Abstract

Functional optical coherence tomography (OCT) imaging based on the decorrelation of the intensity signal has been used extensively in angiography and is finding use in flowmetry and therapy monitoring. In this work, we present a rigorous analysis of the autocorrelation function, introduce the concepts of contrast bias, statistical bias and variability, and identify the optimal definition of the second-order autocorrelation function (ACF) g(2) to improve its estimation from limited data. We benchmark different averaging strategies in reducing statistical bias and variability. We also developed an analytical correction for the noise contributions to the decorrelation of the ACF in OCT that extends the signal-to-noise ratio range in which ACF analysis can be used. We demonstrate the use of all the tools developed in the experimental determination of the lateral speckle size depth dependence in a rotational endoscopic probe with low NA, and we show the ability to more accurately determine the rotational speed of an endoscopic probe to implement NURD detection. We finally present g(2)-based angiography of the finger nailbed, demonstrating the improved results from noise correction and the optimal bias mitigation strategies.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article
OSA Recommended Articles
Quantitative technique for robust and noise-tolerant speed measurements based on speckle decorrelation in optical coherence tomography

Néstor Uribe-Patarroyo, Martin Villiger, and Brett E. Bouma
Opt. Express 22(20) 24411-24429 (2014)

Forward multiple scattering dominates speckle decorrelation in whole-blood flowmetry using optical coherence tomography

Natalie G. Ferris, Taylor M. Cannon, Martin Villiger, Brett E. Bouma, and Néstor Uribe-Patarroyo
Biomed. Opt. Express 11(4) 1947-1966 (2020)

Robust motion tracking based on adaptive speckle decorrelation analysis of OCT signal

Yuewen Wang, Yahui Wang, Ali Akansu, Kevin D. Belfield, Basil Hubbi, and Xuan Liu
Biomed. Opt. Express 6(11) 4302-4316 (2015)

