Abstract

Focusing light through scattering media has broad applications in optical imaging, manipulation and therapy. The contrast of the focus can be quantified by peak-to-background intensity ratio (PBR). Here, we theoretically and numerically show that by using a transmission matrix inversion method to achieve focusing, within a limited field of view and under a low noise condition in transmission matrix measurements, the PBR of the focus can be higher than that achieved by conventional methods such as optical phase conjugation or feedback-based wavefront shaping. Experimentally, using a phase-modulation spatial light modulator, we increase the PBR by 66% over that achieved by conventional methods based on phase conjugation. In addition, we demonstrate that, within a limited field of view and under a low noise condition in transmission matrix measurements, our matrix inversion method enables light focusing to multiple foci with greater fidelity than those of conventional methods.

© 2017 Optical Society of America

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Corrections

24 October 2017: Typographical corrections were made to paragraph 2 of Section 1, paragraph 3 of Section 2.1, paragraph 2 of Section 3, paragraph 1 of Section 3.1, paragraph 1 of Section 3.2, paragraphs 1 and 2 of Section 4, paragraph 1 of the appendix, and the figure captions of Figs. 2 and 6.


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References

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2016 (1)

Y. Shen, Y. Liu, C. Ma, and L. V. Wang, “Focusing light through biological tissue and tissue-mimicking phantoms up to 9.6 cm in thickness with digital optical phase conjugation,” J. Biomed. Opt. 21(8), 85001 (2016).
[PubMed]

2015 (5)

2014 (1)

2012 (2)

Y. M. Wang, B. Judkewitz, C. A. Dimarzio, and C. Yang, “Deep-tissue focal fluorescence imaging with digitally time-reversed ultrasound-encoded light,” Nat. Commun. 3, 928 (2012).
[PubMed]

K. Si, R. Fiolka, and M. Cui, “Fluorescence imaging beyond the ballistic regime by ultrasound pulse guided digital phase conjugation,” Nat. Photonics 6(10), 657–661 (2012).
[PubMed]

2011 (1)

X. Xu, H. Liu, and L. V. Wang, “Time-reversed ultrasonically encoded optical focusing into scattering media,” Nat. Photonics 5(3), 154–157 (2011).
[PubMed]

2010 (5)

N. Ji, D. E. Milkie, and E. Betzig, “Adaptive optics via pupil segmentation for high-resolution imaging in biological tissues,” Nat. Methods 7(2), 141–147 (2010).
[PubMed]

S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque material,” Nat. Commun. 1, 81 (2010).
[PubMed]

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: An approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104(10), 100601 (2010).
[PubMed]

M. Cui and C. Yang, “Implementation of a digital optical phase conjugation system and its application to study the robustness of turbidity suppression by phase conjugation,” Opt. Express 18(4), 3444–3455 (2010).
[PubMed]

C.-L. Hsieh, Y. Pu, R. Grange, G. Laporte, and D. Psaltis, “Imaging through turbid layers by scanning the phase conjugated second harmonic radiation from a nanoparticle,” Opt. Express 18(20), 20723–20731 (2010).
[PubMed]

2008 (2)

I. M. Vellekoop and A. P. Mosk, “Universal optimal transmission of light through disordered materials,” Phys. Rev. Lett. 101(12), 120601 (2008).
[PubMed]

Z. Yaqoob, D. Psaltis, M. S. Feld, and C. Yang, “Optical phase conjugation for turbidity suppression in biological samples,” Nat. Photonics 2(2), 110–115 (2008).
[PubMed]

2007 (1)

2005 (1)

Z. Huang, “A review of progress in clinical photodynamic therapy,” Technol. Cancer Res. Treat. 4(3), 283–293 (2005).
[PubMed]

2004 (1)

A. M. Tulino and S. Verdú, “Random Matrix Theory and Wireless Communications,” Found. TrendsTM Commun. Inf. Theory 1, 1–182 (2004).

1998 (1)

1989 (1)

N. Garcia and A. Z. Genack, “Crossover to strong intensity correlation for microwave radiation in random media,” Phys. Rev. Lett. 63(16), 1678–1681 (1989).
[PubMed]

Betzig, E.

