Abstract

Fluorescence microscopy is a photon-limited imaging modality that allows the study of subcellular objects and processes with high specificity. The best possible accuracy (standard deviation) with which an object of interest can be localized when imaged using a fluorescence microscope is typically calculated using the Cramér-Rao lower bound, that is, the inverse of the Fisher information. However, the current approach for the calculation of the best possible localization accuracy relies on an analytical expression for the image of the object. This can pose practical challenges since it is often difficult to find appropriate analytical models for the images of general objects. In this study, we instead develop an approach that directly uses an experimentally collected image set to calculate the best possible localization accuracy for a general subcellular object. In this approach, we fit splines, i.e. smoothly connected piecewise polynomials, to the experimentally collected image set to provide a continuous model of the object, which can then be used for the calculation of the best possible localization accuracy. Due to its practical importance, we investigate in detail the application of the proposed approach in single molecule fluorescence microscopy. In this case, the object of interest is a point source and, therefore, the acquired image set pertains to an experimental point spread function.

© 2015 Optical Society of America

Full Article  |  PDF Article
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References

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2015 (1)

R. J. Ober, A. Tahmasbi, S. Ram, Z. Lin, and E. S. Ward, “Quantitative aspects of single molecule microscopy: Information-theoretic analysis of single-molecule data,” IEEE Signal Process. Mag. 32(1), 58–69 (2015).
[Crossref]

2014 (5)

2013 (1)

2012 (3)

J. Chao, E. S. Ward, and R. J. Ober, “Fisher information matrix for branching processes with application to electron-multiplying charge-coupled devices,” Multidim. Sys. Sig. Proc. 23, 349–379 (2012).
[Crossref]

S. Quirin, S. R. P. Pavani, and R. Piestun, “Optimal 3D single-molecule localization for superresolution microscopy with aberrations and engineered point spread functions,” Proc. Natl. Acad. Sci. USA 109(3), 675–679 (2012).
[Crossref] [PubMed]

X. Michalet and A. J. Berglund, “Optimal diffusion coefficient estimation in single-particle tracking,” Phys. Rev. E 85, 061916 (2012).
[Crossref]

2010 (2)

C. S. Smith, N. Joseph, B. Rieger, and K. A. Lidke, “Fast, single-molecule localization that achieves theoretically minimum uncertainty,” Nat. Methods 7, 373–375 (2010).
[Crossref] [PubMed]

M. Badieirostami, M. D. Lew, M. A. Thompson, and W. E. Moerner, “Three-dimensional localization precision of the double-helix point spread function versus astigmatism and biplane,” Appl. Phys. Lett. 97, 161103 (2010).
[Crossref] [PubMed]

2009 (3)

2008 (3)

2007 (1)

2006 (1)

S. Ram, E. S. Ward, and R. J. Ober, “A stochastic analysis of performance limits for optical microscopes,” Multidim. Sys. Sig. Proc. 17, 27–57 (2006).
[Crossref]

2005 (2)

M. Arigovindan, M. Suhling, P. Hunziker, and M. Unser, “Variational image reconstruction from arbitrarily spaced samples: a fast multiresolution spline solution,” IEEE Trans. Image Process. 14, 450–460 (2005).
[Crossref] [PubMed]

X. Lai, Z. Lin, E. S. Ward, and R. J. Ober, “Noise suppression of point spread functions and its influence on deconvolution of three-dimensional fluorescence microscopy image sets,” J. Microsc. 217, 93–108 (2005).
[Crossref] [PubMed]

2004 (1)

R. J. Ober, S. Ram, and E. S. Ward, “Localization accuracy in single-molecule microscopy,” Biophys. J. 86, 1185–1200 (2004).
[Crossref] [PubMed]

2003 (1)

W. E. Moerner and D. P. Fromm, “Methods of single-molecule fluorescence spectroscopy and microscopy,” Rev. Sci. Instrum. 74, 3597–3619 (2003).
[Crossref]

2002 (1)

R. E. Campbell, O. Tour, A. E. Palmer, P. A. Steinbach, G. S. Baird, D. A. Zacharias, and R. Y. Tsien, “A monomeric red fluorescent protein,” Proc. Natl. Acad. Sci. USA 99, 7877–7882 (2002).
[Crossref] [PubMed]

1999 (1)

M. Unser, “Splines: a perfect fit for signal and image processing,” IEEE Signal Process. Mag. 16, 22–38 (1999).
[Crossref]

1993 (1)

M. Unser, A. Aldroubi, and M. Eden, “B-spline signal processing: part I–theory,” IEEE Trans. Signal Process.,  41, 821–832 (1993).
[Crossref]

1992 (1)

1969 (1)

I. J. Schoenberg, “Cardinal interpolation and spline functions,” J. Approx. Theory 2, 167–206 (1969).
[Crossref]

1964 (1)

I. J. Schoenberg, “Spline functions and the problem of graduation,” Proc. Natl. Acad. Sci. USA 52, 947–950 (1964).
[Crossref] [PubMed]

Abraham, A. V.

Aguet, F.

Albero, J.

Aldroubi, A.

M. Unser, A. Aldroubi, and M. Eden, “B-spline signal processing: part I–theory,” IEEE Trans. Signal Process.,  41, 821–832 (1993).
[Crossref]

Arigovindan, M.

M. Arigovindan, M. Suhling, P. Hunziker, and M. Unser, “Variational image reconstruction from arbitrarily spaced samples: a fast multiresolution spline solution,” IEEE Trans. Image Process. 14, 450–460 (2005).
[Crossref] [PubMed]

Backer, A. S.

Y. Shechtman, S. J. Sahl, A. S. Backer, and W. E. Moerner, “Optimal point spread function design for 3D imaging,” Phys. Rev. Lett. 113, 133902 (2014).
[Crossref] [PubMed]

Badieirostami, M.

M. Badieirostami, M. D. Lew, M. A. Thompson, and W. E. Moerner, “Three-dimensional localization precision of the double-helix point spread function versus astigmatism and biplane,” Appl. Phys. Lett. 97, 161103 (2010).
[Crossref] [PubMed]

Baird, G. S.

R. E. Campbell, O. Tour, A. E. Palmer, P. A. Steinbach, G. S. Baird, D. A. Zacharias, and R. Y. Tsien, “A monomeric red fluorescent protein,” Proc. Natl. Acad. Sci. USA 99, 7877–7882 (2002).
[Crossref] [PubMed]

Baranski, M.

Bargiel, S.

Berglund, A. J.

X. Michalet and A. J. Berglund, “Optimal diffusion coefficient estimation in single-particle tracking,” Phys. Rev. E 85, 061916 (2012).
[Crossref]

Bewersdorf, J.

Born, M.

M. Born and E. Wolf, Principles of Optics, 7. (Cambridge University, Cambridge, UK, 2002).

Campbell, R. E.

R. E. Campbell, O. Tour, A. E. Palmer, P. A. Steinbach, G. S. Baird, D. A. Zacharias, and R. Y. Tsien, “A monomeric red fluorescent protein,” Proc. Natl. Acad. Sci. USA 99, 7877–7882 (2002).
[Crossref] [PubMed]

Chao, J.

