Abstract

Multifocal plane microscopy (MUM) has made it possible to study subcellular dynamics in 3D at high temporal and spatial resolution by simultaneously imaging distinct planes within the specimen. MUM allows high accuracy localization of a point source along the z-axis since it overcomes the depth discrimination problem of conventional single plane microscopy. An important question in MUM experiments is how the number of focal planes and their spacings should be chosen to achieve the best possible localization accuracy along the z-axis. Here, we propose approaches based on the Fisher information matrix and report spacing scenarios called strong coupling and weak coupling which yield an appropriate 3D localization accuracy. We examine the effect of numerical aperture, magnification, photon count, emission wavelength and extraneous noise on the spacing scenarios. In addition, we investigate the effect of changing the number of focal planes on the 3D localization accuracy. We also introduce a new software package that provides a user-friendly framework to find appropriate plane spacings for a MUM setup. These developments should assist in optimizing MUM experiments.

© 2014 Optical Society of America

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  1. P. Prabhat, S. Ram, E. S. Ward, and R. J. Ober, “Simultaneous imaging of different focal planes in fluorescence microscopy for the study of cellular dynamics in three dimensions,” IEEE Trans. Nanobiosci. 3, 237–242 (2004).
    [CrossRef]
  2. P. M. Blanchard and A. H. Greenaway, “Simultaneous multiplane imaging with a distorted diffraction grating,” Appl. Opt. 38, 6692–6699 (1999).
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  3. S. Ram, J. Chao, P. Prabhat, E. S. Ward, and R. J. Ober, “Overcoming the depth discrimination barrier in widefield microscopes: 3D single molecule tracking with high axial accuracy,” Proc. SPIE 6862, 68620O (2008).
    [CrossRef]
  4. L. Dehmelt and P. I. H. Bastiaens, “Spatial organization of intracellular communication: insights from imaging,” Nature Rev. Mol. Cell Biol. 11, 440–452 (2010).
    [CrossRef]
  5. 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]
  6. A. Jesacher, C. Roider, and M. Ritsch-Marte, “Enhancing diffractive multi-plane microscopy using colored illumination,” Opt. Express 21, 11150–11161 (2013).
    [CrossRef] [PubMed]
  7. S. Wolter, A. Loschberger, T. Holm, S. Aufmkolk, M. Dabauvalle, S. van de Linde, and M. Sauer, “rapidSTORM: accurate, fast open-source software for localization microscopy,” Nat. Methods 9, 1040–1041 (2012).
    [CrossRef] [PubMed]
  8. 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]
  9. H. Kirshner, F. Aguet, D. Sage, and M. Unser, “3-D PSF fitting for fluorescence microscopy: implementation and localization application,” J. Microsc. 249, 13–25 (2013).
    [CrossRef]
  10. P. A. Dalgarno, H. I. C. Dalgarno, A. Putoud, R. Lambert, L. Paterson, D. C. Logan, D. P. Towers, R. J. Warburton, and A. H. Greenaway, “Multiplane imaging and three dimensional nanoscale particle tracking in biological microscopy,” Opt. Express 18, 877–884 (2010).
    [CrossRef] [PubMed]
  11. Z. Gan, S. Ram, R. J. Ober, and E. S. Ward, “Using multifocal plane microscopy to reveal novel trafficking processes in the recycling pathway,” J. Cell Sci. 126, 1176–1188 (2013).
    [CrossRef] [PubMed]
  12. P. Prabhat, Z. Gan, J. Chao, S. Ram, C. Vaccaro, S. Gibbons, R. J. Ober, and E. S. Ward, “Elucidation of intracellular recycling pathways leading to exocytosis of the Fc receptor, FcRn, by using multifocal plane microscopy,” Proc. Natl. Acad. Sci. USA 104, 5889–5894 (2007).
    [CrossRef] [PubMed]
  13. S. Ram, D. Kim, R. J. Ober, and E. S. Ward, “3D single molecule tracking with multifocal plane microscopy reveals rapid intracellular transferrin transport at epithelial cell barriers,” Biophys. J. 103, 1594–1603 (2012).
    [CrossRef] [PubMed]
  14. 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]
  15. S. Abrahamsson, J. Chen, B. Hajj, S. Stallinga, A. Y. Katsov, J. Wisniewski, G. Mizuguchi, P. Soule, F. Mueller, C. D. Darzacq, X. Darzacq, C. Wu, C. I. Bargmann, D. A. Agard, M. Dahan, and M. G. L. Gustafsson, “Fast multicolor 3D imaging using aberration-corrected multifocus microscopy,” Nat. Methods 10, 60–63 (2013).
    [CrossRef]
  16. R. J. Ober, S. Ram, and E. S. Ward, “Localization accuracy in single-molecule microscopy,” Biophys. J. 86, 1185–1200 (2004).
    [CrossRef] [PubMed]
  17. 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]
  18. 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. SPIE 5699, 426–435 (2005).
    [CrossRef]
  19. S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory (Prentice Hall PTR, Upper Saddle River, NJ, 1993).
  20. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, Cambridge, UK, 2002).
  21. C. R. Rao, Linear Statistical Inference and its Applications (John Wiley and Sons, New York, 1965).
  22. D. L. Snyder and M. I. Miller, Random Point Processes in Time and Space, 2nd ed. (Springer Verlag, New York, USA, 1991).
    [CrossRef]
  23. J. Chao, S. Ram, E. S. Ward, and R. J. Ober, “A comparative study of high resolution microscopy imaging modalities using a three-dimensional resolution measure,” Opt. Express 17, 24377–24402 (2009).
    [CrossRef]
  24. L. Tao and C. Nicholson, “The three-dimensional point spread functions of a microscope objective in image and object space,” J. Microsc. 178, 267–271 (1995).
    [CrossRef] [PubMed]
  25. 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]
  26. 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]

