Abstract

We report a differential fluorescence speckle confocal microscope that acquires an image in a fraction of a second by exploiting the very high frame rate of modern digital micromirror devices (DMDs). The DMD projects a sequence of predefined binary speckle patterns to the sample and modulates the intensity of the returning fluorescent light simultaneously. The fluorescent light reflecting from the DMD’s “on” and “off” pixels is modulated by correlated speckle and anticorrelated speckle, respectively, to form two images on two CCD cameras in parallel. The sum of the two images recovers a widefield image, but their difference gives a near-confocal image in real time. Experimental results for both low and high numerical apertures are shown.

© 2010 Optical Society of America

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References

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  1. J. C. Waters, “Live-cell fluorescence imaging,” in Digital Microscopy, 3rd ed. (Elsevier, 2007), pp. 115-140.
    [CrossRef]
  2. A. T. Hammond and B. S. Glick, “Raising the speed limits for 4D fluorescence microscopy,” Traffic 1, 935-940 (2000).
    [CrossRef]
  3. D. Gerlich, J. Beaudouin, M. Gebhard, J. Ellenberg, and R. Eils, “Four-dimensional imaging and quantitative reconstruction to analyse complex spatiotemporal processes in live cells,” Nature Cell Biol. 3, 852-855 (2001).
    [CrossRef] [PubMed]
  4. M. H. Liang, R. L. Stehr, and A. W. Krause, “Confocal pattern period in multiple-aperture confocal imaging systems with coherent illumination,” Opt. Lett. 22, 751-753 (1997).
    [CrossRef] [PubMed]
  5. P. J. Verveer, Q. S. Hanley, P. W. Verbeek, L. J. Van Vliet, and T. M. Jovin, “Theory of confocal fluorescence imaging in the programmable array microscope (PAM),” J. Microsc. (Oxford) 189, 192-198 (1998).
    [CrossRef]
  6. T. Fukano and A. Miyawaki, “Whole-field fluorescence microscope with digital micromirror device: imaging of biological samples,” Appl. Opt. 42, 4119-4124 (2003).
    [CrossRef] [PubMed]
  7. C. H. Wong, N. G. Chen, and C. J. R. Sheppard, “Study on potential of structured illumination microscopy utilizing digital micromirror device for endoscopy purpose,” Proceedings of International Symposium on Biophotonics, Nanophotonics and Metamaterials (IEEE, 2006), pp. 214-217.
  8. G. M. Hagen, W. Caarls, M. Thomas, A. Hill, K. A. Lidke, B. Rieger, C. Fritsch, B. van Geest, T. M. Jovin, and D. J. Arndt-Jovin, “Biological applications of an LCoS-based programmable array microscope (PAM),” Proc. SPIE 6441, 64410S (2007).
    [CrossRef]
  9. V. Bansal, S. Patel, and P. Saggau, “High-speed addressable confocal microscopy for functional imaging of cellular activity,” J. Biomed. Opt. 11 (2006).
    [CrossRef] [PubMed]
  10. V. Bansal, S. Patel, and P. Saggau, “A high-speed confocal laser-scanning microscope based on acousto-optic deflectors and a digital micromirror device,” in Proceedings of the 25th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (IEEE, 2003), Vols. 1- 4, pp. 2124-2127.
  11. J. G. Walker, “Non-scanning confocal fluorescence microscopy using speckle illumination,” Opt. Commun. 189, 221-226(2001).
    [CrossRef]
  12. S. H. Jiang and J. G. Walker, “Experimental confirmation of non-scanning fluorescence confocal microscopy using speckle illumination,” Opt. Commun. 238, 1-12 (2004).
    [CrossRef]
  13. S. Jiang and J. G. Walker, “Non-scanning fluorescence confocal microscopy using speckle illumination and optical data processing,” Opt. Commun. 256, 35-45 (2005).
    [CrossRef]
  14. S. H. Jiang and J. G. Walker, “Speckle-illuminated fluorescence confocal microscopy using a digital micro-mirror device,” Meas. Sci. Technol. 20, 065501 (2009).
    [CrossRef]
  15. C. Ventalon and J. Mertz, “Quasi-confocal fluorescence sectioning with dynamic speckle illumination,” Opt. Lett. 30, 3350-3352 (2005).
    [CrossRef]
  16. C. Ventalon and J. Mertz, “Dynamic speckle illumination microscopy with translated versus randomized speckle patterns,” Opt. Express 14, 7198-7209 (2006).
    [CrossRef] [PubMed]
  17. C. Ventalon, R. Heintzmann, and J. Mertz, “Dynamic speckle illumination microscopy with wavelet prefiltering,” Opt. Lett. 32, 1417-1419 (2007).
    [CrossRef] [PubMed]
  18. D. Lim, K. K. Chu, and J. Mertz, “Wide-field fluorescence sectioning with hybrid speckle and uniform-illumination microscopy,” Opt. Lett. 33, 1819-1821 (2008).
    [CrossRef] [PubMed]
  19. S. Ri, M. Fujigaki, T. Matui, and Y. Morimoto, “Accurate pixel-to-pixel correspondence adjustment in a digital micromirror device camera by using the phase-shifting moire method,” Appl. Opt. 45, 6940-6946 (2006).
    [CrossRef] [PubMed]
  20. S. Jiang, “Non-scanning fluorescence confocal microscopy using laser speckle illumination,” Ph.D. thesis (University of Nottingham, 2005).
  21. ViALUX GmbH, product data sheet, Chemnitz, Germany (2009).