A. Tahmasbi, S. Ram, J. Chao, A. V. Abraham, F. W. Tang, E. S. Ward, and R. J. Ober, “Designing the focal plane spacing for multifocal plane microscopy,” Opt. Express 22(14), 16706–16721 (2014).
[Crossref] [PubMed]

J. Chao, E. S. Ward, and R. J. Ober, “Fisher information matrix for branching processes with application to electron-multiplying charge-coupled devices,” Multidim. Sys. Sig. Proc. 23, 349–379 (2012).
[Crossref]

A. V. Abraham, S. Ram, J. Chao, E. S. Ward, and R. J. Ober, “Quantitative study of single molecule location estimation techniques,” Opt. Express 17, 23352–23373 (2009).
[Crossref]

S. Ram, P. Prabhat, J. Chao, E. S. Ward, and R. J. Ober, “High accuracy 3D quantum dot tracking with multifocal plane microscopy for the study of fast intracellular dynamics in live cells,” Biophys. J. 95, 6025–6043 (2008).
[Crossref] [PubMed]

Claxton, C.

De Boor, C.

C. De Boor, A Practical Guide to Splines (Springer Verlag, NY, USA, 2001), Rev. ed.

Eden, M.

M. Unser, A. Aldroubi, and M. Eden, “B-spline signal processing: part I–theory,” IEEE Trans. Signal Process.,  41, 821–832 (1993).
[Crossref]

Froehly, L.

Fromm, D. P.

W. E. Moerner and D. P. Fromm, “Methods of single-molecule fluorescence spectroscopy and microscopy,” Rev. Sci. Instrum. 74, 3597–3619 (2003).
[Crossref]

Geissbuhler, S.

Gibson, S. F.

Gorecki, C.

Holm, T.

Hunziker, P.

M. Arigovindan, M. Suhling, P. Hunziker, and M. Unser, “Variational image reconstruction from arbitrarily spaced samples: a fast multiresolution spline solution,” IEEE Trans. Image Process. 14, 450–460 (2005).
[Crossref] [PubMed]

Joseph, N.

C. S. Smith, N. Joseph, B. Rieger, and K. A. Lidke, “Fast, single-molecule localization that achieves theoretically minimum uncertainty,” Nat. Methods 7, 373–375 (2010).
[Crossref] [PubMed]

Kao, F. J.

P. Torok and F. J. Kao, Optical Imaging and Microscopy (Springer Verlag, NY, USA, 2003).
[Crossref]

Kay, S. M.

S. M. Kay, Fundamentals of Statistical Signal Processing, Volume I: Estimation Theory (Prentice Hall, Upper Saddle River, NJ, USA, 1993).

Klein, T.

Kromann, E.

Krueger, W.

Lai, X.

X. Lai, Z. Lin, E. S. Ward, and R. J. Ober, “Noise suppression of point spread functions and its influence on deconvolution of three-dimensional fluorescence microscopy image sets,” J. Microsc. 217, 93–108 (2005).
[Crossref] [PubMed]

Lanni, F.

Lasser, T.

Lew, M. D.

M. Badieirostami, M. D. Lew, M. A. Thompson, and W. E. Moerner, “Three-dimensional localization precision of the double-helix point spread function versus astigmatism and biplane,” Appl. Phys. Lett. 97, 161103 (2010).
[Crossref] [PubMed]

Lidke, K.

Lidke, K. A.

C. S. Smith, N. Joseph, B. Rieger, and K. A. Lidke, “Fast, single-molecule localization that achieves theoretically minimum uncertainty,” Nat. Methods 7, 373–375 (2010).
[Crossref] [PubMed]

Lin, Z.

R. J. Ober, A. Tahmasbi, S. Ram, Z. Lin, and E. S. Ward, “Quantitative aspects of single molecule microscopy: Information-theoretic analysis of single-molecule data,” IEEE Signal Process. Mag. 32(1), 58–69 (2015).
[Crossref]

X. Lai, Z. Lin, E. S. Ward, and R. J. Ober, “Noise suppression of point spread functions and its influence on deconvolution of three-dimensional fluorescence microscopy image sets,” J. Microsc. 217, 93–108 (2005).
[Crossref] [PubMed]

Liu, S.

Marki, I.

Michalet, X.

X. Michalet and A. J. Berglund, “Optimal diffusion coefficient estimation in single-particle tracking,” Phys. Rev. E 85, 061916 (2012).
[Crossref]

Miller, M. I.

D. L. Snyder and M. I. Miller, Random Point Processes in Time and Space (Springer Verlag, NY, USA, 1991), 2.
[Crossref]

Moerner, W. E.

Y. Shechtman, S. J. Sahl, A. S. Backer, and W. E. Moerner, “Optimal point spread function design for 3D imaging,” Phys. Rev. Lett. 113, 133902 (2014).
[Crossref] [PubMed]

M. Badieirostami, M. D. Lew, M. A. Thompson, and W. E. Moerner, “Three-dimensional localization precision of the double-helix point spread function versus astigmatism and biplane,” Appl. Phys. Lett. 97, 161103 (2010).
[Crossref] [PubMed]

W. E. Moerner and D. P. Fromm, “Methods of single-molecule fluorescence spectroscopy and microscopy,” Rev. Sci. Instrum. 74, 3597–3619 (2003).
[Crossref]

Ober, R. J.

R. J. Ober, A. Tahmasbi, S. Ram, Z. Lin, and E. S. Ward, “Quantitative aspects of single molecule microscopy: Information-theoretic analysis of single-molecule data,” IEEE Signal Process. Mag. 32(1), 58–69 (2015).
[Crossref]

A. Tahmasbi, S. Ram, J. Chao, A. V. Abraham, F. W. Tang, E. S. Ward, and R. J. Ober, “Designing the focal plane spacing for multifocal plane microscopy,” Opt. Express 22(14), 16706–16721 (2014).
[Crossref] [PubMed]

J. Chao, E. S. Ward, and R. J. Ober, “Fisher information matrix for branching processes with application to electron-multiplying charge-coupled devices,” Multidim. Sys. Sig. Proc. 23, 349–379 (2012).
[Crossref]

A. V. Abraham, S. Ram, J. Chao, E. S. Ward, and R. J. Ober, “Quantitative study of single molecule location estimation techniques,” Opt. Express 17, 23352–23373 (2009).
[Crossref]

S. Ram, P. Prabhat, E. S. Ward, and R. J. Ober, “Improved single particle localization accuracy with dual objective multifocal plane microscopy,” Opt. Express 17, 6881–6898 (2009).
[Crossref] [PubMed]

S. Ram, P. Prabhat, J. Chao, E. S. Ward, and R. J. Ober, “High accuracy 3D quantum dot tracking with multifocal plane microscopy for the study of fast intracellular dynamics in live cells,” Biophys. J. 95, 6025–6043 (2008).
[Crossref] [PubMed]

S. Ram, E. S. Ward, and R. J. Ober, “A stochastic analysis of performance limits for optical microscopes,” Multidim. Sys. Sig. Proc. 17, 27–57 (2006).
[Crossref]

X. Lai, Z. Lin, E. S. Ward, and R. J. Ober, “Noise suppression of point spread functions and its influence on deconvolution of three-dimensional fluorescence microscopy image sets,” J. Microsc. 217, 93–108 (2005).
[Crossref] [PubMed]

R. J. Ober, S. Ram, and E. S. Ward, “Localization accuracy in single-molecule microscopy,” Biophys. J. 86, 1185–1200 (2004).
[Crossref] [PubMed]

S. Ram, E. S. Ward, and R. J. Ober, “How accurately can a single molecule be localized in three dimensions using a fluorescence microscope?” Proc. SPIE5699, 426–435 (2005).
[Crossref]

Olivo-Marin, J.-C.