2013

H. Kirshner, F. Aguet, D. Sage, and M. Unser, “3-D PSF fitting for fluorescence microscopy: implementation and localization application,” J. Microsc. 249, 13–25 (2013).
[CrossRef]

Z. Gan, S. Ram, R. J. Ober, and E. S. Ward, “Using multifocal plane microscopy to reveal novel trafficking processes in the recycling pathway,” J. Cell Sci. 126, 1176–1188 (2013).
[CrossRef] [PubMed]

S. Abrahamsson, J. Chen, B. Hajj, S. Stallinga, A. Y. Katsov, J. Wisniewski, G. Mizuguchi, P. Soule, F. Mueller, C. D. Darzacq, X. Darzacq, C. Wu, C. I. Bargmann, D. A. Agard, M. Dahan, and M. G. L. Gustafsson, “Fast multicolor 3D imaging using aberration-corrected multifocus microscopy,” Nat. Methods 10, 60–63 (2013).
[CrossRef]

A. Jesacher, C. Roider, and M. Ritsch-Marte, “Enhancing diffractive multi-plane microscopy using colored illumination,” Opt. Express 21, 11150–11161 (2013).
[CrossRef] [PubMed]

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]

2012

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, D. Kim, R. J. Ober, and E. S. Ward, “3D single molecule tracking with multifocal plane microscopy reveals rapid intracellular transferrin transport at epithelial cell barriers,” Biophys. J. 103, 1594–1603 (2012).
[CrossRef] [PubMed]

S. Wolter, A. Loschberger, T. Holm, S. Aufmkolk, M. Dabauvalle, S. van de Linde, and M. Sauer, “rapidSTORM: accurate, fast open-source software for localization microscopy,” Nat. Methods 9, 1040–1041 (2012).
[CrossRef] [PubMed]

2010

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]

L. Dehmelt and P. I. H. Bastiaens, “Spatial organization of intracellular communication: insights from imaging,” Nature Rev. Mol. Cell Biol. 11, 440–452 (2010).
[CrossRef]

P. A. Dalgarno, H. I. C. Dalgarno, A. Putoud, R. Lambert, L. Paterson, D. C. Logan, D. P. Towers, R. J. Warburton, and A. H. Greenaway, “Multiplane imaging and three dimensional nanoscale particle tracking in biological microscopy,” Opt. Express 18, 877–884 (2010).
[CrossRef] [PubMed]

2009

2008

S. Ram, J. Chao, P. Prabhat, E. S. Ward, and R. J. Ober, “Overcoming the depth discrimination barrier in widefield microscopes: 3D single molecule tracking with high axial accuracy,” Proc. SPIE 6862, 68620O (2008).
[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]

2007

P. Prabhat, Z. Gan, J. Chao, S. Ram, C. Vaccaro, S. Gibbons, R. J. Ober, and E. S. Ward, “Elucidation of intracellular recycling pathways leading to exocytosis of the Fc receptor, FcRn, by using multifocal plane microscopy,” Proc. Natl. Acad. Sci. USA 104, 5889–5894 (2007).
[CrossRef] [PubMed]

2005

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. SPIE 5699, 426–435 (2005).
[CrossRef]

2004

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

P. Prabhat, S. Ram, E. S. Ward, and R. J. Ober, “Simultaneous imaging of different focal planes in fluorescence microscopy for the study of cellular dynamics in three dimensions,” IEEE Trans. Nanobiosci. 3, 237–242 (2004).
[CrossRef]

1999

1995

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

Abraham, A. V.

Abrahamsson, S.

S. Abrahamsson, J. Chen, B. Hajj, S. Stallinga, A. Y. Katsov, J. Wisniewski, G. Mizuguchi, P. Soule, F. Mueller, C. D. Darzacq, X. Darzacq, C. Wu, C. I. Bargmann, D. A. Agard, M. Dahan, and M. G. L. Gustafsson, “Fast multicolor 3D imaging using aberration-corrected multifocus microscopy,” Nat. Methods 10, 60–63 (2013).
[CrossRef]

Agard, D. A.

S. Abrahamsson, J. Chen, B. Hajj, S. Stallinga, A. Y. Katsov, J. Wisniewski, G. Mizuguchi, P. Soule, F. Mueller, C. D. Darzacq, X. Darzacq, C. Wu, C. I. Bargmann, D. A. Agard, M. Dahan, and M. G. L. Gustafsson, “Fast multicolor 3D imaging using aberration-corrected multifocus microscopy,” Nat. Methods 10, 60–63 (2013).
[CrossRef]

Aguet, F.

