Abstract

A three-dimensional particle tracking technique, based on microscope off-focus images, was introduced in Z. Zhang and C.-H. Menq, Appl. Opt. 47, 2361 (2008) and applied to bright-field imaging. This paper presents two major improvements to the axial localization algorithm of the 3D particle tracking technique. First, it extends the algorithm to measure fluorescent particles in the presence of photobleaching and excitation variation. Second, it enhances the measurement resolution by achieving the best linear unbiased estimation of the particle’s axial position. Similarly to the original algorithm, a radius vector is first converted from the off-focus 2D image of the particle, and the axial position is estimated by comparing the radius vector with an object-specific model, calibrated automatically prior to each experiment. Although it was an intensity-based method, by normalizing the radius vectors the improved algorithm becomes a shape-based method, thus invariant to image intensity change and robust to photobleaching. Moreover, when considering the noise variance of each point in the radius vector and their correlations, the best linear unbiased estimation based on a linearized model is achieved. It is shown that variance equalization and correlation-weighted optimization greatly reduce the estimation variance and lead to near-uniform localization resolution over the entire measurement range. Estimation resolution is theoretically analyzed and validated by experiments. Theoretical analysis enables the prediction of measurement resolution based on calibration data. Finally, experimental results are presented to illustrate the performance of the measurement method in terms of measurement precision and range, as well as its robustness to intensity variation.

© 2009 Optical Society of America

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    [CrossRef]
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2008

2007

2006

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

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

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]

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]

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

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]

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]

2002

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]

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

2001

A. D. Dinsmore, E. R. Weeks, V. Prasad, A. C. Levitt, and D. A. Weitz, “Three-dimensional confocal microscopy of colloids,” Appl. Opt. 40, 4152-4159 (2001).
[CrossRef]

S. R. Buss and J. P. Fillmore, “Spherical averages and applications to spherical splines and interpolation,” ACM Trans. Graphics 20, s95-126 (2001).
[CrossRef]

2000

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

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

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]

1936

M. S. Bartlett, “The square root transformation in analysis of variance,” J. R. Stat. Soc. Suppl. , 3, 68-78 (1936).
[CrossRef]

Aguet, F.

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]

Bartlett, M. S.

M. S. Bartlett, “The square root transformation in analysis of variance,” J. R. Stat. Soc. Suppl. , 3, 68-78 (1936).
[CrossRef]

Bjorck, A.

A. Bjorck, Numerical Methods for Least Squares Problems (SIAM, 1996).
[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, and 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]

Buss, S. R.

S. R. Buss and J. P. Fillmore, “Spherical averages and applications to spherical splines and interpolation,” ACM Trans. Graphics 20, s95-126 (2001).
[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]

Fillmore, J. P.

S. R. Buss and J. P. Fillmore, “Spherical averages and applications to spherical splines and interpolation,” ACM Trans. Graphics 20, s95-126 (2001).
[CrossRef]

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]

Herman, B.

X. F. Wang and B. Herman, Fluorescence Imaging Spectroscopy and Microscopy (Wiley, 1996).

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]

Kariya, T.

T. Kariya and H. Kurata, Generalized Least Squares (Wiley, 2004).
[CrossRef]

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]

Kurata, H.

T. Kariya and H. Kurata, Generalized Least Squares (Wiley, 2004).
[CrossRef]

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).

Mendel, J. M.

J. M. Mendel, Lessons in Estimation Theory for Signal Processing, Communications, and Control (Prentice Hall PTR, 1995).

Menq, C.-H.

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]

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]

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]

Wang, X. F.

X. F. Wang and B. Herman, Fluorescence Imaging Spectroscopy and Microscopy (Wiley, 1996).

Weeks, E. 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]

Zemanek, P.

Zhang, Z.

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]

ACM Trans. Graphics

S. R. Buss and J. P. Fillmore, “Spherical averages and applications to spherical splines and interpolation,” ACM Trans. Graphics 20, s95-126 (2001).
[CrossRef]

Appl. Environ. Microbiol.

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.

Biophys. J.

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]

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]

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]

Exp. Fluids

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]

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).

J. Fluoresc.

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]

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[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

(a) Radial projection scheme, (b) 2D image of 0.75 μ m fluorescent particle with 120 × magnification, (c) radius vector of the image, (d) mesh plot of the calibrated model matrix (the radial distance is the radius index times 53.75 nm ; the axial position is the axial position index times 100 nm ).

