Abstract

Intensity-based techniques in optical coherence tomography (OCT), such as those based on speckle decorrelation, have attracted great interest for biomedical and industrial applications requiring speed or flow information. In this work we present a rigorous analysis of the effects of noise on speckle decorrelation, demonstrate that these effects frustrate accurate speed quantitation, and propose new techniques that achieve quantitative and repeatable measurements. First, we derive the effect of background noise on the speckle autocorrelation function, finding two detrimental effects of noise. We propose a new autocorrelation function that is immune to the main effect of background noise and permits quantitative measurements at high and moderate signal-to-noise ratios. At the same time, this autocorrelation function is able to provide motion contrast information that accurately identifies areas with movement, similar to speckle variance techniques. In order to extend the SNR range, we quantify and model the second effect of background noise on the autocorrelation function through a calibration. By obtaining an explicit expression for the decorrelation time as a function of speed and diffusion, we show how to use our autocorrelation function and noise calibration to measure a flowing liquid. We obtain accurate results, which are validated by Doppler OCT, and demonstrate a very high dynamic range (> 600 mm/s) compared to that of Doppler OCT (±25 mm/s). We also derive the behavior for low flows, and show that there is an inherent non-linearity in speed measurements in the presence of diffusion due to statistical fluctuations of speckle. Our technique allows quantitative and robust measurements of speeds using OCT, and this work delimits precisely the conditions in which it is accurate.

© 2014 Optical Society of America

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References

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2014 (2)

B. Kennedy, K. Kennedy, and D. Sampson, “A review of optical coherence elastography: Fundamentals, techniques and prospects,” IEEE J. Sel. Top. Quantum Electron. 20, 272–288 (2014).
[Crossref]

B. K. Huang and M. A. Choma, “Resolving directional ambiguity in dynamic light scattering-based transverse motion velocimetry in optical coherence tomography,” Opt. Lett. 39, 521–524 (2014).
[Crossref] [PubMed]

2013 (7)

X. Liu, Y. Huang, J. C. Ramella-Roman, S. A. Mathews, and J. U. Kang, “Quantitative transverse flow measurement using optical coherence tomography speckle decorrelation analysis,” Opt. Lett. 38, 805–807 (2013).
[Crossref] [PubMed]

J. Tokayer, Y. Jia, A.-H. Dhalla, and D. Huang, “Blood flow velocity quantification using split-spectrum amplitude-decorrelation angiography with optical coherence tomography,” Biomed. Opt. Express 4, 1909 (2013).
[Crossref] [PubMed]

T. Yonetsu, B. E. Bouma, K. Kato, J. G. Fujimoto, and I.-K. Jang, “Optical coherence tomography,” Circulation J. 77, 1933–1940 (2013).
[Crossref]

D. W. Park, G. H. Kruger, J. M. Rubin, J. Hamilton, P. Gottschalk, R. E. Dodde, A. J. Shih, and W. F. Weitzel, “Quantification of ultrasound correlation-based flow velocity mapping and edge velocity gradient measurement,” J. Ultrasound Med. 32, 1815–1830 (2013).
[Crossref] [PubMed]

N. Weiss, T. G. van Leeuwen, and J. Kalkman, “Localized measurement of longitudinal and transverse flow velocities in colloidal suspensions using optical coherence tomography,” Phys. Rev. E 88, 042312 (2013).
[Crossref]

J. Lee, H. Radhakrishnan, W. Wu, A. Daneshmand, M. Climov, C. Ayata, and D. A. Boas, “Quantitative imaging of cerebral blood flow velocity and intracellular motility using dynamic light scattering–optical coherence tomography,” J. Cereb. Blood Flow Metab. 33, 819–825 (2013).
[Crossref] [PubMed]

J. Senarathna, A. Rege, N. Li, and N. Thakor, “Laser speckle contrast imaging: Theory, instrumentation and applications,” IEEE Rev. Biomed. Eng. 6, 99–110 (2013).
[Crossref] [PubMed]

2012 (3)

2011 (2)

N. Mohan and B. Vakoc, “Principal-component-analysis-based estimation of blood flow velocities using optical coherence tomography intensity signals,” Opt. Lett. 36, 2068–2070 (2011).
[Crossref] [PubMed]

A. M. Forsyth, J. Wan, P. D. Owrutsky, M. Abkarian, and H. A. Stone, “Multiscale approach to link red blood cell dynamics, shear viscosity, and ATP release,” Proc. Natl. Acad. Sci. U. S. A. 108, 10986–10991 (2011).
[Crossref] [PubMed]

2010 (3)

2008 (2)

2006 (2)

K. Wiesauer, M. Pircher, E. Goetzinger, C. K. Hitzenberger, R. Engelke, G. Ahrens, G. Gruetzner, and D. Stifter, “Transversal ultrahigh-resolution polarizationsensitive optical coherence tomography for strain mapping in materials,” Opt. Express 14, 5945–5953 (2006).
[Crossref] [PubMed]

S. H. Yun, G. J. Tearney, B. J. Vakoc, M. Shishkov, W. Y. Oh, A. E. Desjardins, M. J. Suter, R. C. Chan, J. A. Evans, I.-K. Jang, N. S. Nishioka, J. F. de Boer, and B. E. Bouma, “Comprehensive volumetric optical microscopy in vivo,” Nat. Med. 12, 1429–1433 (2006).
[Crossref] [PubMed]

2005 (3)

2004 (1)

2002 (1)

F. A. Lupotti, C. L. De Korte, F. Mastik, and A. F. Van Der Steen, “Dynamic noise correction for IVUS quantitative volume blood flow: methods and numerical validation,” Ultrasound Med. Biol. 28, 1053–1060 (2002).
[Crossref] [PubMed]

2001 (2)

P.-C. Li, C.-J. Cheng, and C.-K. Yeh, “On velocity estimation using speckle decorrelation [blood],” IEEE Trans. Ultrason., Ferroelectr. Freq. Control 48, 1084–1091 (2001).
[Crossref]

X. Li, T. H. Ko, and J. G. Fujimoto, “Intraluminal fiber-optic doppler imaging catheter for structural and functional optical coherence tomography,” Opt. Lett. 26, 1906–1908 (2001).
[Crossref]

2000 (1)

1998 (1)

W. Li, A. F. van der Steen, C. T. Lancée, I. Céspedes, and N. Bom, “Blood flow imaging and volume flow quantitation with intravascular ultrasound,” Ultrasound Med. Biol. 24, 203–214 (1998).
[Crossref] [PubMed]

1997 (2)

W. Li, C. T. Lancee, E. I. Cespedes, A. F. W. v. d. Steen, and N. Bom, “Decorrelation of intravascular echo signals: Potentials for blood velocity estimation,” J. Acoust. Soc. Am. 102, 3785–3794 (1997).
[Crossref]

J. F. de Boer, T. E. Milner, M. J. C. van Gemert, and J. S. Nelson, “Two-dimensional birefringence imaging in biological tissue by polarization-sensitive optical coherence tomography,” Opt. Lett. 22, 934–936 (1997).
[Crossref] [PubMed]

1987 (1)

K. Schatzel, “Correlation techniques in dynamic light scattering,” Appl. Phys. B 42, 193–213 (1987).
[Crossref]

Abkarian, M.