References

  • View by:
  • |
  • |
  • |

  1. D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and A. Et, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
    [Crossref]
  2. J. Fujimoto and W. Drexler, “Introduction to Optical Coherence Tomography,” in Optical Coherence Tomography: Technology and Applications, W. Drexler and J. G. Fujimoto, eds., Biological and Medical Physics, Biomedical Engineering (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008), pp. 1–45.
  3. A. Curatolo, B. F. Kennedy, D. D. Sampson, and T. R. Hillman, “Speckle in Optical Coherence Tomography,” Adv. Biophotonics p. 68 (2016).
  4. J. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts & Company, 2007).
  5. A. Mariampillai, B. A. Standish, E. H. Moriyama, M. Khurana, N. R. Munce, M. 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(13), 1530–1532 (2008).
    [Crossref]
  6. 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(4), 4710–4725 (2012).
    [Crossref]
  7. Y. Hong, S. Makita, M. Yamanari, M. Miura, S. Kim, T. Yatagai, and Y. Yasuno, “Three-dimensional visualization of choroidal vessels by using standard and ultra-high resolution scattering optical coherence angiography,” Opt. Express 15(12), 7538 (2007).
    [Crossref]
  8. J. Barton and S. Stromski, “Flow measurement without phase information in optical coherence tomography images,” Opt. Express 13(14), 5234–5239 (2005).
    [Crossref]
  9. A. S. Nam, I. Chico-Calero, and B. J. Vakoc, “Complex differential variance algorithm for optical coherence tomography angiography,” Biomed. Opt. Express 5(11), 3822 (2014).
    [Crossref]
  10. J. Lee, W. Wu, J. Y. Jiang, B. Zhu, and D. A. Boas, “Dynamic light scattering optical coherence tomography,” Opt. Express 20(20), 22262–22277 (2012).
    [Crossref]
  11. 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(3), 521–524 (2014).
    [Crossref]
  12. X. Liu, Y. Huang, J. C. Ramella-Roman, S. A. Mathews, and J. U. Kang, “Quantitative transverse flow measurement using OCT speckle decorrelation analysis,” Opt. Lett. 38(5), 805–807 (2013).
    [Crossref]
  13. 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(4), 042312 (2013).
    [Crossref]
  14. N. Uribe-Patarroyo, M. Villiger, and B. E. Bouma, “Quantitative technique for robust and noise-tolerant speed measurements based on speckle decorrelation in optical coherence tomography,” Opt. Express 22(20), 24411–24429 (2014).
    [Crossref]
  15. N. Uribe-Patarroyo and B. E. Bouma, “Velocity gradients in spatially-resolved laser Doppler flowmetry and dynamic light scattering with confocal and coherence gating,” Phys. Rev. E 94(2), 022604 (2016).
    [Crossref]
  16. B. J. Vakoc, G. J. T. M.d, and B. E. Bouma, “Real-time microscopic visualization of tissue response to laser thermal therapy,” J. Biomed. Opt. 12(2), 020501 (2007).
    [Crossref]
  17. W. C. Y. Lo, N. Uribe-Patarroyo, A. S. Nam, M. Villiger, B. J. Vakoc, and B. E. Bouma, “Laser thermal therapy monitoring using complex differential variance in optical coherence tomography,” J. Biophotonics 10(1), 84–91 (2017).
    [Crossref]
  18. W. C. Y. Lo, N. Uribe-Patarroyo, K. Hoebel, K. Beaudette, M. Villiger, N. S. Nishioka, B. J. Vakoc, and B. E. Bouma, “Balloon catheter-based radiofrequency ablation monitoring in porcine esophagus using optical coherence tomography,” Biomed. Opt. Express 10(4), 2067–2089 (2019).
    [Crossref]
  19. N. Weiss, T. G. van Leeuwen, and J. Kalkman, “Doppler-based lateral motion tracking for optical coherence tomography,” Opt. Lett. 37(12), 2220–2222 (2012).
    [Crossref]
  20. N. Uribe-Patarroyo and B. E. Bouma, “Rotational distortion correction in endoscopic optical coherence tomography based on speckle decorrelation,” Opt. Lett. 40(23), 5518 (2015).
    [Crossref]
  21. B. J. Berne and R. Pecora, Dynamic Light Scattering: With Applications to Chemistry, Biology, and Physics (Dover Publications, 2000).
  22. K. Schätzel, M. Drewel, and S. Stimac, “Photon Correlation Measurements at Large Lag Times: Improving Statistical Accuracy,” J. Mod. Opt. 35(4), 711–718 (1988).
    [Crossref]
  23. K. Schatzel, “Noise on photon correlation data. I. Autocorrelation functions,” Quantum Opt. 2(4), 287–305 (1990).
    [Crossref]
  24. S. Makita, K. Kurokawa, Y.-J. Hong, M. Miura, and Y. Yasuno, “Noise-immune complex correlation for optical coherence angiography based on standard and Jones matrix optical coherence tomography,” Biomed. Opt. Express 7(4), 1525–1548 (2016).
    [Crossref]
  25. I. Popov, A. S. Weatherbee, and A. Vitkin, “Impact of velocity gradient in Poiseuille flow on the statistics of coherent radiation scattered by flowing Brownian particles in optical coherence tomography,” J. Biomed. Opt. 24(9), 097001 (2019).
    [Crossref]
  26. M. Almasian, T. G. van Leeuwen, and D. J. Faber, “OCT Amplitude and Speckle Statistics of Discrete Random Media,” Sci. Rep. 7(1), 14873 (2017).
    [Crossref]
  27. N. Mohan and B. Vakoc, “Principal-component-analysis-based estimation of blood flow velocities using optical coherence tomography intensity signals,” Opt. Lett. 36(11), 2068–2070 (2011).
    [Crossref]
  28. V. J. Srinivasan, H. Radhakrishnan, E. H. Lo, E. T. Mandeville, J. Y. Jiang, S. Barry, and A. E. Cable, “OCT methods for capillary velocimetry,” Biomed. Opt. Express 3(3), 612 (2012).
    [Crossref]
  29. P. Zakharov and F. Scheffold, “Advances in dynamic light scattering techniques,” in Light Scattering Reviews 4: Single Light Scattering and Radiative Transfer, A. A. Kokhanovsky, ed., Springer Praxis Books (Springer, Berlin, Heidelberg, 2009), pp. 433–467.
  30. Y. Bromberg, Y. Lahini, E. Small, and Y. Silberberg, “Hanbury Brown and Twiss interferometry with interacting photons,” Nat. Photonics 4(10), 721–726 (2010).
    [Crossref]
  31. D. D. Duncan, S. J. Kirkpatrick, and R. K. Wang, “Statistics of local speckle contrast,” J. Opt. Soc. Am. A 25(1), 9–15 (2008).
    [Crossref]
  32. N. Weiss, T. G. van Leeuwen, and J. Kalkman, “Simultaneous and localized measurement of diffusion and flow using optical coherence tomography,” Opt. Express 23(3), 3448 (2015).
    [Crossref]
  33. 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(6), 819–825 (2013).
    [Crossref]
  34. T. R. Hillman and D. D. Sampson, “The effect of water dispersion and absorption on axial resolution in ultrahigh-resolution optical coherence tomography,” Opt. Express 13(6), 1860–1874 (2005).
    [Crossref]
  35. I. Popov, A. Weatherbee, and I. A. Vitkin, “Statistical properties of dynamic speckles from flowing Brownian scatterers in the vicinity of the image plane in optical coherence tomography,” Biomed. Opt. Express 8(4), 2004–2017 (2017).
    [Crossref]
  36. N. G. Ferris, T. M. Cannon, M. Villiger, B. E. Bouma, and N. Uribe-Patarroyo, “Forward multiple scattering dominates speckle decorrelation in whole-blood flowmetry using optical coherence tomography,” Biomed. Opt. Express 11(4), 1947–1966 (2020).
    [Crossref]
  37. D. L. Marks, T. S. Ralston, P. S. Carney, and S. A. Boppart, “Inverse scattering for rotationally scanned optical coherence tomography,” J. Opt. Soc. Am. A 23(10), 2433–2439 (2006).
    [Crossref]
  38. M. L. Villiger and B. E. Bouma, “Physics of Cardiovascular OCT,” in Cardiovascular OCT Imaging, I.-K. Jang, ed. (Springer International Publishing, Cham, 2015), pp. 23–38.
  39. 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(20), 4822–4828 (2004).
    [Crossref]
  40. M. S. Bartlett, “PERIODOGRAM ANALYSIS AND CONTINUOUS SPECTRA,” Biometrika 37(1-2), 1–16 (1950).
    [Crossref]
  41. 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. de Boer, “In vivo dynamic human retinal blood flow imaging using ultra-high-speed spectral domain optical Doppler tomography,” Opt. Express 11(25), 3490 (2003).
    [Crossref]
  42. M. Geissbuehler and T. Lasser, “How to display data by color schemes compatible with red-green color perception deficiencies,” Opt. Express 21(8), 9862–9874 (2013).
    [Crossref]
  43. P. Kovesi, “Good Colour Maps: How to Design Them,” arXiv:1509.03700 [cs] (2015).
  44. M. Zhang, T. S. Hwang, J. P. Campbell, S. T. Bailey, D. J. Wilson, D. Huang, and Y. Jia, “Projection-resolved optical coherence tomographic angiography,” Biomed. Opt. Express 7(3), 816–828 (2016).
    [Crossref]
  45. K. C. Zhou, B. K. Huang, H. Tagare, and M. A. Choma, “Improved velocimetry in optical coherence tomography using Bayesian analysis,” Biomed. Opt. Express 6(12), 4796–4811 (2015).
    [Crossref]

2020 (1)

2019 (2)

I. Popov, A. S. Weatherbee, and A. Vitkin, “Impact of velocity gradient in Poiseuille flow on the statistics of coherent radiation scattered by flowing Brownian particles in optical coherence tomography,” J. Biomed. Opt. 24(9), 097001 (2019).
[Crossref]

W. C. Y. Lo, N. Uribe-Patarroyo, K. Hoebel, K. Beaudette, M. Villiger, N. S. Nishioka, B. J. Vakoc, and B. E. Bouma, “Balloon catheter-based radiofrequency ablation monitoring in porcine esophagus using optical coherence tomography,” Biomed. Opt. Express 10(4), 2067–2089 (2019).
[Crossref]