Palmer, A. E.

R. E. Campbell, O. Tour, A. E. Palmer, P. A. Steinbach, G. S. Baird, D. A. Zacharias, and R. Y. Tsien, “A monomeric red fluorescent protein,” Proc. Natl. Acad. Sci. USA 99, 7877–7882 (2002).
[Crossref] [PubMed]

Passilly, N.

Pavani, S. R. P.

S. Quirin, S. R. P. Pavani, and R. Piestun, “Optimal 3D single-molecule localization for superresolution microscopy with aberrations and engineered point spread functions,” Proc. Natl. Acad. Sci. USA 109(3), 675–679 (2012).
[Crossref] [PubMed]

S. R. P. Pavani and R. Piestun, “Three dimensional tracking of fluorescent microparticles using a photon-limited double-helix response system,” Opt. Express 16, 22048–22057 (2008).
[Crossref] [PubMed]

Perrin, S.

Piestun, R.

S. Quirin, S. R. P. Pavani, and R. Piestun, “Optimal 3D single-molecule localization for superresolution microscopy with aberrations and engineered point spread functions,” Proc. Natl. Acad. Sci. USA 109(3), 675–679 (2012).
[Crossref] [PubMed]

S. R. P. Pavani and R. Piestun, “Three dimensional tracking of fluorescent microparticles using a photon-limited double-helix response system,” Opt. Express 16, 22048–22057 (2008).
[Crossref] [PubMed]

Prabhat, P.

S. Ram, P. Prabhat, E. S. Ward, and R. J. Ober, “Improved single particle localization accuracy with dual objective multifocal plane microscopy,” Opt. Express 17, 6881–6898 (2009).
[Crossref] [PubMed]

S. Ram, P. Prabhat, J. Chao, E. S. Ward, and R. J. Ober, “High accuracy 3D quantum dot tracking with multifocal plane microscopy for the study of fast intracellular dynamics in live cells,” Biophys. J. 95, 6025–6043 (2008).
[Crossref] [PubMed]

Proppert, S.

Quirin, S.

S. Quirin, S. R. P. Pavani, and R. Piestun, “Optimal 3D single-molecule localization for superresolution microscopy with aberrations and engineered point spread functions,” Proc. Natl. Acad. Sci. USA 109(3), 675–679 (2012).
[Crossref] [PubMed]

Ram, S.

R. J. Ober, A. Tahmasbi, S. Ram, Z. Lin, and E. S. Ward, “Quantitative aspects of single molecule microscopy: Information-theoretic analysis of single-molecule data,” IEEE Signal Process. Mag. 32(1), 58–69 (2015).
[Crossref]

A. Tahmasbi, S. Ram, J. Chao, A. V. Abraham, F. W. Tang, E. S. Ward, and R. J. Ober, “Designing the focal plane spacing for multifocal plane microscopy,” Opt. Express 22(14), 16706–16721 (2014).
[Crossref] [PubMed]

A. V. Abraham, S. Ram, J. Chao, E. S. Ward, and R. J. Ober, “Quantitative study of single molecule location estimation techniques,” Opt. Express 17, 23352–23373 (2009).
[Crossref]

S. Ram, P. Prabhat, E. S. Ward, and R. J. Ober, “Improved single particle localization accuracy with dual objective multifocal plane microscopy,” Opt. Express 17, 6881–6898 (2009).
[Crossref] [PubMed]

S. Ram, P. Prabhat, J. Chao, E. S. Ward, and R. J. Ober, “High accuracy 3D quantum dot tracking with multifocal plane microscopy for the study of fast intracellular dynamics in live cells,” Biophys. J. 95, 6025–6043 (2008).
[Crossref] [PubMed]

S. Ram, E. S. Ward, and R. J. Ober, “A stochastic analysis of performance limits for optical microscopes,” Multidim. Sys. Sig. Proc. 17, 27–57 (2006).
[Crossref]

R. J. Ober, S. Ram, and E. S. Ward, “Localization accuracy in single-molecule microscopy,” Biophys. J. 86, 1185–1200 (2004).
[Crossref] [PubMed]

S. Ram, E. S. Ward, and R. J. Ober, “How accurately can a single molecule be localized in three dimensions using a fluorescence microscope?” Proc. SPIE5699, 426–435 (2005).
[Crossref]

Rieger, B.

C. S. Smith, N. Joseph, B. Rieger, and K. A. Lidke, “Fast, single-molecule localization that achieves theoretically minimum uncertainty,” Nat. Methods 7, 373–375 (2010).
[Crossref] [PubMed]

Sahl, S. J.

Y. Shechtman, S. J. Sahl, A. S. Backer, and W. E. Moerner, “Optimal point spread function design for 3D imaging,” Phys. Rev. Lett. 113, 133902 (2014).
[Crossref] [PubMed]

Sauer, M.

Schoenberg, I. J.

I. J. Schoenberg, “Cardinal interpolation and spline functions,” J. Approx. Theory 2, 167–206 (1969).
[Crossref]

I. J. Schoenberg, “Spline functions and the problem of graduation,” Proc. Natl. Acad. Sci. USA 52, 947–950 (1964).
[Crossref] [PubMed]

Shechtman, Y.

Y. Shechtman, S. J. Sahl, A. S. Backer, and W. E. Moerner, “Optimal point spread function design for 3D imaging,” Phys. Rev. Lett. 113, 133902 (2014).
[Crossref] [PubMed]

Small, A.

A. Small and S. Stahlheber, “Fluorophore localization algorithms for super-resolution microscopy,” Nat. Methods 11(3), 267–279 (2014).
[Crossref] [PubMed]

Smith, C. S.

C. S. Smith, N. Joseph, B. Rieger, and K. A. Lidke, “Fast, single-molecule localization that achieves theoretically minimum uncertainty,” Nat. Methods 7, 373–375 (2010).
[Crossref] [PubMed]

Snyder, D. L.

D. L. Snyder and M. I. Miller, Random Point Processes in Time and Space (Springer Verlag, NY, USA, 1991), 2.
[Crossref]

Stahlheber, S.

A. Small and S. Stahlheber, “Fluorophore localization algorithms for super-resolution microscopy,” Nat. Methods 11(3), 267–279 (2014).
[Crossref] [PubMed]

Staunton, R.

Steinbach, P. A.

R. E. Campbell, O. Tour, A. E. Palmer, P. A. Steinbach, G. S. Baird, D. A. Zacharias, and R. Y. Tsien, “A monomeric red fluorescent protein,” Proc. Natl. Acad. Sci. USA 99, 7877–7882 (2002).
[Crossref] [PubMed]

Suhling, M.

M. Arigovindan, M. Suhling, P. Hunziker, and M. Unser, “Variational image reconstruction from arbitrarily spaced samples: a fast multiresolution spline solution,” IEEE Trans. Image Process. 14, 450–460 (2005).
[Crossref] [PubMed]

Tahmasbi, A.

