossRef] [PubMed]

Jonas, A.

Kao, H. P.

H. P. Kao and A. S. Verkman, “Tracking of single fluorescent particles in three dimensions: use of cylindrical optics to encode particle position,” Biophys. J. 67, 1291-1300 (1994).
[CrossRef] [PubMed]

Kihm, K. D.

K. D. Kihm, A. Banerjee, C. K. Choi, and T. Takagi, “Near-wall hindered Brownian diffusion of nanoparticles examined by three-dimensional ratiometric total internal reflection fluorescence microscopy (3-D R-TIRFM),” Exp. Fluids 37, 811-824 (2004).
[CrossRef]

Kuhl, M.

R. Thar, N. Blackburn, and M. Kuhl, “A new system for three-dimensional tracking of motile microorganisms,” Appl. Environ. Microbiol. 66, 2238-2242 (2000).
[CrossRef] [PubMed]

Levi, V.

V. Levi, Q. Ruan, and E. Gratton, “3-D particle tracking in a two-photon microscope: application to the study of molecular dynamics in cells,” Biophys. J. 88, 2919-2928 (2005).
[CrossRef] [PubMed]

Levitt, A. C.

Li, D.

D. Li, J. Xiong, A. Qu, and T. Xu, “Three-dimensional tracking of single secretory granules in live pc12 cells,” Biophys. J. 87, 1991-2001 (2004).
[CrossRef] [PubMed]

Luo, R.

R. Luo, X. Y. Yang, X. F. Peng, and Y. F. Sun, “Three-dimensional tracking of fluorescent particles applied to micro-fluidic measurements,” J. Micromech. Microeng. 16, 1689-1699 (2006).
[CrossRef]

Meinhart, C. D.

S. Devasenathipathy, J. G. Santiago, S. T. Wereley, C. D. Meinhart, and K. Takehara, “ Particle imaging techniques for microfabricated fluidic systems,” Exp. Fluids 34, 504-514 (2003).

Milanfar, P.

D. Robinson and P. Milanfar, “Fundamental performance limits in image registration,” IEEE Trans. Image Process. 13, 1185-1199 (2004).
[CrossRef]

Neumayer, D.

A. Rohrbach, C. Tischer, D. Neumayer, E.-L. Florin, and E. H. K Stelzer, “Trapping and tracking a local probe with a photonic force microscope,” Rev. Sci. Instrum. 75, 2197-2210 (2004).
[CrossRef]

Ng, K. C.

B. F. Alexander and K. C. Ng, “Elimination of systematic error in subpixel accuracy centroid estimation,” Opt. Eng. 30, 1320-1331 (1991).
[CrossRef]

Peng, X. F.

R. Luo, X. Y. Yang, X. F. Peng, and Y. F. Sun, “Three-dimensional tracking of fluorescent particles applied to micro-fluidic measurements,” J. Micromech. Microeng. 16, 1689-1699 (2006).
[CrossRef]

Prasad, V.

Qu, A.

D. Li, J. Xiong, A. Qu, and T. Xu, “Three-dimensional tracking of single secretory granules in live pc12 cells,” Biophys. J. 87, 1991-2001 (2004).
[CrossRef] [PubMed]

Ragan, T.

T. Ragan, H. Huang, P. So, and E. Gratton, “3D particle tracking on a two-photon microscope,” J. Fluoresc. 16, 325-336(2006).
[CrossRef] [PubMed]

Rines, D. R.

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

Roberts, J. W.

M. Wu, J. W. Roberts, and M. Buckley, “Three-dimensional fluorescent particle tracking at micro-scale using a single camera,” Exp. Fluids 38, 461-465 (2005).
[CrossRef]

Robinson, D.

D. Robinson and P. Milanfar, “Fundamental performance limits in image registration,” IEEE Trans. Image Process. 13, 1185-1199 (2004).
[CrossRef]

Rohrbach, A.

A. Rohrbach, C. Tischer, D. Neumayer, E.-L. Florin, and E. H. K Stelzer, “Trapping and tracking a local probe with a photonic force microscope,” Rev. Sci. Instrum. 75, 2197-2210 (2004).
[CrossRef]

Ruan, Q.