2017 (3)

W. C. Y. Lo, N. Uribe-Patarroyo, A. S. Nam, M. Villiger, B. J. Vakoc, and B. E. Bouma, “Laser thermal therapy monitoring using complex differential variance in optical coherence tomography,” J. Biophotonics 10(1), 84–91 (2017).
[Crossref]

M. Almasian, T. G. van Leeuwen, and D. J. Faber, “OCT Amplitude and Speckle Statistics of Discrete Random Media,” Sci. Rep. 7(1), 14873 (2017).
[Crossref]

I. Popov, A. Weatherbee, and I. A. Vitkin, “Statistical properties of dynamic speckles from flowing Brownian scatterers in the vicinity of the image plane in optical coherence tomography,” Biomed. Opt. Express 8(4), 2004–2017 (2017).
[Crossref]

2016 (3)

2015 (3)

2014 (3)

2013 (4)

X. Liu, Y. Huang, J. C. Ramella-Roman, S. A. Mathews, and J. U. Kang, “Quantitative transverse flow measurement using OCT speckle decorrelation analysis,” Opt. Lett. 38(5), 805–807 (2013).
[Crossref]

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(4), 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(6), 819–825 (2013).
[Crossref]

M. Geissbuehler and T. Lasser, “How to display data by color schemes compatible with red-green color perception deficiencies,” Opt. Express 21(8), 9862–9874 (2013).
[Crossref]

2012 (4)

2011 (1)

2010 (1)

Y. Bromberg, Y. Lahini, E. Small, and Y. Silberberg, “Hanbury Brown and Twiss interferometry with interacting photons,” Nat. Photonics 4(10), 721–726 (2010).
[Crossref]

2008 (2)

2007 (2)

2006 (1)

2005 (2)

2004 (1)

2003 (1)

1991 (1)

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and A. Et, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref]

1990 (1)

K. Schatzel, “Noise on photon correlation data. I. Autocorrelation functions,” Quantum Opt. 2(4), 287–305 (1990).
[Crossref]

1988 (1)

K. Schätzel, M. Drewel, and S. Stimac, “Photon Correlation Measurements at Large Lag Times: Improving Statistical Accuracy,” J. Mod. Opt. 35(4), 711–718 (1988).
[Crossref]

1950 (1)

M. S. Bartlett, “PERIODOGRAM ANALYSIS AND CONTINUOUS SPECTRA,” Biometrika 37(1-2), 1–16 (1950).
[Crossref]

Almasian, M.

M. Almasian, T. G. van Leeuwen, and D. J. Faber, “OCT Amplitude and Speckle Statistics of Discrete Random Media,” Sci. Rep. 7(1), 14873 (2017).
[Crossref]

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(6), 819–825 (2013).
[Crossref]

Bailey, S. T.

Barry, S.

Bartlett, M. S.

M. S. Bartlett, “PERIODOGRAM ANALYSIS AND CONTINUOUS SPECTRA,” Biometrika 37(1-2), 1–16 (1950).
[Crossref]

Barton, J.

Beaudette, K.

Berne, B. J.

B. J. Berne and R. Pecora, Dynamic Light Scattering: With Applications to Chemistry, Biology, and Physics (Dover Publications, 2000).

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(6), 819–825 (2013).
[Crossref]

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

Boppart, S. A.

Bouma, B.

Bouma, B. E.

N. G. Ferris, T. M. Cannon, M. Villiger, B. E. Bouma, and N. Uribe-Patarroyo, “Forward multiple scattering dominates speckle decorrelation in whole-blood flowmetry using optical coherence tomography,” Biomed. Opt. Express 11(4), 1947–1966 (2020).
[Crossref]

W. C. Y. Lo, N. Uribe-Patarroyo, K. Hoebel, K. Beaudette, M. Villiger, N. S. Nishioka, B. J. Vakoc, and B. E. Bouma, “Balloon catheter-based radiofrequency ablation monitoring in porcine esophagus using optical coherence tomography,” Biomed. Opt. Express 10(4), 2067–2089 (2019).
[Crossref]

W. C. Y. Lo, N. Uribe-Patarroyo, A. S. Nam, M. Villiger, B. J. Vakoc, and B. E. Bouma, “Laser thermal therapy monitoring using complex differential variance in optical coherence tomography,” J. Biophotonics 10(1), 84–91 (2017).
[Crossref]

N. Uribe-Patarroyo and B. E. Bouma, “Velocity gradients in spatially-resolved laser Doppler flowmetry and dynamic light scattering with confocal and coherence gating,” Phys. Rev. E 94(2), 022604 (2016).
[Crossref]

N. Uribe-Patarroyo and B. E. Bouma, “Rotational distortion correction in endoscopic optical coherence tomography based on speckle decorrelation,” Opt. Lett. 40(23), 5518 (2015).
[Crossref]

N. Uribe-Patarroyo, M. Villiger, and B. E. Bouma, “Quantitative technique for robust and noise-tolerant speed measurements based on speckle decorrelation in optical coherence tomography,” Opt. Express 22(20), 24411–24429 (2014).
[Crossref]

B. J. Vakoc, G. J. T. M.d, and B. E. Bouma, “Real-time microscopic visualization of tissue response to laser thermal therapy,” J. Biomed. Opt. 12(2), 020501 (2007).
[Crossref]

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. de Boer, “In vivo dynamic human retinal blood flow imaging using ultra-high-speed spectral domain optical Doppler tomography,” Opt. Express 11(25), 3490 (2003).
[Crossref]

M. L. Villiger and B. E. Bouma, “Physics of Cardiovascular OCT,” in Cardiovascular OCT Imaging, I.-K. Jang, ed. (Springer International Publishing, Cham, 2015), pp. 23–38.

Bromberg, Y.

Y. Bromberg, Y. Lahini, E. Small, and Y. Silberberg, “Hanbury Brown and Twiss interferometry with interacting photons,” Nat. Photonics 4(10), 721–726 (2010).
[Crossref]

Cable, A.

Cable, A. E.

Campbell, J. P.

