Noise statistics of phase-resolved optical coherence tomography imaging: single-and dual-beam-scan Doppler optical coherence tomography

Shuichi Makita; Franck Jaillon; Israt Jahan; Yoshiaki Yasuno

doi:10.1364/OE.22.004830

Noise statistics of phase-resolved optical coherence tomography imaging: single-and dual-beam-scan Doppler optical coherence tomography

Shuichi Makita, Franck Jaillon, Israt Jahan, and Yoshiaki Yasuno

Optics Express
Vol. 22,
Issue 4,
pp. 4830-4848
(2014)
•https://doi.org/10.1364/OE.22.004830

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Shuichi Makita, Franck Jaillon, Israt Jahan, and Yoshiaki Yasuno, "Noise statistics of phase-resolved optical coherence tomography imaging: single-and dual-beam-scan Doppler optical coherence tomography," Opt. Express 22, 4830-4848 (2014)

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Related Topics
Table of Contents Category
- Fourier Optics and Signal Processing
Optics & Photonics Topics
?

The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Optical coherence tomography (110.4500)
- Phase measurement (120.5050)
- Laser Doppler velocimetry (170.3340)
- Medical and biological imaging (170.3880)

About this Article
History
- Original Manuscript: January 10, 2014
- Manuscript Accepted: February 10, 2014
- Published: February 21, 2014
Virtual Issues
Virtual Journal for Biomedical Optics Vol. 9, Iss. 4

Accessible

Open Access

Abstract

Noise statistics of phase-resolved optical coherence tomography (OCT) imaging are complicated and involve noises of OCT, correlation of signals, and speckles. In this paper, the statistical properties of phase shift between two OCT signals that contain additive random noises and speckle noises are presented. Experimental results obtained with a scattering tissue phantom are in good agreement with theoretical predictions. The performances of the dual-beam method and conventional single-beam method are compared. As expected, phase shift noise in the case of the dual-beam-scan method is less than that for the single-beam method when the transversal sampling step is large.

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References

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[Crossref] [PubMed]

2008 (2)

D. C. Adler, S.-W. Huang, R. Huber, and J. G. Fujimoto, “Photothermal detection of gold nanoparticles using phase-sensitive optical coherence tomography,” Opt. Express 16, 4376–4393 (2008).
[Crossref] [PubMed]

A. Szkulmowska, M. Szkulmowski, A. Kowalczyk, and M. Wojtkowski, “Phase-resolved Doppler optical coherence tomographylimitations and improvements,” Opt. Lett. 33, 1425–1427 (2008).
[Crossref] [PubMed]

2007 (1)

R. K. Wang, S. Kirkpatrick, and M. Hinds, “Phase-sensitive optical coherence elastography for mapping tissue microstrains in real time,” Appl. Phys. Lett. 90, 164105, 2007).
[Crossref]

2006 (1)

S. Makita, Y. Hong, M. Yamanari, T. Yatagai, and Y. Yasuno, “Optical coherence angiography,” Opt. Express 14, 7821–7840 (2006).
[Crossref] [PubMed]

2005 (2)

S. Yazdanfar, C. Yang, M. Sarunic, and J. Izatt, “Frequency estimation precision in Doppler optical coherence tomography using the Cramer-Rao lower bound,” Opt. Express 13, 410–416 (2005).
[Crossref] [PubMed]

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[Crossref] [PubMed]

2004 (2)

S. H. Yun, G. J. Tearney, J. F. d. Boer, and B. E. Bouma, “Motion artifacts in optical coherence tomography with frequency-domain ranging,” Opt. Express 12, 2977–2998 (2004).
[Crossref] [PubMed]

S. A. Telenkov, D. P. Dave, S. Sethuraman, T. Akkin, and T. E. Milner, “Differential phase optical coherence probe for depth-resolved detection of photothermal response in tissue,” Phys. Med. Biol. 49, 111–119 (2004).
[Crossref] [PubMed]

2003 (3)

T. Akkin, D. P. Dav, J.-I. Youn, S. A. Telenkov, H. G. R., and T. E. Milner, “Imaging tissue response to electrical and photothermal stimulation with nanometer sensitivity,” Lasers Surg. Med. 33, 219–225 (2003).
[Crossref] [PubMed]