V. Levi, Q. Ruan, and E. Gratton, “3-D particle tracking in a two-photon microscope: application to the study of molecular dynamics in cells,” Biophys. J. 88, 2919-2928 (2005).
[CrossRef] [PubMed]

Santiago, J. G.

S. Devasenathipathy, J. G. Santiago, S. T. Wereley, C. D. Meinhart, and K. Takehara, “ Particle imaging techniques for microfabricated fluidic systems,” Exp. Fluids 34, 504-514 (2003).

Shivashankar, G. V.

G. V. Soni, B. M. Jaffar Ali, Y. Hatwalne, and G. V. Shivashankar, “Single particle tracking of correlated bacterial dynamics,” Biophys. J. 84, 2634-2637 (2003).
[CrossRef] [PubMed]

So, P.

T. Ragan, H. Huang, P. So, and E. Gratton, “3D particle tracking on a two-photon microscope,” J. Fluoresc. 16, 325-336(2006).
[CrossRef] [PubMed]

Soni, G. V.

G. V. Soni, B. M. Jaffar Ali, Y. Hatwalne, and G. V. Shivashankar, “Single particle tracking of correlated bacterial dynamics,” Biophys. J. 84, 2634-2637 (2003).
[CrossRef] [PubMed]

Sorger, P. K.

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

Speidel, M.

Stelzer, E. H. K

A. Rohrbach, C. Tischer, D. Neumayer, E.-L. Florin, and E. H. K Stelzer, “Trapping and tracking a local probe with a photonic force microscope,” Rev. Sci. Instrum. 75, 2197-2210 (2004).
[CrossRef]

Sun, Y. F.

R. Luo, X. Y. Yang, X. F. Peng, and Y. F. Sun, “Three-dimensional tracking of fluorescent particles applied to micro-fluidic measurements,” J. Micromech. Microeng. 16, 1689-1699 (2006).
[CrossRef]

Takagi, T.

K. D. Kihm, A. Banerjee, C. K. Choi, and T. Takagi, “Near-wall hindered Brownian diffusion of nanoparticles examined by three-dimensional ratiometric total internal reflection fluorescence microscopy (3-D R-TIRFM),” Exp. Fluids 37, 811-824 (2004).
[CrossRef]

Takehara, K.

S. Devasenathipathy, J. G. Santiago, S. T. Wereley, C. D. Meinhart, and K. Takehara, “ Particle imaging techniques for microfabricated fluidic systems,” Exp. Fluids 34, 504-514 (2003).

Thar, R.

R. Thar, N. Blackburn, and M. Kuhl, “A new system for three-dimensional tracking of motile microorganisms,” Appl. Environ. Microbiol. 66, 2238-2242 (2000).
[CrossRef] [PubMed]

Thomann, D.

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

Tischer, C.

A. Rohrbach, C. Tischer, D. Neumayer, E.-L. Florin, and E. H. K Stelzer, “Trapping and tracking a local probe with a photonic force microscope,” Rev. Sci. Instrum. 75, 2197-2210 (2004).
[CrossRef]

Unser, M.

Van De Ville, D.

Verkman, A. S.

H. P. Kao and A. S. Verkman, “Tracking of single fluorescent particles in three dimensions: use of cylindrical optics to encode particle position,” Biophys. J. 67, 1291-1300 (1994).
[CrossRef] [PubMed]

Weeks, Eric R.

Weitz, D. A.

Wereley, S. T.

S. Devasenathipathy, J. G. Santiago, S. T. Wereley, C. D. Meinhart, and K. Takehara, “ Particle imaging techniques for microfabricated fluidic systems,” Exp. Fluids 34, 504-514 (2003).

Wu, M.

M. Wu, J. W. Roberts, and M. Buckley, “Three-dimensional fluorescent particle tracking at micro-scale using a single camera,” Exp. Fluids 38, 461-465 (2005).
[CrossRef]

Xiong, J.

D. Li, J. Xiong, A. Qu, and T. Xu, “Three-dimensional tracking of single secretory granules in live pc12 cells,” Biophys. J. 87, 1991-2001 (2004).
[CrossRef] [PubMed]

Xu, T.

D. Li, J. Xiong, A. Qu, and T. Xu, “Three-dimensional tracking of single secretory granules in live pc12 cells,” Biophys. J. 87, 1991-2001 (2004).
[CrossRef] [PubMed]

Yang, X. Y.