2009

S. H. Jiang and J. G. Walker, “Speckle-illuminated fluorescence confocal microscopy using a digital micro-mirror device,” Meas. Sci. Technol. 20, 065501 (2009).
[CrossRef]

2008

2007

C. Ventalon, R. Heintzmann, and J. Mertz, “Dynamic speckle illumination microscopy with wavelet prefiltering,” Opt. Lett. 32, 1417-1419 (2007).
[CrossRef] [PubMed]

G. M. Hagen, W. Caarls, M. Thomas, A. Hill, K. A. Lidke, B. Rieger, C. Fritsch, B. van Geest, T. M. Jovin, and D. J. Arndt-Jovin, “Biological applications of an LCoS-based programmable array microscope (PAM),” Proc. SPIE 6441, 64410S (2007).
[CrossRef]

2006

2005

C. Ventalon and J. Mertz, “Quasi-confocal fluorescence sectioning with dynamic speckle illumination,” Opt. Lett. 30, 3350-3352 (2005).
[CrossRef]

S. Jiang and J. G. Walker, “Non-scanning fluorescence confocal microscopy using speckle illumination and optical data processing,” Opt. Commun. 256, 35-45 (2005).
[CrossRef]

2004

S. H. Jiang and J. G. Walker, “Experimental confirmation of non-scanning fluorescence confocal microscopy using speckle illumination,” Opt. Commun. 238, 1-12 (2004).
[CrossRef]

2003

2001

J. G. Walker, “Non-scanning confocal fluorescence microscopy using speckle illumination,” Opt. Commun. 189, 221-226(2001).
[CrossRef]

D. Gerlich, J. Beaudouin, M. Gebhard, J. Ellenberg, and R. Eils, “Four-dimensional imaging and quantitative reconstruction to analyse complex spatiotemporal processes in live cells,” Nature Cell Biol. 3, 852-855 (2001).
[CrossRef] [PubMed]

2000

A. T. Hammond and B. S. Glick, “Raising the speed limits for 4D fluorescence microscopy,” Traffic 1, 935-940 (2000).
[CrossRef]

1998

P. J. Verveer, Q. S. Hanley, P. W. Verbeek, L. J. Van Vliet, and T. M. Jovin, “Theory of confocal fluorescence imaging in the programmable array microscope (PAM),” J. Microsc. (Oxford) 189, 192-198 (1998).
[CrossRef]

1997

Arndt-Jovin, D. J.

G. M. Hagen, W. Caarls, M. Thomas, A. Hill, K. A. Lidke, B. Rieger, C. Fritsch, B. van Geest, T. M. Jovin, and D. J. Arndt-Jovin, “Biological applications of an LCoS-based programmable array microscope (PAM),” Proc. SPIE 6441, 64410S (2007).
[CrossRef]

Bansal, V.