R. Leitgeb, L. Schmetterer, W. Drexler, A. Fercher, R. Zawadzki, and T. Bajraszewski, “Real-time assessment of retinal blood flow with ultrafast acquisition by color Doppler Fourier domain optical coherence tomography,” Opt. Express 11, 3116–3121 (2003).
[Crossref] [PubMed]

B. R. White, M. C. Pierce, N. Nassif, B. Cense, B. H. Park, G. J. Tearney, B. E. Bouma, T. C. Chen, and J. F. d. Boer, “In vivo dynamic human retinal blood flow imaging using ultra-high-speed spectral domain optical Doppler tomography,” Opt. Express 11, 3490–3497 (2003).
[Crossref] [PubMed]

2002 (1)

V. X. Yang, M. L. Gordon, A. Mok, Y. Zhao, Z. Chen, R. S. Cobbold, B. C. Wilson, and I. A. Vitkin, “Improved phase-resolved optical Doppler tomography using Kasai velocity estimator and histogram segmentation,” Opt. Commun. 208, 209–214 (2002).
[Crossref]

2000 (1)

Y. Zhao, Z. Chen, C. Saxer, S. Xiang, J. F. d. Boer, and J. S. Nelson, “Phase-resolved optical coherence tomography and optical Doppler tomography for imaging blood flow in human skin with fast scanning speed and high velocity sensitivity,” Opt. Lett. 25, 114–116 (2000).
[Crossref]

1995 (1)

R. J. A. Tough, D. Blacknell, and S. Quegan, “A statistical description of polarimetric and interferometric synthetic aperture radar data,” Proc. R. Soc. Lond. A 449, 567–589 (1995).
[Crossref]

1994 (1)

L. Jong-Sen, K. Hoppel, S. Mango, and A. Miller, “Intensity and phase statistics of multilook polarimetric and interferometric SAR imagery,” IEEE Trans. Geosci. Remote Sens. 32, 1017–1028 (1994).
[Crossref]

1991 (1)

A. Moreira, “Improved multilook techniques applied to SAR and SCANSAR imagery,” IEEE Trans. Geosci. Remote Sens. 29, 529–534, 1991).
[Crossref]

Adie, S. G.

S. G. Adie, X. Liang, B. F. Kennedy, R. John, D. D. Sampson, and S. A. Boppart, “Spectroscopic optical coherence elastography,” Opt. Express 18, 25519–25534 (2010).
[Crossref] [PubMed]

Adler, D. C.

Akkin, T.

An, L.

Bajraszewski, T.

Bartlett, L. A.

Bendat, J. S.

J. S. Bendat and A. G. Piersol, Random Data: Analysis and Measurement Procedures (John Wiley and Sons, 2010).
[Crossref]

Bever, M.

Birngruber, R.

Blacknell, D.

R. J. A. Tough, D. Blacknell, and S. Quegan, “A statistical description of polarimetric and interferometric synthetic aperture radar data,” Proc. R. Soc. Lond. A 449, 567–589 (1995).
[Crossref]

Boas, D. A.

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]

Boer, J. F. d.

Bonesi, M.

Boppart, S. A.

S. G. Adie, X. Liang, B. F. Kennedy, R. John, D. D. Sampson, and S. A. Boppart, “Spectroscopic optical coherence elastography,” Opt. Express 18, 25519–25534 (2010).
[Crossref] [PubMed]

Bouma, B. E.

Braaf, B.

Brinkmann, R.

Cense, B.

Chen, T. C.

Chen, Z.

Cobbold, R. S.

Dav, D. P.

Dave, D. P.

de Boer, J. F.

Debbeler, C.

Drexler, W.

W. Drexler and J. G. Fujimoto, Optical Coherence Tomography: Technology and Applications (Springer, 2008).
[Crossref]

Fercher, A.

Fingler, J.

Fraser, S. E.

Fujimoto, J. G.

W. Drexler and J. G. Fujimoto, Optical Coherence Tomography: Technology and Applications (Springer, 2008).
[Crossref]

Fukumura, D.

Gorczynska, I.

Gordon, M. L.

Grulkowski, I.

Gtzinger, E.

H. G. R.,

Hinds, M.

R. K. Wang, S. Kirkpatrick, and M. Hinds, “Phase-sensitive optical coherence elastography for mapping tissue microstrains in real time,” Appl. Phys. Lett. 90, 164105, 2007).
[Crossref]

Hitzenberger, C. K.