N. Ji, D. E. Milkie, and E. Betzig, “Adaptive optics via pupil segmentation for high-resolution imaging in biological tissues,” Nat. Methods 7(2), 141–147 (2010).
[PubMed]

Boccara, A. C.

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: An approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104(10), 100601 (2010).
[PubMed]

S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque material,” Nat. Commun. 1, 81 (2010).
[PubMed]

Carminati, R.

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: An approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104(10), 100601 (2010).
[PubMed]

Choi, C.

J. Yoon, M. Lee, K. Lee, N. Kim, J. M. Kim, J. Park, H. Yu, C. Choi, W. D. Heo, and Y. Park, “Optogenetic control of cell signaling pathway through scattering skull using wavefront shaping,” Sci. Rep. 5, 13289 (2015).
[PubMed]

Choi, W.

Choi, Y.

Cui, M.

K. Si, R. Fiolka, and M. Cui, “Fluorescence imaging beyond the ballistic regime by ultrasound pulse guided digital phase conjugation,” Nat. Photonics 6(10), 657–661 (2012).
[PubMed]

M. Cui and C. Yang, “Implementation of a digital optical phase conjugation system and its application to study the robustness of turbidity suppression by phase conjugation,” Opt. Express 18(4), 3444–3455 (2010).
[PubMed]

Daudet, L.

Dimarzio, C. A.

Y. M. Wang, B. Judkewitz, C. A. Dimarzio, and C. Yang, “Deep-tissue focal fluorescence imaging with digitally time-reversed ultrasound-encoded light,” Nat. Commun. 3, 928 (2012).
[PubMed]

Drémeau, A.

Feld, M. S.

Z. Yaqoob, D. Psaltis, M. S. Feld, and C. Yang, “Optical phase conjugation for turbidity suppression in biological samples,” Nat. Photonics 2(2), 110–115 (2008).
[PubMed]

Fink, M.

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: An approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104(10), 100601 (2010).
[PubMed]

S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque material,” Nat. Commun. 1, 81 (2010).
[PubMed]

Fiolka, R.

K. Si, R. Fiolka, and M. Cui, “Fluorescence imaging beyond the ballistic regime by ultrasound pulse guided digital phase conjugation,” Nat. Photonics 6(10), 657–661 (2012).
[PubMed]

Garcia, N.

N. Garcia and A. Z. Genack, “Crossover to strong intensity correlation for microwave radiation in random media,” Phys. Rev. Lett. 63(16), 1678–1681 (1989).
[PubMed]

Genack, A. Z.

N. Garcia and A. Z. Genack, “Crossover to strong intensity correlation for microwave radiation in random media,” Phys. Rev. Lett. 63(16), 1678–1681 (1989).
[PubMed]

Gigan, S.

A. Drémeau, A. Liutkus, D. Martina, O. Katz, C. Schülke, F. Krzakala, S. Gigan, and L. Daudet, “Reference-less measurement of the transmission matrix of a highly scattering material using a DMD and phase retrieval techniques,” Opt. Express 23(9), 11898–11911 (2015).
[PubMed]

S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque material,” Nat. Commun. 1, 81 (2010).
[PubMed]

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: An approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104(10), 100601 (2010).
[PubMed]

Grange, R.

Heo, W. D.

J. Yoon, M. Lee, K. Lee, N. Kim, J. M. Kim, J. Park, H. Yu, C. Choi, W. D. Heo, and Y. Park, “Optogenetic control of cell signaling pathway through scattering skull using wavefront shaping,” Sci. Rep. 5, 13289 (2015).
[PubMed]

Horstmeyer, R.

R. Horstmeyer, H. Ruan, and C. Yang, “Guidestar-assisted wavefront-shaping methods for focusing light into biological tissue,” Nat. Photonics 9, 563–571 (2015).
[PubMed]

Hsieh, C.-L.

Huang, Z.

Z. Huang, “A review of progress in clinical photodynamic therapy,” Technol. Cancer Res. Treat. 4(3), 283–293 (2005).
[PubMed]

Ji, N.