V. Bansal, S. Patel, and P. Saggau, “High-speed addressable confocal microscopy for functional imaging of cellular activity,” J. Biomed. Opt. 11 (2006).
[CrossRef] [PubMed]

V. Bansal, S. Patel, and P. Saggau, “A high-speed confocal laser-scanning microscope based on acousto-optic deflectors and a digital micromirror device,” in Proceedings of the 25th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (IEEE, 2003), Vols. 1- 4, pp. 2124-2127.

Beaudouin, J.

D. Gerlich, J. Beaudouin, M. Gebhard, J. Ellenberg, and R. Eils, “Four-dimensional imaging and quantitative reconstruction to analyse complex spatiotemporal processes in live cells,” Nature Cell Biol. 3, 852-855 (2001).
[CrossRef] [PubMed]

Caarls, W.

G. M. Hagen, W. Caarls, M. Thomas, A. Hill, K. A. Lidke, B. Rieger, C. Fritsch, B. van Geest, T. M. Jovin, and D. J. Arndt-Jovin, “Biological applications of an LCoS-based programmable array microscope (PAM),” Proc. SPIE 6441, 64410S (2007).
[CrossRef]

Chen, N. G.

C. H. Wong, N. G. Chen, and C. J. R. Sheppard, “Study on potential of structured illumination microscopy utilizing digital micromirror device for endoscopy purpose,” Proceedings of International Symposium on Biophotonics, Nanophotonics and Metamaterials (IEEE, 2006), pp. 214-217.

Chu, K. K.

Eils, R.

D. Gerlich, J. Beaudouin, M. Gebhard, J. Ellenberg, and R. Eils, “Four-dimensional imaging and quantitative reconstruction to analyse complex spatiotemporal processes in live cells,” Nature Cell Biol. 3, 852-855 (2001).
[CrossRef] [PubMed]

Ellenberg, J.

D. Gerlich, J. Beaudouin, M. Gebhard, J. Ellenberg, and R. Eils, “Four-dimensional imaging and quantitative reconstruction to analyse complex spatiotemporal processes in live cells,” Nature Cell Biol. 3, 852-855 (2001).
[CrossRef] [PubMed]

Fritsch, C.

G. M. Hagen, W. Caarls, M. Thomas, A. Hill, K. A. Lidke, B. Rieger, C. Fritsch, B. van Geest, T. M. Jovin, and D. J. Arndt-Jovin, “Biological applications of an LCoS-based programmable array microscope (PAM),” Proc. SPIE 6441, 64410S (2007).
[CrossRef]

Fujigaki, M.

Fukano, T.

Gebhard, M.

D. Gerlich, J. Beaudouin, M. Gebhard, J. Ellenberg, and R. Eils, “Four-dimensional imaging and quantitative reconstruction to analyse complex spatiotemporal processes in live cells,” Nature Cell Biol. 3, 852-855 (2001).
[CrossRef] [PubMed]

Gerlich, D.

D. Gerlich, J. Beaudouin, M. Gebhard, J. Ellenberg, and R. Eils, “Four-dimensional imaging and quantitative reconstruction to analyse complex spatiotemporal processes in live cells,” Nature Cell Biol. 3, 852-855 (2001).
[CrossRef] [PubMed]

Glick, B. S.

A. T. Hammond and B. S. Glick, “Raising the speed limits for 4D fluorescence microscopy,” Traffic 1, 935-940 (2000).
[CrossRef]

Hagen, G. M.

G. M. Hagen, W. Caarls, M. Thomas, A. Hill, K. A. Lidke, B. Rieger, C. Fritsch, B. van Geest, T. M. Jovin, and D. J. Arndt-Jovin, “Biological applications of an LCoS-based programmable array microscope (PAM),” Proc. SPIE 6441, 64410S (2007).
[CrossRef]

Hammond, A. T.

A. T. Hammond and B. S. Glick, “Raising the speed limits for 4D fluorescence microscopy,” Traffic 1, 935-940 (2000).
[CrossRef]

Hanley, Q. S.

P. J. Verveer, Q. S. Hanley, P. W. Verbeek, L. J. Van Vliet, and T. M. Jovin, “Theory of confocal fluorescence imaging in the programmable array microscope (PAM),” J. Microsc. (Oxford) 189, 192-198 (1998).
[CrossRef]

Heintzmann, R.