Hong, Y.

S. Makita, Y. Hong, M. Yamanari, T. Yatagai, and Y. Yasuno, “Optical coherence angiography,” Opt. Express 14, 7821–7840 (2006).
[Crossref] [PubMed]

Hong, Y.-J.

Hoppel, K.

Httmann, G.

Huang, S.-W.

Huber, R.

Izatt, J.

Jaillon, F.

F. Jaillon, S. Makita, and Y. Yasuno, “Variable velocity range imaging of the choroid with dual-beam optical coherence angiography,” Opt. Express 20, 385–396 (2012).
[Crossref] [PubMed]

Jain, R. K.

Jiang, J. Y.

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]

John, R.

S. G. Adie, X. Liang, B. F. Kennedy, R. John, D. D. Sampson, and S. A. Boppart, “Spectroscopic optical coherence elastography,” Opt. Express 18, 25519–25534 (2010).
[Crossref] [PubMed]

Jong-Sen, L.

Kennedy, B. F.

S. G. Adie, X. Liang, B. F. Kennedy, R. John, D. D. Sampson, and S. A. Boppart, “Spectroscopic optical coherence elastography,” Opt. Express 18, 25519–25534 (2010).
[Crossref] [PubMed]

Kennedy, K. M.

Kim, D. Y.

Kirkpatrick, S.

R. K. Wang, S. Kirkpatrick, and M. Hinds, “Phase-sensitive optical coherence elastography for mapping tissue microstrains in real time,” Appl. Phys. Lett. 90, 164105, 2007).
[Crossref]

Koch, E.

Koh, S. H.

Koinzer, S.

Kowalczyk, A.

Kurokawa, K.

Lanning, R. M.

Lee, B. H.

Lee, J.

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]

Leitgeb, R.

Leitgeb, R. A.

Liang, X.

S. G. Adie, X. Liang, B. F. Kennedy, R. John, D. D. Sampson, and S. A. Boppart, “Spectroscopic optical coherence elastography,” Opt. Express 18, 25519–25534 (2010).
[Crossref] [PubMed]

Makita, S.

F. Jaillon, S. Makita, and Y. Yasuno, “Variable velocity range imaging of the choroid with dual-beam optical coherence angiography,” Opt. Express 20, 385–396 (2012).
[Crossref] [PubMed]

S. Makita, Y. Hong, M. Yamanari, T. Yatagai, and Y. Yasuno, “Optical coherence angiography,” Opt. Express 14, 7821–7840 (2006).
[Crossref] [PubMed]

Mango, S.

McLaughlin, R. A.

Miller, A.

Milner, T. E.

Min, E.-J.

Miura, M.

Mller, H. H.

Mok, A.

Moreira, A.

A. Moreira, “Improved multilook techniques applied to SAR and SCANSAR imagery,” IEEE Trans. Geosci. Remote Sens. 29, 529–534, 1991).
[Crossref]

Mujat, M.

Munn, L. L.

Munro, P. R. T.

Nassif, N.

Nelson, J. S.

Padera, T. P.

Park, B. H.

Pierce, M. C.

Piersol, A. G.

J. S. Bendat and A. G. Piersol, Random Data: Analysis and Measurement Procedures (John Wiley and Sons, 2010).
[Crossref]

Pircher, M.

Ptaszynski, L.

Qin, J.

Quegan, S.

R. J. A. Tough, D. Blacknell, and S. Quegan, “A statistical description of polarimetric and interferometric synthetic aperture radar data,” Proc. R. Soc. Lond. A 449, 567–589 (1995).
[Crossref]

Ruvinskaya, S.

Sakadi, S.

Sampson, D. D.

S. G. Adie, X. Liang, B. F. Kennedy, R. John, D. D. Sampson, and S. A. Boppart, “Spectroscopic optical coherence elastography,” Opt. Express 18, 25519–25534 (2010).
[Crossref] [PubMed]

Sarunic, M.

Sasaki, K.

Saxer, C.

Schlott, K.

Schmetterer, L.

Schwartz, D. M.

Sethuraman, S.

Sicam, V. A. D.

Srinivasan, V. J.

Stylianopoulos, T.

Szkulmowska, A.

Szkulmowski, M.

Szlag, D.

Tearney, G. J.

Telenkov, S. A.

Torzicky, T.

Tough, R. J. A.

