Abstract

We introduce a fast and robust technique for single-particle tracking with nanometer accuracy. We extract the center-of-mass of the image of a single particle with a simple, iterative algorithm that efficiently suppresses background-induced bias in a simplistic centroid estimator. Unlike many commonly used algorithms, our position estimator requires no prior information about the shape or size of the tracked particle image and uses only simple arithmetic operations, making it appropriate for future hardware implementation and real-time feedback applications. We demonstrate it both numerically and experimentally, using an inexpensive CCD camera to localize 190 nm fluorescent microspheres to better than 5 nm.

© 2008 Optical Society of America

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  1. M. J. Saxton and K. Jacobson, "Single-particle tracking: applications to membrane dynamics," Annu. Rev. Biophys. Biomol. Struct. 26, 373-399 (1997).
    [CrossRef] [PubMed]
  2. A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goodman, and P. R. Selvin, "Myosin V walks hand-overhand: Single fluorophore imaging with 1.5-nm localization," Science 300, 2061-2065 (2003).
    [CrossRef] [PubMed]
  3. X. Michalet, F. F. Pinaud, L. A. Bentolila, J. M. Tsay, S. Doose, J. J. Li, G. Sundaresan, A. M. Wu, S. S. Gambhir, and S. Weiss, "Quantum Dots for Live Cells, in Vivo Imaging, and Diagnostics," Science 307, 538-544 (2005).
    [CrossRef] [PubMed]
  4. D. Weihs, T. G. Mason, and M. A. Teitell, "Bio-Microrheology: A Frontier in Microrheology," Biophys. J. 91, 4296-4305 (2006).
    [CrossRef] [PubMed]
  5. P. Bahukudumbi and M. A. Bevan, "Imaging energy landscapes with concentrated diffusing colloidal probes," J. Chem. Phys. 126, 244702 (2007).
    [CrossRef] [PubMed]
  6. H.-J. Wu, W. Everett, S. Anekal, and M. Bevan, "Mapping Patterned Potential Energy Landscapes with Diffusing Colloidal Probes," Langmuir 22, 6826-6836 (2006).
    [CrossRef] [PubMed]
  7. S. K. Sainis, V. Germain, and E. R. Dufresne, "Statistics of Particle Trajectories at Short Time Intervals Reveal fN-Scale Colloidal Forces," Phys. Rev. Lett. 99, 018303 (2007).
    [CrossRef] [PubMed]
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    [CrossRef]
  9. M. Cheezum, W. Walker, and W. Guilford, "Quantitative Comparison of Algorithms for Tracking Single Fluorescent Particles," Biophys. J. 81, 2378-2388 (2001).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
  21. M. Armani, S. Chaudhary, R. Probst, and B. Shapiro, "Using feedback control of microflows to independently steer multiple particles," IEEE J. Microelectromech. Syst. 15, 945-956 (2006).
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2007

P. Bahukudumbi and M. A. Bevan, "Imaging energy landscapes with concentrated diffusing colloidal probes," J. Chem. Phys. 126, 244702 (2007).
[CrossRef] [PubMed]

S. K. Sainis, V. Germain, and E. R. Dufresne, "Statistics of Particle Trajectories at Short Time Intervals Reveal fN-Scale Colloidal Forces," Phys. Rev. Lett. 99, 018303 (2007).
[CrossRef] [PubMed]

K. McHale, A. J. Berglund, and H. Mabuchi, "Quantum dot photon statistics measured by three-dimensional particle tracking," Nano Lett. 7, 3535-3539 (2007).
[CrossRef] [PubMed]

2006

M. Armani, S. Chaudhary, R. Probst, and B. Shapiro, "Using feedback control of microflows to independently steer multiple particles," IEEE J. Microelectromech. Syst. 15, 945-956 (2006).
[CrossRef]

H. Cang, C. M. Wong, C. S. Xu, A. H. Rizvi, and H. Yang, "Confocal three dimensional tracking of a single nanoparticle with concurrent spectroscopic readout," Appl. Phys. Lett. 88, 223901 (2006).
[CrossRef]

H.-J. Wu, W. Everett, S. Anekal, and M. Bevan, "Mapping Patterned Potential Energy Landscapes with Diffusing Colloidal Probes," Langmuir 22, 6826-6836 (2006).
[CrossRef] [PubMed]

D. Weihs, T. G. Mason, and M. A. Teitell, "Bio-Microrheology: A Frontier in Microrheology," Biophys. J. 91, 4296-4305 (2006).
[CrossRef] [PubMed]

