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.

© 2014 Optical Society of America

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2012 (7)

B. F. Kennedy, S. H. Koh, R. A. McLaughlin, K. M. Kennedy, P. R. T. Munro, and D. D. Sampson, “Strain estimation in phase-sensitive optical coherence elastography,” Biomed. Opt. Express 3, 1865–1879 (2012).
[Crossref] [PubMed]

H. H. Mller, L. Ptaszynski, K. Schlott, C. Debbeler, M. Bever, S. Koinzer, R. Birngruber, R. Brinkmann, and G. Httmann, “Imaging thermal expansion and retinal tissue changes during photocoagulation by high speed OCT,” Biomed. Opt. Express 3, 1025–1046 (2012).
[Crossref]

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]

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, F. Jaillon, M. Yamanari, and Y. Yasuno, “Dual-beam-scan Doppler optical coherence angiography for birefringence-artifact-free vasculature imaging,” Opt. Express 20, 2681–2692 (2012).
[Crossref] [PubMed]

K. Kurokawa, K. Sasaki, S. Makita, Y.-J. Hong, and Y. Yasuno, “Three-dimensional retinal and choroidal capillary imaging by power Doppler optical coherence angiography with adaptive optics,” Opt. Express 20, 22796–22812 (2012).
[Crossref] [PubMed]

B. Braaf, K. A. Vermeer, K. V. Vienola, and J. F. de Boer, “Angiography of the retina and the choroid with phase-resolved OCT using interval-optimized backstitched b-scans,” Opt. Express 20, 20516–20534 (2012).
[Crossref] [PubMed]

2011 (5)

2010 (3)

2009 (4)

I. Grulkowski, I. Gorczynska, M. Szkulmowski, D. Szlag, A. Szkulmowska, R. A. Leitgeb, A. Kowalczyk, and M. Wojtkowski, “Scanning protocols dedicated to smart velocity ranging in spectral OCT,” Opt. Express 17, 23736–23754 (2009).
[Crossref]

J. Walther and E. Koch, “Transverse motion as a source of noise and reduced correlation of the Doppler phase shift in spectral domain OCT,” Opt. Express 17, 19698–19713 (2009).
[Crossref] [PubMed]

B. J. Vakoc, G. J. Tearney, and B. E. Bouma, “Statistical properties of phase-decorrelation in phase-resolved Doppler optical coherence tomography,” IEEE Trans. Med. Imaging 28, 814–821 (2009).
[Crossref] [PubMed]

B. J. Vakoc, R. M. Lanning, J. A. Tyrrell, T. P. Padera, L. A. Bartlett, T. Stylianopoulos, L. L. Munn, G. J. Tearney, D. Fukumura, R. K. Jain, and B. E. Bouma, “Three-dimensional microscopy of the tumor microenvironment in vivo using optical frequency domain imaging,” Nat. Med. 15, 1219–1223 (2009).
[Crossref] [PubMed]

2008 (2)

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)

2005 (2)

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)

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)

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.

Adler, D. C.

Akkin, T.

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]

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]

An, L.

Bajraszewski, T.

Bartlett, L. A.

B. J. Vakoc, R. M. Lanning, J. A. Tyrrell, T. P. Padera, L. A. Bartlett, T. Stylianopoulos, L. L. Munn, G. J. Tearney, D. Fukumura, R. K. Jain, and B. E. Bouma, “Three-dimensional microscopy of the tumor microenvironment in vivo using optical frequency domain imaging,” Nat. Med. 15, 1219–1223 (2009).
[Crossref] [PubMed]

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.

Boer, J. F. d.

Bonesi, M.

Boppart, S. A.

Bouma, B. E.

Braaf, B.

Brinkmann, R.

Cense, B.

Chen, T. C.

Chen, Z.

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]

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]

Cobbold, R. S.

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]

Dav, D. P.

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]

Dave, D. P.

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]

de Boer, J. F.

Debbeler, C.

Drexler, W.

Fercher, A.

Fingler, J.

Fraser, S. E.

Fujimoto, J. G.

Fukumura, D.