A. M. Forsyth, J. Wan, P. D. Owrutsky, M. Abkarian, and H. A. Stone, “Multiscale approach to link red blood cell dynamics, shear viscosity, and ATP release,” Proc. Natl. Acad. Sci. U. S. A. 108, 10986–10991 (2011).
[Crossref] [PubMed]

Ahrens, G.

Ayata, C.

J. Lee, H. Radhakrishnan, W. Wu, A. Daneshmand, M. Climov, C. Ayata, and D. A. Boas, “Quantitative imaging of cerebral blood flow velocity and intracellular motility using dynamic light scattering–optical coherence tomography,” J. Cereb. Blood Flow Metab. 33, 819–825 (2013).
[Crossref] [PubMed]

Barton, J.

Bauer, S.

Baumann, B.

Boas, D. A.

J. Lee, H. Radhakrishnan, W. Wu, A. Daneshmand, M. Climov, C. Ayata, and D. A. Boas, “Quantitative imaging of cerebral blood flow velocity and intracellular motility using dynamic light scattering–optical coherence tomography,” J. Cereb. Blood Flow Metab. 33, 819–825 (2013).
[Crossref] [PubMed]

J. Lee, W. Wu, J. Y. Jiang, B. Zhu, and D. A. Boas, “Dynamic light scattering optical coherence tomography,” Opt. Express 20, 22262–22277 (2012).
[Crossref] [PubMed]

Bom, N.

W. Li, A. F. van der Steen, C. T. Lancée, I. Céspedes, and N. Bom, “Blood flow imaging and volume flow quantitation with intravascular ultrasound,” Ultrasound Med. Biol. 24, 203–214 (1998).
[Crossref] [PubMed]

W. Li, C. T. Lancee, E. I. Cespedes, A. F. W. v. d. Steen, and N. Bom, “Decorrelation of intravascular echo signals: Potentials for blood velocity estimation,” J. Acoust. Soc. Am. 102, 3785–3794 (1997).
[Crossref]

Bouma, B.

Bouma, B. E.

T. Yonetsu, B. E. Bouma, K. Kato, J. G. Fujimoto, and I.-K. Jang, “Optical coherence tomography,” Circulation J. 77, 1933–1940 (2013).
[Crossref]

S. H. Yun, G. J. Tearney, B. J. Vakoc, M. Shishkov, W. Y. Oh, A. E. Desjardins, M. J. Suter, R. C. Chan, J. A. Evans, I.-K. Jang, N. S. Nishioka, J. F. de Boer, and B. E. Bouma, “Comprehensive volumetric optical microscopy in vivo,” Nat. Med. 12, 1429–1433 (2006).
[Crossref] [PubMed]

B. J. Vakoc, S. H. Yun, J. F. de Boer, G. J. Tearney, and B. E. Bouma, “Phase-resolved optical frequency domain imaging,” Opt. Express 13, 5483 (2005).
[Crossref] [PubMed]

Bromberg, Y.

Y. Bromberg, Y. Lahini, E. Small, and Y. Silberberg, “Hanbury brown and twiss interferometry with interacting photons,” Nat. Photonics 4, 721–726 (2010).
[Crossref]

Cable, A.

Cespedes, E. I.

W. Li, C. T. Lancee, E. I. Cespedes, A. F. W. v. d. Steen, and N. Bom, “Decorrelation of intravascular echo signals: Potentials for blood velocity estimation,” J. Acoust. Soc. Am. 102, 3785–3794 (1997).
[Crossref]

Céspedes, I.

W. Li, A. F. van der Steen, C. T. Lancée, I. Céspedes, and N. Bom, “Blood flow imaging and volume flow quantitation with intravascular ultrasound,” Ultrasound Med. Biol. 24, 203–214 (1998).
[Crossref] [PubMed]

Chan, R. C.

S. H. Yun, G. J. Tearney, B. J. Vakoc, M. Shishkov, W. Y. Oh, A. E. Desjardins, M. J. Suter, R. C. Chan, J. A. Evans, I.-K. Jang, N. S. Nishioka, J. F. de Boer, and B. E. Bouma, “Comprehensive volumetric optical microscopy in vivo,” Nat. Med. 12, 1429–1433 (2006).
[Crossref] [PubMed]

Chang, E. W.

Cheng, C.-J.

P.-C. Li, C.-J. Cheng, and C.-K. Yeh, “On velocity estimation using speckle decorrelation [blood],” IEEE Trans. Ultrason., Ferroelectr. Freq. Control 48, 1084–1091 (2001).
[Crossref]

Choma, M. A.

Climov, M.

J. Lee, H. Radhakrishnan, W. Wu, A. Daneshmand, M. Climov, C. Ayata, and D. A. Boas, “Quantitative imaging of cerebral blood flow velocity and intracellular motility using dynamic light scattering–optical coherence tomography,” J. Cereb. Blood Flow Metab. 33, 819–825 (2013).
[Crossref] [PubMed]

Creath, K.

J. C. Wyant and K. Creath, “Basic Wavefront Aberration Theory for Optical Metrology,” in Applied Optics and Optical Engineering XI, R. R. Shannon and J. C. Wyant, eds. (Academic, 1992), pp. 2.

Daneshmand, A.

J. Lee, H. Radhakrishnan, W. Wu, A. Daneshmand, M. Climov, C. Ayata, and D. A. Boas, “Quantitative imaging of cerebral blood flow velocity and intracellular motility using dynamic light scattering–optical coherence tomography,” J. Cereb. Blood Flow Metab. 33, 819–825 (2013).
[Crossref] [PubMed]

de Boer, J.

de Boer, J. F.

De Korte, C. L.

F. A. Lupotti, C. L. De Korte, F. Mastik, and A. F. Van Der Steen, “Dynamic noise correction for IVUS quantitative volume blood flow: methods and numerical validation,” Ultrasound Med. Biol. 28, 1053–1060 (2002).
[Crossref] [PubMed]

Desjardins, A. E.