R. J. A. Tough, D. Blacknell, and S. Quegan, “A statistical description of polarimetric and interferometric synthetic aperture radar data,” Proc. R. Soc. Lond. A 449, 567–589 (1995).
[Crossref]

Tyrrell, J. A.

Vakoc, B. J.

van Meurs, J. C.

van Zeeburg, E.

Vermeer, K. A.

Vienola, K. V.

Vitkin, I. A.

Walther, J.

Wang, R. K.

R. K. Wang, S. Kirkpatrick, and M. Hinds, “Phase-sensitive optical coherence elastography for mapping tissue microstrains in real time,” Appl. Phys. Lett. 90, 164105, 2007).
[Crossref]

Werner, J. S.

White, B. R.

Wilson, B. C.

Wojtkowski, M.

Wu, W.

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]

Xiang, S.

Yamanari, M.

S. Makita, Y. Hong, M. Yamanari, T. Yatagai, and Y. Yasuno, “Optical coherence angiography,” Opt. Express 14, 7821–7840 (2006).
[Crossref] [PubMed]

Yang, C.

Yang, V. X.

Yasuno, Y.

F. Jaillon, S. Makita, and Y. Yasuno, “Variable velocity range imaging of the choroid with dual-beam optical coherence angiography,” Opt. Express 20, 385–396 (2012).
[Crossref] [PubMed]

S. Makita, Y. Hong, M. Yamanari, T. Yatagai, and Y. Yasuno, “Optical coherence angiography,” Opt. Express 14, 7821–7840 (2006).
[Crossref] [PubMed]

Yatagai, T.

S. Makita, Y. Hong, M. Yamanari, T. Yatagai, and Y. Yasuno, “Optical coherence angiography,” Opt. Express 14, 7821–7840 (2006).
[Crossref] [PubMed]

Yazdanfar, S.

Youn, J.-I.

Yun, S. H.

Yun, S.-H.

Zawadzki, R.

Zawadzki, R. J.

Zhao, Y.

Zhu, B.

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]

Zotter, S.

Appl. Phys. Lett. (1)

R. K. Wang, S. Kirkpatrick, and M. Hinds, “Phase-sensitive optical coherence elastography for mapping tissue microstrains in real time,” Appl. Phys. Lett. 90, 164105, 2007).
[Crossref]

Biomed. Opt. Express (4)

IEEE Trans. Geosci. Remote Sens. (2)

A. Moreira, “Improved multilook techniques applied to SAR and SCANSAR imagery,” IEEE Trans. Geosci. Remote Sens. 29, 529–534, 1991).
[Crossref]

IEEE Trans. Med. Imaging (1)

Lasers Surg. Med. (1)

Nat. Med. (1)

Opt. Commun. (1)

Opt. Express (20)

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Fig. 1 A cross-sectional OCT image of the tissue phantom. The yellow box indicates the ROI of phase shift analysis.

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Fig. 2 Phase-resolved images with dual-beam-scan (right column; a, c, e) and emulated conventional phase-resolved (left column; b, d, f) methods. The fractional sampling step was set to be (a), (b) 0.84, (c), (d) 0.43, (e), (f) and 0.1.

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Fig. 3 Scatter plot of phase shift noise vs correlation coefficient. The solid curve shows the population standard deviation of phase shift (Eq. (7)).

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Fig. 4 Phase shift noise vs fractional sampling step δx.

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Fig. 5 The transitional point δx_c (Eq. (38)) is plotted. In the upper region, the dual-beam method exhibits less phase shift noise than that of the single-beam method.

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Fig. 6 Phase shift noises vs ESNR.

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Fig. 7 Profiles of the correlation coefficients of the Hermitian products with displacement along (a) the lateral direction and (b) the axial direction. In each figure, the horizontal axis is fractional displacement iδx = iΔx/w and lδz = lΔz/ζ, where i and l are the displacements in the number of pixels along lateral and axial directions, respectively. δx = 0.22 and δz = 0.52. Solid curves are the expected correlation coefficients from Eq. (29).

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Fig. 8 Phase shift noise vs effective number of independent samples.

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Fig. 9 Phase shift noise vs fractional sampling step with averaging using a constant window size.