R. Luo, X. Y. Yang, X. F. Peng, and Y. F. Sun, “Three-dimensional tracking of fluorescent particles applied to micro-fluidic measurements,” J. Micromech. Microeng. 16, 1689-1699 (2006).
[CrossRef]

Yu, X.

G. Cao and X. Yu, “Accuracy analysis of a Hartmann-Shack wavefront sensor operated with a faint object,” Opt. Eng. 33, 2331-2335 (1994).
[CrossRef]

Zemanek, P.

Zink, D.

H. Bornfleth, P. Edelmann, D. Zink, T. Cremer, and C. Cremer, “Quantitative motion analysis of subchromosomal foci in living cells using four-dimensional microscopy,” Biophys. J. 77, 2871-2886 (1999).
[CrossRef] [PubMed]

Appl. Environ. Microbiol. (1)

R. Thar, N. Blackburn, and M. Kuhl, “A new system for three-dimensional tracking of motile microorganisms,” Appl. Environ. Microbiol. 66, 2238-2242 (2000).
[CrossRef] [PubMed]

Appl. Opt. (2)

Biophys. J. (5)

G. V. Soni, B. M. Jaffar Ali, Y. Hatwalne, and G. V. Shivashankar, “Single particle tracking of correlated bacterial dynamics,” Biophys. J. 84, 2634-2637 (2003).
[CrossRef] [PubMed]

H. P. Kao and A. S. Verkman, “Tracking of single fluorescent particles in three dimensions: use of cylindrical optics to encode particle position,” Biophys. J. 67, 1291-1300 (1994).
[CrossRef] [PubMed]

V. Levi, Q. Ruan, and E. Gratton, “3-D particle tracking in a two-photon microscope: application to the study of molecular dynamics in cells,” Biophys. J. 88, 2919-2928 (2005).
[CrossRef] [PubMed]

H. Bornfleth, P. Edelmann, D. Zink, T. Cremer, and C. Cremer, “Quantitative motion analysis of subchromosomal foci in living cells using four-dimensional microscopy,” Biophys. J. 77, 2871-2886 (1999).
[CrossRef] [PubMed]

D. Li, J. Xiong, A. Qu, and T. Xu, “Three-dimensional tracking of single secretory granules in live pc12 cells,” Biophys. J. 87, 1991-2001 (2004).
[CrossRef] [PubMed]

Exp. Fluids (3)

M. Wu, J. W. Roberts, and M. Buckley, “Three-dimensional fluorescent particle tracking at micro-scale using a single camera,” Exp. Fluids 38, 461-465 (2005).
[CrossRef]

S. Devasenathipathy, J. G. Santiago, S. T. Wereley, C. D. Meinhart, and K. Takehara, “ Particle imaging techniques for microfabricated fluidic systems,” Exp. Fluids 34, 504-514 (2003).

K. D. Kihm, A. Banerjee, C. K. Choi, and T. Takagi, “Near-wall hindered Brownian diffusion of nanoparticles examined by three-dimensional ratiometric total internal reflection fluorescence microscopy (3-D R-TIRFM),” Exp. Fluids 37, 811-824 (2004).
[CrossRef]

IEEE Trans. Image Process. (1)

D. Robinson and P. Milanfar, “Fundamental performance limits in image registration,” IEEE Trans. Image Process. 13, 1185-1199 (2004).
[CrossRef]

J. Fluoresc. (1)

T. Ragan, H. Huang, P. So, and E. Gratton, “3D particle tracking on a two-photon microscope,” J. Fluoresc. 16, 325-336(2006).
[CrossRef] [PubMed]

J. Micromech. Microeng. (1)

R. Luo, X. Y. Yang, X. F. Peng, and Y. F. Sun, “Three-dimensional tracking of fluorescent particles applied to micro-fluidic measurements,” J. Micromech. Microeng. 16, 1689-1699 (2006).
[CrossRef]

J. Microsc. (1)

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

Opt. Eng. (2)

G. Cao and X. Yu, “Accuracy analysis of a Hartmann-Shack wavefront sensor operated with a faint object,” Opt. Eng. 33, 2331-2335 (1994).
[CrossRef]