S. H. Yun, G. J. Tearney, B. J. Vakoc, M. Shishkov, W. Y. Oh, A. E. Desjardins, M. J. Suter, R. C. Chan, J. A. Evans, I.-K. Jang, N. S. Nishioka, J. F. de Boer, and B. E. Bouma, “Comprehensive volumetric optical microscopy in vivo,” Nat. Med. 12, 1429–1433 (2006).
[Crossref] [PubMed]

Dhalla, A.-H.

Dodde, R. E.

D. W. Park, G. H. Kruger, J. M. Rubin, J. Hamilton, P. Gottschalk, R. E. Dodde, A. J. Shih, and W. F. Weitzel, “Quantification of ultrasound correlation-based flow velocity mapping and edge velocity gradient measurement,” J. Ultrasound Med. 32, 1815–1830 (2013).
[Crossref] [PubMed]

Drexler, W.

Duncan, D. D.

Engelke, R.

Evans, J. A.

S. H. Yun, G. J. Tearney, B. J. Vakoc, M. Shishkov, W. Y. Oh, A. E. Desjardins, M. J. Suter, R. C. Chan, J. A. Evans, I.-K. Jang, N. S. Nishioka, J. F. de Boer, and B. E. Bouma, “Comprehensive volumetric optical microscopy in vivo,” Nat. Med. 12, 1429–1433 (2006).
[Crossref] [PubMed]

Forsyth, A. M.

A. M. Forsyth, J. Wan, P. D. Owrutsky, M. Abkarian, and H. A. Stone, “Multiscale approach to link red blood cell dynamics, shear viscosity, and ATP release,” Proc. Natl. Acad. Sci. U. S. A. 108, 10986–10991 (2011).
[Crossref] [PubMed]

Fujimoto, J. G.

Goetzinger, E.

Gottschalk, P.

D. W. Park, G. H. Kruger, J. M. Rubin, J. Hamilton, P. Gottschalk, R. E. Dodde, A. J. Shih, and W. F. Weitzel, “Quantification of ultrasound correlation-based flow velocity mapping and edge velocity gradient measurement,” J. Ultrasound Med. 32, 1815–1830 (2013).
[Crossref] [PubMed]

Götzinger, E.

Gruetzner, G.

Grützner, G.

Hamilton, J.

D. W. Park, G. H. Kruger, J. M. Rubin, J. Hamilton, P. Gottschalk, R. E. Dodde, A. J. Shih, and W. F. Weitzel, “Quantification of ultrasound correlation-based flow velocity mapping and edge velocity gradient measurement,” J. Ultrasound Med. 32, 1815–1830 (2013).
[Crossref] [PubMed]

Harding, K.

G. Song and K. Harding, “OCT for industrial applications,” in Optical Metrology and Inspection for Industrial Applications II, Proc. SPIE8563, 85630N (2012).
[Crossref]

Heise, B.

Hitzenberger, C.

Hitzenberger, C. K.

Hornegger, J.

Huang, B. K.

Huang, D.

Huang, Y.

Ippen, E. P.

Jang, I.-K.

T. Yonetsu, B. E. Bouma, K. Kato, J. G. Fujimoto, and I.-K. Jang, “Optical coherence tomography,” Circulation J. 77, 1933–1940 (2013).
[Crossref]

S. H. Yun, G. J. Tearney, B. J. Vakoc, M. Shishkov, W. Y. Oh, A. E. Desjardins, M. J. Suter, R. C. Chan, J. A. Evans, I.-K. Jang, N. S. Nishioka, J. F. de Boer, and B. E. Bouma, “Comprehensive volumetric optical microscopy in vivo,” Nat. Med. 12, 1429–1433 (2006).
[Crossref] [PubMed]

Jia, Y.

Jiang, J.

Jiang, J. Y.

Kalkman, J.

N. Weiss, T. G. van Leeuwen, and J. Kalkman, “Localized measurement of longitudinal and transverse flow velocities in colloidal suspensions using optical coherence tomography,” Phys. Rev. E 88, 042312 (2013).
[Crossref]

Kang, J. U.

Kärtner, F. X.

Kato, K.

T. Yonetsu, B. E. Bouma, K. Kato, J. G. Fujimoto, and I.-K. Jang, “Optical coherence tomography,” Circulation J. 77, 1933–1940 (2013).
[Crossref]

Kennedy, B.

B. Kennedy, K. Kennedy, and D. Sampson, “A review of optical coherence elastography: Fundamentals, techniques and prospects,” IEEE J. Sel. Top. Quantum Electron. 20, 272–288 (2014).
[Crossref]

Kennedy, K.

B. Kennedy, K. Kennedy, and D. Sampson, “A review of optical coherence elastography: Fundamentals, techniques and prospects,” IEEE J. Sel. Top. Quantum Electron. 20, 272–288 (2014).
[Crossref]

Khurana, M.

Kirkpatrick, S. J.

Ko, T. H.

Kobler, J. B.

Kraus, M. F.

Kruger, G. H.

D. W. Park, G. H. Kruger, J. M. Rubin, J. Hamilton, P. Gottschalk, R. E. Dodde, A. J. Shih, and W. F. Weitzel, “Quantification of ultrasound correlation-based flow velocity mapping and edge velocity gradient measurement,” J. Ultrasound Med. 32, 1815–1830 (2013).
[Crossref] [PubMed]

Lahini, Y.

Y. Bromberg, Y. Lahini, E. Small, and Y. Silberberg, “Hanbury brown and twiss interferometry with interacting photons,” Nat. Photonics 4, 721–726 (2010).
[Crossref]

Lancee, C. T.

W. Li, C. T. Lancee, E. I. Cespedes, A. F. W. v. d. Steen, and N. Bom, “Decorrelation of intravascular echo signals: Potentials for blood velocity estimation,” J. Acoust. Soc. Am. 102, 3785–3794 (1997).
[Crossref]

Lancée, C. T.

W. Li, A. F. van der Steen, C. T. Lancée, I. Céspedes, and N. Bom, “Blood flow imaging and volume flow quantitation with intravascular ultrasound,” Ultrasound Med. Biol. 24, 203–214 (1998).
[Crossref] [PubMed]

Lee, J.

J. Lee, H. Radhakrishnan, W. Wu, A. Daneshmand, M. Climov, C. Ayata, and D. A. Boas, “Quantitative imaging of cerebral blood flow velocity and intracellular motility using dynamic light scattering–optical coherence tomography,” J. Cereb. Blood Flow Metab. 33, 819–825 (2013).
[Crossref] [PubMed]

J. Lee, W. Wu, J. Y. Jiang, B. Zhu, and D. A. Boas, “Dynamic light scattering optical coherence tomography,” Opt. Express 20, 22262–22277 (2012).
[Crossref] [PubMed]

Leeuwen, T. G. V.