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Tables (1)

Table 1 List of notations

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Equations (44)

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Δ ϕ = arg [g_{1}^{*} g_{2}],

G_{1} = S_{1} + N_{1}, G_{2} = S_{2} + N_{2} .

s = a_{R} \sum_{m} a_{m} exp [- i 2 k_{c} (z_{m} - z_{0})],

p_{Δ Φ} (Δ ϕ | ρ, Δ ϕ_{0}) = \frac{1 - ρ^{2}}{2 π} {\frac{β (\frac{π}{2} + {sin}^{- 1} β)}{{[1 - β^{2}]}^{3 / 2}} + \frac{1}{1 - β^{2}}},

ρ e^{i Δ ϕ_{0}} = \frac{E [G_{1}^{*} G_{2}]}{\sqrt{E [{| G_{1} |}^{2}] E [{| G_{2} |}^{2}]}},

E [Δ ϕ] = \frac{ρ sin Δ ϕ_{0}}{\sqrt{1 - ρ^{2} {cos}^{2} Δ ϕ_{0}}} {cos}^{- 1} (ρ cos Δ ϕ_{0}),

σ_{Δ ϕ} = \sqrt{\frac{1 - ρ^{2}}{1 - {β^{'}}^{2}} [\frac{π^{2}}{4} - π {sin}^{- 1} β^{'} + {({sin}^{- 1} β^{'})}^{2}] + \frac{π^{2}}{12} - \frac{{Li}_{2} (ρ^{2})}{2}},

\bar{Δ ϕ} = \frac{1}{N} \sum_{n = 1}^{N} Δ ϕ_{n},

S_{Δ ϕ} = \sqrt{\frac{1}{N} \sum_{n = 1}^{N} {(Δ ϕ_{n} - \bar{Δ ϕ})}^{2}},

ρ = \frac{ρ_{s}}{\sqrt{(1 + {SNR}_{1}^{- 1}) (1 + {SNR}_{2}^{- 1})}},

ρ_{s} = \frac{| E [S_{1}^{*} S_{2}] |}{\sqrt{E [{| S_{1} |}^{2}] E [{| S_{2} |}^{2}]}} .

\frac{1}{1 + {ESNR}^{- 1}} \equiv \frac{1}{\sqrt{(1 + {SNR}_{1}^{- 1}) (1 + {SNR}_{2}^{- 1})}} .

s_{1} (t_{1}) = η_{1} * h_{1} (r_{1} (t_{1})), s_{2} (t_{2}) = η_{2} * h_{2} (r_{2} (t_{2})) .

ρ_{s} e^{i Δ ϕ_{0}} \equiv ρ_{S_{1}, S_{2}} (t_{1}, t_{2}) = \frac{E [s_{1}^{*} (t_{1}) s_{2} (t_{2})]}{\sqrt{E [{| s_{1} (t_{1}) |}^{2}] E [{| s_{2} (t_{2}) |}^{2}]}} .

h_{χ} (x_{L}, y_{L}, z_{L}) \propto e^{- 2 \frac{x_{L}^{2}}{w_{χ}^{2}}} e^{- 2 \frac{y_{L}^{2}}{w_{χ}^{2}}} e^{- \frac{Δ k_{χ}^{2} z_{L}^{2}}{8}} e^{- 2 i k_{c χ} (z_{L} - z_{0})} (χ = 1, 2),

\begin{array}{l} ρ_{s} \approx & ρ_{η_{1}, η_{2}} \\ \times \frac{2 w_{1} w_{2}}{w_{1}^{2} + w_{2}^{2}} \sqrt{\frac{2 Δ k_{1} Δ k_{2}}{Δ k_{1}^{2} + Δ k_{2}^{2}}} \\ \times e^{- 2 \frac{x^{'} {(t_{1}, t_{2})}^{2} + y^{'} {(t_{1}, t_{2})}^{2}}{w_{1}^{2} + w_{2}^{2}}} e^{- \frac{Δ k_{1}^{2} Δ k_{2}^{2}}{Δ k_{1}^{2} + Δ k_{2}^{2}} \frac{z^{'} {(t_{1}, t_{2})}^{2}}{8}} \\ \times e^{- \frac{8 {(k_{c 1} - k_{c 2})}^{2}}{Δ k_{1}^{2} + Δ k_{2}^{2}}} \\ \times e^{- 4 \frac{k_{c 1}^{2} Δ k_{2}^{2} + k_{c 2}^{2} Δ k_{1}^{2}}{Δ k_{1}^{2} + Δ k_{2}^{2}} D (\frac{t_{2} + t_{1}}{2}) (t_{2} - t_{1})}, \end{array}