N. Ji, D. E. Milkie, and E. Betzig, “Adaptive optics via pupil segmentation for high-resolution imaging in biological tissues,” Nat. Methods 7(2), 141–147 (2010).
[PubMed]

Judkewitz, B.

E. H. Zhou, H. Ruan, C. Yang, and B. Judkewitz, “Focusing on moving targets through scattering samples,” Optica 1(4), 227–232 (2014).
[PubMed]

Y. M. Wang, B. Judkewitz, C. A. Dimarzio, and C. Yang, “Deep-tissue focal fluorescence imaging with digitally time-reversed ultrasound-encoded light,” Nat. Commun. 3, 928 (2012).
[PubMed]

Katz, O.

Kim, J. M.

J. Yoon, M. Lee, K. Lee, N. Kim, J. M. Kim, J. Park, H. Yu, C. Choi, W. D. Heo, and Y. Park, “Optogenetic control of cell signaling pathway through scattering skull using wavefront shaping,” Sci. Rep. 5, 13289 (2015).
[PubMed]

Kim, M.

Kim, N.

J. Yoon, M. Lee, K. Lee, N. Kim, J. M. Kim, J. Park, H. Yu, C. Choi, W. D. Heo, and Y. Park, “Optogenetic control of cell signaling pathway through scattering skull using wavefront shaping,” Sci. Rep. 5, 13289 (2015).
[PubMed]

Krzakala, F.

Laporte, G.

Lee, K.

J. Yoon, K. Lee, J. Park, and Y. Park, “Measuring optical transmission matrices by wavefront shaping,” Opt. Express 23(8), 10158–10167 (2015).
[PubMed]

J. Yoon, M. Lee, K. Lee, N. Kim, J. M. Kim, J. Park, H. Yu, C. Choi, W. D. Heo, and Y. Park, “Optogenetic control of cell signaling pathway through scattering skull using wavefront shaping,” Sci. Rep. 5, 13289 (2015).
[PubMed]

Lee, M.

J. Yoon, M. Lee, K. Lee, N. Kim, J. M. Kim, J. Park, H. Yu, C. Choi, W. D. Heo, and Y. Park, “Optogenetic control of cell signaling pathway through scattering skull using wavefront shaping,” Sci. Rep. 5, 13289 (2015).
[PubMed]

Lerosey, G.

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: An approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104(10), 100601 (2010).
[PubMed]

S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque material,” Nat. Commun. 1, 81 (2010).
[PubMed]

Liu, H.

X. Xu, H. Liu, and L. V. Wang, “Time-reversed ultrasonically encoded optical focusing into scattering media,” Nat. Photonics 5(3), 154–157 (2011).
[PubMed]

Liu, Y.

Y. Shen, Y. Liu, C. Ma, and L. V. Wang, “Focusing light through biological tissue and tissue-mimicking phantoms up to 9.6 cm in thickness with digital optical phase conjugation,” J. Biomed. Opt. 21(8), 85001 (2016).
[PubMed]

Liutkus, A.

Ma, C.

Y. Shen, Y. Liu, C. Ma, and L. V. Wang, “Focusing light through biological tissue and tissue-mimicking phantoms up to 9.6 cm in thickness with digital optical phase conjugation,” J. Biomed. Opt. 21(8), 85001 (2016).
[PubMed]

Martina, D.

Milkie, D. E.

N. Ji, D. E. Milkie, and E. Betzig, “Adaptive optics via pupil segmentation for high-resolution imaging in biological tissues,” Nat. Methods 7(2), 141–147 (2010).
[PubMed]

Mosk, A. P.

I. M. Vellekoop and A. P. Mosk, “Universal optimal transmission of light through disordered materials,” Phys. Rev. Lett. 101(12), 120601 (2008).
[PubMed]

I. M. Vellekoop and A. P. Mosk, “Focusing coherent light through opaque strongly scattering media,” Opt. Lett. 32(16), 2309–2311 (2007).
[PubMed]

Park, J.