Hill, A.

G. M. Hagen, W. Caarls, M. Thomas, A. Hill, K. A. Lidke, B. Rieger, C. Fritsch, B. van Geest, T. M. Jovin, and D. J. Arndt-Jovin, “Biological applications of an LCoS-based programmable array microscope (PAM),” Proc. SPIE 6441, 64410S (2007).
[CrossRef]

Jiang, S.

S. Jiang and J. G. Walker, “Non-scanning fluorescence confocal microscopy using speckle illumination and optical data processing,” Opt. Commun. 256, 35-45 (2005).
[CrossRef]

S. Jiang, “Non-scanning fluorescence confocal microscopy using laser speckle illumination,” Ph.D. thesis (University of Nottingham, 2005).

Jiang, S. H.

S. H. Jiang and J. G. Walker, “Speckle-illuminated fluorescence confocal microscopy using a digital micro-mirror device,” Meas. Sci. Technol. 20, 065501 (2009).
[CrossRef]

S. H. Jiang and J. G. Walker, “Experimental confirmation of non-scanning fluorescence confocal microscopy using speckle illumination,” Opt. Commun. 238, 1-12 (2004).
[CrossRef]

Jovin, T. M.

G. M. Hagen, W. Caarls, M. Thomas, A. Hill, K. A. Lidke, B. Rieger, C. Fritsch, B. van Geest, T. M. Jovin, and D. J. Arndt-Jovin, “Biological applications of an LCoS-based programmable array microscope (PAM),” Proc. SPIE 6441, 64410S (2007).
[CrossRef]

P. J. Verveer, Q. S. Hanley, P. W. Verbeek, L. J. Van Vliet, and T. M. Jovin, “Theory of confocal fluorescence imaging in the programmable array microscope (PAM),” J. Microsc. (Oxford) 189, 192-198 (1998).
[CrossRef]

Krause, A. W.

Liang, M. H.

Lidke, K. A.

G. M. Hagen, W. Caarls, M. Thomas, A. Hill, K. A. Lidke, B. Rieger, C. Fritsch, B. van Geest, T. M. Jovin, and D. J. Arndt-Jovin, “Biological applications of an LCoS-based programmable array microscope (PAM),” Proc. SPIE 6441, 64410S (2007).
[CrossRef]

Lim, D.

Matui, T.

Mertz, J.

Miyawaki, A.

Morimoto, Y.

Patel, S.

V. Bansal, S. Patel, and P. Saggau, “High-speed addressable confocal microscopy for functional imaging of cellular activity,” J. Biomed. Opt. 11 (2006).
[CrossRef] [PubMed]

V. Bansal, S. Patel, and P. Saggau, “A high-speed confocal laser-scanning microscope based on acousto-optic deflectors and a digital micromirror device,” in Proceedings of the 25th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (IEEE, 2003), Vols. 1- 4, pp. 2124-2127.

Ri, S.

Rieger, B.

G. M. Hagen, W. Caarls, M. Thomas, A. Hill, K. A. Lidke, B. Rieger, C. Fritsch, B. van Geest, T. M. Jovin, and D. J. Arndt-Jovin, “Biological applications of an LCoS-based programmable array microscope (PAM),” Proc. SPIE 6441, 64410S (2007).
[CrossRef]

Saggau, P.

V. Bansal, S. Patel, and P. Saggau, “High-speed addressable confocal microscopy for functional imaging of cellular activity,” J. Biomed. Opt. 11 (2006).
[CrossRef] [PubMed]

V. Bansal, S. Patel, and P. Saggau, “A high-speed confocal laser-scanning microscope based on acousto-optic deflectors and a digital micromirror device,” in Proceedings of the 25th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (IEEE, 2003), Vols. 1- 4, pp. 2124-2127.

Sheppard, C. J. R.

C. H. Wong, N. G. Chen, and C. J. R. Sheppard, “Study on potential of structured illumination microscopy utilizing digital micromirror device for endoscopy purpose,” Proceedings of International Symposium on Biophotonics, Nanophotonics and Metamaterials (IEEE, 2006), pp. 214-217.

Stehr, R. L.

Thomas, M.