2005

X. Michalet, F. F. Pinaud, L. A. Bentolila, J. M. Tsay, S. Doose, J. J. Li, G. Sundaresan, A. M. Wu, S. S. Gambhir, and S. Weiss, "Quantum Dots for Live Cells, in Vivo Imaging, and Diagnostics," Science 307, 538-544 (2005).
[CrossRef] [PubMed]

A. J. Berglund and H. Mabuchi, "Tracking-FCS: Fluorescence Correlation Spectroscopy of Individual Particles," Opt. Express 13, 8069-8082 (2005).
[CrossRef] [PubMed]

B. Carter, G. Shubeita, and S. Gross, "Tracking single particles: a user-friendly quantitative evaluation," Phys. Biol. 2, 60-72 (2005).
[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]

A. E. Cohen and W. E. Moerner, "Method for Trapping and Manipulating Nanoscale Objects in Solution," Appl. Phys. Lett. 86, 093109 (2005).
[CrossRef]

2003

A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goodman, and P. R. Selvin, "Myosin V walks hand-overhand: Single fluorophore imaging with 1.5-nm localization," Science 300, 2061-2065 (2003).
[CrossRef] [PubMed]

2002

R. Thompson, D. Larson, and W. Webb, "Precise Nanometer Localization Analysis for Individual Fluorescent Probes," Biophys. J. 82, 2775-2783 (2002).
[CrossRef] [PubMed]

2001

M. Cheezum, W. Walker, and W. Guilford, "Quantitative Comparison of Algorithms for Tracking Single Fluorescent Particles," Biophys. J. 81, 2378-2388 (2001).
[CrossRef] [PubMed]

1997

M. J. Saxton and K. Jacobson, "Single-particle tracking: applications to membrane dynamics," Annu. Rev. Biophys. Biomol. Struct. 26, 373-399 (1997).
[CrossRef] [PubMed]

1996

J. Crocker and D. Grier, "Methods of Digital Video Microscopy for Colloidal Studies," J. Colloid Interface Sci. 179, 298-310 (1996).
[CrossRef]

1986

N. Bobroff, "Position measurement with a resolution and noise-limited instrument," Rev. Sci. Instrum. 57, 1152 (1986).
[CrossRef]

Anekal, S.

H.-J. Wu, W. Everett, S. Anekal, and M. Bevan, "Mapping Patterned Potential Energy Landscapes with Diffusing Colloidal Probes," Langmuir 22, 6826-6836 (2006).
[CrossRef] [PubMed]

Armani, M.

M. Armani, S. Chaudhary, R. Probst, and B. Shapiro, "Using feedback control of microflows to independently steer multiple particles," IEEE J. Microelectromech. Syst. 15, 945-956 (2006).
[CrossRef]

Bahukudumbi, P.

P. Bahukudumbi and M. A. Bevan, "Imaging energy landscapes with concentrated diffusing colloidal probes," J. Chem. Phys. 126, 244702 (2007).
[CrossRef] [PubMed]

Bentolila, L. A.

X. Michalet, F. F. Pinaud, L. A. Bentolila, J. M. Tsay, S. Doose, J. J. Li, G. Sundaresan, A. M. Wu, S. S. Gambhir, and S. Weiss, "Quantum Dots for Live Cells, in Vivo Imaging, and Diagnostics," Science 307, 538-544 (2005).
[CrossRef] [PubMed]

Berglund, A. J.

K. McHale, A. J. Berglund, and H. Mabuchi, "Quantum dot photon statistics measured by three-dimensional particle tracking," Nano Lett. 7, 3535-3539 (2007).
[CrossRef] [PubMed]

A. J. Berglund and H. Mabuchi, "Tracking-FCS: Fluorescence Correlation Spectroscopy of Individual Particles," Opt. Express 13, 8069-8082 (2005).
[CrossRef] [PubMed]

Bevan, M.

H.-J. Wu, W. Everett, S. Anekal, and M. Bevan, "Mapping Patterned Potential Energy Landscapes with Diffusing Colloidal Probes," Langmuir 22, 6826-6836 (2006).
[CrossRef] [PubMed]

Bevan, M. A.

P. Bahukudumbi and M. A. Bevan, "Imaging energy landscapes with concentrated diffusing colloidal probes," J. Chem. Phys. 126, 244702 (2007).
[CrossRef] [PubMed]

Bobroff, N.