H. Kirshner, F. Aguet, D. Sage, and M. Unser, “3-D PSF fitting for fluorescence microscopy: implementation and localization application,” J. Microsc. 249, 13–25 (2013).
[CrossRef]

Aufmkolk, S.

S. Wolter, A. Loschberger, T. Holm, S. Aufmkolk, M. Dabauvalle, S. van de Linde, and M. Sauer, “rapidSTORM: accurate, fast open-source software for localization microscopy,” Nat. Methods 9, 1040–1041 (2012).
[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]

Bargmann, C. I.

S. Abrahamsson, J. Chen, B. Hajj, S. Stallinga, A. Y. Katsov, J. Wisniewski, G. Mizuguchi, P. Soule, F. Mueller, C. D. Darzacq, X. Darzacq, C. Wu, C. I. Bargmann, D. A. Agard, M. Dahan, and M. G. L. Gustafsson, “Fast multicolor 3D imaging using aberration-corrected multifocus microscopy,” Nat. Methods 10, 60–63 (2013).
[CrossRef]

Bastiaens, P. I. H.

L. Dehmelt and P. I. H. Bastiaens, “Spatial organization of intracellular communication: insights from imaging,” Nature Rev. Mol. Cell Biol. 11, 440–452 (2010).
[CrossRef]

Bewersdorf, J.

Blanchard, P. M.

Born, M.

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

Chao, J.

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]

J. Chao, S. Ram, E. S. Ward, and R. J. Ober, “A comparative study of high resolution microscopy imaging modalities using a three-dimensional resolution measure,” Opt. Express 17, 24377–24402 (2009).
[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, J. Chao, P. Prabhat, E. S. Ward, and R. J. Ober, “Overcoming the depth discrimination barrier in widefield microscopes: 3D single molecule tracking with high axial accuracy,” Proc. SPIE 6862, 68620O (2008).
[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]

P. Prabhat, Z. Gan, J. Chao, S. Ram, C. Vaccaro, S. Gibbons, R. J. Ober, and E. S. Ward, “Elucidation of intracellular recycling pathways leading to exocytosis of the Fc receptor, FcRn, by using multifocal plane microscopy,” Proc. Natl. Acad. Sci. USA 104, 5889–5894 (2007).
[CrossRef] [PubMed]

Chen, J.

S. Abrahamsson, J. Chen, B. Hajj, S. Stallinga, A. Y. Katsov, J. Wisniewski, G. Mizuguchi, P. Soule, F. Mueller, C. D. Darzacq, X. Darzacq, C. Wu, C. I. Bargmann, D. A. Agard, M. Dahan, and M. G. L. Gustafsson, “Fast multicolor 3D imaging using aberration-corrected multifocus microscopy,” Nat. Methods 10, 60–63 (2013).
[CrossRef]

Dabauvalle, M.

S. Wolter, A. Loschberger, T. Holm, S. Aufmkolk, M. Dabauvalle, S. van de Linde, and M. Sauer, “rapidSTORM: accurate, fast open-source software for localization microscopy,” Nat. Methods 9, 1040–1041 (2012).
[CrossRef] [PubMed]

Dahan, M.

S. Abrahamsson, J. Chen, B. Hajj, S. Stallinga, A. Y. Katsov, J. Wisniewski, G. Mizuguchi, P. Soule, F. Mueller, C. D. Darzacq, X. Darzacq, C. Wu, C. I. Bargmann, D. A. Agard, M. Dahan, and M. G. L. Gustafsson, “Fast multicolor 3D imaging using aberration-corrected multifocus microscopy,” Nat. Methods 10, 60–63 (2013).
[CrossRef]

Dalgarno, H. I. C.

Dalgarno, P. A.

Darzacq, C. D.

S. Abrahamsson, J. Chen, B. Hajj, S. Stallinga, A. Y. Katsov, J. Wisniewski, G. Mizuguchi, P. Soule, F. Mueller, C. D. Darzacq, X. Darzacq, C. Wu, C. I. Bargmann, D. A. Agard, M. Dahan, and M. G. L. Gustafsson, “Fast multicolor 3D imaging using aberration-corrected multifocus microscopy,” Nat. Methods 10, 60–63 (2013).
[CrossRef]

Darzacq, X.

S. Abrahamsson, J. Chen, B. Hajj, S. Stallinga, A. Y. Katsov, J. Wisniewski, G. Mizuguchi, P. Soule, F. Mueller, C. D. Darzacq, X. Darzacq, C. Wu, C. I. Bargmann, D. A. Agard, M. Dahan, and M. G. L. Gustafsson, “Fast multicolor 3D imaging using aberration-corrected multifocus microscopy,” Nat. Methods 10, 60–63 (2013).
[CrossRef]

Dehmelt, L.

L. Dehmelt and P. I. H. Bastiaens, “Spatial organization of intracellular communication: insights from imaging,” Nature Rev. Mol. Cell Biol. 11, 440–452 (2010).
[CrossRef]

Gan, Z.