R. J. Ober, A. Tahmasbi, S. Ram, Z. Lin, and E. S. Ward, “Quantitative aspects of single molecule microscopy: Information-theoretic analysis of single-molecule data,” IEEE Signal Process. Mag. 32(1), 58–69 (2015).
[Crossref]

A. Tahmasbi, S. Ram, J. Chao, A. V. Abraham, F. W. Tang, E. S. Ward, and R. J. Ober, “Designing the focal plane spacing for multifocal plane microscopy,” Opt. Express 22(14), 16706–16721 (2014).
[Crossref] [PubMed]

Tang, F. W.

Thompson, M. A.

M. Badieirostami, M. D. Lew, M. A. Thompson, and W. E. Moerner, “Three-dimensional localization precision of the double-helix point spread function versus astigmatism and biplane,” Appl. Phys. Lett. 97, 161103 (2010).
[Crossref] [PubMed]

Torok, P.

P. Torok and F. J. Kao, Optical Imaging and Microscopy (Springer Verlag, NY, USA, 2003).
[Crossref]

Tour, O.

R. E. Campbell, O. Tour, A. E. Palmer, P. A. Steinbach, G. S. Baird, D. A. Zacharias, and R. Y. Tsien, “A monomeric red fluorescent protein,” Proc. Natl. Acad. Sci. USA 99, 7877–7882 (2002).
[Crossref] [PubMed]

Tsien, R. Y.

R. E. Campbell, O. Tour, A. E. Palmer, P. A. Steinbach, G. S. Baird, D. A. Zacharias, and R. Y. Tsien, “A monomeric red fluorescent protein,” Proc. Natl. Acad. Sci. USA 99, 7877–7882 (2002).
[Crossref] [PubMed]

Unser, M.

F. Aguet, S. Geissbuhler, I. Marki, T. Lasser, and M. Unser, “Super-resolution orientation estimation and localization of fluorescent dipoles using 3-D steerable filters,” Opt. Express 17, 6829–6848 (2009).
[Crossref] [PubMed]

M. Arigovindan, M. Suhling, P. Hunziker, and M. Unser, “Variational image reconstruction from arbitrarily spaced samples: a fast multiresolution spline solution,” IEEE Trans. Image Process. 14, 450–460 (2005).
[Crossref] [PubMed]

M. Unser, “Splines: a perfect fit for signal and image processing,” IEEE Signal Process. Mag. 16, 22–38 (1999).
[Crossref]

M. Unser, A. Aldroubi, and M. Eden, “B-spline signal processing: part I–theory,” IEEE Trans. Signal Process.,  41, 821–832 (1993).
[Crossref]

van de Linde, S.

Ward, E. S.

R. J. Ober, A. Tahmasbi, S. Ram, Z. Lin, and E. S. Ward, “Quantitative aspects of single molecule microscopy: Information-theoretic analysis of single-molecule data,” IEEE Signal Process. Mag. 32(1), 58–69 (2015).
[Crossref]

A. Tahmasbi, S. Ram, J. Chao, A. V. Abraham, F. W. Tang, E. S. Ward, and R. J. Ober, “Designing the focal plane spacing for multifocal plane microscopy,” Opt. Express 22(14), 16706–16721 (2014).
[Crossref] [PubMed]

J. Chao, E. S. Ward, and R. J. Ober, “Fisher information matrix for branching processes with application to electron-multiplying charge-coupled devices,” Multidim. Sys. Sig. Proc. 23, 349–379 (2012).
[Crossref]

A. V. Abraham, S. Ram, J. Chao, E. S. Ward, and R. J. Ober, “Quantitative study of single molecule location estimation techniques,” Opt. Express 17, 23352–23373 (2009).
[Crossref]

S. Ram, P. Prabhat, E. S. Ward, and R. J. Ober, “Improved single particle localization accuracy with dual objective multifocal plane microscopy,” Opt. Express 17, 6881–6898 (2009).
[Crossref] [PubMed]

S. Ram, P. Prabhat, J. Chao, E. S. Ward, and R. J. Ober, “High accuracy 3D quantum dot tracking with multifocal plane microscopy for the study of fast intracellular dynamics in live cells,” Biophys. J. 95, 6025–6043 (2008).
[Crossref] [PubMed]

S. Ram, E. S. Ward, and R. J. Ober, “A stochastic analysis of performance limits for optical microscopes,” Multidim. Sys. Sig. Proc. 17, 27–57 (2006).
[Crossref]

X. Lai, Z. Lin, E. S. Ward, and R. J. Ober, “Noise suppression of point spread functions and its influence on deconvolution of three-dimensional fluorescence microscopy image sets,” J. Microsc. 217, 93–108 (2005).
[Crossref] [PubMed]

R. J. Ober, S. Ram, and E. S. Ward, “Localization accuracy in single-molecule microscopy,” Biophys. J. 86, 1185–1200 (2004).
[Crossref] [PubMed]

S. Ram, E. S. Ward, and R. J. Ober, “How accurately can a single molecule be localized in three dimensions using a fluorescence microscope?” Proc. SPIE5699, 426–435 (2005).
[Crossref]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7. (Cambridge University, Cambridge, UK, 2002).

Wolter, S.

Zacharias, D. A.

R. E. Campbell, O. Tour, A. E. Palmer, P. A. Steinbach, G. S. Baird, D. A. Zacharias, and R. Y. Tsien, “A monomeric red fluorescent protein,” Proc. Natl. Acad. Sci. USA 99, 7877–7882 (2002).
[Crossref] [PubMed]

Zerubia, J.

Zhang, B.

Appl. Opt. (1)

Appl. Phys. Lett. (1)

M. Badieirostami, M. D. Lew, M. A. Thompson, and W. E. Moerner, “Three-dimensional localization precision of the double-helix point spread function versus astigmatism and biplane,” Appl. Phys. Lett. 97, 161103 (2010).
[Crossref] [PubMed]

Biophys. J. (2)

S. Ram, P. Prabhat, J. Chao, E. S. Ward, and R. J. Ober, “High accuracy 3D quantum dot tracking with multifocal plane microscopy for the study of fast intracellular dynamics in live cells,” Biophys. J. 95, 6025–6043 (2008).
[Crossref] [PubMed]

R. J. Ober, S. Ram, and E. S. Ward, “Localization accuracy in single-molecule microscopy,” Biophys. J. 86, 1185–1200 (2004).
[Crossref] [PubMed]

IEEE Signal Process. Mag. (2)

R. J. Ober, A. Tahmasbi, S. Ram, Z. Lin, and E. S. Ward, “Quantitative aspects of single molecule microscopy: Information-theoretic analysis of single-molecule data,” IEEE Signal Process. Mag. 32(1), 58–69 (2015).
[Crossref]

M. Unser, “Splines: a perfect fit for signal and image processing,” IEEE Signal Process. Mag. 16, 22–38 (1999).
[Crossref]

IEEE Trans. Image Process. (1)

M. Arigovindan, M. Suhling, P. Hunziker, and M. Unser, “Variational image reconstruction from arbitrarily spaced samples: a fast multiresolution spline solution,” IEEE Trans. Image Process. 14, 450–460 (2005).
[Crossref] [PubMed]

IEEE Trans. Signal Process. (1)

M. Unser, A. Aldroubi, and M. Eden, “B-spline signal processing: part I–theory,” IEEE Trans. Signal Process.,  41, 821–832 (1993).
[Crossref]