B. J. Vakoc, R. M. Lanning, J. A. Tyrrell, T. P. Padera, L. A. Bartlett, T. Stylianopoulos, L. L. Munn, G. J. Tearney, D. Fukumura, R. K. Jain, and B. E. Bouma, “Three-dimensional microscopy of the tumor microenvironment in vivo using optical frequency domain imaging,” Nat. Med. 15, 1219–1223 (2009).
[Crossref] [PubMed]

Gorczynska, I.

Gordon, M. L.

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]

Grulkowski, I.

Gtzinger, E.

H. G. R.,

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]

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.

Hong, Y.-J.

Hoppel, K.

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]

Httmann, G.

Huang, S.-W.

Huber, R.

Izatt, J.

Jaillon, F.

Jain, R. K.

B. J. Vakoc, R. M. Lanning, J. A. Tyrrell, T. P. Padera, L. A. Bartlett, T. Stylianopoulos, L. L. Munn, G. J. Tearney, D. Fukumura, R. K. Jain, and B. E. Bouma, “Three-dimensional microscopy of the tumor microenvironment in vivo using optical frequency domain imaging,” Nat. Med. 15, 1219–1223 (2009).
[Crossref] [PubMed]

Jiang, J. Y.

John, R.

Jong-Sen, L.

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]

Kennedy, B. F.

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.

B. J. Vakoc, R. M. Lanning, J. A. Tyrrell, T. P. Padera, L. A. Bartlett, T. Stylianopoulos, L. L. Munn, G. J. Tearney, D. Fukumura, R. K. Jain, and B. E. Bouma, “Three-dimensional microscopy of the tumor microenvironment in vivo using optical frequency domain imaging,” Nat. Med. 15, 1219–1223 (2009).
[Crossref] [PubMed]

Lee, B. H.

Lee, J.

Leitgeb, R.

Leitgeb, R. A.

Liang, X.

Makita, S.

Mango, S.

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]

McLaughlin, R. A.

Miller, A.

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]

Milner, T. E.

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]

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]

Min, E.-J.

Miura, M.

Mller, H. H.

Mok, A.

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]

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.

B. J. Vakoc, R. M. Lanning, J. A. Tyrrell, T. P. Padera, L. A. Bartlett, T. Stylianopoulos, L. L. Munn, G. J. Tearney, D. Fukumura, R. K. Jain, and B. E. Bouma, “Three-dimensional microscopy of the tumor microenvironment in vivo using optical frequency domain imaging,” Nat. Med. 15, 1219–1223 (2009).
[Crossref] [PubMed]

Munro, P. R. T.

Nassif, N.

Nelson, J. S.

Padera, T. P.

B. J. Vakoc, R. M. Lanning, J. A. Tyrrell, T. P. Padera, L. A. Bartlett, T. Stylianopoulos, L. L. Munn, G. J. Tearney, D. Fukumura, R. K. Jain, and B. E. Bouma, “Three-dimensional microscopy of the tumor microenvironment in vivo using optical frequency domain imaging,” Nat. Med. 15, 1219–1223 (2009).
[Crossref] [PubMed]

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.

Sarunic, M.

Sasaki, K.

Saxer, C.

Schlott, K.

Schmetterer, L.

Schwartz, D. M.

Sethuraman, S.

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]

Sicam, V. A. D.

Srinivasan, V. J.

Stylianopoulos, T.

B. J. Vakoc, R. M. Lanning, J. A. Tyrrell, T. P. Padera, L. A. Bartlett, T. Stylianopoulos, L. L. Munn, G. J. Tearney, D. Fukumura, R. K. Jain, and B. E. Bouma, “Three-dimensional microscopy of the tumor microenvironment in vivo using optical frequency domain imaging,” Nat. Med. 15, 1219–1223 (2009).
[Crossref] [PubMed]

Szkulmowska, A.

Szkulmowski, M.

Szlag, D.

Tearney, G. J.

Telenkov, S. A.