E. Regar, P. W. Serruys, and T. G. V. Leeuwen, Optical Coherence Tomography in Cardiovascular Research (CRC Press, 2007).
[Crossref]

Leiss-Holzinger, E.

Leung, M. K. K.

Li, N.

J. Senarathna, A. Rege, N. Li, and N. Thakor, “Laser speckle contrast imaging: Theory, instrumentation and applications,” IEEE Rev. Biomed. Eng. 6, 99–110 (2013).
[Crossref] [PubMed]

Li, P.-C.

P.-C. Li, C.-J. Cheng, and C.-K. Yeh, “On velocity estimation using speckle decorrelation [blood],” IEEE Trans. Ultrason., Ferroelectr. Freq. Control 48, 1084–1091 (2001).
[Crossref]

Li, W.

W. Li, A. F. van der Steen, C. T. Lancée, I. Céspedes, and N. Bom, “Blood flow imaging and volume flow quantitation with intravascular ultrasound,” Ultrasound Med. Biol. 24, 203–214 (1998).
[Crossref] [PubMed]

W. Li, C. T. Lancee, E. I. Cespedes, A. F. W. v. d. Steen, and N. Bom, “Decorrelation of intravascular echo signals: Potentials for blood velocity estimation,” J. Acoust. Soc. Am. 102, 3785–3794 (1997).
[Crossref]

Li, X.

Li, X. D.

Liu, J. J.

Liu, X.

Lupotti, F. A.

F. A. Lupotti, C. L. De Korte, F. Mastik, and A. F. Van Der Steen, “Dynamic noise correction for IVUS quantitative volume blood flow: methods and numerical validation,” Ultrasound Med. Biol. 28, 1053–1060 (2002).
[Crossref] [PubMed]

Major, Z.

Mariampillai, A.

Mastik, F.

F. A. Lupotti, C. L. De Korte, F. Mastik, and A. F. Van Der Steen, “Dynamic noise correction for IVUS quantitative volume blood flow: methods and numerical validation,” Ultrasound Med. Biol. 28, 1053–1060 (2002).
[Crossref] [PubMed]

Mathews, S. A.

Milner, T. E.

Mohan, N.

Morgner, U.

Moriyama, E. H.

Munce, N. R.

Nelson, J. S.

Nishioka, N. S.

S. H. Yun, G. J. Tearney, B. J. Vakoc, M. Shishkov, W. Y. Oh, A. E. Desjardins, M. J. Suter, R. C. Chan, J. A. Evans, I.-K. Jang, N. S. Nishioka, J. F. de Boer, and B. E. Bouma, “Comprehensive volumetric optical microscopy in vivo,” Nat. Med. 12, 1429–1433 (2006).
[Crossref] [PubMed]

Oh, W. Y.

S. H. Yun, G. J. Tearney, B. J. Vakoc, M. Shishkov, W. Y. Oh, A. E. Desjardins, M. J. Suter, R. C. Chan, J. A. Evans, I.-K. Jang, N. S. Nishioka, J. F. de Boer, and B. E. Bouma, “Comprehensive volumetric optical microscopy in vivo,” Nat. Med. 12, 1429–1433 (2006).
[Crossref] [PubMed]

Owrutsky, P. D.

A. M. Forsyth, J. Wan, P. D. Owrutsky, M. Abkarian, and H. A. Stone, “Multiscale approach to link red blood cell dynamics, shear viscosity, and ATP release,” Proc. Natl. Acad. Sci. U. S. A. 108, 10986–10991 (2011).
[Crossref] [PubMed]

Park, D. W.

D. W. Park, G. H. Kruger, J. M. Rubin, J. Hamilton, P. Gottschalk, R. E. Dodde, A. J. Shih, and W. F. Weitzel, “Quantification of ultrasound correlation-based flow velocity mapping and edge velocity gradient measurement,” J. Ultrasound Med. 32, 1815–1830 (2013).
[Crossref] [PubMed]

Pircher, M.

Pitris, C.

Potsaid, B.

Radhakrishnan, H.

J. Lee, H. Radhakrishnan, W. Wu, A. Daneshmand, M. Climov, C. Ayata, and D. A. Boas, “Quantitative imaging of cerebral blood flow velocity and intracellular motility using dynamic light scattering–optical coherence tomography,” J. Cereb. Blood Flow Metab. 33, 819–825 (2013).
[Crossref] [PubMed]

Ramella-Roman, J. C.

Regar, E.

E. Regar, P. W. Serruys, and T. G. V. Leeuwen, Optical Coherence Tomography in Cardiovascular Research (CRC Press, 2007).
[Crossref]

Rege, A.

J. Senarathna, A. Rege, N. Li, and N. Thakor, “Laser speckle contrast imaging: Theory, instrumentation and applications,” IEEE Rev. Biomed. Eng. 6, 99–110 (2013).
[Crossref] [PubMed]

Rubin, J. M.

D. W. Park, G. H. Kruger, J. M. Rubin, J. Hamilton, P. Gottschalk, R. E. Dodde, A. J. Shih, and W. F. Weitzel, “Quantification of ultrasound correlation-based flow velocity mapping and edge velocity gradient measurement,” J. Ultrasound Med. 32, 1815–1830 (2013).
[Crossref] [PubMed]

Sampson, D.

B. Kennedy, K. Kennedy, and D. Sampson, “A review of optical coherence elastography: Fundamentals, techniques and prospects,” IEEE J. Sel. Top. Quantum Electron. 20, 272–288 (2014).
[Crossref]

Schatzel, K.

K. Schatzel, “Correlation techniques in dynamic light scattering,” Appl. Phys. B 42, 193–213 (1987).
[Crossref]

Senarathna, J.

J. Senarathna, A. Rege, N. Li, and N. Thakor, “Laser speckle contrast imaging: Theory, instrumentation and applications,” IEEE Rev. Biomed. Eng. 6, 99–110 (2013).
[Crossref] [PubMed]

Serruys, P. W.

E. Regar, P. W. Serruys, and T. G. V. Leeuwen, Optical Coherence Tomography in Cardiovascular Research (CRC Press, 2007).
[Crossref]

Shih, A. J.

D. W. Park, G. H. Kruger, J. M. Rubin, J. Hamilton, P. Gottschalk, R. E. Dodde, A. J. Shih, and W. F. Weitzel, “Quantification of ultrasound correlation-based flow velocity mapping and edge velocity gradient measurement,” J. Ultrasound Med. 32, 1815–1830 (2013).
[Crossref] [PubMed]

Shishkov, M.