G. M. Hagen, W. Caarls, M. Thomas, A. Hill, K. A. Lidke, B. Rieger, C. Fritsch, B. van Geest, T. M. Jovin, and D. J. Arndt-Jovin, “Biological applications of an LCoS-based programmable array microscope (PAM),” Proc. SPIE 6441, 64410S (2007).
[CrossRef]

van Geest, B.

G. M. Hagen, W. Caarls, M. Thomas, A. Hill, K. A. Lidke, B. Rieger, C. Fritsch, B. van Geest, T. M. Jovin, and D. J. Arndt-Jovin, “Biological applications of an LCoS-based programmable array microscope (PAM),” Proc. SPIE 6441, 64410S (2007).
[CrossRef]

Van Vliet, L. J.

P. J. Verveer, Q. S. Hanley, P. W. Verbeek, L. J. Van Vliet, and T. M. Jovin, “Theory of confocal fluorescence imaging in the programmable array microscope (PAM),” J. Microsc. (Oxford) 189, 192-198 (1998).
[CrossRef]

Ventalon, C.

Verbeek, P. W.

P. J. Verveer, Q. S. Hanley, P. W. Verbeek, L. J. Van Vliet, and T. M. Jovin, “Theory of confocal fluorescence imaging in the programmable array microscope (PAM),” J. Microsc. (Oxford) 189, 192-198 (1998).
[CrossRef]

Verveer, P. J.

P. J. Verveer, Q. S. Hanley, P. W. Verbeek, L. J. Van Vliet, and T. M. Jovin, “Theory of confocal fluorescence imaging in the programmable array microscope (PAM),” J. Microsc. (Oxford) 189, 192-198 (1998).
[CrossRef]

Walker, J. G.

S. H. Jiang and J. G. Walker, “Speckle-illuminated fluorescence confocal microscopy using a digital micro-mirror device,” Meas. Sci. Technol. 20, 065501 (2009).
[CrossRef]

S. Jiang and J. G. Walker, “Non-scanning fluorescence confocal microscopy using speckle illumination and optical data processing,” Opt. Commun. 256, 35-45 (2005).
[CrossRef]

S. H. Jiang and J. G. Walker, “Experimental confirmation of non-scanning fluorescence confocal microscopy using speckle illumination,” Opt. Commun. 238, 1-12 (2004).
[CrossRef]

J. G. Walker, “Non-scanning confocal fluorescence microscopy using speckle illumination,” Opt. Commun. 189, 221-226(2001).
[CrossRef]

Waters, J. C.

J. C. Waters, “Live-cell fluorescence imaging,” in Digital Microscopy, 3rd ed. (Elsevier, 2007), pp. 115-140.
[CrossRef]

Wong, C. H.

C. H. Wong, N. G. Chen, and C. J. R. Sheppard, “Study on potential of structured illumination microscopy utilizing digital micromirror device for endoscopy purpose,” Proceedings of International Symposium on Biophotonics, Nanophotonics and Metamaterials (IEEE, 2006), pp. 214-217.

Appl. Opt.

J. Biomed. Opt.

V. Bansal, S. Patel, and P. Saggau, “High-speed addressable confocal microscopy for functional imaging of cellular activity,” J. Biomed. Opt. 11 (2006).
[CrossRef] [PubMed]

J. Microsc. (Oxford)

P. J. Verveer, Q. S. Hanley, P. W. Verbeek, L. J. Van Vliet, and T. M. Jovin, “Theory of confocal fluorescence imaging in the programmable array microscope (PAM),” J. Microsc. (Oxford) 189, 192-198 (1998).
[CrossRef]

Meas. Sci. Technol.

S. H. Jiang and J. G. Walker, “Speckle-illuminated fluorescence confocal microscopy using a digital micro-mirror device,” Meas. Sci. Technol. 20, 065501 (2009).
[CrossRef]

Nature Cell Biol.

D. Gerlich, J. Beaudouin, M. Gebhard, J. Ellenberg, and R. Eils, “Four-dimensional imaging and quantitative reconstruction to analyse complex spatiotemporal processes in live cells,” Nature Cell Biol. 3, 852-855 (2001).
[CrossRef] [PubMed]

Opt. Commun.