J. Yoon, K. Lee, J. Park, and Y. Park, “Measuring optical transmission matrices by wavefront shaping,” Opt. Express 23(8), 10158–10167 (2015).
[PubMed]

J. Yoon, M. Lee, K. Lee, N. Kim, J. M. Kim, J. Park, H. Yu, C. Choi, W. D. Heo, and Y. Park, “Optogenetic control of cell signaling pathway through scattering skull using wavefront shaping,” Sci. Rep. 5, 13289 (2015).
[PubMed]

Park, Y.

J. Yoon, M. Lee, K. Lee, N. Kim, J. M. Kim, J. Park, H. Yu, C. Choi, W. D. Heo, and Y. Park, “Optogenetic control of cell signaling pathway through scattering skull using wavefront shaping,” Sci. Rep. 5, 13289 (2015).
[PubMed]

J. Yoon, K. Lee, J. Park, and Y. Park, “Measuring optical transmission matrices by wavefront shaping,” Opt. Express 23(8), 10158–10167 (2015).
[PubMed]

Popoff, S.

S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque material,” Nat. Commun. 1, 81 (2010).
[PubMed]

Popoff, S. M.

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: An approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104(10), 100601 (2010).
[PubMed]

Psaltis, D.

C.-L. Hsieh, Y. Pu, R. Grange, G. Laporte, and D. Psaltis, “Imaging through turbid layers by scanning the phase conjugated second harmonic radiation from a nanoparticle,” Opt. Express 18(20), 20723–20731 (2010).
[PubMed]

Z. Yaqoob, D. Psaltis, M. S. Feld, and C. Yang, “Optical phase conjugation for turbidity suppression in biological samples,” Nat. Photonics 2(2), 110–115 (2008).
[PubMed]

Pu, Y.

Ruan, H.

R. Horstmeyer, H. Ruan, and C. Yang, “Guidestar-assisted wavefront-shaping methods for focusing light into biological tissue,” Nat. Photonics 9, 563–571 (2015).
[PubMed]

E. H. Zhou, H. Ruan, C. Yang, and B. Judkewitz, “Focusing on moving targets through scattering samples,” Optica 1(4), 227–232 (2014).
[PubMed]

Schülke, C.

Shen, Y.

Y. Shen, Y. Liu, C. Ma, and L. V. Wang, “Focusing light through biological tissue and tissue-mimicking phantoms up to 9.6 cm in thickness with digital optical phase conjugation,” J. Biomed. Opt. 21(8), 85001 (2016).
[PubMed]

Si, K.

K. Si, R. Fiolka, and M. Cui, “Fluorescence imaging beyond the ballistic regime by ultrasound pulse guided digital phase conjugation,” Nat. Photonics 6(10), 657–661 (2012).
[PubMed]

Tulino, A. M.

A. M. Tulino and S. Verdú, “Random Matrix Theory and Wireless Communications,” Found. TrendsTM Commun. Inf. Theory 1, 1–182 (2004).

Vellekoop, I. M.

I. M. Vellekoop and A. P. Mosk, “Universal optimal transmission of light through disordered materials,” Phys. Rev. Lett. 101(12), 120601 (2008).
[PubMed]

I. M. Vellekoop and A. P. Mosk, “Focusing coherent light through opaque strongly scattering media,” Opt. Lett. 32(16), 2309–2311 (2007).
[PubMed]

Verdú, S.

A. M. Tulino and S. Verdú, “Random Matrix Theory and Wireless Communications,” Found. TrendsTM Commun. Inf. Theory 1, 1–182 (2004).

Wang, L. V.

Y. Shen, Y. Liu, C. Ma, and L. V. Wang, “Focusing light through biological tissue and tissue-mimicking phantoms up to 9.6 cm in thickness with digital optical phase conjugation,” J. Biomed. Opt. 21(8), 85001 (2016).
[PubMed]

X. Xu, H. Liu, and L. V. Wang, “Time-reversed ultrasonically encoded optical focusing into scattering media,” Nat. Photonics 5(3), 154–157 (2011).
[PubMed]

Wang, Y. M.

Y. M. Wang, B. Judkewitz, C. A. Dimarzio, and C. Yang, “Deep-tissue focal fluorescence imaging with digitally time-reversed ultrasound-encoded light,” Nat. Commun. 3, 928 (2012).
[PubMed]

Xu, X.