S. H. Yun, G. J. Tearney, B. J. Vakoc, M. Shishkov, W. Y. Oh, A. E. Desjardins, M. J. Suter, R. C. Chan, J. A. Evans, I.-K. Jang, N. S. Nishioka, J. F. de Boer, and B. E. Bouma, “Comprehensive volumetric optical microscopy in vivo,” Nat. Med. 12, 1429–1433 (2006).
[Crossref] [PubMed]

Silberberg, Y.

Y. Bromberg, Y. Lahini, E. Small, and Y. Silberberg, “Hanbury brown and twiss interferometry with interacting photons,” Nat. Photonics 4, 721–726 (2010).
[Crossref]

Small, E.

Y. Bromberg, Y. Lahini, E. Small, and Y. Silberberg, “Hanbury brown and twiss interferometry with interacting photons,” Nat. Photonics 4, 721–726 (2010).
[Crossref]

Song, G.

G. Song and K. Harding, “OCT for industrial applications,” in Optical Metrology and Inspection for Industrial Applications II, Proc. SPIE8563, 85630N (2012).
[Crossref]

Standish, B. A.

Steen, A. F. W. v. d.

W. Li, C. T. Lancee, E. I. Cespedes, A. F. W. v. d. Steen, and N. Bom, “Decorrelation of intravascular echo signals: Potentials for blood velocity estimation,” J. Acoust. Soc. Am. 102, 3785–3794 (1997).
[Crossref]

Stifter, D.

Stone, H. A.

A. M. Forsyth, J. Wan, P. D. Owrutsky, M. Abkarian, and H. A. Stone, “Multiscale approach to link red blood cell dynamics, shear viscosity, and ATP release,” Proc. Natl. Acad. Sci. U. S. A. 108, 10986–10991 (2011).
[Crossref] [PubMed]

Stromski, S.

Subhash, H.

Suter, M. J.

S. H. Yun, G. J. Tearney, B. J. Vakoc, M. Shishkov, W. Y. Oh, A. E. Desjardins, M. J. Suter, R. C. Chan, J. A. Evans, I.-K. Jang, N. S. Nishioka, J. F. de Boer, and B. E. Bouma, “Comprehensive volumetric optical microscopy in vivo,” Nat. Med. 12, 1429–1433 (2006).
[Crossref] [PubMed]

Tan, O.

Tearney, G.

Tearney, G. J.

S. H. Yun, G. J. Tearney, B. J. Vakoc, M. Shishkov, W. Y. Oh, A. E. Desjardins, M. J. Suter, R. C. Chan, J. A. Evans, I.-K. Jang, N. S. Nishioka, J. F. de Boer, and B. E. Bouma, “Comprehensive volumetric optical microscopy in vivo,” Nat. Med. 12, 1429–1433 (2006).
[Crossref] [PubMed]

B. J. Vakoc, S. H. Yun, J. F. de Boer, G. J. Tearney, and B. E. Bouma, “Phase-resolved optical frequency domain imaging,” Opt. Express 13, 5483 (2005).
[Crossref] [PubMed]

Thakor, N.

J. Senarathna, A. Rege, N. Li, and N. Thakor, “Laser speckle contrast imaging: Theory, instrumentation and applications,” IEEE Rev. Biomed. Eng. 6, 99–110 (2013).
[Crossref] [PubMed]

Tokayer, J.

Vakoc, B.

Vakoc, B. J.

S. H. Yun, G. J. Tearney, B. J. Vakoc, M. Shishkov, W. Y. Oh, A. E. Desjardins, M. J. Suter, R. C. Chan, J. A. Evans, I.-K. Jang, N. S. Nishioka, J. F. de Boer, and B. E. Bouma, “Comprehensive volumetric optical microscopy in vivo,” Nat. Med. 12, 1429–1433 (2006).
[Crossref] [PubMed]

B. J. Vakoc, S. H. Yun, J. F. de Boer, G. J. Tearney, and B. E. Bouma, “Phase-resolved optical frequency domain imaging,” Opt. Express 13, 5483 (2005).
[Crossref] [PubMed]

Van Der Steen, A. F.

F. A. Lupotti, C. L. De Korte, F. Mastik, and A. F. Van Der Steen, “Dynamic noise correction for IVUS quantitative volume blood flow: methods and numerical validation,” Ultrasound Med. Biol. 28, 1053–1060 (2002).
[Crossref] [PubMed]

W. Li, A. F. van der Steen, C. T. Lancée, I. Céspedes, and N. Bom, “Blood flow imaging and volume flow quantitation with intravascular ultrasound,” Ultrasound Med. Biol. 24, 203–214 (1998).
[Crossref] [PubMed]

van Gemert, M. J. C.

van Leeuwen, T. G.

N. Weiss, T. G. van Leeuwen, and J. Kalkman, “Localized measurement of longitudinal and transverse flow velocities in colloidal suspensions using optical coherence tomography,” Phys. Rev. E 88, 042312 (2013).
[Crossref]

Vaughan, M.

M. Vaughan, The Fabry-Perot Interferometer: History, Theory, Practice and Applications (CRC Press, 1989).

Vitkin, I. A.

Wan, J.

A. M. Forsyth, J. Wan, P. D. Owrutsky, M. Abkarian, and H. A. Stone, “Multiscale approach to link red blood cell dynamics, shear viscosity, and ATP release,” Proc. Natl. Acad. Sci. U. S. A. 108, 10986–10991 (2011).
[Crossref] [PubMed]

Wang, R.

Wang, Y.

Weiss, N.

N. Weiss, T. G. van Leeuwen, and J. Kalkman, “Localized measurement of longitudinal and transverse flow velocities in colloidal suspensions using optical coherence tomography,” Phys. Rev. E 88, 042312 (2013).
[Crossref]

Weitzel, W. F.

D. W. Park, G. H. Kruger, J. M. Rubin, J. Hamilton, P. Gottschalk, R. E. Dodde, A. J. Shih, and W. F. Weitzel, “Quantification of ultrasound correlation-based flow velocity mapping and edge velocity gradient measurement,” J. Ultrasound Med. 32, 1815–1830 (2013).
[Crossref] [PubMed]

Wiesauer, K.

Wilson, B. C.

Wu, W.

J. Lee, H. Radhakrishnan, W. Wu, A. Daneshmand, M. Climov, C. Ayata, and D. A. Boas, “Quantitative imaging of cerebral blood flow velocity and intracellular motility using dynamic light scattering–optical coherence tomography,” J. Cereb. Blood Flow Metab. 33, 819–825 (2013).
[Crossref] [PubMed]

J. Lee, W. Wu, J. Y. Jiang, B. Zhu, and D. A. Boas, “Dynamic light scattering optical coherence tomography,” Opt. Express 20, 22262–22277 (2012).
[Crossref] [PubMed]

Wyant, J. C.