Z. Gan, S. Ram, R. J. Ober, and E. S. Ward, “Using multifocal plane microscopy to reveal novel trafficking processes in the recycling pathway,” J. Cell Sci. 126, 1176–1188 (2013).
[CrossRef] [PubMed]

P. Prabhat, Z. Gan, J. Chao, S. Ram, C. Vaccaro, S. Gibbons, R. J. Ober, and E. S. Ward, “Elucidation of intracellular recycling pathways leading to exocytosis of the Fc receptor, FcRn, by using multifocal plane microscopy,” Proc. Natl. Acad. Sci. USA 104, 5889–5894 (2007).
[CrossRef] [PubMed]

Gibbons, S.

P. Prabhat, Z. Gan, J. Chao, S. Ram, C. Vaccaro, S. Gibbons, R. J. Ober, and E. S. Ward, “Elucidation of intracellular recycling pathways leading to exocytosis of the Fc receptor, FcRn, by using multifocal plane microscopy,” Proc. Natl. Acad. Sci. USA 104, 5889–5894 (2007).
[CrossRef] [PubMed]

Greenaway, A. H.

Gustafsson, M. G. L.

S. Abrahamsson, J. Chen, B. Hajj, S. Stallinga, A. Y. Katsov, J. Wisniewski, G. Mizuguchi, P. Soule, F. Mueller, C. D. Darzacq, X. Darzacq, C. Wu, C. I. Bargmann, D. A. Agard, M. Dahan, and M. G. L. Gustafsson, “Fast multicolor 3D imaging using aberration-corrected multifocus microscopy,” Nat. Methods 10, 60–63 (2013).
[CrossRef]

Hajj, B.

S. Abrahamsson, J. Chen, B. Hajj, S. Stallinga, A. Y. Katsov, J. Wisniewski, G. Mizuguchi, P. Soule, F. Mueller, C. D. Darzacq, X. Darzacq, C. Wu, C. I. Bargmann, D. A. Agard, M. Dahan, and M. G. L. Gustafsson, “Fast multicolor 3D imaging using aberration-corrected multifocus microscopy,” Nat. Methods 10, 60–63 (2013).
[CrossRef]

Holm, T.

S. Wolter, A. Loschberger, T. Holm, S. Aufmkolk, M. Dabauvalle, S. van de Linde, and M. Sauer, “rapidSTORM: accurate, fast open-source software for localization microscopy,” Nat. Methods 9, 1040–1041 (2012).
[CrossRef] [PubMed]

Jesacher, A.

Katsov, A. Y.

S. Abrahamsson, J. Chen, B. Hajj, S. Stallinga, A. Y. Katsov, J. Wisniewski, G. Mizuguchi, P. Soule, F. Mueller, C. D. Darzacq, X. Darzacq, C. Wu, C. I. Bargmann, D. A. Agard, M. Dahan, and M. G. L. Gustafsson, “Fast multicolor 3D imaging using aberration-corrected multifocus microscopy,” Nat. Methods 10, 60–63 (2013).
[CrossRef]

Kay, S. M.

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

Kim, D.

S. Ram, D. Kim, R. J. Ober, and E. S. Ward, “3D single molecule tracking with multifocal plane microscopy reveals rapid intracellular transferrin transport at epithelial cell barriers,” Biophys. J. 103, 1594–1603 (2012).
[CrossRef] [PubMed]

Kirshner, H.

H. Kirshner, F. Aguet, D. Sage, and M. Unser, “3-D PSF fitting for fluorescence microscopy: implementation and localization application,” J. Microsc. 249, 13–25 (2013).
[CrossRef]

Kromann, E.

Krueger, W.

Lambert, R.

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.

Liu, S.

Logan, D. C.

Loschberger, A.

S. Wolter, A. Loschberger, T. Holm, S. Aufmkolk, M. Dabauvalle, S. van de Linde, and M. Sauer, “rapidSTORM: accurate, fast open-source software for localization microscopy,” Nat. Methods 9, 1040–1041 (2012).
[CrossRef] [PubMed]

Miller, M. I.

D. L. Snyder and M. I. Miller, Random Point Processes in Time and Space, 2nd ed. (Springer Verlag, New York, USA, 1991).
[CrossRef]

Mizuguchi, G.

S. Abrahamsson, J. Chen, B. Hajj, S. Stallinga, A. Y. Katsov, J. Wisniewski, G. Mizuguchi, P. Soule, F. Mueller, C. D. Darzacq, X. Darzacq, C. Wu, C. I. Bargmann, D. A. Agard, M. Dahan, and M. G. L. Gustafsson, “Fast multicolor 3D imaging using aberration-corrected multifocus microscopy,” Nat. Methods 10, 60–63 (2013).
[CrossRef]

Moerner, W. E.

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]

Mueller, F.

S. Abrahamsson, J. Chen, B. Hajj, S. Stallinga, A. Y. Katsov, J. Wisniewski, G. Mizuguchi, P. Soule, F. Mueller, C. D. Darzacq, X. Darzacq, C. Wu, C. I. Bargmann, D. A. Agard, M. Dahan, and M. G. L. Gustafsson, “Fast multicolor 3D imaging using aberration-corrected multifocus microscopy,” Nat. Methods 10, 60–63 (2013).
[CrossRef]

Nicholson, C.

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

Ober, R. J.