J. G. Walker, “Non-scanning confocal fluorescence microscopy using speckle illumination,” Opt. Commun. 189, 221-226(2001).
[CrossRef]

S. H. Jiang and J. G. Walker, “Experimental confirmation of non-scanning fluorescence confocal microscopy using speckle illumination,” Opt. Commun. 238, 1-12 (2004).
[CrossRef]

S. Jiang and J. G. Walker, “Non-scanning fluorescence confocal microscopy using speckle illumination and optical data processing,” Opt. Commun. 256, 35-45 (2005).
[CrossRef]

Opt. Express

Opt. Lett.

Proc. SPIE

G. M. Hagen, W. Caarls, M. Thomas, A. Hill, K. A. Lidke, B. Rieger, C. Fritsch, B. van Geest, T. M. Jovin, and D. J. Arndt-Jovin, “Biological applications of an LCoS-based programmable array microscope (PAM),” Proc. SPIE 6441, 64410S (2007).
[CrossRef]

Traffic

A. T. Hammond and B. S. Glick, “Raising the speed limits for 4D fluorescence microscopy,” Traffic 1, 935-940 (2000).
[CrossRef]

Other

S. Jiang, “Non-scanning fluorescence confocal microscopy using laser speckle illumination,” Ph.D. thesis (University of Nottingham, 2005).

ViALUX GmbH, product data sheet, Chemnitz, Germany (2009).

C. H. Wong, N. G. Chen, and C. J. R. Sheppard, “Study on potential of structured illumination microscopy utilizing digital micromirror device for endoscopy purpose,” Proceedings of International Symposium on Biophotonics, Nanophotonics and Metamaterials (IEEE, 2006), pp. 214-217.

V. Bansal, S. Patel, and P. Saggau, “A high-speed confocal laser-scanning microscope based on acousto-optic deflectors and a digital micromirror device,” in Proceedings of the 25th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (IEEE, 2003), Vols. 1- 4, pp. 2124-2127.

J. C. Waters, “Live-cell fluorescence imaging,” in Digital Microscopy, 3rd ed. (Elsevier, 2007), pp. 115-140.
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Schematic diagram for the optical arrangement.

Fig. 2
Fig. 2

Example of a computer-generated binary speckle pattern.

Fig. 3
Fig. 3

Comparison of response to a multipoint object at different focal positions for conventional, scanning confocal, and DMD-based systems. The images are shown for the in-focus case (top row) and for the object defocused by 20, 40, 60, and 80 μm , respectively (rows two to five).

Fig. 4
Fig. 4

Depth discrimination property.

Fig. 5
Fig. 5

Intensity nonuniformity σ against the number of frames averaged.

Fig. 6
Fig. 6

Type 1 and confocal images of 2.5 μm fluorescent microspheres: (a)–(e) direct type 1 images; (f)–(j) I 1 + I 2 , (k)–(o) I 1 I 2 . The images in each row are obtained with the sample at the focal position and 20 μm , 40 μm , 60 μm , and 80 μm out of focus.

Fig. 7
Fig. 7

Mesh plot of the processed images of 2.5 μm fluorescent microspheres: (a) in focus, (b)  20 μm out of focus, (c)  40 μm out of focus, (d)  60 μm out of focus, (e)  80 μm out of focus.

Fig. 8
Fig. 8

Experimental results for BPAE cells: (a)–(c) direct type 1 images, (d)–(f) processed images. The images in each row are obtained with the sample at different focal positions in 1 μm increments.

Equations (13)

Equations on this page are rendered with MathJax. Learn more.

S = | S B * h 1 | 2 ,
I = ( S · O ) * | h 2 | 2 ,
I 1 = I · S B ,
I 1 = S B S I I + S B S I I I ,
I I = O * | h 2 | 2 ,
I I I = O * ( P S · | h 2 | 2 ) ,
P S ( x x , y y , z ) = S B ( x , y ) · S ( x , y , z ) S B S S B S ,
I 2 = I · ( 1 S B ) .
I 2 = S ( 1 S B ) I I S B S I I I ,
I 1 + I 2 = S I I ,
I p = I 1 S B ( I 1 + I 2 ) = S B S I I I .
I p = I 1 I 2 ,
σ = I p 2 I p 2 / I p .

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