J. Approx. Theory (1)

I. J. Schoenberg, “Cardinal interpolation and spline functions,” J. Approx. Theory 2, 167–206 (1969).
[Crossref]

J. Microsc. (1)

X. Lai, Z. Lin, E. S. Ward, and R. J. Ober, “Noise suppression of point spread functions and its influence on deconvolution of three-dimensional fluorescence microscopy image sets,” J. Microsc. 217, 93–108 (2005).
[Crossref] [PubMed]

J. Opt. Soc. Am. A (2)

Multidim. Sys. Sig. Proc. (2)

J. Chao, E. S. Ward, and R. J. Ober, “Fisher information matrix for branching processes with application to electron-multiplying charge-coupled devices,” Multidim. Sys. Sig. Proc. 23, 349–379 (2012).
[Crossref]

S. Ram, E. S. Ward, and R. J. Ober, “A stochastic analysis of performance limits for optical microscopes,” Multidim. Sys. Sig. Proc. 17, 27–57 (2006).
[Crossref]

Nat. Methods (2)

C. S. Smith, N. Joseph, B. Rieger, and K. A. Lidke, “Fast, single-molecule localization that achieves theoretically minimum uncertainty,” Nat. Methods 7, 373–375 (2010).
[Crossref] [PubMed]

A. Small and S. Stahlheber, “Fluorophore localization algorithms for super-resolution microscopy,” Nat. Methods 11(3), 267–279 (2014).
[Crossref] [PubMed]

Opt. Express (8)

S. R. P. Pavani and R. Piestun, “Three dimensional tracking of fluorescent microparticles using a photon-limited double-helix response system,” Opt. Express 16, 22048–22057 (2008).
[Crossref] [PubMed]

F. Aguet, S. Geissbuhler, I. Marki, T. Lasser, and M. Unser, “Super-resolution orientation estimation and localization of fluorescent dipoles using 3-D steerable filters,” Opt. Express 17, 6829–6848 (2009).
[Crossref] [PubMed]

S. Ram, P. Prabhat, E. S. Ward, and R. J. Ober, “Improved single particle localization accuracy with dual objective multifocal plane microscopy,” Opt. Express 17, 6881–6898 (2009).
[Crossref] [PubMed]

A. V. Abraham, S. Ram, J. Chao, E. S. Ward, and R. J. Ober, “Quantitative study of single molecule location estimation techniques,” Opt. Express 17, 23352–23373 (2009).
[Crossref]

S. Liu, E. Kromann, W. Krueger, J. Bewersdorf, and K. Lidke, “Three dimensional single molecule localization using a phase retrieved pupil function,” Opt. Express 21, 29462–29487 (2013).
[Crossref]

S. Proppert, S. Wolter, T. Holm, T. Klein, S. van de Linde, and M. Sauer, “Cubic B-spline calibration for 3D super-resolution measurements using astigmatic imaging,” Opt. Express 22(9), 10304–10316 (2014).
[Crossref] [PubMed]

M. Baranski, S. Perrin, N. Passilly, L. Froehly, J. Albero, S. Bargiel, and C. Gorecki, “A simple method for quality evaluation of micro-optical components based on 3D IPSF measurement,” Opt. Express 22, 13202–13212 (2014).
[Crossref] [PubMed]

A. Tahmasbi, S. Ram, J. Chao, A. V. Abraham, F. W. Tang, E. S. Ward, and R. J. Ober, “Designing the focal plane spacing for multifocal plane microscopy,” Opt. Express 22(14), 16706–16721 (2014).
[Crossref] [PubMed]

Phys. Rev. E (1)

X. Michalet and A. J. Berglund, “Optimal diffusion coefficient estimation in single-particle tracking,” Phys. Rev. E 85, 061916 (2012).
[Crossref]

Phys. Rev. Lett. (1)

Y. Shechtman, S. J. Sahl, A. S. Backer, and W. E. Moerner, “Optimal point spread function design for 3D imaging,” Phys. Rev. Lett. 113, 133902 (2014).
[Crossref] [PubMed]

Proc. Natl. Acad. Sci. USA (3)

S. Quirin, S. R. P. Pavani, and R. Piestun, “Optimal 3D single-molecule localization for superresolution microscopy with aberrations and engineered point spread functions,” Proc. Natl. Acad. Sci. USA 109(3), 675–679 (2012).
[Crossref] [PubMed]

R. E. Campbell, O. Tour, A. E. Palmer, P. A. Steinbach, G. S. Baird, D. A. Zacharias, and R. Y. Tsien, “A monomeric red fluorescent protein,” Proc. Natl. Acad. Sci. USA 99, 7877–7882 (2002).
[Crossref] [PubMed]

I. J. Schoenberg, “Spline functions and the problem of graduation,” Proc. Natl. Acad. Sci. USA 52, 947–950 (1964).
[Crossref] [PubMed]

Rev. Sci. Instrum. (1)

W. E. Moerner and D. P. Fromm, “Methods of single-molecule fluorescence spectroscopy and microscopy,” Rev. Sci. Instrum. 74, 3597–3619 (2003).
[Crossref]

Other (6)

S. M. Kay, Fundamentals of Statistical Signal Processing, Volume I: Estimation Theory (Prentice Hall, Upper Saddle River, NJ, USA, 1993).

D. L. Snyder and M. I. Miller, Random Point Processes in Time and Space (Springer Verlag, NY, USA, 1991), 2.
[Crossref]

P. Torok and F. J. Kao, Optical Imaging and Microscopy (Springer Verlag, NY, USA, 2003).
[Crossref]

C. De Boor, A Practical Guide to Splines (Springer Verlag, NY, USA, 2001), Rev. ed.

M. Born and E. Wolf, Principles of Optics, 7. (Cambridge University, Cambridge, UK, 2002).

S. Ram, E. S. Ward, and R. J. Ober, “How accurately can a single molecule be localized in three dimensions using a fluorescence microscope?” Proc. SPIE5699, 426–435 (2005).
[Crossref]

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Figures (8)

Fig. 1
Fig. 1

A plot of the symmetrical B-spline of degree d for d = 0,1,2,3, with unit element spacing, which will lead to nearest neighborhood, linear, quadratic and cubic interpolation of the experimental image set, respectively. The vertical dotted lines show the pixel boundaries and the B-splines are located at the center of pixel 3.

Fig. 2
Fig. 2

Verification of the approach in idealized imaging conditions for a 2D PSF. (a) The image of a point source simulated using the Airy profile for a 100x, NA 1.4 objective lens with an emission wavelength of 690 nm. (b) An experimental PSF simulated using the model image in (a) in the absence of stochasticity and noise. The pixel size and the detector size are 7 μm × 7 μm and 21 × 21 pixels, respectively. (c) The bicubic spline fit of the simulated experimental PSF in (b). (d) The line profiles that pass through the peaks of the analytical PSF and the spline fit, and the error between the line profiles. (e) The deduced experimental x0-PLAM and its corresponding analytical PLAM as a function of the effective pixel size in the object space (the results for y0-PLAM are similar due to the radial symmetry of the PSF and are omitted). The absolute difference error is also shown.