Cannon, T. M.

Carney, P. S.

Cense, B.

Chang, W.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and A. Et, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref]

Chen, T. C.

Chico-Calero, I.

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(6), 819–825 (2013).
[Crossref]

Curatolo, A.

A. Curatolo, B. F. Kennedy, D. D. Sampson, and T. R. Hillman, “Speckle in Optical Coherence Tomography,” Adv. Biophotonics p. 68 (2016).

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(6), 819–825 (2013).
[Crossref]

de Boer, J.

de Boer, J. F.

Drewel, M.

K. Schätzel, M. Drewel, and S. Stimac, “Photon Correlation Measurements at Large Lag Times: Improving Statistical Accuracy,” J. Mod. Opt. 35(4), 711–718 (1988).
[Crossref]

Drexler, W.

J. Fujimoto and W. Drexler, “Introduction to Optical Coherence Tomography,” in Optical Coherence Tomography: Technology and Applications, W. Drexler and J. G. Fujimoto, eds., Biological and Medical Physics, Biomedical Engineering (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008), pp. 1–45.

Duncan, D. D.

Et, A.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and A. Et, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref]

Faber, D. J.

M. Almasian, T. G. van Leeuwen, and D. J. Faber, “OCT Amplitude and Speckle Statistics of Discrete Random Media,” Sci. Rep. 7(1), 14873 (2017).
[Crossref]

Ferris, N. G.

Flotte, T.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and A. Et, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref]

Fujimoto, J.

J. Fujimoto and W. Drexler, “Introduction to Optical Coherence Tomography,” in Optical Coherence Tomography: Technology and Applications, W. Drexler and J. G. Fujimoto, eds., Biological and Medical Physics, Biomedical Engineering (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008), pp. 1–45.

Fujimoto, J. G.

Geissbuehler, M.

Goodman, J.

J. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts & Company, 2007).

Gregory, K.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and A. Et, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref]

Hee, M. R.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and A. Et, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref]

Hillman, T. R.

T. R. Hillman and D. D. Sampson, “The effect of water dispersion and absorption on axial resolution in ultrahigh-resolution optical coherence tomography,” Opt. Express 13(6), 1860–1874 (2005).
[Crossref]

A. Curatolo, B. F. Kennedy, D. D. Sampson, and T. R. Hillman, “Speckle in Optical Coherence Tomography,” Adv. Biophotonics p. 68 (2016).

Hoebel, K.

Hong, Y.

Hong, Y.-J.

Hornegger, J.

Huang, B. K.

Huang, D.

Huang, Y.

Hwang, T. S.

Jia, Y.

Jiang, J.

Jiang, J. Y.

Kalkman, J.

Kang, J. U.

Kennedy, B. F.

A. Curatolo, B. F. Kennedy, D. D. Sampson, and T. R. Hillman, “Speckle in Optical Coherence Tomography,” Adv. Biophotonics p. 68 (2016).

Khurana, M.

Kim, S.

Kirkpatrick, S. J.

Kovesi, P.

P. Kovesi, “Good Colour Maps: How to Design Them,” arXiv:1509.03700 [cs] (2015).

Kraus, M. F.

Kurokawa, K.

Lahini, Y.

Y. Bromberg, Y. Lahini, E. Small, and Y. Silberberg, “Hanbury Brown and Twiss interferometry with interacting photons,” Nat. Photonics 4(10), 721–726 (2010).
[Crossref]

Lasser, T.

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(6), 819–825 (2013).
[Crossref]

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

Leung, M. K.

Lin, C. P.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and A. Et, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref]

Liu, J. J.

Liu, X.

Lo, E. H.

Lo, W. C. Y.

W. C. Y. Lo, N. Uribe-Patarroyo, K. Hoebel, K. Beaudette, M. Villiger, N. S. Nishioka, B. J. Vakoc, and B. E. Bouma, “Balloon catheter-based radiofrequency ablation monitoring in porcine esophagus using optical coherence tomography,” Biomed. Opt. Express 10(4), 2067–2089 (2019).
[Crossref]

W. C. Y. Lo, N. Uribe-Patarroyo, A. S. Nam, M. Villiger, B. J. Vakoc, and B. E. Bouma, “Laser thermal therapy monitoring using complex differential variance in optical coherence tomography,” J. Biophotonics 10(1), 84–91 (2017).
[Crossref]

M.d, G. J. T.

B. J. Vakoc, G. J. T. M.d, and B. E. Bouma, “Real-time microscopic visualization of tissue response to laser thermal therapy,” J. Biomed. Opt. 12(2), 020501 (2007).
[Crossref]

Makita, S.

Mandeville, E. T.

Mariampillai, A.

Marks, D. L.

Mathews, S. A.

Miura, M.

Mohan, N.

Moriyama, E. H.

Munce, N. R.

Nam, A. S.

W. C. Y. Lo, N. Uribe-Patarroyo, A. S. Nam, M. Villiger, B. J. Vakoc, and B. E. Bouma, “Laser thermal therapy monitoring using complex differential variance in optical coherence tomography,” J. Biophotonics 10(1), 84–91 (2017).
[Crossref]

A. S. Nam, I. Chico-Calero, and B. J. Vakoc, “Complex differential variance algorithm for optical coherence tomography angiography,” Biomed. Opt. Express 5(11), 3822 (2014).
[Crossref]

Nassif, N.

Nishioka, N. S.

Park, B. H.

Pecora, R.

B. J. Berne and R. Pecora, Dynamic Light Scattering: With Applications to Chemistry, Biology, and Physics (Dover Publications, 2000).

Pierce, M. C.

Popov, I.

I. Popov, A. S. Weatherbee, and A. Vitkin, “Impact of velocity gradient in Poiseuille flow on the statistics of coherent radiation scattered by flowing Brownian particles in optical coherence tomography,” J. Biomed. Opt. 24(9), 097001 (2019).
[Crossref]

I. Popov, A. Weatherbee, and I. A. Vitkin, “Statistical properties of dynamic speckles from flowing Brownian scatterers in the vicinity of the image plane in optical coherence tomography,” Biomed. Opt. Express 8(4), 2004–2017 (2017).
[Crossref]

Potsaid, B.