Fig. 2
Fig. 2

(a) Normalized model matrix, (b) norm curve ψ ( z ) (the radial distance is the radius index times 53.75 nm ; the axial position is the axial position index times 100 nm ).

Fig. 3
Fig. 3

(a) Variance versus mean value of the intensities of two arbitrary pixels at different illumination intensity levels, (b) variance versus mean value of the square root of the pixel intensities at different illumination intensity levels.

Fig. 4
Fig. 4

Variance-equalized model matrix.

Fig. 5
Fig. 5

(a) Mesh plot of the normalized variance-equalized model matrix, (b) general norm curve ψ ¯ ( z ) (the radial distance is the radius index times 53.75 nm ; the axial position is the axial position index times 100 nm ).

Fig. 6
Fig. 6

Performance comparison of the NBLUE algorithm and the normalized algorithm.

Fig. 7
Fig. 7

Nanostepping along three axes.

Fig. 8
Fig. 8

Large-range triangular motion along three axes.

Fig. 9
Fig. 9

(a) Axial position estimation fluctuation, (b) estimated gain, (c) standard deviation of the fluctuation versus estimated gain. The blue dots are the standard deviations calculated using a moving window of 2000 points of the data shown in (a), and the red line is the fitted line through the origin.

Fig. 10
Fig. 10

Measured resolution divided by gain (dotted curve) and predicted resolution (solid curve).

Equations (21)

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m i = 1 k = 1 M i c i , k k = 1 M i c i , k I i , k , c i , 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 ) = 1 2 i = 0 N [ ρ i ( z ) m i ] 2 .
ρ i ( z ) = g m i ,
J ̂ ( z ) = 1 2 i = 0 N [ ρ ̂ i ( z ) m ̂ i ] 2 ,
g = ψ ( z * ) m ( z * ) .
σ m i 2 = k = 1 M i c i , k 2 ζ I i , k ( k = 1 M i c i , k ) 2 [ k = 1 M i c i , k 2 ( k = 1 M i c i , k ) 2 ] ζ m i .
σ I 2 = ζ 4 .
m ¯ i = 1 k = 1 M i c i , k 2 k = 1 M i c i , k I i , k , c i , 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 }
σ m ¯ i 2 = σ I 2 = ζ 4 ,
cov ( m ¯ ) = Ω ¯ σ m ¯ i 2 = ζ 4 Ω ¯ .
J ¯ ( z ) = 1 2 [ ρ ¯ ( z ) m ¯ ] T Ω ¯ 1 [ ρ ¯ ( z ) m ¯ ] ,
cov ( m ¯ i , m ¯ i + 1 ) ζ 4 = 1 k = 1 M i c i , k 2 1 l = 1 M i + 1 c i + 1 , l 2 k = 1 M i , i + 1 c i , k ( 1 c i , k ) ,
Ω ¯ [ 1 0.354 0.354 1 0.25 0.25 0.25 0.25 1 0.354 0.354 1 ] .
J ̃ ( z ) = 1 2 [ ρ ̃ ( z ) m ̃ ] T Ω ̃ 1 [ ρ ̃ ( z ) m ̃ ] ,
σ m ̃ i 2 σ m ¯ i 2 m ¯ Ω ¯ 1 2 = m ¯ Ω ¯ 1 2 σ I 2 ,
δ z n ̃ T Ω ̃ 1 ρ ̃ ( z ) ρ ̃ ( z ) T Ω ̃ 1 ρ ̃ ( z ) .
σ z 2 ( z ) 1 ρ ̃ ( z ) T Ω ̃ 1 ρ ̃ ( z ) σ n ̃ 2 ρ ̃ ( z ) Ω ¯ 1 2 m ¯ Ω ¯ 1 2 σ I 2 .
g ̃ = ψ ¯ ( z * ) m ¯ ( z * ) Ω ¯ 1 ,
σ z ( z ) = g ̃ S ̃ ( z ) σ I ,
S ̃ ( z ) = [ ρ ¯ ( z ) Ω ¯ 1 ρ ̃ ( z ) Ω ¯ 1 ] 1 .
S ̂ ( z ) = ρ ̂ ( z ) Ω ρ ( z ) 1 ρ ̂ ( z ) 2 ,

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