Δ ϕ_{0} \approx - 2 \frac{k_{c 1} Δ k_{2}^{2} + k_{c 2} Δ k_{1}^{2}}{Δ k_{1}^{2} + Δ k_{2}^{2}} z^{'} (t_{1}, t_{2}),

ρ_{s, SS} \equiv ρ_{η_{1}, η_{2}} | ρ_{h_{1}, h_{2}} (r^{'} (Δ t)) |,

ρ_{h_{1}, h_{2}} (r^{'}) = \frac{2 w_{1} w_{2}}{w_{1}^{2} + w_{2}^{2}} \sqrt{\frac{2 Δ k_{1} Δ k_{2}}{Δ k_{1}^{2} + Δ k_{2}^{2}}} e^{- 2 \frac{{x^{'}}^{2} + {y^{'}}^{2}}{w_{1}^{2} + w_{2}^{2}}} e^{- \frac{Δ k_{1}^{2} Δ k_{2}^{2}}{Δ k_{1}^{2} + Δ k_{2}^{2}} \frac{{z^{'}}^{2}}{8}} e^{- \frac{8 {(k_{c 1} - k_{c 2})}^{2}}{Δ k_{1}^{2} + Δ k_{1}^{2}}} e^{- 2 i \frac{k_{c 1} Δ k_{2}^{2} + k_{c 2} Δ k_{1}^{2}}{Δ k_{1}^{2} + Δ k_{2}^{2}} z^{'}} .

r = \frac{| \bar{g_{1}^{*} (t_{1}) g_{2} (t_{2})} |}{\sqrt{\bar{{| g_{1} (t_{1}) |}^{2}} \bar{{| g_{2} (t_{2}) |}^{2}}}} .

\begin{array}{l} {\hat{ρ}}_{s} & = (1 + {\hat{ESNR}}^{- 1}) r \\ = \frac{| \bar{g_{1}^{*} (t_{1}) g_{2} (t_{2})} |}{\sqrt{[\bar{{| g_{1} (t_{1}) |}^{2}} - \bar{{| n_{1} |}^{2}}] [\bar{{| g_{2} (t_{2}) |}^{2}} - \bar{{| n_{2} |}^{2}}]}}, \end{array}

1 + {\hat{ESNR}}^{- 1} = \sqrt{\frac{\bar{{| g_{1} (t_{1}) |}^{2}} \bar{{| g_{2} (t_{2}) |}^{2}}}{[\bar{{| g_{1} (t_{1}) |}^{2}} - \bar{{| n_{1} |}^{2}}] [\bar{{| g_{2} (t_{2}) |}^{2}} - \bar{{| n_{2} |}^{2}}]} .}

Δ {\hat{ϕ}}_{0} = arg [\sum_{κ = 1}^{ν} g_{1}^{(κ) *} g_{2}^{(κ)}] .

\begin{array}{l} p_{Δ {\hat{Φ}}_{0}} (Δ {\hat{ϕ}}_{0} | ρ, Δ ϕ_{0}) = & \frac{Γ (ν + \frac{1}{2}) {(1 - ρ^{2})}^{ν} ρ cos (Δ {\hat{ϕ}}_{0} - Δ ϕ_{0})}{2 \sqrt{π} Γ (ν) {[1 - ρ^{2} {cos}^{2} (Δ {\hat{ϕ}}_{0} - Δ ϕ_{0})]}^{ν + 1 / 2}} \\ + \frac{{(1 - ρ^{2})}^{ν}}{2 π} {}_{2}{F_{1}} (ν, 1; \frac{1}{2}; ρ^{2} {cos}^{2} (Δ {\hat{ϕ}}_{0} - Δ ϕ_{0})), \end{array}

σ_{Δ {\hat{ϕ}}_{0}} = \sqrt{E [{Δ {\hat{ϕ}}_{0}}^{2}] - E {[Δ {\hat{ϕ}}_{0}]}^{2}} .