Puliafito, C. A.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and A. Et, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref]

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(6), 819–825 (2013).
[Crossref]

V. J. Srinivasan, H. Radhakrishnan, E. H. Lo, E. T. Mandeville, J. Y. Jiang, S. Barry, and A. E. Cable, “OCT methods for capillary velocimetry,” Biomed. Opt. Express 3(3), 612 (2012).
[Crossref]

Ralston, T. S.

Ramella-Roman, J. C.

Sampson, D. D.

T. R. Hillman and D. D. Sampson, “The effect of water dispersion and absorption on axial resolution in ultrahigh-resolution optical coherence tomography,” Opt. Express 13(6), 1860–1874 (2005).
[Crossref]

A. Curatolo, B. F. Kennedy, D. D. Sampson, and T. R. Hillman, “Speckle in Optical Coherence Tomography,” Adv. Biophotonics p. 68 (2016).

Schatzel, K.

K. Schatzel, “Noise on photon correlation data. I. Autocorrelation functions,” Quantum Opt. 2(4), 287–305 (1990).
[Crossref]

Schätzel, K.

K. Schätzel, M. Drewel, and S. Stimac, “Photon Correlation Measurements at Large Lag Times: Improving Statistical Accuracy,” J. Mod. Opt. 35(4), 711–718 (1988).
[Crossref]

Scheffold, F.

P. Zakharov and F. Scheffold, “Advances in dynamic light scattering techniques,” in Light Scattering Reviews 4: Single Light Scattering and Radiative Transfer, A. A. Kokhanovsky, ed., Springer Praxis Books (Springer, Berlin, Heidelberg, 2009), pp. 433–467.

Schuman, J. S.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and A. Et, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref]

Silberberg, Y.

Y. Bromberg, Y. Lahini, E. Small, and Y. Silberberg, “Hanbury Brown and Twiss interferometry with interacting photons,” Nat. Photonics 4(10), 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(10), 721–726 (2010).
[Crossref]

Srinivasan, V. J.

Standish, B. A.

Stimac, S.

K. Schätzel, M. Drewel, and S. Stimac, “Photon Correlation Measurements at Large Lag Times: Improving Statistical Accuracy,” J. Mod. Opt. 35(4), 711–718 (1988).
[Crossref]

Stinson, W. G.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and A. Et, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref]

Stromski, S.

Subhash, H.

Swanson, E. A.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and A. Et, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref]

Tagare, H.

Tan, O.

Tearney, G.

Tearney, G. J.

Tokayer, J.

Uribe-Patarroyo, N.

Vakoc, B.

Vakoc, B. J.

W. C. Y. Lo, N. Uribe-Patarroyo, K. Hoebel, K. Beaudette, M. Villiger, N. S. Nishioka, B. J. Vakoc, and B. E. Bouma, “Balloon catheter-based radiofrequency ablation monitoring in porcine esophagus using optical coherence tomography,” Biomed. Opt. Express 10(4), 2067–2089 (2019).
[Crossref]

W. C. Y. Lo, N. Uribe-Patarroyo, A. S. Nam, M. Villiger, B. J. Vakoc, and B. E. Bouma, “Laser thermal therapy monitoring using complex differential variance in optical coherence tomography,” J. Biophotonics 10(1), 84–91 (2017).
[Crossref]

A. S. Nam, I. Chico-Calero, and B. J. Vakoc, “Complex differential variance algorithm for optical coherence tomography angiography,” Biomed. Opt. Express 5(11), 3822 (2014).
[Crossref]

B. J. Vakoc, G. J. T. M.d, and B. E. Bouma, “Real-time microscopic visualization of tissue response to laser thermal therapy,” J. Biomed. Opt. 12(2), 020501 (2007).
[Crossref]

van Leeuwen, T. G.

M. Almasian, T. G. van Leeuwen, and D. J. Faber, “OCT Amplitude and Speckle Statistics of Discrete Random Media,” Sci. Rep. 7(1), 14873 (2017).
[Crossref]

N. Weiss, T. G. van Leeuwen, and J. Kalkman, “Simultaneous and localized measurement of diffusion and flow using optical coherence tomography,” Opt. Express 23(3), 3448 (2015).
[Crossref]

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(4), 042312 (2013).
[Crossref]

N. Weiss, T. G. van Leeuwen, and J. Kalkman, “Doppler-based lateral motion tracking for optical coherence tomography,” Opt. Lett. 37(12), 2220–2222 (2012).
[Crossref]

Villiger, M.

Villiger, M. L.

M. L. Villiger and B. E. Bouma, “Physics of Cardiovascular OCT,” in Cardiovascular OCT Imaging, I.-K. Jang, ed. (Springer International Publishing, Cham, 2015), pp. 23–38.

Vitkin, A.

I. Popov, A. S. Weatherbee, and A. Vitkin, “Impact of velocity gradient in Poiseuille flow on the statistics of coherent radiation scattered by flowing Brownian particles in optical coherence tomography,” J. Biomed. Opt. 24(9), 097001 (2019).
[Crossref]

Vitkin, I. A.

Wang, R. K.

Wang, Y.

Weatherbee, A.

Weatherbee, A. S.

I. Popov, A. S. Weatherbee, and A. Vitkin, “Impact of velocity gradient in Poiseuille flow on the statistics of coherent radiation scattered by flowing Brownian particles in optical coherence tomography,” J. Biomed. Opt. 24(9), 097001 (2019).
[Crossref]

Weiss, N.

White, B. R.

Wilson, B. C.

Wilson, D. J.

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(6), 819–825 (2013).
[Crossref]

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

Yamanari, M.

Yang, V. X. D.

Yasuno, Y.

Yatagai, T.

Yun, S.

Zakharov, P.

P. Zakharov and F. Scheffold, “Advances in dynamic light scattering techniques,” in Light Scattering Reviews 4: Single Light Scattering and Radiative Transfer, A. A. Kokhanovsky, ed., Springer Praxis Books (Springer, Berlin, Heidelberg, 2009), pp. 433–467.