Fig. 3
Fig. 3

Verification of the approach in idealized imaging conditions for a 3D PSF. (a), (b) xy-(at z0 = 0.8 μm) and xz-projections of an experimental PSF simulated using the Born and Wolf 3D PSF model for a 100x, NA 1.4 objective lens and an emission wavelength of 690 nm in the absence of stochasticity and noise. The refractive index of the immersion oil noil is set to 1.515. The pixel size, the z-step size and the detector size are 13 μm 13 μm, 50 nm and 13 × 13 pixels, respectively. (c), (d) xy and xz projections of the cubic × volume spline fit of the simulated experimental PSF. (e) The error between the analytical PSF and the spline fit evaluated at the pixels and at the z-steps. (f) The deduced experimental x0-PLAM and its corresponding analytical PLAM as a function of the z0 position of the single molecule in the object space (the results for y0-PLAM are similar and are omitted). The error is also shown. (g) The deduced experimental z0-PLAM and its corresponding analytical PLAM as a function of z0, and the error between them. For the calculation of the PLAM, we assumed a background level of b = 10 photons/pixel and a photon count of N = 500 photons.

Fig. 4
Fig. 4

The performance of the approach in the presence of stochasticity and noise. (a), (a′) An xy-projection (at z0 = 0.9 μm) of the Born and Wolf 3D PSF model for a 100x, NA 1.4 objective lens and an emission wavelength of 690 nm. The refractive index of the immersion oil noil is set to 1.515. (b), (b′) The corresponding xy-projection of the experimental PSF simulated using the model in panels (a) and (a′) in the presence of stochasticity and noise, where the standard deviation σ c is 10 e/pixel and the photon count Nc is 10000 photons. The pixel size, the z-step size and the detector size are the same as those in Fig. 3. (c), (c′) The xy-projection of the cubic volume spline fit of the simulated experimental PSF, where the smoothing factor γ is set to 0.01. (d), (d′) The error between xy-projections of the analytical PSF model and the simulated experimental PSF (i.e. data) and between the analytical PSF model and the spline fit, respectively, evaluated at the pixels. (e), (f) The deduced experimental x0-PLAM and z0-PLAM and their corresponding analytical PLAMs as functions of the z0 position of the single molecule in the object space. The errors are also shown. For the calculation of the PLAM, we assumed a background level of b = 10 photons/pixel and a photon count of N = 500 photons.

Fig. 5
Fig. 5

The effect of the B-spline degree and the smoothing factor on the error between analytical and experimental PLAMs in the presence of noise. (a), (b) The root mean square percentage error (see Eq. (17) for the definition) between the experimental and analytical PLAMs as functions of the standard deviation σ c of the readout noise and background level bc, respectively, for different B-spline degrees d. (c), (d) The same for different smoothing factors γ. In (a) and (b) the smoothing factor is set to 0.01 and in (c) and (d), the B-spline degree is set to 3. Each data point is the average of the errors for x0-PLAM, y0-PLAM and z0-PLAM calculated over a z-range of [0.2, 0.9] μm. For the calculation of the PLAMs, we assumed a background level of b = 10 photons/pixel and a photon count of N = 500 photons. Other parameters are the same as those used in Fig. 4.

Fig. 6
Fig. 6

A practical example. (a), (b) The yz-projection and the xy-projection (at z0 = 2.6 μm) of a deliberately aberrated experimentally collected PSF from a practical microscopy setup, respectively, where the ROI size is 33 × 33 pixels (for information regarding other parameters see Section 2.1). (a′), (b′) The corresponding yz- and xy-projections of the cubic volume spline fit with a smoothing factor of γ = 0.01, which is evaluated on a finer grid (color scale bars are in photons). The vertical dashed lines show the location of the plane of focus and the size bars are 1.5 μm (panels (a) and (a′) are stretched in the z-direction for better visualization while their scale in the y-direction is the same as panels (b) and (b′)). The estimated photon count and background level of the bead sample are approximately Nc = 4500 photons and bc = 16 photons/pixel, respectively. (c), (d) The experimental x0-PLAM and z0-PLAM, respectively, along the z-axis (the reported results are the average of the results for multiple beads). For the calculation of the experimental PLAMs we assumed N = 500 photons and b = 10 photons/pixel.

Fig. 7
Fig. 7

The experimental PLAM for a spherical shell. (a) The xy-projection at z0 = 0 μm, (b) the yz-projection (z0 ∈ [−0.1, 2.2] μm) and (c) the xy-projection at z0 = 1.8 μm of the simulated 3D image set of a spherical shell with internal and external radii of 1.5 μm and 1.8 μm, respectively, where the ROI size is 49 × 49 pixels. The image set was obtained by convolving the simulated object with the Born and Wolf 3D PSF model. We considered Poisson statistics, background and readout noise, where the standard deviation σ c is 4 e/pixel, the background level bc is 10 photons/pixel and the photon count Nc is 60000 photons. We assumed a 100x, NA 1.4 objective lens with noil = 1.515. The pixel size, the z-step size and the emission wavelength are 13 μm × 13 μm, 50 nm and 690 nm, respectively. (a′), (b′) and (c′) The corresponding xy-, yz- and xy-projections of the cubic volume spline fit with a smoothing factor of γ = 0.015, which is evaluated on a finer grid (color scale bars are in photons). The focal plane is located at 0 μm and the size bars are 1 μm (panels (b) and (b′) are stretched in the z-direction for better visualization while their scale in the y-direction is the same as panel (a)). (d), (e) The experimental x0-PLAM (y0-PLAM) and z0-PLAM, respectively, along the z-axis. For the calculation of the experimental PLAMs, we assumed a background level of b = 20 photons/pixel and a photon count of N = 5000 photons.

Fig. 8
Fig. 8

The experimental PLAM for lysosomes. (a) The image of a 22Rv1 cell transfected with mRFP-LAMP-1 which was acquired as described in Section 2.2. (b) and (c) The images of two individual lysosomal compartments marked by arrows in panel (a). All of the imaging parameters are reported in Section 2.2. (b’) and (c’) The corresponding cubic surface spline fit of the lysosomes with a smoothing factor of γ = 0.01, which is evaluated on a finer grid (color scale bars are in photons). The size bar in panel (a) is 3.8 μm and other size bars are 0.645 μm. (d) The experimental x0-PLAM and y0-PLAM for the lysosomes. For the calculation of the experimental PLAMs, we assumed a background level of b = 20 photons/pixel and a photon count of N = 5000 photons.