N. Bobroff, "Position measurement with a resolution and noise-limited instrument," Rev. Sci. Instrum. 57, 1152 (1986).
[CrossRef]

Cang, H.

H. Cang, C. M. Wong, C. S. Xu, A. H. Rizvi, and H. Yang, "Confocal three dimensional tracking of a single nanoparticle with concurrent spectroscopic readout," Appl. Phys. Lett. 88, 223901 (2006).
[CrossRef]

Carter, B.

B. Carter, G. Shubeita, and S. Gross, "Tracking single particles: a user-friendly quantitative evaluation," Phys. Biol. 2, 60-72 (2005).
[CrossRef] [PubMed]

Chaudhary, S.

M. Armani, S. Chaudhary, R. Probst, and B. Shapiro, "Using feedback control of microflows to independently steer multiple particles," IEEE J. Microelectromech. Syst. 15, 945-956 (2006).
[CrossRef]

Cheezum, M.

M. Cheezum, W. Walker, and W. Guilford, "Quantitative Comparison of Algorithms for Tracking Single Fluorescent Particles," Biophys. J. 81, 2378-2388 (2001).
[CrossRef] [PubMed]

Cohen, A. E.

A. E. Cohen and W. E. Moerner, "Method for Trapping and Manipulating Nanoscale Objects in Solution," Appl. Phys. Lett. 86, 093109 (2005).
[CrossRef]

Crocker, J.

J. Crocker and D. Grier, "Methods of Digital Video Microscopy for Colloidal Studies," J. Colloid Interface Sci. 179, 298-310 (1996).
[CrossRef]

Doose, S.

X. Michalet, F. F. Pinaud, L. A. Bentolila, J. M. Tsay, S. Doose, J. J. Li, G. Sundaresan, A. M. Wu, S. S. Gambhir, and S. Weiss, "Quantum Dots for Live Cells, in Vivo Imaging, and Diagnostics," Science 307, 538-544 (2005).
[CrossRef] [PubMed]

Dufresne, E. R.

S. K. Sainis, V. Germain, and E. R. Dufresne, "Statistics of Particle Trajectories at Short Time Intervals Reveal fN-Scale Colloidal Forces," Phys. Rev. Lett. 99, 018303 (2007).
[CrossRef] [PubMed]

Everett, W.

H.-J. Wu, W. Everett, S. Anekal, and M. Bevan, "Mapping Patterned Potential Energy Landscapes with Diffusing Colloidal Probes," Langmuir 22, 6826-6836 (2006).
[CrossRef] [PubMed]

Forkey, J. N.

A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goodman, and P. R. Selvin, "Myosin V walks hand-overhand: Single fluorophore imaging with 1.5-nm localization," Science 300, 2061-2065 (2003).
[CrossRef] [PubMed]

Gambhir, S. S.

X. Michalet, F. F. Pinaud, L. A. Bentolila, J. M. Tsay, S. Doose, J. J. Li, G. Sundaresan, A. M. Wu, S. S. Gambhir, and S. Weiss, "Quantum Dots for Live Cells, in Vivo Imaging, and Diagnostics," Science 307, 538-544 (2005).
[CrossRef] [PubMed]

Germain, V.

S. K. Sainis, V. Germain, and E. R. Dufresne, "Statistics of Particle Trajectories at Short Time Intervals Reveal fN-Scale Colloidal Forces," Phys. Rev. Lett. 99, 018303 (2007).
[CrossRef] [PubMed]

Goodman, Y. E.

A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goodman, and P. R. Selvin, "Myosin V walks hand-overhand: Single fluorophore imaging with 1.5-nm localization," Science 300, 2061-2065 (2003).
[CrossRef] [PubMed]

Gratton, E.

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]

Grier, D.

J. Crocker and D. Grier, "Methods of Digital Video Microscopy for Colloidal Studies," J. Colloid Interface Sci. 179, 298-310 (1996).
[CrossRef]

Gross, S.

B. Carter, G. Shubeita, and S. Gross, "Tracking single particles: a user-friendly quantitative evaluation," Phys. Biol. 2, 60-72 (2005).
[CrossRef] [PubMed]

Guilford, W.

M. Cheezum, W. Walker, and W. Guilford, "Quantitative Comparison of Algorithms for Tracking Single Fluorescent Particles," Biophys. J. 81, 2378-2388 (2001).
[CrossRef] [PubMed]

Ha, T.

A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goodman, and P. R. Selvin, "Myosin V walks hand-overhand: Single fluorophore imaging with 1.5-nm localization," Science 300, 2061-2065 (2003).
[CrossRef] [PubMed]

Jacobson, K.