Zhang, M.

Zhou, K. C.

Zhu, B.

Biomed. Opt. Express (8)

V. J. Srinivasan, H. Radhakrishnan, E. H. Lo, E. T. Mandeville, J. Y. Jiang, S. Barry, and A. E. Cable, “OCT methods for capillary velocimetry,” Biomed. Opt. Express 3(3), 612 (2012).
[Crossref]

A. S. Nam, I. Chico-Calero, and B. J. Vakoc, “Complex differential variance algorithm for optical coherence tomography angiography,” Biomed. Opt. Express 5(11), 3822 (2014).
[Crossref]

K. C. Zhou, B. K. Huang, H. Tagare, and M. A. Choma, “Improved velocimetry in optical coherence tomography using Bayesian analysis,” Biomed. Opt. Express 6(12), 4796–4811 (2015).
[Crossref]

M. Zhang, T. S. Hwang, J. P. Campbell, S. T. Bailey, D. J. Wilson, D. Huang, and Y. Jia, “Projection-resolved optical coherence tomographic angiography,” Biomed. Opt. Express 7(3), 816–828 (2016).
[Crossref]

S. Makita, K. Kurokawa, Y.-J. Hong, M. Miura, and Y. Yasuno, “Noise-immune complex correlation for optical coherence angiography based on standard and Jones matrix optical coherence tomography,” Biomed. Opt. Express 7(4), 1525–1548 (2016).
[Crossref]

I. Popov, A. Weatherbee, and I. A. Vitkin, “Statistical properties of dynamic speckles from flowing Brownian scatterers in the vicinity of the image plane in optical coherence tomography,” Biomed. Opt. Express 8(4), 2004–2017 (2017).
[Crossref]

W. C. Y. Lo, N. Uribe-Patarroyo, K. Hoebel, K. Beaudette, M. Villiger, N. S. Nishioka, B. J. Vakoc, and B. E. Bouma, “Balloon catheter-based radiofrequency ablation monitoring in porcine esophagus using optical coherence tomography,” Biomed. Opt. Express 10(4), 2067–2089 (2019).
[Crossref]

N. G. Ferris, T. M. Cannon, M. Villiger, B. E. Bouma, and N. Uribe-Patarroyo, “Forward multiple scattering dominates speckle decorrelation in whole-blood flowmetry using optical coherence tomography,” Biomed. Opt. Express 11(4), 1947–1966 (2020).
[Crossref]

Biometrika (1)

M. S. Bartlett, “PERIODOGRAM ANALYSIS AND CONTINUOUS SPECTRA,” Biometrika 37(1-2), 1–16 (1950).
[Crossref]

J. Biomed. Opt. (2)

I. Popov, A. S. Weatherbee, and A. Vitkin, “Impact of velocity gradient in Poiseuille flow on the statistics of coherent radiation scattered by flowing Brownian particles in optical coherence tomography,” J. Biomed. Opt. 24(9), 097001 (2019).
[Crossref]

B. J. Vakoc, G. J. T. M.d, and B. E. Bouma, “Real-time microscopic visualization of tissue response to laser thermal therapy,” J. Biomed. Opt. 12(2), 020501 (2007).
[Crossref]

J. Biophotonics (1)

W. C. Y. Lo, N. Uribe-Patarroyo, A. S. Nam, M. Villiger, B. J. Vakoc, and B. E. Bouma, “Laser thermal therapy monitoring using complex differential variance in optical coherence tomography,” J. Biophotonics 10(1), 84–91 (2017).
[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(6), 819–825 (2013).
[Crossref]

J. Mod. Opt. (1)

K. Schätzel, M. Drewel, and S. Stimac, “Photon Correlation Measurements at Large Lag Times: Improving Statistical Accuracy,” J. Mod. Opt. 35(4), 711–718 (1988).
[Crossref]

J. Opt. Soc. Am. A (2)

Nat. Photonics (1)

Y. Bromberg, Y. Lahini, E. Small, and Y. Silberberg, “Hanbury Brown and Twiss interferometry with interacting photons,” Nat. Photonics 4(10), 721–726 (2010).
[Crossref]

Opt. Express (10)

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(4), 4710–4725 (2012).
[Crossref]

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

M. Geissbuehler and T. Lasser, “How to display data by color schemes compatible with red-green color perception deficiencies,” Opt. Express 21(8), 9862–9874 (2013).
[Crossref]

Y. Hong, S. Makita, M. Yamanari, M. Miura, S. Kim, T. Yatagai, and Y. Yasuno, “Three-dimensional visualization of choroidal vessels by using standard and ultra-high resolution scattering optical coherence angiography,” Opt. Express 15(12), 7538 (2007).
[Crossref]

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. de Boer, “In vivo dynamic human retinal blood flow imaging using ultra-high-speed spectral domain optical Doppler tomography,” Opt. Express 11(25), 3490 (2003).
[Crossref]

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(20), 4822–4828 (2004).
[Crossref]

T. R. Hillman and D. D. Sampson, “The effect of water dispersion and absorption on axial resolution in ultrahigh-resolution optical coherence tomography,” Opt. Express 13(6), 1860–1874 (2005).
[Crossref]

J. Barton and S. Stromski, “Flow measurement without phase information in optical coherence tomography images,” Opt. Express 13(14), 5234–5239 (2005).
[Crossref]

N. Weiss, T. G. van Leeuwen, and J. Kalkman, “Simultaneous and localized measurement of diffusion and flow using optical coherence tomography,” Opt. Express 23(3), 3448 (2015).
[Crossref]

N. Uribe-Patarroyo, M. Villiger, and B. E. Bouma, “Quantitative technique for robust and noise-tolerant speed measurements based on speckle decorrelation in optical coherence tomography,” Opt. Express 22(20), 24411–24429 (2014).
[Crossref]

Opt. Lett. (6)

Phys. Rev. E (2)

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(4), 042312 (2013).
[Crossref]

N. Uribe-Patarroyo and B. E. Bouma, “Velocity gradients in spatially-resolved laser Doppler flowmetry and dynamic light scattering with confocal and coherence gating,” Phys. Rev. E 94(2), 022604 (2016).
[Crossref]

Quantum Opt. (1)

K. Schatzel, “Noise on photon correlation data. I. Autocorrelation functions,” Quantum Opt. 2(4), 287–305 (1990).
[Crossref]

Sci. Rep. (1)

M. Almasian, T. G. van Leeuwen, and D. J. Faber, “OCT Amplitude and Speckle Statistics of Discrete Random Media,” Sci. Rep. 7(1), 14873 (2017).
[Crossref]

Science (1)

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and A. Et, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref]

Other (7)

J. Fujimoto and W. Drexler, “Introduction to Optical Coherence Tomography,” in Optical Coherence Tomography: Technology and Applications, W. Drexler and J. G. Fujimoto, eds., Biological and Medical Physics, Biomedical Engineering (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008), pp. 1–45.