X. Xu, H. Liu, and L. V. Wang, “Time-reversed ultrasonically encoded optical focusing into scattering media,” Nat. Photonics 5(3), 154–157 (2011).
[PubMed]

Yamaguchi, I.

Yang, C.

R. Horstmeyer, H. Ruan, and C. Yang, “Guidestar-assisted wavefront-shaping methods for focusing light into biological tissue,” Nat. Photonics 9, 563–571 (2015).
[PubMed]

E. H. Zhou, H. Ruan, C. Yang, and B. Judkewitz, “Focusing on moving targets through scattering samples,” Optica 1(4), 227–232 (2014).
[PubMed]

Y. M. Wang, B. Judkewitz, C. A. Dimarzio, and C. Yang, “Deep-tissue focal fluorescence imaging with digitally time-reversed ultrasound-encoded light,” Nat. Commun. 3, 928 (2012).
[PubMed]

M. Cui and C. Yang, “Implementation of a digital optical phase conjugation system and its application to study the robustness of turbidity suppression by phase conjugation,” Opt. Express 18(4), 3444–3455 (2010).
[PubMed]

Z. Yaqoob, D. Psaltis, M. S. Feld, and C. Yang, “Optical phase conjugation for turbidity suppression in biological samples,” Nat. Photonics 2(2), 110–115 (2008).
[PubMed]

Yaqoob, Z.

Z. Yaqoob, D. Psaltis, M. S. Feld, and C. Yang, “Optical phase conjugation for turbidity suppression in biological samples,” Nat. Photonics 2(2), 110–115 (2008).
[PubMed]

Yoon, C.

Yoon, J.

J. Yoon, M. Lee, K. Lee, N. Kim, J. M. Kim, J. Park, H. Yu, C. Choi, W. D. Heo, and Y. Park, “Optogenetic control of cell signaling pathway through scattering skull using wavefront shaping,” Sci. Rep. 5, 13289 (2015).
[PubMed]

J. Yoon, K. Lee, J. Park, and Y. Park, “Measuring optical transmission matrices by wavefront shaping,” Opt. Express 23(8), 10158–10167 (2015).
[PubMed]

Yu, H.

J. Yoon, M. Lee, K. Lee, N. Kim, J. M. Kim, J. Park, H. Yu, C. Choi, W. D. Heo, and Y. Park, “Optogenetic control of cell signaling pathway through scattering skull using wavefront shaping,” Sci. Rep. 5, 13289 (2015).
[PubMed]

Zhang, T.

Zhou, E. H.

Found. TrendsTM Commun. Inf. Theory (1)

A. M. Tulino and S. Verdú, “Random Matrix Theory and Wireless Communications,” Found. TrendsTM Commun. Inf. Theory 1, 1–182 (2004).

J. Biomed. Opt. (1)

Y. Shen, Y. Liu, C. Ma, and L. V. Wang, “Focusing light through biological tissue and tissue-mimicking phantoms up to 9.6 cm in thickness with digital optical phase conjugation,” J. Biomed. Opt. 21(8), 85001 (2016).
[PubMed]

Nat. Commun. (2)

S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque material,” Nat. Commun. 1, 81 (2010).
[PubMed]

Y. M. Wang, B. Judkewitz, C. A. Dimarzio, and C. Yang, “Deep-tissue focal fluorescence imaging with digitally time-reversed ultrasound-encoded light,” Nat. Commun. 3, 928 (2012).
[PubMed]

Nat. Methods (1)

N. Ji, D. E. Milkie, and E. Betzig, “Adaptive optics via pupil segmentation for high-resolution imaging in biological tissues,” Nat. Methods 7(2), 141–147 (2010).
[PubMed]

Nat. Photonics (4)

X. Xu, H. Liu, and L. V. Wang, “Time-reversed ultrasonically encoded optical focusing into scattering media,” Nat. Photonics 5(3), 154–157 (2011).
[PubMed]

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Figures (10)