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]

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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).
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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).
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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).
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S. Makita, F. Jaillon, M. Yamanari, and Y. Yasuno, “Dual-beam-scan Doppler optical coherence angiography for birefringence-artifact-free vasculature imaging,” Opt. Express 20, 2681–2692 (2012).
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K. Kurokawa, K. Sasaki, S. Makita, Y.-J. Hong, and Y. Yasuno, “Three-dimensional retinal and choroidal capillary imaging by power Doppler optical coherence angiography with adaptive optics,” Opt. Express 20, 22796–22812 (2012).
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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).
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I. Grulkowski, I. Gorczynska, M. Szkulmowski, D. Szlag, A. Szkulmowska, R. A. Leitgeb, A. Kowalczyk, and M. Wojtkowski, “Scanning protocols dedicated to smart velocity ranging in spectral OCT,” Opt. Express 17, 23736–23754 (2009).
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L. An, J. Qin, and R. K. Wang, “Ultrahigh sensitive optical microangiography for in vivo imaging of microcirculations within human skin tissue beds,” Opt. Express 18, 8220–8228 (2010).
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B. Braaf, K. A. Vermeer, K. V. Vienola, and J. F. de Boer, “Angiography of the retina and the choroid with phase-resolved OCT using interval-optimized backstitched b-scans,” Opt. Express 20, 20516–20534 (2012).
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S. Makita, Y. Hong, M. Yamanari, T. Yatagai, and Y. Yasuno, “Optical coherence angiography,” Opt. Express 14, 7821–7840 (2006).
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S. Makita, F. Jaillon, M. Yamanari, M. Miura, and Y. Yasuno, “Comprehensive in vivo micro-vascular imaging of the human eye by dual-beam-scan Doppler optical coherence angiography,” Opt. Express 19, 1271–1283 (2011).
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S. Zotter, M. Pircher, T. Torzicky, M. Bonesi, E. Gtzinger, R. A. Leitgeb, and C. K. Hitzenberger, “Visualization of microvasculature by dual-beam phase-resolved Doppler optical coherence tomography,” Opt. Express 19, 1217–1227 (2011).
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Opt. Lett. (2)

Phys. Med. Biol. (1)

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]

Proc. R. Soc. Lond. A (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]

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J. S. Bendat and A. G. Piersol, Random Data: Analysis and Measurement Procedures (John Wiley and Sons, 2010).
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W. Drexler and J. G. Fujimoto, Optical Coherence Tomography: Technology and Applications (Springer, 2008).
[Crossref]

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Figures (9)

Fig. 1
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
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
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
Fig. 4

Phase shift noise vs fractional sampling step δx.

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Fig. 5
Fig. 5

The transitional point δxc (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
Fig. 6

Phase shift noises vs ESNR.

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Fig. 7
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
Fig. 8

Phase shift noise vs effective number of independent samples.

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Fig. 9
Fig. 9

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

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Tables (1)