Z. Gan, S. Ram, R. J. Ober, and E. S. Ward, “Using multifocal plane microscopy to reveal novel trafficking processes in the recycling pathway,” J. Cell Sci. 126, 1176–1188 (2013).
[CrossRef] [PubMed]

S. Ram, D. Kim, R. J. Ober, and E. S. Ward, “3D single molecule tracking with multifocal plane microscopy reveals rapid intracellular transferrin transport at epithelial cell barriers,” Biophys. J. 103, 1594–1603 (2012).
[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]

J. Chao, S. Ram, E. S. Ward, and R. J. Ober, “A comparative study of high resolution microscopy imaging modalities using a three-dimensional resolution measure,” Opt. Express 17, 24377–24402 (2009).
[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, J. Chao, P. Prabhat, E. S. Ward, and R. J. Ober, “Overcoming the depth discrimination barrier in widefield microscopes: 3D single molecule tracking with high axial accuracy,” Proc. SPIE 6862, 68620O (2008).
[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]

P. Prabhat, Z. Gan, J. Chao, S. Ram, C. Vaccaro, S. Gibbons, R. J. Ober, and E. S. Ward, “Elucidation of intracellular recycling pathways leading to exocytosis of the Fc receptor, FcRn, by using multifocal plane microscopy,” Proc. Natl. Acad. Sci. USA 104, 5889–5894 (2007).
[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. SPIE 5699, 426–435 (2005).
[CrossRef]

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

P. Prabhat, S. Ram, E. S. Ward, and R. J. Ober, “Simultaneous imaging of different focal planes in fluorescence microscopy for the study of cellular dynamics in three dimensions,” IEEE Trans. Nanobiosci. 3, 237–242 (2004).
[CrossRef]

Paterson, L.

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]

S. Ram, J. Chao, P. Prabhat, E. S. Ward, and R. J. Ober, “Overcoming the depth discrimination barrier in widefield microscopes: 3D single molecule tracking with high axial accuracy,” Proc. SPIE 6862, 68620O (2008).
[CrossRef]

P. Prabhat, Z. Gan, J. Chao, S. Ram, C. Vaccaro, S. Gibbons, R. J. Ober, and E. S. Ward, “Elucidation of intracellular recycling pathways leading to exocytosis of the Fc receptor, FcRn, by using multifocal plane microscopy,” Proc. Natl. Acad. Sci. USA 104, 5889–5894 (2007).
[CrossRef] [PubMed]

P. Prabhat, S. Ram, E. S. Ward, and R. J. Ober, “Simultaneous imaging of different focal planes in fluorescence microscopy for the study of cellular dynamics in three dimensions,” IEEE Trans. Nanobiosci. 3, 237–242 (2004).
[CrossRef]

Putoud, A.

Ram, S.

Z. Gan, S. Ram, R. J. Ober, and E. S. Ward, “Using multifocal plane microscopy to reveal novel trafficking processes in the recycling pathway,” J. Cell Sci. 126, 1176–1188 (2013).
[CrossRef] [PubMed]

S. Ram, D. Kim, R. J. Ober, and E. S. Ward, “3D single molecule tracking with multifocal plane microscopy reveals rapid intracellular transferrin transport at epithelial cell barriers,” Biophys. J. 103, 1594–1603 (2012).
[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]

J. Chao, S. Ram, E. S. Ward, and R. J. Ober, “A comparative study of high resolution microscopy imaging modalities using a three-dimensional resolution measure,” Opt. Express 17, 24377–24402 (2009).
[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, J. Chao, P. Prabhat, E. S. Ward, and R. J. Ober, “Overcoming the depth discrimination barrier in widefield microscopes: 3D single molecule tracking with high axial accuracy,” Proc. SPIE 6862, 68620O (2008).
[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]

P. Prabhat, Z. Gan, J. Chao, S. Ram, C. Vaccaro, S. Gibbons, R. J. Ober, and E. S. Ward, “Elucidation of intracellular recycling pathways leading to exocytosis of the Fc receptor, FcRn, by using multifocal plane microscopy,” Proc. Natl. Acad. Sci. USA 104, 5889–5894 (2007).
[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. SPIE 5699, 426–435 (2005).
[CrossRef]

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

P. Prabhat, S. Ram, E. S. Ward, and R. J. Ober, “Simultaneous imaging of different focal planes in fluorescence microscopy for the study of cellular dynamics in three dimensions,” IEEE Trans. Nanobiosci. 3, 237–242 (2004).
[CrossRef]

Rao, C. R.

C. R. Rao, Linear Statistical Inference and its Applications (John Wiley and Sons, New York, 1965).

Ritsch-Marte, M.

Roider, C.

Sage, D.

H. Kirshner, F. Aguet, D. Sage, and M. Unser, “3-D PSF fitting for fluorescence microscopy: implementation and localization application,” J. Microsc. 249, 13–25 (2013).
[CrossRef]

Sauer, M.

S. Wolter, A. Loschberger, T. Holm, S. Aufmkolk, M. Dabauvalle, S. van de Linde, and M. Sauer, “rapidSTORM: accurate, fast open-source software for localization microscopy,” Nat. Methods 9, 1040–1041 (2012).
[CrossRef] [PubMed]

Snyder, D. L.