A. Curatolo, B. F. Kennedy, D. D. Sampson, and T. R. Hillman, “Speckle in Optical Coherence Tomography,” Adv. Biophotonics p. 68 (2016).

J. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts & Company, 2007).

B. J. Berne and R. Pecora, Dynamic Light Scattering: With Applications to Chemistry, Biology, and Physics (Dover Publications, 2000).

P. Zakharov and F. Scheffold, “Advances in dynamic light scattering techniques,” in Light Scattering Reviews 4: Single Light Scattering and Radiative Transfer, A. A. Kokhanovsky, ed., Springer Praxis Books (Springer, Berlin, Heidelberg, 2009), pp. 433–467.

M. L. Villiger and B. E. Bouma, “Physics of Cardiovascular OCT,” in Cardiovascular OCT Imaging, I.-K. Jang, ed. (Springer International Publishing, Cham, 2015), pp. 23–38.

P. Kovesi, “Good Colour Maps: How to Design Them,” arXiv:1509.03700 [cs] (2015).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (11)

Fig. 1.
Fig. 1. Autocorrelation analysis in OCT is used to determine local dynamics of scatterers, which adds functional contrast such as blood flowmetry. The evolution of the complex-amplitude OCT signal reveals presence of moving scatterers [upper blue curve in (a)], which can be quantified using the first-order ACF $g^{(1)}$ [blue curve in (b)]. When analyzing the OCT intensity signal [lower blue curve in (a)], the second-order ACF $g^{(2)}$ is used for quantification [blue curve in (c)]. In both cases, the decay of the ACF can be related to the scatterer dynamics —including diffusive and translational motion, indicated generically as $v$ in the figure. Noise in the OCT signal [green curves in (a)] is known to affect $g^{(1)}$ and $g^{(2)}$ [green curves in (b) and (c)], most commonly seen as a sharp initial drop in the ACF. Furthermore, $g^{(1)}(\tau =0)=1$ by definition, but $g^{(2)}(\tau =0)$ value is dependent on the contrast of the OCT intensity signal time series. Statistical bias manifests as a different estimated ACF decay for the same process when using time series with different lengths.
Fig. 2.
Fig. 2. Illustration of the operation of averaging strategies to obtain a more robust estimation of the ACF of the times series in (a) where three temporal members are used for the temporal ensemble average. For the averaging strategies, we consider the goal of having six members of the ensemble in total: in (b) the number of time samples is higher ($t$-EA averaging), and in (c) and (d) there are time series for two spatial locations $x_1$ and $x_2$ to calculate the ACF using (c) $s$-EA and (d) ACC. As we are considering only the numerator of $g^{(2)}$ for illustration purposes, $\Pi (\tau )$ represents the product of the elements in the intersection of the time series with and without time-lag $\tau$: six elements for (b) and three elements for (a), (c) and (d).
Fig. 3.
Fig. 3. Experimentally determined RSD in an M-mode acquisition of a static sample and comparison with the model prediction. The good match validates our noise model on which the rest of our results rely on. The dashed line corresponds to the expected behavior for a system with depth-independent noise floor, which our benchtop system follows closely.
Fig. 4.
Fig. 4. Average ACFs using global and contrast normalization of $\tilde {g}^{(2)}_{\textrm {DLS}}$ and $\tilde {g}^{(2)}_{\textrm {HBT}}$ for a signal with 28 dB SNR using a correlation window size $n_{\omega }$ of (a) 64 and (b) 256 samples. Variability of $\tilde {g}^{(2)}_{\textrm {DLS}}$ and $\tilde {g}^{(2)}_{\textrm {HBT}}$ with (c) global and (d) contrast normalization for $n_{\omega }=64$. Shaded area encloses the average value $\pm$ standard deviation.
Fig. 5.
Fig. 5. Average ACFs for the three definitions of $g^{(2)}$ at two depths corresponding to two 28 dB and 38 dB SNR (see colorbar) using a correlation window size $n_{\omega }$ of (a) 64 and (b) 256 samples. Variability each definition for (c) 28 dB and (d) 38 dB using $n_{\omega }=64$. Shaded area encloses the mean value $\pm$ standard deviation. To allow a direct comparison, we plot $g^{(2)}_{\textrm {P}}+1$ to match the range of the other ACFs.
Fig. 6.
Fig. 6. Average ACFs before [$\tilde {g}^{(2)}_{\textrm {HBT,g}}$, left sides] and after noise correction [$g^{(2)}_{\textrm {HBT,g}}$, right sides] calculated using $s$-EA and ACC with $N_z = 8$ for five depths with corresponding SNR values coded in color (see colorbar) with a correlation window size $n_{\omega }$ of (a–b, top) 64 and (c–d, bottom) 256 samples. (e) Variability for $n_{\omega } = 64$. Shaded area encloses the mean value $\pm$ standard deviation.
Fig. 7.
Fig. 7. Exemplary autocorrelation analysis mimicking a flowmetry application; before [$\tilde {g}^{(2)}_{\textrm {HBT,g}}$, (a), left sides] and after [$\tilde {g}^{(2)}_{\textrm {HBT,g}}$, (b), right sides] noise correction for depths with different SNR values (see colorbar), using $s$-EA (top) and ACC (bottom) averaging with $N_z = 8$. We reproduced a typical flowmetry with a total of 128 time samples using $N_{\Omega } = 2$ and $n_{\omega } = 64$.
Fig. 8.
Fig. 8. Average ACFs $\tilde {g}^{(2)}_{\textrm {HBT,g}}$ calculated using combinations of $t$- and $s$-EA averaging for a single depth with 28 dB SNR with different total ensemble sizes $n_{ts}$ of 95 (a), 160 (b) and 472 (c) at $\tau =30$ A-lines. The corresponding variabilities are in (d), (e) and (f); therein the shaded area encloses the mean value $\pm$ standard deviation. Each specific average has a combinations of $n_{\omega }$ (for $t$-EA) and $N_z$ (for $s$-EA) indicated in the legend in the form $n_{\omega }$;$N_z$. The black curve used as a reference ACF with no bias was calculated using $n_{\omega }=512$ and $N_z=80$ and is the same for all cases.
Fig. 9.
Fig. 9. Lateral speckle size analysis in a rotational endoscopic probe using the decorrelation rate determined from the depth-resolved ACFs and $g^{(2)}_{\textrm {HBT,g}}$. Decorrelation rate as a function of depth for $\tilde {g}^{(2)}_{\textrm {HBT,g}}$ [before noise correction, (a)]; noise-corrected decorrelation rate for $g^{(2)}_{\textrm {HBT,g}}$ [(b)]; and depth-averaged noise-corrected decorrelation rate as a function of rotational speed for different correlation window sizes $n_{\omega }$ [see colorbar, (c)]. Errorbars in $c$ denote the standard deviation in depth, seen as the fluctuations in (b). The near-constant decorrelation rate (i.e. lateral speckle size in angular units) as a function of depth is only evident after noise correction.
Fig. 10.
Fig. 10. Improvement of NURD detection in a tomogram of chicken breast obtained with an endoscopic system. A tomogram is presented with overlays of the decorrelation rates calculated before (a) and after (b) noise correction (the luminance is given by the tomogram intensity and the hue of the nearly-perceptually-isoluminant colormap by the decorrelation rate). After noise correction, the decorrelation rate as a function of depth varies much less with depth, as shown here for (c) a region with high rotation speed (violet squares in tomograms). The standard deviation $(\sigma )$ over the mean $(\mu )$ of the decorrelation rate ($\tau ^{-1}$) decreases due to the noise correction, as visible in the histogram over all the correlation windows (d). Depth-averaging of rotation speed as a function of A-line (e). Without noise correction, rotation speed is positively skewed with respect to the nominal value; after noise correction rotation speed exhibits the expected fluctuation around the nominal 50 RPS value.
Fig. 11.
Fig. 11. Example of $g^{(2)}$-based angiography with and without SNR correction, using $s$-EA and ACC averaging strategies. All views correspond to overlays where the luminance is given by the tomogram intensity and the hue of the nearly-perceptually-isoluminant colormap by the decorrelation rate. B-scan views [(a)–(d)] and en-face views [(e)–(h)]. The dashed line in the B-scans (en-face views) show the depth (out-of-plane location) corresponding to the en-face views (B-scans). Visually, $s$-EA shows a higher contrast between static tissue and blood vessels, such as in the tissue surrounding the two large blood vessels near the center of the tissue. Low-SNR tissue regions exhibit some degree of decorrelation presumably due to noise, which became significantly lower after SNR correction; visually, the $s$-EA approach is more robust. Plots at bottom show $g^{(2)}$ at the four locations A–F marked in (a) with high (A–C) and low (D–F) SNR; full decorrelation (A, D), partial decorrelation (D, E), and static tissue (C, F).