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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 ∑ m a m exp [ − i 2 k c ( z m − z 0 ) ] ,
p Δ Φ ( Δ ϕ | ρ , Δ ϕ 0 ) = 1 − ρ 2 2 π { β ( π 2 + sin − 1 β ) [ 1 − β 2 ] 3 / 2 + 1 1 − β 2 } ,
ρ e i Δ ϕ 0 = E [ G 1 * G 2 ] E [ | G 1 | 2 ] E [ | G 2 | 2 ] ,
E [ Δ ϕ ] = ρ sin Δ ϕ 0 1 − ρ 2 cos 2 Δ ϕ 0 cos − 1 ( ρ cos Δ ϕ 0 ) ,
σ Δ ϕ = 1 − ρ 2 1 − β ′ 2 [ π 2 4 − π sin − 1 β ′ + ( sin − 1 β ′ ) 2 ] + π 2 12 − Li 2 ( ρ 2 ) 2 ,
Δ ϕ ¯ = 1 N ∑ n = 1 N Δ ϕ n ,
S Δ ϕ = 1 N ∑ n = 1 N ( Δ ϕ n − Δ ϕ ¯ ) 2 ,
ρ = ρ s ( 1 + SNR 1 − 1 ) ( 1 + SNR 2 − 1 ) ,
ρ s = | E [ S 1 * S 2 ] | E [ | S 1 | 2 ] E [ | S 2 | 2 ] .
1 1 + ESNR − 1 ≡ 1 ( 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 ≡ ρ S 1 , S 2 ( t 1 , t 2 ) = E [ s 1 * ( t 1 ) s 2 ( t 2 ) ] E [ | s 1 ( t 1 ) | 2 ] E [ | s 2 ( t 2 ) | 2 ] .
h χ ( x L , y L , z L ) ∝ e − 2 x L 2 w χ 2 e − 2 y L 2 w χ 2 e − Δ k χ 2 z L 2 8 e − 2 i k c χ ( z L − z 0 )           ( χ = 1 , 2 ) ,
ρ s ≈ ρ η 1 , η 2 × 2 w 1 w 2 w 1 2 + w 2 2 2 Δ k 1 Δ k 2 Δ k 1 2 + Δ k 2 2 × e − 2 x ′ ( t 1 , t 2 ) 2 + y ′ ( t 1 , t 2 ) 2 w 1 2 + w 2 2 e − Δ k 1 2 Δ k 2 2 Δ k 1 2 + Δ k 2 2 z ′ ( t 1 , t 2 ) 2 8 × e − 8 ( k c 1 − k c 2 ) 2 Δ k 1 2 + Δ k 2 2 × e − 4 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 ( t 2 + t 1 2 ) ( t 2 − t 1 ) ,
Δ ϕ 0 ≈ − 2 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 ≡ ρ η 1 , η 2 | ρ h 1 , h 2 ( r ′ ( Δ t ) ) | ,
ρ h 1 , h 2 ( r ′ ) = 2 w 1 w 2 w 1 2 + w 2 2 2 Δ k 1 Δ k 2 Δ k 1 2 + Δ k 2 2 e − 2 x ′ 2 + y ′ 2 w 1 2 + w 2 2 e − Δ k 1 2 Δ k 2 2 Δ k 1 2 + Δ k 2 2 z ′ 2 8 e − 8 ( k c 1 − k c 2 ) 2 Δ k 1 2 + Δ k 1 2 e − 2 i k c 1 Δ k 2 2 + k c 2 Δ k 1 2 Δ k 1 2 + Δ k 2 2 z ′ .
r = | g 1 * ( t 1 ) g 2 ( t 2 ) ¯ | | g 1 ( t 1 ) | 2 ¯ | g 2 ( t 2 ) | 2 ¯ .
ρ ^ s = ( 1 + ESNR ^ − 1 ) r = | g 1 * ( t 1 ) g 2 ( t 2 ) ¯ | [ | g 1 ( t 1 ) | 2 ¯ − | n 1 | 2 ¯ ] [ | g 2 ( t 2 ) | 2 ¯ − | n 2 | 2 ¯ ] ,
1 + ESNR ^ − 1 = | g 1 ( t 1 ) | 2 ¯ | g 2 ( t 2 ) | 2 ¯ [ | g 1 ( t 1 ) | 2 ¯ − | n 1 | 2 ¯ ] [ | g 2 ( t 2 ) | 2 ¯ − | n 2 | 2 ¯ ] .
Δ ϕ ^ 0 = arg [ ∑ κ = 1 ν g 1 ( κ ) * g 2 ( κ ) ] .
p Δ Φ ^ 0 ( Δ ϕ ^ 0 | ρ , Δ ϕ 0 ) = Γ ( ν + 1 2 ) ( 1 − ρ 2 ) ν ρ cos ( Δ ϕ ^ 0 − Δ ϕ 0 ) 2 π Γ ( ν ) [ 1 − ρ 2 cos 2 ( Δ ϕ ^ 0 − Δ ϕ 0 ) ] ν + 1 / 2 + ( 1 − ρ 2 ) ν 2 π F 1 2 ( ν , 1 ; 1 2 ; ρ 2 cos 2 ( Δ ϕ ^ 0 − Δ ϕ 0 ) ) ,