D. L. Snyder and M. I. Miller, Random Point Processes in Time and Space, 2nd ed. (Springer Verlag, New York, USA, 1991).
[CrossRef]

Soule, P.

S. Abrahamsson, J. Chen, B. Hajj, S. Stallinga, A. Y. Katsov, J. Wisniewski, G. Mizuguchi, P. Soule, F. Mueller, C. D. Darzacq, X. Darzacq, C. Wu, C. I. Bargmann, D. A. Agard, M. Dahan, and M. G. L. Gustafsson, “Fast multicolor 3D imaging using aberration-corrected multifocus microscopy,” Nat. Methods 10, 60–63 (2013).
[CrossRef]

Stallinga, S.

S. Abrahamsson, J. Chen, B. Hajj, S. Stallinga, A. Y. Katsov, J. Wisniewski, G. Mizuguchi, P. Soule, F. Mueller, C. D. Darzacq, X. Darzacq, C. Wu, C. I. Bargmann, D. A. Agard, M. Dahan, and M. G. L. Gustafsson, “Fast multicolor 3D imaging using aberration-corrected multifocus microscopy,” Nat. Methods 10, 60–63 (2013).
[CrossRef]

Tao, L.

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

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]

Towers, D. P.

Unser, M.

H. Kirshner, F. Aguet, D. Sage, and M. Unser, “3-D PSF fitting for fluorescence microscopy: implementation and localization application,” J. Microsc. 249, 13–25 (2013).
[CrossRef]

Vaccaro, C.

P. Prabhat, Z. Gan, J. Chao, S. Ram, C. Vaccaro, S. Gibbons, R. J. Ober, and E. S. Ward, “Elucidation of intracellular recycling pathways leading to exocytosis of the Fc receptor, FcRn, by using multifocal plane microscopy,” Proc. Natl. Acad. Sci. USA 104, 5889–5894 (2007).
[CrossRef] [PubMed]

van de Linde, S.

S. Wolter, A. Loschberger, T. Holm, S. Aufmkolk, M. Dabauvalle, S. van de Linde, and M. Sauer, “rapidSTORM: accurate, fast open-source software for localization microscopy,” Nat. Methods 9, 1040–1041 (2012).
[CrossRef] [PubMed]

Warburton, R. J.

Ward, E. S.

Z. Gan, S. Ram, R. J. Ober, and E. S. Ward, “Using multifocal plane microscopy to reveal novel trafficking processes in the recycling pathway,” J. Cell Sci. 126, 1176–1188 (2013).
[CrossRef] [PubMed]

S. Ram, D. Kim, R. J. Ober, and E. S. Ward, “3D single molecule tracking with multifocal plane microscopy reveals rapid intracellular transferrin transport at epithelial cell barriers,” Biophys. J. 103, 1594–1603 (2012).
[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]

J. Chao, S. Ram, E. S. Ward, and R. J. Ober, “A comparative study of high resolution microscopy imaging modalities using a three-dimensional resolution measure,” Opt. Express 17, 24377–24402 (2009).
[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, J. Chao, P. Prabhat, E. S. Ward, and R. J. Ober, “Overcoming the depth discrimination barrier in widefield microscopes: 3D single molecule tracking with high axial accuracy,” Proc. SPIE 6862, 68620O (2008).
[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]

P. Prabhat, Z. Gan, J. Chao, S. Ram, C. Vaccaro, S. Gibbons, R. J. Ober, and E. S. Ward, “Elucidation of intracellular recycling pathways leading to exocytosis of the Fc receptor, FcRn, by using multifocal plane microscopy,” Proc. Natl. Acad. Sci. USA 104, 5889–5894 (2007).
[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. SPIE 5699, 426–435 (2005).
[CrossRef]

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

P. Prabhat, S. Ram, E. S. Ward, and R. J. Ober, “Simultaneous imaging of different focal planes in fluorescence microscopy for the study of cellular dynamics in three dimensions,” IEEE Trans. Nanobiosci. 3, 237–242 (2004).
[CrossRef]

Wisniewski, J.

S. Abrahamsson, J. Chen, B. Hajj, S. Stallinga, A. Y. Katsov, J. Wisniewski, G. Mizuguchi, P. Soule, F. Mueller, C. D. Darzacq, X. Darzacq, C. Wu, C. I. Bargmann, D. A. Agard, M. Dahan, and M. G. L. Gustafsson, “Fast multicolor 3D imaging using aberration-corrected multifocus microscopy,” Nat. Methods 10, 60–63 (2013).
[CrossRef]

Wolf, E.

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

Wolter, S.

S. Wolter, A. Loschberger, T. Holm, S. Aufmkolk, M. Dabauvalle, S. van de Linde, and M. Sauer, “rapidSTORM: accurate, fast open-source software for localization microscopy,” Nat. Methods 9, 1040–1041 (2012).
[CrossRef] [PubMed]

Wu, C.

S. Abrahamsson, J. Chen, B. Hajj, S. Stallinga, A. Y. Katsov, J. Wisniewski, G. Mizuguchi, P. Soule, F. Mueller, C. D. Darzacq, X. Darzacq, C. Wu, C. I. Bargmann, D. A. Agard, M. Dahan, and M. G. L. Gustafsson, “Fast multicolor 3D imaging using aberration-corrected multifocus microscopy,” Nat. Methods 10, 60–63 (2013).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

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.