Δ {\hat{ϕ}}_{0} = arg [\sum_{i}^{I} \sum_{j}^{J} \sum_{l}^{L} g_{1}^{*} (x_{1 + i}, y_{1 + j}, z_{1 + l}) g_{2} (x_{1 + i}, y_{1 + j}, z_{1 + l})],

\begin{array}{l} ENIS = & \frac{I}{1 + 2 \sum_{i = 1}^{I - 1} \frac{I - l}{I} ρ_{g^{2}}^{2} (i Δ x, 0, 0)} \frac{J}{1 + 2 \sum_{j = 1}^{J - 1} \frac{J - l}{J} ρ_{g^{2}}^{2} (0, j Δ y, 0)} \\ \times \frac{L}{1 + 2 \sum_{l = 1}^{L - 1} \frac{L - l}{L} ρ_{g^{2}}^{2} (0, 0, l Δ z)}, \end{array}

\begin{array}{l} ρ_{g^{2}} (i Δ x, 0, 0) = \\ \frac{| E [G_{1}^{*} (x_{1}) G_{2} (x_{1}) G_{1} (x_{1 + i}) G_{2}^{*} (x_{1 + i})] - E [G_{1}^{*} (x_{1}) G_{2} (x_{1})] E [G_{1} (x_{1 + i}) G_{2}^{*} (x_{1 + i})] |}{\sqrt{E [{| G_{1}^{*} (x_{1}) G_{2} (x_{1}) - E [G_{1}^{*} (x_{1}) G_{2} (x_{1})] |}^{2}]} \sqrt{E [{| G_{1}^{*} (x_{1 + i}) G_{2} (x_{1 + i}) - E [G_{1}^{*} (x_{1 + i}) G_{2} (x_{1 + i})] |}^{2}]}} . \end{array}

ρ_{g^{2}, S S} (i Δ x, j Δ y, l Δ z) = \frac{1}{{(1 + {ESNR}^{- 1})}^{2}} | ρ_{h_{1}, h_{1}}^{*} (i Δ x, j Δ y, l Δ z) ρ_{h_{2}, h_{2}} (i Δ x, j Δ y, l Δ z) |,

| ρ_{h_{1}, h_{1}}^{*} (i Δ x, j Δ y, l Δ z) ρ_{h_{2}, h_{2}} (i Δ x, j Δ y, l Δ z) | = e^{- \frac{w_{1}^{2} + w_{2}^{2}}{w_{1}^{2} w_{2}^{2}} [{(i Δ x)}^{2} + {(j Δ y)}^{2}]} e^{- \frac{{(l Δ z)}^{2}}{16} (Δ k_{1}^{2} + Δ k_{2}^{2})} .

Δ ϕ_{flow} = 2 n k_{c} V Δ t cos θ,

v_{\min} = K \frac{σ_{Δ ϕ_{SS}}}{Δ t},

ρ_{s, SS}^{(SB)} = e^{- \frac{x_{b}^{' 2} + y_{b}^{' 2}}{w^{2}}},

σ_{Δ ϕ_{SS}}^{(SB)} = {\begin{array}{l} {σ Δ ϕ |}_{ρ = \frac{e^{- δ x^{2}}}{1 + 1 / {ESNR}^{(SB)}}} & (I J L = 1) \\ {σ Δ {\hat{ϕ}}_{0} |}_{ρ = \frac{e^{- δ x^{2}}}{1 + 1 / {ESNR}^{(SB)}}, ν = {ENIS}_{I - 1, J, L}} & (I J L > = 2) \end{array},

ρ_{s, SS}^{(DB)} = ρ_{Pol .} \frac{2 w_{1} w_{2}}{w_{1}^{2} + w_{2}^{2}} \sqrt{\frac{2 Δ k_{1} Δ k_{2}}{Δ k_{1}^{2} + Δ k_{2}^{2}}} e^{- \frac{8 {(k_{c 1} - k_{c 2})}^{2}}{Δ k_{1}^{2} + Δ k_{2}^{2}}},

σ_{Δ ϕ_{SS}}^{(DB)} = {\begin{array}{l} {σ_{Δ ϕ} |}_{ρ = \frac{ρ_{s, SS}^{(DB)}}{1 + 1 / {ESNR}^{(DB)}}} & (I J L = 1) \\ {σ_{Δ {\hat{ϕ}}_{0}} |}_{ρ = \frac{ρ_{s, SS}^{(DB)}}{1 + 1 / {ESNR}^{(DB)}}, ν = {ENIS}_{I, J, L}} & (I J L > = 2) \end{array} .