J. C. Wyant and K. Creath, “Basic Wavefront Aberration Theory for Optical Metrology,” in Applied Optics and Optical Engineering XI, R. R. Shannon and J. C. Wyant, eds. (Academic, 1992), pp. 2.

Yang, V. X. D.

Yeh, C.-K.

P.-C. Li, C.-J. Cheng, and C.-K. Yeh, “On velocity estimation using speckle decorrelation [blood],” IEEE Trans. Ultrason., Ferroelectr. Freq. Control 48, 1084–1091 (2001).
[Crossref]

Yonetsu, T.

T. Yonetsu, B. E. Bouma, K. Kato, J. G. Fujimoto, and I.-K. Jang, “Optical coherence tomography,” Circulation J. 77, 1933–1940 (2013).
[Crossref]

Yun, S.

Yun, S. H.

E. W. Chang, J. B. Kobler, and S. H. Yun, “Subnanometer optical coherence tomographic vibrography,” Opt. Lett. 37, 3678–3680 (2012).
[Crossref] [PubMed]

S. H. Yun, G. J. Tearney, B. J. Vakoc, M. Shishkov, W. Y. Oh, A. E. Desjardins, M. J. Suter, R. C. Chan, J. A. Evans, I.-K. Jang, N. S. Nishioka, J. F. de Boer, and B. E. Bouma, “Comprehensive volumetric optical microscopy in vivo,” Nat. Med. 12, 1429–1433 (2006).
[Crossref] [PubMed]

B. J. Vakoc, S. H. Yun, J. F. de Boer, G. J. Tearney, and B. E. Bouma, “Phase-resolved optical frequency domain imaging,” Opt. Express 13, 5483 (2005).
[Crossref] [PubMed]

Zhu, B.

Appl. Phys. B (1)

K. Schatzel, “Correlation techniques in dynamic light scattering,” Appl. Phys. B 42, 193–213 (1987).
[Crossref]

Biomed. Opt. Express (1)

Circulation J. (1)

T. Yonetsu, B. E. Bouma, K. Kato, J. G. Fujimoto, and I.-K. Jang, “Optical coherence tomography,” Circulation J. 77, 1933–1940 (2013).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

B. Kennedy, K. Kennedy, and D. Sampson, “A review of optical coherence elastography: Fundamentals, techniques and prospects,” IEEE J. Sel. Top. Quantum Electron. 20, 272–288 (2014).
[Crossref]

IEEE Rev. Biomed. Eng. (1)

J. Senarathna, A. Rege, N. Li, and N. Thakor, “Laser speckle contrast imaging: Theory, instrumentation and applications,” IEEE Rev. Biomed. Eng. 6, 99–110 (2013).
[Crossref] [PubMed]

IEEE Trans. Ultrason., Ferroelectr. Freq. Control (1)

P.-C. Li, C.-J. Cheng, and C.-K. Yeh, “On velocity estimation using speckle decorrelation [blood],” IEEE Trans. Ultrason., Ferroelectr. Freq. Control 48, 1084–1091 (2001).
[Crossref]

J. Acoust. Soc. Am. (1)

W. Li, C. T. Lancee, E. I. Cespedes, A. F. W. v. d. Steen, and N. Bom, “Decorrelation of intravascular echo signals: Potentials for blood velocity estimation,” J. Acoust. Soc. Am. 102, 3785–3794 (1997).
[Crossref]

J. Cereb. Blood Flow Metab. (1)

J. Lee, H. Radhakrishnan, W. Wu, A. Daneshmand, M. Climov, C. Ayata, and D. A. Boas, “Quantitative imaging of cerebral blood flow velocity and intracellular motility using dynamic light scattering–optical coherence tomography,” J. Cereb. Blood Flow Metab. 33, 819–825 (2013).
[Crossref] [PubMed]

J. Opt. Soc. Am. A (1)

J. Ultrasound Med. (1)

D. W. Park, G. H. Kruger, J. M. Rubin, J. Hamilton, P. Gottschalk, R. E. Dodde, A. J. Shih, and W. F. Weitzel, “Quantification of ultrasound correlation-based flow velocity mapping and edge velocity gradient measurement,” J. Ultrasound Med. 32, 1815–1830 (2013).
[Crossref] [PubMed]

Nat. Med. (1)

S. H. Yun, G. J. Tearney, B. J. Vakoc, M. Shishkov, W. Y. Oh, A. E. Desjardins, M. J. Suter, R. C. Chan, J. A. Evans, I.-K. Jang, N. S. Nishioka, J. F. de Boer, and B. E. Bouma, “Comprehensive volumetric optical microscopy in vivo,” Nat. Med. 12, 1429–1433 (2006).
[Crossref] [PubMed]

Nat. Photonics (1)

Y. Bromberg, Y. Lahini, E. Small, and Y. Silberberg, “Hanbury brown and twiss interferometry with interacting photons,” Nat. Photonics 4, 721–726 (2010).
[Crossref]

Opt. Express (8)

S. Yun, G. Tearney, J. de Boer, and B. Bouma, “Removing the depth-degeneracy in optical frequency domain imaging with frequency shifting,” Opt. Express 12, 4822–4828 (2004).
[Crossref] [PubMed]

K. Wiesauer, M. Pircher, E. Götzinger, S. Bauer, R. Engelke, G. Ahrens, G. Grützner, C. Hitzenberger, and D. Stifter, “En-face scanning optical coherence tomography with ultra-high resolution for material investigation,” Opt. Express 13, 1015–1024 (2005).
[Crossref] [PubMed]

J. Barton and S. Stromski, “Flow measurement without phase information in optical coherence tomography images,” Opt. Express 13, 5234–5239 (2005).
[Crossref] [PubMed]

B. J. Vakoc, S. H. Yun, J. F. de Boer, G. J. Tearney, and B. E. Bouma, “Phase-resolved optical frequency domain imaging,” Opt. Express 13, 5483 (2005).
[Crossref] [PubMed]

K. Wiesauer, M. Pircher, E. Goetzinger, C. K. Hitzenberger, R. Engelke, G. Ahrens, G. Gruetzner, and D. Stifter, “Transversal ultrahigh-resolution polarizationsensitive optical coherence tomography for strain mapping in materials,” Opt. Express 14, 5945–5953 (2006).
[Crossref] [PubMed]

J. Lee, W. Wu, J. Y. Jiang, B. Zhu, and D. A. Boas, “Dynamic light scattering optical coherence tomography,” Opt. Express 20, 22262–22277 (2012).
[Crossref] [PubMed]