M. J. Saxton and K. Jacobson, "Single-particle tracking: applications to membrane dynamics," Annu. Rev. Biophys. Biomol. Struct. 26, 373-399 (1997).
[CrossRef] [PubMed]

Larson, D.

R. Thompson, D. Larson, and W. Webb, "Precise Nanometer Localization Analysis for Individual Fluorescent Probes," Biophys. J. 82, 2775-2783 (2002).
[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]

Li, J. J.

X. Michalet, F. F. Pinaud, L. A. Bentolila, J. M. Tsay, S. Doose, J. J. Li, G. Sundaresan, A. M. Wu, S. S. Gambhir, and S. Weiss, "Quantum Dots for Live Cells, in Vivo Imaging, and Diagnostics," Science 307, 538-544 (2005).
[CrossRef] [PubMed]

Mabuchi, H.

K. McHale, A. J. Berglund, and H. Mabuchi, "Quantum dot photon statistics measured by three-dimensional particle tracking," Nano Lett. 7, 3535-3539 (2007).
[CrossRef] [PubMed]

A. J. Berglund and H. Mabuchi, "Tracking-FCS: Fluorescence Correlation Spectroscopy of Individual Particles," Opt. Express 13, 8069-8082 (2005).
[CrossRef] [PubMed]

Mason, T. G.

D. Weihs, T. G. Mason, and M. A. Teitell, "Bio-Microrheology: A Frontier in Microrheology," Biophys. J. 91, 4296-4305 (2006).
[CrossRef] [PubMed]

McHale, K.

K. McHale, A. J. Berglund, and H. Mabuchi, "Quantum dot photon statistics measured by three-dimensional particle tracking," Nano Lett. 7, 3535-3539 (2007).
[CrossRef] [PubMed]

McKinney, S. A.

A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goodman, and P. R. Selvin, "Myosin V walks hand-overhand: Single fluorophore imaging with 1.5-nm localization," Science 300, 2061-2065 (2003).
[CrossRef] [PubMed]

Michalet, X.

X. Michalet, F. F. Pinaud, L. A. Bentolila, J. M. Tsay, S. Doose, J. J. Li, G. Sundaresan, A. M. Wu, S. S. Gambhir, and S. Weiss, "Quantum Dots for Live Cells, in Vivo Imaging, and Diagnostics," Science 307, 538-544 (2005).
[CrossRef] [PubMed]

Moerner, W. E.

A. E. Cohen and W. E. Moerner, "Method for Trapping and Manipulating Nanoscale Objects in Solution," Appl. Phys. Lett. 86, 093109 (2005).
[CrossRef]

Pinaud, F. F.

X. Michalet, F. F. Pinaud, L. A. Bentolila, J. M. Tsay, S. Doose, J. J. Li, G. Sundaresan, A. M. Wu, S. S. Gambhir, and S. Weiss, "Quantum Dots for Live Cells, in Vivo Imaging, and Diagnostics," Science 307, 538-544 (2005).
[CrossRef] [PubMed]

Probst, R.

M. Armani, S. Chaudhary, R. Probst, and B. Shapiro, "Using feedback control of microflows to independently steer multiple particles," IEEE J. Microelectromech. Syst. 15, 945-956 (2006).
[CrossRef]

Rizvi, A. H.

H. Cang, C. M. Wong, C. S. Xu, A. H. Rizvi, and H. Yang, "Confocal three dimensional tracking of a single nanoparticle with concurrent spectroscopic readout," Appl. Phys. Lett. 88, 223901 (2006).
[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]

Sainis, S. K.

S. K. Sainis, V. Germain, and E. R. Dufresne, "Statistics of Particle Trajectories at Short Time Intervals Reveal fN-Scale Colloidal Forces," Phys. Rev. Lett. 99, 018303 (2007).
[CrossRef] [PubMed]

Saxton, M. J.

M. J. Saxton and K. Jacobson, "Single-particle tracking: applications to membrane dynamics," Annu. Rev. Biophys. Biomol. Struct. 26, 373-399 (1997).
[CrossRef] [PubMed]

Selvin, P. R.

A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goodman, and P. R. Selvin, "Myosin V walks hand-overhand: Single fluorophore imaging with 1.5-nm localization," Science 300, 2061-2065 (2003).
[CrossRef] [PubMed]

Shapiro, B.