Equations (52)

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μ θ ( k ) = N M 2 C k q z 0 ( x M x 0 , y M y 0 ) d r , k = 1 , , K pix ,
q z 0 ( x , y ) = A | 0 1 J 0 ( 2 π n a λ ρ x 2 + y 2 ) e j π n a 2 z 0 λ n oil ρ 2 ρ d ρ | 2 , ( x , y ) 2 ,
I ( θ ) = k = 1 K pix α ( k ) ν θ ( k ) ( ν θ ( k ) θ ) T ν θ ( k ) θ , θ Θ ,
α ( k ) : = ν θ ( k ) ( e ν θ ( k ) 2 π σ k ( l = 1 ν θ l 1 ( k ) ( l 1 ) ! e ( z l η k ) 2 2 σ k 2 ) 2 l = 0 ν θ l ( k ) l ! e ( z l η k ) 2 2 σ k 2 d z 1 ) , k = 1 , , K pix ,
H k , p ~ P o i s s o n ( N c M 2 C k q z 0 , z p ( x M x 0 , y M y 0 ) d x d y + b k , p c ) * N ( 0 , σ k 2 , c ) ,
s a d ( x , y , z ) : = m = 1 K row n = 1 K col p = 1 K stk a m , n , p β d ( x Δ x 0 n ) β d ( y Δ y 0 m ) β d ( z Δ z 0 p ) , ( x , y , z ) 3 ,
β d ( x ) : = i = 0 d + 1 ( 1 ) i d ! ( d + 1 i ) ( x + d + 1 2 i ) d u ( x + d + 1 2 i ) , x ,
u ( x ) = { 1 , x 0 0 , x < 0 .
C k s a d ( x M x 0 , y M y 0 , z p z 0 ) d x d y h k , p b k , p c , k = 1 , , K pix , p = 1 , , K stk ,
k = 1 K pix p = 1 K stk | C k s a d ( x M x 0 , y M y 0 , z p z 0 ) d x d y ( h k , p b k , p c ) | 2 .
h : = ( h 1 , 1 b 1 , 1 c , , h K pix , 1 b K pix , 1 c , h 1 , 2 b 1 , 2 c , , h K pix , K stk b K pix , K stk c ) T K ,
a : = ( a 1 , 1 , 1 , , a K row , 1 , 1 , a 1 , 2 , 1 , , a K row , K col , 1 , a 1 , 1 , 2 , , a K row , K col , K stk ) T K ,
S k + ( i 1 ) K pix , m + ( n 1 ) K row + ( p 1 ) K pix = C k β d ( x M x 0 Δ x 0 n ) β d ( y M y 0 Δ y 0 m ) d r β d ( z i z 0 Δ z 0 p ) , k = 1 , , K pix , i , p = 1 , , K stk , m = 1 , , K row , n = 1 , , K col ,
ε ( a ) : = h S a 2 , a K ,
ϕ l d ( a ) : = 3 D l s a d ( x , y , z ) 2 d x d y d z , a K , d 0 , l d ,
ϕ 1 d ( a ) = 3 [ ( s a d ( x , y , z ) x ) 2 + ( s a d ( x , y , z ) y ) 2 + ( s a d ( x , y , z ) z ) 2 ] d x d y d z , a K .
ϕ l d ( a ) = 3 D l s d ( x , y , z ) 2 d x d y d z = q 1 + q 2 + q 3 = l ( l q 1 , q 2 , q 3 ) 3 ( l s a d ( x , y , z ) x q 1 y q 2 z q 3 ) 2 d x d y d z .
ϕ l d ( a ) = m , m = 1 K row n , n = 1 K col p , p = 1 K stk a m , n , p a m , n , p × q 1 + q 2 + q 3 = l ( l q 1 , q 2 , q 3 ) B Δ x 0 q 1 ( n , n ) B Δ y 0 q 2 ( m , m ) B Δ z 0 q 3 ( p , p ) , a K ,
B Δ q ( n , n ) : = q t q β d ( t Δ n ) q t q β d ( t Δ n ) d t , n , n , q = 1 , , l , Δ > 0.
B ( p 1 ) K pix + ( n 1 ) K row + m , ( p 1 ) K pix + ( n 1 ) K row + m = q 1 + q 2 + q 3 = l ( l q 1 , q 2 , q 3 ) B Δ x 0 q 1 ( n , n ) B Δ y 0 q 2 ( m , m ) B Δ z 0 q 3 ( p , p ) .
ϕ ( a ) = a T B a , a K ,
a ^ = arg min a K ( ε ( a ) + γ ϕ ( a ) ) = arg min a K ( h S a 2 + γ a T B a ) ,
( S T S + γ B ) a ^ = S T h ,
C ( z 0 ) : = 2 s ^ a d ( x , y , z 0 ) d r = m , n , p a ^ m , n , p 2 β d ( x Δ x 0 n ) β d ( y Δ y 0 m ) d r β d ( z 0 Δ z 0 p ) = Δ x 0 Δ y 0 m , n , p a ^ m , n , p β d ( z 0 Δ z 0 p ) , z 0 ,
q ^ z 0 ( x , y ) : = s ^ a d ( x , y , z 0 ) C ( z 0 ) = m . n , p a ˜ m , n , p z 0 β d ( x Δ x 0 n ) β d ( y Δ y 0 m ) β d ( z 0 Δ z 0 p ) ,
μ θ ( k ) N M 2 C k q ^ z 0 ( x M x 0 , y M y 0 ) = N M 2 m , n , p a ˜ m , n , p z 0 C k β d ( x M x 0 Δ x 0 n ) β d ( y M y 0 Δ y 0 m ) d r β d ( z 0 Δ z 0 p ) .
β d ( x ) x = β d 1 ( x + 1 2 ) β d 1 ( x 1 2 ) , x , d .
μ θ ( k ) x 0 N M 2 m = 1 K row n = 1 K col + 1 p = 1 K stk a ˜ m , n , p z 0 a ˜ m , n 1 , p z 0 Δ x 0 β d ( z 0 Δ z 0 p ) × C k β d 1 ( x M x 0 Δ x 0 n + 1 2 ) β d ( y M y 0 Δ y 0 m ) d r ,
μ θ ( k ) y 0 N M 2 m = 1 K row + 1 n = 1 K col p = 1 K stk a ˜ m , n , p z 0 a ˜ m 1 , n , p z 0 Δ y 0 β d ( z 0 Δ z 0 p ) × C k β d ( x M x 0 Δ x 0 n ) β d 1 ( y M y 0 Δ y 0 m + 1 2 ) d r ,
μ θ ( k ) z 0 N M 2 m = 1 K row n = 1 K col p = 1 K stk + 1 a ˜ m , n , p z 0 a ˜ m , n , p 1 z 0 Δ y 0 β d 1 ( z 0 Δ z 0 p + 1 2 ) × C k β d ( x M x 0 Δ x 0 n ) β d ( y M y 0 Δ y 0 m ) d r μ θ ( k ) ξ ( z 0 ) ,
ξ ( z 0 ) : = 1 C ( z 0 ) C ( z 0 ) z 0 = Δ x 0 Δ y 0 Δ z 0 m = 1 K row n = 1 K col p = 1 K stk + 1 ( a ˜ m , n , p z 0 a ˜ m , n , p 1 z 0 ) β d 1 ( z 0 Δ z 0 p + 1 2 ) .
ν θ ( k ) N = N ( μ θ ( k ) + b ) = μ θ ( k ) N , k = 1 , , K pix .
ν θ ( k ) b = b ( μ θ ( k ) + b ) = 1 , k = 1 , , K pix .