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, D. Kim, R. J. Ober, and E. S. Ward, “3D single molecule tracking with multifocal plane microscopy reveals rapid intracellular transferrin transport at epithelial cell barriers,” Biophys. J. 103, 1594–1603 (2012).
[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]

IEEE Trans. Nanobiosci.

P. Prabhat, S. Ram, E. S. Ward, and R. J. Ober, “Simultaneous imaging of different focal planes in fluorescence microscopy for the study of cellular dynamics in three dimensions,” IEEE Trans. Nanobiosci. 3, 237–242 (2004).
[CrossRef]

J. Cell Sci.

Z. Gan, S. Ram, R. J. Ober, and E. S. Ward, “Using multifocal plane microscopy to reveal novel trafficking processes in the recycling pathway,” J. Cell Sci. 126, 1176–1188 (2013).
[CrossRef] [PubMed]

J. Microsc.

H. Kirshner, F. Aguet, D. Sage, and M. Unser, “3-D PSF fitting for fluorescence microscopy: implementation and localization application,” J. Microsc. 249, 13–25 (2013).
[CrossRef]

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

Multidim. Sys. Sig. Proc.

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]

Nat. Methods

S. Abrahamsson, J. Chen, B. Hajj, S. Stallinga, A. Y. Katsov, J. Wisniewski, G. Mizuguchi, P. Soule, F. Mueller, C. D. Darzacq, X. Darzacq, C. Wu, C. I. Bargmann, D. A. Agard, M. Dahan, and M. G. L. Gustafsson, “Fast multicolor 3D imaging using aberration-corrected multifocus microscopy,” Nat. Methods 10, 60–63 (2013).
[CrossRef]

S. Wolter, A. Loschberger, T. Holm, S. Aufmkolk, M. Dabauvalle, S. van de Linde, and M. Sauer, “rapidSTORM: accurate, fast open-source software for localization microscopy,” Nat. Methods 9, 1040–1041 (2012).
[CrossRef] [PubMed]

Nature Rev. Mol. Cell Biol.

L. Dehmelt and P. I. H. Bastiaens, “Spatial organization of intracellular communication: insights from imaging,” Nature Rev. Mol. Cell Biol. 11, 440–452 (2010).
[CrossRef]

Opt. Express

Proc. Natl. Acad. Sci. USA

P. Prabhat, Z. Gan, J. Chao, S. Ram, C. Vaccaro, S. Gibbons, R. J. Ober, and E. S. Ward, “Elucidation of intracellular recycling pathways leading to exocytosis of the Fc receptor, FcRn, by using multifocal plane microscopy,” Proc. Natl. Acad. Sci. USA 104, 5889–5894 (2007).
[CrossRef] [PubMed]

Proc. SPIE

S. Ram, J. Chao, P. Prabhat, E. S. Ward, and R. J. Ober, “Overcoming the depth discrimination barrier in widefield microscopes: 3D single molecule tracking with high axial accuracy,” Proc. SPIE 6862, 68620O (2008).
[CrossRef]

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. SPIE 5699, 426–435 (2005).
[CrossRef]

Other

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

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

C. R. Rao, Linear Statistical Inference and its Applications (John Wiley and Sons, New York, 1965).

D. L. Snyder and M. I. Miller, Random Point Processes in Time and Space, 2nd ed. (Springer Verlag, New York, USA, 1991).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

The behavior of the axial-FIM and the axial-PLAM for conventional single plane microscopy and MUM. (a) The axial-FIM and (b) the axial-PLAM for a 2-plane MUM setup and two conventional single plane setups (Plane 1 and Plane 2) as a function of the z-position of the point source. The zero, local maxima and shoulders of the axial-FIM of Plane 1, and the depth discrimination problem can be seen in panels (a) and (b), respectively. The results are calculated for a 100x, NA 1.3 objective lens. The plane spacing (Δz12) is 0.5 μm, the photon count is 250 photons/plane and the emission wavelength is 520 nm. The background level is 1 photon/pixel/plane and the standard deviation of the readout noise is 2 e/pixel. The ROI size is 11 × 11 pixels.

Fig. 2
Fig. 2

The effect of plane spacing on the axial-PLAM and the axial-FIM for a MUM setup. The left-hand (right-hand) side plots show the axial-PLAM (axial-FIM) for a 2-plane MUM setup as a function of the z-position of the point source. The figure also shows the axial-FIMs and axial-PLAMs of focal planes 1 and 2. The plane spacings (Δz12) are 0.1, 0.5, 1 and 3 μm from top to bottom. (c) and (d) show the strong coupling spacing whereas (e) and (f) show the weak coupling spacing. (f’) shows the shoulder of the axial-FIM1 more clearly. The results are calculated for a 100x, NA 1.3 oil immersion objective lens where the photon count is 250 photons/plane and the emission wavelength is 520 nm. The background level is 1 photon/pixel/plane and the standard deviation of the readout noise is 2 e/pixel. The ROI size is 11 × 11 pixels.