M. Armani, S. Chaudhary, R. Probst, and B. Shapiro, "Using feedback control of microflows to independently steer multiple particles," IEEE J. Microelectromech. Syst. 15, 945-956 (2006).
[CrossRef]

Shubeita, G.

B. Carter, G. Shubeita, and S. Gross, "Tracking single particles: a user-friendly quantitative evaluation," Phys. Biol. 2, 60-72 (2005).
[CrossRef] [PubMed]

Sundaresan, G.

X. Michalet, F. F. Pinaud, L. A. Bentolila, J. M. Tsay, S. Doose, J. J. Li, G. Sundaresan, A. M. Wu, S. S. Gambhir, and S. Weiss, "Quantum Dots for Live Cells, in Vivo Imaging, and Diagnostics," Science 307, 538-544 (2005).
[CrossRef] [PubMed]

Teitell, M. A.

D. Weihs, T. G. Mason, and M. A. Teitell, "Bio-Microrheology: A Frontier in Microrheology," Biophys. J. 91, 4296-4305 (2006).
[CrossRef] [PubMed]

Thompson, R.

R. Thompson, D. Larson, and W. Webb, "Precise Nanometer Localization Analysis for Individual Fluorescent Probes," Biophys. J. 82, 2775-2783 (2002).
[CrossRef] [PubMed]

Tsay, J. M.

X. Michalet, F. F. Pinaud, L. A. Bentolila, J. M. Tsay, S. Doose, J. J. Li, G. Sundaresan, A. M. Wu, S. S. Gambhir, and S. Weiss, "Quantum Dots for Live Cells, in Vivo Imaging, and Diagnostics," Science 307, 538-544 (2005).
[CrossRef] [PubMed]

Walker, W.

M. Cheezum, W. Walker, and W. Guilford, "Quantitative Comparison of Algorithms for Tracking Single Fluorescent Particles," Biophys. J. 81, 2378-2388 (2001).
[CrossRef] [PubMed]

Webb, W.

R. Thompson, D. Larson, and W. Webb, "Precise Nanometer Localization Analysis for Individual Fluorescent Probes," Biophys. J. 82, 2775-2783 (2002).
[CrossRef] [PubMed]

Weihs, D.

D. Weihs, T. G. Mason, and M. A. Teitell, "Bio-Microrheology: A Frontier in Microrheology," Biophys. J. 91, 4296-4305 (2006).
[CrossRef] [PubMed]

Weiss, S.

X. Michalet, F. F. Pinaud, L. A. Bentolila, J. M. Tsay, S. Doose, J. J. Li, G. Sundaresan, A. M. Wu, S. S. Gambhir, and S. Weiss, "Quantum Dots for Live Cells, in Vivo Imaging, and Diagnostics," Science 307, 538-544 (2005).
[CrossRef] [PubMed]

Wong, C. M.

H. Cang, C. M. Wong, C. S. Xu, A. H. Rizvi, and H. Yang, "Confocal three dimensional tracking of a single nanoparticle with concurrent spectroscopic readout," Appl. Phys. Lett. 88, 223901 (2006).
[CrossRef]

Wu, A. M.

X. Michalet, F. F. Pinaud, L. A. Bentolila, J. M. Tsay, S. Doose, J. J. Li, G. Sundaresan, A. M. Wu, S. S. Gambhir, and S. Weiss, "Quantum Dots for Live Cells, in Vivo Imaging, and Diagnostics," Science 307, 538-544 (2005).
[CrossRef] [PubMed]

Wu, H.-J.

H.-J. Wu, W. Everett, S. Anekal, and M. Bevan, "Mapping Patterned Potential Energy Landscapes with Diffusing Colloidal Probes," Langmuir 22, 6826-6836 (2006).
[CrossRef] [PubMed]

Xu, C. S.

H. Cang, C. M. Wong, C. S. Xu, A. H. Rizvi, and H. Yang, "Confocal three dimensional tracking of a single nanoparticle with concurrent spectroscopic readout," Appl. Phys. Lett. 88, 223901 (2006).
[CrossRef]

Yang, H.

H. Cang, C. M. Wong, C. S. Xu, A. H. Rizvi, and H. Yang, "Confocal three dimensional tracking of a single nanoparticle with concurrent spectroscopic readout," Appl. Phys. Lett. 88, 223901 (2006).
[CrossRef]

Yildiz, A.

A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goodman, and P. R. Selvin, "Myosin V walks hand-overhand: Single fluorophore imaging with 1.5-nm localization," Science 300, 2061-2065 (2003).
[CrossRef] [PubMed]

Annu. Rev. Biophys. Biomol. Struct.