RMSPE : = 100 × ( 1 3 P i = 1 P ( x 0 - PLAM E ( Z i ) x 0 - PLAM A ( Z i ) x 0 - PLAM A ( Z i ) ) 2 + ( y 0 - PLAM E ( Z i ) y 0 - PLAM A ( Z i ) y 0 - PLAM A ( Z i ) ) 2 + ( z 0 - PLAM E ( Z i ) z 0 - PLAM A ( Z i ) z 0 - PLAM A ( Z i ) ) 2 ) 1 / 2 ,
ϕ l d ( a ) = 3 D l s a d ( x , y , z ) 2 d x d y d z = q 1 + q 2 + q 3 = 1 ( l q 1 , q 2 , q 3 ) 3 ( l s a d ( x , y , z ) z q 1 y q 2 z q 3 ) 2 d x d y d z = q 1 + q 2 + q 3 = l ( l q 1 , q 2 , q 3 ) 3 ( l x q 1 y q 2 z q 3 m , n , p a m , n , p × β d ( x Δ x 0 n ) β d ( y Δ y 0 m ) β d ( z Δ z 0 p ) ) 2 d x d y d z = q 1 + q 2 + q 3 = l ( l q 1 , q 2 , q 3 ) 3 ( m , n , p a m , n , p × q 1 x q 1 β d ( x Δ x 0 n ) q 2 y q 2 β d ( y Δ y 0 m ) q 3 z q 3 β d ( z Δ z 0 p ) ) 2 d x d y d z = q 1 + q 2 + q 3 = l ( l q 1 , q 2 , q 3 ) 3 ( m , n , p a m , n , p × q 1 x q 1 β d ( x Δ x 0 n ) q 2 y q 2 β d ( y Δ y 0 m ) q 3 z q 3 β d ( z Δ z 0 p ) ) 2 × ( m , n , p a m , n , p q 1 x q 1 β d ( x Δ x 0 n ) q 2 y q 2 β d ( y Δ y 0 m ) q 3 z q 3 β d ( z Δ z 0 p ) ) d x d y d z ,
ϕ l d ( a ) = m , m = 1 K row n , n = 1 K col p , p = 1 K stk a m , n , p a m , n , p q 1 , q 2 , q 3 ( l q 1 , q 2 , q 3 ) B Δ x 0 q 1 ( n , n ) B Δ y 0 q 2 ( m , m ) B Δ z 0 q 3 ( p , p ) ,
B Δ q ( n , n ) : = q t q β d ( t Δ n ) q t q β d ( t Δ n ) d t , n , n , q = 1 , , l , Δ > 0.
β d ( x Δ x 0 n ) d x = Δ x 0 β d ( v ) d v = Δ x 0 , n = 1 , , K col ,
β 0 ( x ) d x = j = 0 1 ( 1 ) j 1 ! ( 1 j ) ( x + 1 2 j ) 0 u ( x + 1 2 j ) d x = u ( x + 1 2 ) u ( x 1 2 ) d x = 1 2 1 2 d x = 1 .
β k + 1 ( x ) d x = β k ( x ) β 0 ( x ) d x = β k ( x ) d x × β 0 ( x ) d x = 1 × 1 = 1 ,
μ θ ( k ) x 0 x 0 N M 2 C k m , n , p a ˜ m , n , p z 0 β d ( x M x 0 Δ x 0 n ) β d ( y M y 0 Δ y 0 m ) β d ( z 0 Δ z 0 p ) d r = N M 2 C k m , n , p a ˜ m , n , p z 0 β d ( x M x 0 Δ x 0 n ) x 0 β d ( y M y 0 Δ y 0 m ) β d ( z 0 Δ z 0 p ) d r = N M 2 Δ x 0 p = 1 K s t k β d ( z 0 Δ z 0 p ) C k m = 1 K row β d ( y M y 0 Δ y 0 m ) × n = 1 K c o l a ˜ m , n , p z 0 ( β d 1 ( x M x 0 Δ x 0 n + 1 2 ) β d 1 ( x M x 0 Δ x 0 n 1 2 ) ) d r , T 1
T 1 = n = 1 K col a ˜ m , n , p z 0 β d 1 ( u n + 1 2 ) A n = 1 K col a ˜ m , n , p z 0 β d 1 ( u n 1 2 ) B ,
A = n = 1 K col a ˜ m , n , p z 0 β d 1 ( u n + 1 2 ) = n = 1 K col + 1 a ˜ m , n , p z 0 β d 1 ( u n + 1 2 ) .
B = j = 2 K col + 1 a ˜ m , j 1 , p z 0 β d 1 ( u ( j 1 ) 1 2 ) = j = 1 K col + 1 a ˜ m , j 1 , p z 0 β d 1 ( u j + 1 2 ) ,
T 1 = n = 1 K col + 1 ( a ˜ m , n , p z 0 a ˜ m , n 1 , p z 0 ) β d 1 ( u n + 1 2 ) .
μ θ ( k ) x 0 N M 2 m = 1 K row n = 1 K col + 1 p = 1 K stk a ˜ m , n , p z 0 a ˜ m , n 1 , p z 0 Δ x 0 β d ( z 0 Δ z 0 p ) × C k β d 1 ( x M x 0 Δ x 0 n + 1 2 ) β d ( y M y 0 Δ y 0 m ) d r ,
μ θ ( k ) z 0 z 0 N M 2 C k s ^ a d ( x M x 0 , y M y 0 , z 0 ) d r C ( z 0 ) = G θ ( k ) 1 C 2 ( z 0 ) C ( z 0 ) z 0 N C k s ^ a d ( x M x 0 , y M y 0 , z 0 ) d r M 2 = G θ ( k ) 1 C 2 ( z 0 ) C ( z 0 ) z 0 μ θ ( k ) , k = 1 , , K pix ,
G θ ( k ) : = N M 2 C ( z 0 ) C k z 0 s ^ a d ( x M x 0 , y M y 0 , z 0 ) d r , k = 1 , , K pix .
G θ ( k ) = N M 2 C ( z 0 ) C k m , n , p a ^ m , n , p β d ( x M x 0 Δ x 0 n ) β d ( y M y 0 Δ y 0 m ) β d ( z 0 Δ z 0 p ) z 0 d r = N M 2 C ( z 0 ) Δ z 0 C k m = 1 K row n = 1 K col β d ( x M x 0 Δ x 0 n ) β d ( y M y 0 Δ y 0 m ) d r × p = 1 K stk a ^ m , n , p ( β d 1 ( z 0 Δ z 0 p + 1 2 ) β d 1 ( z 0 Δ z 0 p 1 2 ) ) , T 2
T 2 = p = 1 K stk + 1 ( a ^ m , n , p a ^ m , n , p 1 ) β d 1 ( z 0 Δ z 0 p + 1 2 ) .
G θ ( k ) = N M 2 m = 1 K row n = 1 K col p = 1 K stk + 1 a ˜ m , n , p z 0 a ˜ m , n , p 1 z 0 Δ z 0 β d 1 ( z 0 Δ z 0 p + 1 2 ) × C k β d ( x M x 0 Δ x 0 n ) β d ( y M y 0 Δ y 0 m ) d r ,
1 C ( z 0 ) C ( z 0 ) z 0 = Δ x 0 Δ y 0 C ( z 0 ) m , n , p a ^ m , n , p z 0 β d ( z 0 Δ z 0 p ) = Δ x 0 Δ y 0 m , n , p a ˜ m , n , p z 0 z 0 β d ( z 0 Δ z 0 p ) = Δ x 0 Δ y 0 Δ z 0 m = 1 K row n = 1 K col p = 1 K stk a ˜ m , n , p z 0 ( β d 1 ( z 0 Δ z 0 p + 1 2 ) β d 1 ( z 0 Δ z 0 p 1 2 ) ) = Δ x 0 Δ y 0 Δ z 0 m = 1 K row n = 1 K col p = 1 K stk + 1 ( a ˜ m , n , p z 0 a ˜ m , n , p 1 z 0 ) β d 1 ( z 0 Δ z 0 p + 1 2 ) , z 0 ,

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