B. F. Alexander and K. C. Ng, “Elimination of systematic error in subpixel accuracy centroid estimation,” Opt. Eng. 30, 1320-1331 (1991).
[CrossRef]

Opt. Express (2)

Opt. Lett. (2)

Rev. Sci. Instrum. (1)

A. Rohrbach, C. Tischer, D. Neumayer, E.-L. Florin, and E. H. K Stelzer, “Trapping and tracking a local probe with a photonic force microscope,” Rev. Sci. Instrum. 75, 2197-2210 (2004).
[CrossRef]

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Figures (11)

Fig. 1
Fig. 1

In-focus and out-of-focus images of a 4.5 μm bead.

Fig. 2
Fig. 2

Radial projection.

Fig. 3
Fig. 3

Upsampling.

Fig. 4
Fig. 4

(a) Original image, (b) radius vector of the image (the radius index times 222.2 nm is the actual radial distance).

Fig. 5
Fig. 5

Mesh plot of the model matrix. The radial distance is the radius index times 222.2 nm ; the axial position is the axial position index times 200 nm .

Fig. 6
Fig. 6

Intensity change along the z axis of the center point

Fig. 7
Fig. 7

Nanostepping in three axes.

Fig. 8
Fig. 8

10 μm triangular motion in three axes.

Fig. 9
Fig. 9

K curve.

Fig. 10
Fig. 10

S curve.

Fig. 11
Fig. 11

Measured variances (dotted curve) and predicted variances (solid curve).

Equations (22)

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( x c , y c ) = ( I i x i I i , I i y i I i ) ,
m i = 1 k = 1 M i c k k = 1 M i c k I k , c k = { d r i 1 Δ , if     r i 1 d < r i r i + 1 d Δ , if     r i d < r i + 1 ,
J ( z ) = i = 0 N [ ρ i ( z ) m i ] 2 .
f ( z ) = 1 2 d J d z = i = 0 N [ ρ i ( z ) m i ] ρ i ( z ) .
z j + 1 = z j [ f ( z j ) ] 1 f ( z j ) ,
f ( z ) = i = 0 N [ ρ i ( z ) ρ i ( z ) + ρ i ( z ) ρ i ( z ) ] i = 0 N m i ρ i ( z ) .
σ n i 2 = [ k = 1 M i c k 2 ( k = 1 M i c k ) 2 ] σ η 2 = σ η 2 L i ,
where L i { 2 3 π Δ 2 w 2 , i = 0 3 π i Δ 2 w 2 , i 1
J n ( z ) = i = 0 N [ ρ i ( z ) ( m i + n i ) ] 2 ,
f n ( z ) = f ( z ) i = 0 N n i ρ i ( z ) = f ( z ) n ,
δ z n f ( z a ) = i = 0 N n i ρ i ( z ) f ( z a ) .
f ( z a ) = i = 0 N ρ i ( z a ) ρ i ( z a ) .
σ δ z 2 | z a = i = 0 N [ ρ i ( z a ) ] 2 1 L i [ i = 0 N [ ρ i ( z a ) ] 2 ] 2 σ η 2 = S ( z a ) 2 σ η 2
σ δ z | z a = S ( z a ) σ η ,
S ( z a ) = [ i = 0 N [ ρ i ( z a ) ] 2 1 L i ] 1 / 2 i = 0 N [ ρ i ( z a ) ] 2 .
q ^ z ( r ) = 0 2 π q z ( ( r cos θ δ r ) 2 + ( r sin θ ) 2 ) d θ 2 π .
( r cos θ δ r ) 2 + ( r sin θ ) 2 r δ r cos θ .
q ^ z ( r ) 0 2 π q z ( r δ r cos θ ) d θ 2 π q z ( r ) + q z ( r ) 4 δ r 2 .
e i = q z ( r i ) 4 δ r 2 ,
δ z n f ( z a ) = i = 0 N ρ i ( z a ) q z a ( r i ) 4 δ r 2 i = 0 N ρ i ( z a ) ρ i ( z a ) = 2 2 K δ r 2 ,
K = 2 4 i = 0 N ρ i ( z a ) q z a ( r i ) i = 0 N ρ i ( z a ) ρ i ( z a ) .
σ δ z | z a = | K ( z a ) | σ r 2

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