D. Stifter, E. Leiss-Holzinger, Z. Major, B. Baumann, M. Pircher, E. Götzinger, C. K. Hitzenberger, and B. Heise, “Dynamic optical studies in materials testing with spectral-domain polarization-sensitive optical coherence tomography,” Opt. Express 18, 25712–25725 (2010).
[Crossref] [PubMed]

Y. Jia, O. Tan, J. Tokayer, B. Potsaid, Y. Wang, J. J. Liu, M. F. Kraus, H. Subhash, J. G. Fujimoto, J. Hornegger, and D. Huang, “Split-spectrum amplitude-decorrelation angiography with optical coherence tomography,” Opt. Express 20, 4710–4725 (2012).
[Crossref] [PubMed]

Opt. Lett. (9)

E. W. Chang, J. B. Kobler, and S. H. Yun, “Subnanometer optical coherence tomographic vibrography,” Opt. Lett. 37, 3678–3680 (2012).
[Crossref] [PubMed]

N. Mohan and B. Vakoc, “Principal-component-analysis-based estimation of blood flow velocities using optical coherence tomography intensity signals,” Opt. Lett. 36, 2068–2070 (2011).
[Crossref] [PubMed]

X. Liu, Y. Huang, J. C. Ramella-Roman, S. A. Mathews, and J. U. Kang, “Quantitative transverse flow measurement using optical coherence tomography speckle decorrelation analysis,” Opt. Lett. 38, 805–807 (2013).
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B. K. Huang and M. A. Choma, “Resolving directional ambiguity in dynamic light scattering-based transverse motion velocimetry in optical coherence tomography,” Opt. Lett. 39, 521–524 (2014).
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A. Mariampillai, B. A. Standish, E. H. Moriyama, M. Khurana, N. R. Munce, M. K. K. Leung, J. Jiang, A. Cable, B. C. Wilson, I. A. Vitkin, and V. X. D. Yang, “Speckle variance detection of microvasculature using swept-source optical coherence tomography,” Opt. Lett. 33, 1530–1532 (2008).
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Y. Wang and R. Wang, “Autocorrelation optical coherence tomography for mapping transverse particle-flow velocity,” Opt. Lett. 35, 3538–3540 (2010).
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U. Morgner, W. Drexler, F. X. Kärtner, X. D. Li, C. Pitris, E. P. Ippen, and J. G. Fujimoto, “Spectroscopic optical coherence tomography,” Opt. Lett. 25, 111–113 (2000).
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J. F. de Boer, T. E. Milner, M. J. C. van Gemert, and J. S. Nelson, “Two-dimensional birefringence imaging in biological tissue by polarization-sensitive optical coherence tomography,” Opt. Lett. 22, 934–936 (1997).
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X. Li, T. H. Ko, and J. G. Fujimoto, “Intraluminal fiber-optic doppler imaging catheter for structural and functional optical coherence tomography,” Opt. Lett. 26, 1906–1908 (2001).
[Crossref]

Phys. Rev. E (1)

N. Weiss, T. G. van Leeuwen, and J. Kalkman, “Localized measurement of longitudinal and transverse flow velocities in colloidal suspensions using optical coherence tomography,” Phys. Rev. E 88, 042312 (2013).
[Crossref]

Proc. Natl. Acad. Sci. U. S. A. (1)

A. M. Forsyth, J. Wan, P. D. Owrutsky, M. Abkarian, and H. A. Stone, “Multiscale approach to link red blood cell dynamics, shear viscosity, and ATP release,” Proc. Natl. Acad. Sci. U. S. A. 108, 10986–10991 (2011).
[Crossref] [PubMed]

Ultrasound Med. Biol. (2)

W. Li, A. F. van der Steen, C. T. Lancée, I. Céspedes, and N. Bom, “Blood flow imaging and volume flow quantitation with intravascular ultrasound,” Ultrasound Med. Biol. 24, 203–214 (1998).
[Crossref] [PubMed]

F. A. Lupotti, C. L. De Korte, F. Mastik, and A. F. Van Der Steen, “Dynamic noise correction for IVUS quantitative volume blood flow: methods and numerical validation,” Ultrasound Med. Biol. 28, 1053–1060 (2002).
[Crossref] [PubMed]

Other (5)

B. E. Bouma and G. J. Tearney, eds., Handbook of optical coherence tomography (Marcel Dekker Inc., 2002).

E. Regar, P. W. Serruys, and T. G. V. Leeuwen, Optical Coherence Tomography in Cardiovascular Research (CRC Press, 2007).
[Crossref]

G. Song and K. Harding, “OCT for industrial applications,” in Optical Metrology and Inspection for Industrial Applications II, Proc. SPIE8563, 85630N (2012).
[Crossref]

J. C. Wyant and K. Creath, “Basic Wavefront Aberration Theory for Optical Metrology,” in Applied Optics and Optical Engineering XI, R. R. Shannon and J. C. Wyant, eds. (Academic, 1992), pp. 2.

M. Vaughan, The Fabry-Perot Interferometer: History, Theory, Practice and Applications (CRC Press, 1989).

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Figures (8)