g (x_{i}, z_{l}) = g_{H} (x_{i}, z_{l}) + g_{V} (x_{i}, z_{l}) exp [- i Δ ϕ_{ch} (x_{i})],

\begin{array}{l} δ x_{c} & = \sqrt{- ln [ρ_{Pol .} \frac{{ESNR}^{(DB)}}{{ESNR}^{(SB)}} \frac{{ESNR}^{(SB)} + 1}{{ESNR}^{(DB)} + 1}]} \\ = \sqrt{- ln [ρ_{Pol .} \frac{{ESNR}^{(SB)} + 1}{{ESNR}^{(SB)} + 2}] .} \end{array}

{ESNR}_{ρ_{s}} = \frac{ρ_{s}}{1 - ρ_{s}} .

I^{(DB)} = {\begin{array}{l} ⌊ \frac{2}{δ x} ⌋ & \frac{2}{δ x} \geq 1, \\ 1 & otherwise \end{array}

I^{(SB)} = {\begin{array}{l} ⌊ \frac{2}{δ x} ⌋ - 1 & \frac{2}{δ x} \geq 2, \\ 1 & otherwise . \end{array}

\begin{array}{l} E [Δ {\hat{ϕ}}_{0}^{n}] = & {(- 1)}^{n / 2} \frac{π^{n} cos (\frac{n π}{2})}{n + 1} \\ + 2 \sum_{l = 1}^{\infty} {\frac{{(- 1)}^{l} ρ^{l} Γ (\frac{l}{2} + 1) Γ (\frac{l}{2} + ν) {}_{2}{F_{1}} (\frac{l}{2}, \frac{l}{2} - ν + 1; l + 1; ρ^{2})}{π Γ (ν) Γ (l + 1)} \\ {\times [\frac{\partial^{n}}{\partial s^{n}} \frac{sinh (π s) [s cos (l Δ ϕ_{0}) - l sin (l Δ ϕ_{0})]}{l^{2} + s^{2}}] |}_{s \to 0}} . \end{array}

E [Δ {\hat{ϕ}}_{0}] = - \sum_{l = 1}^{\infty} \frac{2 {(- ρ)}^{l} sin (l Δ ϕ_{0}) Γ (\frac{l}{2} + 1) Γ (\frac{l}{2} + ν) {}_{2}{F_{1}} [\frac{l}{2}, \frac{l}{2} - ν + 1; l + 1; ρ^{2}]}{l Γ (ν) Γ (l + 1)},

E [{Δ {\hat{ϕ}}_{0}}^{2}] = \frac{π^{2}}{3} + \sum_{l = 1}^{\infty} \frac{4 {(- ρ)}^{l} cos (l Δ ϕ_{0}) Γ (\frac{l}{2} + 1) Γ (\frac{l}{2} + ν) {}_{2}{F_{1}} [\frac{l}{2}, \frac{l}{2} - ν + 1; l + 1; ρ^{2}]}{l^{2} Γ (ν) Γ (l + 1)} .

Symbols	Descriptions
Δϕ₀	population phase shift between two OCT signals
Δϕ	realization of phase shift
σ_Δ_ϕ	population standard deviation of phase shift
σ_{Δϕ_SS}	population standard deviation of phase shift with a static tissue; i.e., noise of phase-resolved flow imaging
S_Δ_ϕ	sample standard deviation of phase shift
Δϕ̂₀	Maximum likelihood estimation (MLE) of phase shift
ν	number of independent realizations
σ_Δϕ̂₀	population standard deviation of phase shift (MLE)
ρ	population correlation coefficient of detected OCT signals
ρ_s	population correlation coefficient of OCT signals
ρ_s_,SS	population correlation coefficient of OCT signals with a static tissue
ρ_η₁,η₂	population correlation coefficient of scattering process between two channels
ρ_h₁,h₂	population correlation coefficient of point spread functions between two channels
ρ_g²	population correlation coefficient of 2nd order detected OCT signals
ρ_g²,SS	population correlation coefficient of 2nd order detected OCT signals with a static tissue
r	sample correlation coefficient
ρ̂_s	estimation for correlation coefficient of OCT signals
h	the point spread function of OCT system
SNR	signal-to-noise ratio of OCT system
ESNR	representative SNR of two OCT signals
ENIS	effective number of independent realizations

Abstract

References

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