Fig. 3
Fig. 3

The behavior of the lateral-PLAM. The left and middle columns show the axial-PLAM and the lateral-PLAM, respectively, for a 2-plane MUM setup as a function of the z-position of the point source. The plane spacings (Δz12) are 0.1, 0.5, 1 and 3 μm from top to bottom. The right column shows the mesh plots of the simulated images of point sources located at z-positions shown by the red circles on the design plane. The simulation parameters are identical to those used in Fig. 2.

Fig. 4
Fig. 4

The effect of changing imaging parameters on the spacing scenarios. A plot of the strong coupling spacing (Δzsc) and weak coupling spacing (Δzwc) versus (a) the numerical aperture na, (b) magnification M and (c) emission wavelength λ. (d), (e) and (f) show the axial-PLAM for a 2-plane MUM setup as a function of the z-position of the point source for different numerical apertures, magnifications and emission wavelengths, respectively. In (d), (e) and (f), the plane spacings are adjusted based on the strong coupling scenario. The photon count is 250 photons/plane. The ROI size is 32 × 32 pixels. The magnification is 100, the emission wavelength is 520 nm and the numerical aperture is 1.4. The background level and the standard deviation of the readout noise are 2.5 photons/pixel/plane and 8 e/pixel, respectively.

Fig. 5
Fig. 5

The effect of photon count and extraneous noise on the spacing scenarios. A plot of the strong coupling spacing (Δzsc) and weak coupling spacing (Δzwc) as a function of (a) the photon count (N) and (b) the standard deviation of the readout noise (σ). The ROI size is 32 × 32 pixels. The magnification is 100, the emission wavelength is 520 nm and the numerical aperture is 1.4. In (a), the background level and the standard deviation of the readout noise are 2.5 photons/pixel and 8 e/pixel, respectively. In (b), the photon count and the background level are 500 photons and 0 photons/pixel, respectively.

Fig. 6
Fig. 6

The strong and weak coupling spacings for a 4-plane MUM setup. (a) The axial-PLAM and (b) the lateral-PLAM as a function of the z-position of the point source for the strong coupling spacing for a 4-plane MUM setup. The planes are placed at 0, 0.45, 0.9 and 1.35 μm. (c) and (d) show the same for the weak coupling spacing for a 4-plane MUM setup where the planes are located at 0, 1, 2 and 3 μm. The magnification is 100, the numerical aperture is 1.3, the photon count is 250 photons/plane and the ROI size is 11 × 11 pixels. The emission wavelength is 520 nm. The background level and the standard deviation of the readout noise are 20 photons/pixel/plane and 3 e/pixel, respectively.

Fig. 7
Fig. 7

The effect of changing the number of focal planes on the 3D localization accuracy. (a) The axial-PLAM and (b) the lateral-PLAM for a MUM setup with different numbers of focal planes over a range of [−1, 1] μm. The magnification is 100, the numerical aperture is 1.3 and the ROI size is 11 × 11 pixels. The emission wavelength is 520 nm. The focal planes are located based on the strong coupling spacing. The total photon count is 1000 photons and is split equally among the focal planes. The background level and the standard deviation of the readout noise are 25 photons/pixel/plane and 8 e/pixel, respectively.

Equations (7)

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μ θ , n ( k ) : = N n μ ˜ θ , n ( k ) = N n M n 2 C k , n q z 0 Δ z 1 n ( x M n x 0 , y M n y 0 ) d x d y , θ Θ 3 , n = 1 , , K pln , k = 1 , , K pix ,
M n : M ( Δ z 1 n ) = M 1 L LM 1 2 Δ z 1 n n oil L + M 1 2 Δ z 1 n L , n = 1 , , K pln , Δ z 1 n ,
I n ( θ ) = k = 1 K pix ψ n ( k ) ν θ , n ( k ) ( μ θ , n ( k ) θ ) T μ θ , n ( k ) θ , θ Θ , n = 1 , , K pln ,
ψ n ( k ) = ν θ , n ( k ) × ( e ν θ , n ( k ) 2 π σ k , n ( l = 1 ν θ , n l 1 ( k ) ( l 1 ) ! e ( z l η k , n ) 2 2 σ k , n 2 ) 2 l = 0 ν θ , n l ( k ) l ! e ( z l η k , n ) 2 2 σ k , n 2 d z 1 ) ,
I MUM ( θ ) = I 1 ( θ ) + I 2 ( θ ) + + I K pln ( θ ) , θ = ( x 0 , y 0 , z 0 ) Θ .
I n ( θ ) = k = 1 K pix 1 N n μ ˜ θ , n ( k ) ( ( N n μ ˜ θ , n ( k ) ) θ ) T ( N n μ ˜ θ , n ( k ) ) θ = N n k = 1 K pix 1 μ ˜ θ , n ( k ) ( μ ˜ θ , n ( k ) θ ) T μ ˜ θ , n ( k ) θ : = N n I ˜ n ( θ ) , θ Θ , n = 1 , , K pln .
I ˜ MUM ( θ ) : = N 1 I ˜ 1 ( x 0 , y 0 , z 0 ) + N 2 I ˜ 1 ( x 0 , y 0 , z 0 Δ z 12 ) + + N K pln I ˜ 1 ( x 0 , y 0 , z 0 Δ z 1 K pln ) , θ = ( x 0 , y 0 , z 0 ) Θ ,

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