Fig. 1
Fig. 1 Measurement setup. The xyz coordinate system was defined by the light propagation direction z and its associated perpendicular plane xy (y towards the reader). The solid rubber phantom traveled in the x direction on a motorized linear stage. Inset: configuration for measuring signals from flowing intralipid. The beam was reflected almost perpendicular to the fiber after internal reflection (a rotation around y). The angle between the beam and the flow was θ = 82.5°.
Fig. 2
Fig. 2 (left) M-mode tomogram of phantom fluid at 2.5 mL/min with low flow velocities and large horizontal speckle size, (right) the same sample at 80 mL/min yielding high flow speeds and small horizontal speckle size. The strong reflections at the top correspond to the lens and other surfaces of the probe, the wall of the tube is the weak reflection at the bottom. The depth scale in mm is approximate as the index of refraction of the different materials has been considered equal to the fluid’s index 1.33.
Fig. 3
Fig. 3 (a–b) Mean intensity of the structural image, (c–d) inverse correlation time profiles for moving solid sample using the traditional autocorrelation function and (e–f) using the newly developed normalized autocorrelation function. (g–h) Profile comparing the inverse decorrelation time using the new and the traditional autocorrelation function at a selected column. Top row corresponds to 120 mm/s, bottom row to 240 mm/s.
Fig. 4
Fig. 4 (top) Traditional autocorrelation at selected depths for the solid sample moving at (left) 120 mm/s, (center) 200 mm/s and (right) 400 mm/s. The autocorrelation for 20 px at 400 mm/s is out of the range. (bottom) The new normalized autocorrelation for the same speeds.
Fig. 5
Fig. 5 (a) Scatter plot of τ−1 and SNR for different sample speeds as calculated using Eq. (18). (b) Scatter plot of the same data using the Pearson autocorrelation function. (c) Parametrization F of the effective inverse correlation time as a function of inverse correlation time and SNR, Eq. (23). (d) τ eff 1 as a function of τ−1 in the parametrization for several representative SNRs. Note the undefined value for low SNRs and low τ−1, as well as a unity slope line 1 : 1 as a visual guide.
Fig. 6
Fig. 6 Speed profiles for moving solid sample using τ eff 1. (a) 40 mm/s, (b) 100 mm/s, (c) 200 mm/s and (d) 300 mm/s. The areas in dark gray that denote where it is not possible to uniquely determine the speed due to the low SNR.
Fig. 7
Fig. 7 (a) Scatter plot of the time-averaged effective inverse speckle-decorrelation time τeff and the measured Doppler velocity vx for different data sets in the liquid setup. The colors denote the depth, from black to green (yellow) for the experimental data (model fit). (b) Fitting parameters as a function of depth. (c) Flow speed as determined by speckle correlation of the calibration dataset. Each flow rate has 50 flow profiles in time, and all flow rates are shown in sequence. (d) Reference Doppler velocity of the calibration dataset.
Fig. 8
Fig. 8 Structural image of the first 2048 A-lines (first row), speckle decorrelation flow speed (second row), line-of-sight-corrected Doppler flow speed (third row) and time-average flow speed profiles (fourth row) for intralipid solution at 20 mL/min, 40 mL/min, and 80 mL/min flow rates in 3.2 mm-diameter tubing. The depth is measured from the surface of the sheath, which has a 0.4 mm radius. Doppler* indicates the signal after unwrapping.

Equations (35)

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g ( 1 ) ( τ ) = E ( t ) E * ( t + τ ) d t ,
g ( 1 ) ( τ ) = M s + M m e i 2 n k 0 v z τ e 4 n 2 k 0 2 D τ e v z 2 τ 2 / w z 2 e v t 2 τ 2 / w t 2 .
g ( 2 ) ( τ ) = I ( t ) I ( t + τ ) d t .
g ( 2 ) ( τ ) = 1 + | g ( 1 ) ( τ ) | 2 = 1 + e 8 n 2 k o 2 D τ e 2 v z 2 τ 2 / w z 2 e 2 v t 2 τ 2 / w t 2 .
g ( 2 ) ( τ ) = I ( t ) I ( t + τ ) d t I ( t ) I ( t + τ ) d t ,
g ( P ) ( τ ) = [ I ( t ) I ¯ ] [ I ( t + τ ) I ¯ ] σ I σ I d t ,
C g ( 2 ) ( 0 ) g ( 2 ) ( τ ) .
g ˜ ( 2 ) ( τ ) = 1 + γ | g ( τ ) |
I ( t ) = w S ( t ) + ( 1 w ) N ( t ) ,
SNR = 10 * log 10 w 1 w .
R II = w 2 R S S + ( 1 w ) 2 R N N + w ( 1 w ) ( R S N + R N S ) .
| I ( t ) | 2 = | w [ S R ( t ) + i S I ( t ) ] + ( 1 w ) [ N R ( t ) + i N I ( t ) ] | 2 ,
R II = w 2 R S S + ( 1 w ) 2 [ 1 + δ ( τ ) ] + 2 w ( 1 w ) ,
R II ( P ) = w 2 R S S ( P ) + ( 1 w ) 2 δ ( τ ) w 2 + ( 1 w 2 ) .
R N N = { 2 : τ = 0 1 + f τ ( τ ) : otherwise
R II = { 2 ( 1 w ) 2 + w 2 R S S ( 0 ) + 2 w ( 1 w ) : τ = 0 ( 1 w ) 2 [ 1 + f τ ] + w 2 R S S ( τ ) + 2 w ( 1 w ) : otherwise ,
Δ g = [ R II ( 0 ) R II ( Δ t ) ] [ 2 R S S ( Δ t ) ] = ( 1 w ) 2 + ( w 2 1 ) [ 2 R S S ( Δ t ) ] .
g new ( 2 ) ( τ ) g ( 2 ) ( τ Δ t ) median { g ( 2 ) ( τ > n s / 2 ) } C new ,
C new = g ( 2 ) ( τ = Δ t ) median { g ( 2 ) ( τ > n s / 2 ) } .
τ c 1 = 2 g ^ c K D + 4 g ^ c 2 K D 2 + 2 g ^ c v 2 / w t 2 ,
τ c 1 = 2 g ^ c v w t v ,
τ 1 = K B M + K B M 2 + k 2 v x 2 T ( v x , K B M ) ,
v x , c = k c 1 F ( τ 1 , SNR ) = k c 1 [ F τ ( τ 1 ) + F SNR ( SNR ) ] ,
v x , c = k n 1 F ( τ 1 , SNR ) k n 1 τ eff 1
v x , c = τ eff 1 k n 1 = τ corr 1 k c 1
F τ ( τ 1 ) = k 0 + k 1 τ 1 + k 5 τ 5 ,
F SNR ( SNR ) = α 1 ( SNR α 2 α 3 τ 1 ) α 4 .
k 0 = 24.7 ms 1 k 1 = 1.09 k 5 = 1.95 10 8 ms 4 α 1 = 34.5 ms 1 α 2 = 9 dB α 3 = 1.46 10 2 dB ms α 4 = 9.12 10 2 ,
k 0 = 2.4 ms 1 k 1 = 1.28 k 5 = 3.6 10 8 ms 4 α 1 = 5.56 ms 1 α 2 = 15.5 dB α 3 = 3.51 10 2 dB ms α 4 = 5.55 10 1 ,
τ corr 1 = K B M + K B M 2 + k n 2 v x 2 ,
τ eff 1 = F ( τ 1 , SNR ) = K ˜ B M + K ˜ B M 2 + k ˜ 2 v x 2 ,
v x = 1 k ˜ ( τ eff 1 K ˜ B M ) 2 K ˜ B M 2 .
k ˜ = 0.074 μ m K ˜ B M = 1.45 ms 1 ,
τ eff 1 ( j ) = 2 K ˜ B M + ξ ( j ) .
v x ( j ) = 1 k ˜ ( K ˜ B M + ξ ( j ) ) 2 K ˜ B M 2 = 1 k ˜ { ξ ( j ) 2 + 2 K ˜ B M ξ ( j ) } = 1 k ˜ | ξ ( j ) | { 1 + 2 K ˜ B M ξ ( j ) } ,

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