Fig. 1
Fig. 1 Illustration of focusing light to a target location by (a) the transmission matrix inversion method, and (b) the phase conjugation method. The red box in (a) denotes the selected field of view.
Fig. 2
Fig. 2 Two-dimensional simulations of focusing light through a scattering medium to a target location by (a) phase conjugation, and (b) transmission matrix inversion.
Fig. 3
Fig. 3 Illustration of focusing light to two target locations by (a) phase conjugation and (b) transmission matrix inversion.
Fig. 4
Fig. 4 Two-dimensional simulations of focusing light to two positions by (a) phase conjugation and (b) transmission matrix inversion.
Fig. 5
Fig. 5 Schematic of the experimental setup. B, beam splitter; L, lens; P, polarizer; S, shutter; SLM, spatial light modulator.
Fig. 6
Fig. 6 (a) Workflow of realizing a high PBR focusing by matrix inversion. Optical focus achieved by (b) phase conjugation and (c) matrix inversion. Over-exposure images of the focus achieved by (d) phase conjugation and (e) matrix inversion to see the background speckle grains. Scale bar, 50 μm. β = 0.1 in our experiment.
Fig. 7
Fig. 7 Focusing light to two target locations achieved by phase conjugation ((a) – (c)) and by matrix inversion ((d) – (f)). (g) – (i) Line profiles of the foci shown in (a) – (f) achieved by phase conjugation and matrix inversion. Scale bar, 50 μm.
Fig. 8
Fig. 8 PBR versus (a) β and (b) σ2 as n is fixed as 200. Solid lines: theoretical calculation. Dashed lines: numerical simulations.
Fig. 9
Fig. 9 Experimental and simulated ratios between the PBRs achieved by matrix inversion and phase conjugation, considering noise and phase-only modulation (black dots and green curve, respectively). Red curve shows the simulation result considering amplitude & phase modulation. In the simulation, n = 128 and σ n = 1 .
Fig. 10
Fig. 10 Results of speckle darkening. The original speckle patterns before darkening are shown in (a) – (c). After using the matrix inversion method, we can selectively darken the speckle grains enclosed in the red circles ((d) – (f)). Scale bar, 50 μm.

Equations (14)

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E o u t = [ E o u t , 1 E o u t , 2 ... E o u t , m ] = [ t 11 ... t m 1 t 12 t m 2 ... ... ... ... ... ... t 1 n ... t m n ] m × n [ E i n , 1 E i n , 2 ... E i n , n ] = T E i n , m < n .
E o u t c o n j = T × T * × [ 1 0 ... 0 ] .
I o u t c o n j = | E o u t c o n j | 2 = [ | E o u t , 1 c o n j | 2 | E o u t , 2 c o n j | 2 ... | E o u t , m c o n j | 2 ] = [ I o u t , 1 c o n j I o u t , 2 c o n j ... I o u t , m c o n j ] α 2 [ 1 n 1 ... n 1 ] .
E o u t i n v = T × E i n i n v = T × T + × [ 1 0 ... 0 ] = [ 1 0 ... 0 ] .
PBR = n m σ n 2 .
E o u t i n v = T 0 × E i n i n v = T 0 × ( T 0 + e ) + × [ 1 0 ... 0 ] ,
T 0 × ( T 0 + e ) + = I - e ( T 0 + e ) + .
I - e ( T 0 + e ) + = I - e ( U D V * ) + = I - e V D - 1 U * = I - e ˜ D - 1 U * .
| b i j ' | 2 = 1 m 2 | j = 1 m i = 1 m e ˜ i j γ i i | 2 = 1 m | i = 1 m e ˜ i j γ i i | 2 = | e ˜ i j | 2 γ i i 2 .
| b i j ' | 2 = | e ˜ i j | 2 γ i i 2 = | e ˜ i j | 2 γ i i 2 = 2 σ n 2 γ i i 2 .
p d ( x ) = ( x 2 2 n ( 1 + β ) 2 ) ( x 2 2 n ( 1 β ) 2 ) π β x , β = m n < 1.
p γ 2 ( y ) = ( 1 2 n y ( 1 + β ) 2 ) ( 1 2 n y ( 1 β ) 2 ) 2 π β y , β = m n < 1.
| b i j ' | 2 = σ n 2 n ( 1 β ) .
P B R = | p i | 2 | b i j ' | 2 = n m σ n 2 + 1 n m σ n 2 .

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