σ Δ ϕ ^ 0 = E [ Δ ϕ ^ 0 2 ] − E [ Δ ϕ ^ 0 ] 2 .
Δ ϕ ^ 0 = arg [ ∑ i I ∑ j J ∑ l L g 1 * ( x 1 + i , y 1 + j , z 1 + l ) g 2 ( x 1 + i , y 1 + j , z 1 + l ) ] ,
ENIS = I 1 + 2 ∑ i = 1 I − 1 I − l I ρ g 2 2 ( i Δ x , 0 , 0 ) J 1 + 2 ∑ j = 1 J − 1 J − l J ρ g 2 2 ( 0 , j Δ y , 0 ) × L 1 + 2 ∑ l = 1 L − 1 L − l L ρ g 2 2 ( 0 , 0 , l Δ z ) ,
ρ g 2 ( i Δ x , 0 , 0 ) = | 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 ) ] | E [ | G 1 * ( x 1 ) G 2 ( x 1 ) − E [ G 1 * ( x 1 ) G 2 ( x 1 ) ] | 2 ] E [ | G 1 * ( x 1 + i ) G 2 ( x 1 + i ) − E [ G 1 * ( x 1 + i ) G 2 ( x 1 + i ) ] | 2 ] .
ρ g 2 , S S ( i Δ x , j Δ y , l Δ z ) = 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 − w 1 2 + w 2 2 w 1 2 w 2 2 [ ( i Δ x ) 2 + ( j Δ y ) 2 ] e − ( 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 σ Δ ϕ SS Δ t ,
ρ s , SS ( SB ) = e − x b ′ 2 + y b ′ 2 w 2 ,
σ Δ ϕ SS ( SB ) = { σ Δ ϕ | ρ = e − δ x 2 1 + 1 / ESNR ( SB ) ( I J L = 1 ) σ Δ ϕ ^ 0 | ρ = e − δ x 2 1 + 1 / ESNR ( SB ) , ν = ENIS I − 1 , J , L ( I J L > = 2 ) ,
ρ s , SS ( DB ) = ρ Pol . 2 w 1 w 2 w 1 2 + w 2 2 2 Δ k 1 Δ k 2 Δ k 1 2 + Δ k 2 2 e − 8 ( k c 1 − k c 2 ) 2 Δ k 1 2 + Δ k 2 2 ,
σ Δ ϕ SS ( DB ) = { σ Δ ϕ | ρ = ρ s , SS ( DB ) 1 + 1 / ESNR ( DB ) ( I J L = 1 ) σ Δ ϕ ^ 0 | ρ = ρ s , SS ( DB ) 1 + 1 / ESNR ( DB ) , ν = ENIS I , J , L ( I J L > = 2 ) .
g ( x i , z l ) = g H ( x i , z l ) + g V ( x i , z l ) exp [ − i Δ ϕ ch ( x i ) ] ,
δ x c = − ln [ ρ Pol . ESNR ( DB ) ESNR ( SB ) ESNR ( SB ) + 1 ESNR ( DB ) + 1 ] = − ln [ ρ Pol . ESNR ( SB ) + 1 ESNR ( SB ) + 2 ] .
ESNR ρ s = ρ s 1 − ρ s .
I ( DB ) = { ⌊ 2 δ x ⌋ 2 δ x ≥ 1 , 1 otherwise
I ( SB ) = { ⌊ 2 δ x ⌋ − 1 2 δ x ≥ 2 , 1 otherwise .
E [ Δ ϕ ^ 0 n ] = ( − 1 ) n / 2 π n cos ( n π 2 ) n + 1 + 2 ∑ l = 1 ∞ { ( − 1 ) l ρ l Γ ( l 2 + 1 ) Γ ( l 2 + ν ) F 1 2 ( l 2 , l 2 − ν + 1 ; l + 1 ; ρ 2 ) π Γ ( ν ) Γ ( l + 1 ) × [ ∂ n ∂ s n sinh ( π s ) [ s cos ( l Δ ϕ 0 ) − l sin ( l Δ ϕ 0 ) ] l 2 + s 2 ] | s → 0 } .
E [ Δ ϕ ^ 0 ] = − ∑ l = 1 ∞ 2 ( − ρ ) l sin ( l Δ ϕ 0 ) Γ ( l 2 + 1 ) Γ ( l 2 + ν ) F 1 2 [ l 2 , l 2 − ν + 1 ; l + 1 ; ρ 2 ] l Γ ( ν ) Γ ( l + 1 ) ,
E [ Δ ϕ ^ 0 2 ] = π 2 3 + ∑ l = 1 ∞ 4 ( − ρ ) l cos ( l Δ ϕ 0 ) Γ ( l 2 + 1 ) Γ ( l 2 + ν ) F 1 2 [ l 2 , l 2 − ν + 1 ; l + 1 ; ρ 2 ] l 2 Γ ( ν ) Γ ( l + 1 ) .

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