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Figures (4)

Fig. 1.
Fig. 1.

Cartoon illustration of the iterative VWCM algorithm operating on an image of a particle with a constant background. The background biases the center-of-mass (CM) estimate towards the center of the array. The first, biased centroid estimate (yellow) is offset by δ 1, x from the array center. At the second iteration, the window is truncated by an amount 2δ 1, x along x and a new centroid (blue) is calculated within this window. Where part of a pixel is truncated by the “virtual window,” its value is scaled proportionally to the relative area. At each iteration, the window is further adjusted until the center of the window and the CM estimate coincide, giving a bias-free estimate of the particle position.

Fig. 2.
Fig. 2.

Simulation results showing the bias and localization accuracy achieved for 21×21 pixel images using the Gaussian fit (GF), Gaussian mask (GM), CM and VWCM algorithms. For each image, the underlying position was varied over a grid of positions spanning the center pixel (Δ=123 nm, dashed squares). The resulting distribution of estimated positions is displayed as a circle centered at the mean with 1σ radius as defined in the text. “N.C.” denotes instances when the Gaussian fit and mask algorithms either failed to converge or had large errors with the resulting 1σ circles larger than the pixel. Image details (top row): The first four images represent the (non-paraxial) point-spread function of a dipole emitter [24] with wavelength λ=550 nm, at depths z=0 nm, 500 nm, 750 nm and 1000 nm imaged through a microscope with magnification M=60 and numerical aperture 1.2. The final image is a simple rod shape. Each image has 〈�� S 〉=〈�� B 〉=5000 photons.

Fig. 3.
Fig. 3.

Positions of a single 190 nm fluorescent microparticle stepped in 100 nm increments as estimated by the Gaussian fit (GF, red), CM (green), and VWCM (blue). Results are plotted as in Fig. 2, but with 2σ radii for better visibility. Note the extreme distortion introduced by bias in the CM algorithm. As a typical example, we quote the following results for the upper left position: the bias along x (measured from the Gaussian fit estimate) Bx , standard deviations along x x ) and algorithm execution time (T). GF: Bx =0 nm, σ x =1.0 nm, T=66 ms. CM: Bx =127 nm, σ x =0.6 nm, T=0.2 ms. VWCM: Bx =-1.3 nm, σ x =2.2 nm, T=2.7 ms.

Fig. 4.
Fig. 4.

VWCM position estimates (blue squares) for a binary aggregate of two microparticles stepped in 100 nm increments parallel to the image plane and 1 µm increments out of the focal plane. Approximately 30 images were taken at each position. Error bars represent one standard deviation in the estimated positions, and are inside the data points for some cases. The standard deviation along x ranged between 4.0 nm and 5.6 nm for this series. The red circles are a guide to the eye, spaced at 100 nm intervals.

Equations (12)

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S jk = P jk dxdy Δ 2 [ N S ( x x 0 , y y 0 ) + N B ( x , y ) ]
S jk S j k = S jk S j k + δ jj δ kk P j k dxdy Δ 2 [ N S ( x x 0 , y y 0 ) + σ B 2 ( x , y ) ] .
x ̂ CM = jk x jk S jk jk S jk .
x ̂ C M = 1 𝓝 j k x j k S j k , 𝓝 = j k S j k .
x ̂ C M = 1 𝓝 jk { x jk P jk dxdy Δ 2 [ N S ( x x 0 , y y 0 ) + N B ( x, y ) ] }
x ̂ C M 2 x ̂ C M 2 = 1 𝓝 2 jk { x jk 2 P jk dxdy Δ 2 [ N S ( x− x 0 , y y 0 ) + σ B 2 ( x , y ) ] } .
B CM x ̂ CM x 0
= 1 jk N S Δ ( x jk x 0 , y jk y 0 ) [ jk ( x jk x 0 ) N S Δ ( x jk x 0 , y jk y 0 ) + N B jk ( x jk x ̂ CM ) ] .
jk ( x jk x 0 ) N S Δ ( x jk x 0 , y jk y 0 ) dxdy Δ 2 x N S ( x , y ) = 0 .
B CM 𝓝 B 𝓝 S ( x ¯ x ̂ CM )
B CM ( n ) = ( 1 1 + 𝓝 S 𝓝 B ) n 1 B CM ( 1 )
f ( x , y ) = A exp [ ( x x 0 ) 2 2 σ x 2 ( y y 0 ) 2 2 σ y 2 ] + B

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