Equations (82)

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

F = S + N .
S N = S N .
T = | S | 2 , Q = | N | 2 , I = | F | 2 ;
I = T + Q + 2 Re { S N } = T + Q + 2 | S | | N | cos φ ,
ρ = 2 | N | t | S | t .
g ( 1 ) ( τ ) = S 1 S 2 | S 1 | 2 | S 2 | 2 ; g ~ ( 1 ) ( τ ) = F 1 F 2 | F 1 | 2 | F 2 | 2 ,
g ~ ( 1 ) ( τ ) = g ( 1 ) ( τ ) 1 ( 1 + R 1 1 ) ( 1 + R 2 1 ) ,
g DLS ( 2 ) ( τ ) = T 1 T 2 T 1 2 ; g ~ DLS ( 2 ) ( τ ) = I 1 I 2 I 1 2 ,
g DLS,c ( 2 ) ( τ ) = g DLS ( 2 ) ( τ ) 1 g DLS ( 2 ) ( τ = 0 ) 1 C 2 + 1 ,
g DLS,g ( 2 ) ( τ ) = g DLS ( 2 ) ( τ ) g DLS ( 2 ) ( τ = 0 ) [ C 2 + 1 ] ,
g ~ DLS ( 2 ) ( τ > 0 ) = g DLS ( 2 ) ( τ > 0 ) + R 12 1 + L 21 R 1 1 + R 1 1 R 12 1 1 + R 1 2 + 2 R 1 1 .
g DLS ( 2 ) ( τ > 0 ) = g ~ DLS ( 2 ) ( τ > 0 ) [ 1 + G ] G ,
G = 1 + 2 R R 2 .
g DLS ( 2 ) ( τ = 0 ) = g ~ DLS ( 2 ) ( τ = 0 ) + 2 R 1 2 + 4 R 1 1 1 + R 1 2 + 2 R 1 1 ,
g DLS ( 2 ) ( τ = 0 ) = g ~ DLS ( 2 ) ( τ = 0 ) [ 1 + G ] 2 G .
g HBT ( 2 ) ( τ ) = T 1 T 2 T 1 T 2 ; g ~ HBT ( 2 ) ( τ ) = I 1 I 2 I 1 I 2 ,
g ~ HBT ( 2 ) ( τ > 0 ) = g HBT ( 2 ) ( τ > 0 ) + R 2 1 + R 1 1 + R 1 1 R 2 1 1 + R 2 1 + R 1 1 + R 1 1 R 2 1 .
g HBT ( 2 ) ( τ > 0 ) = g ~ HBT ( 2 ) ( τ > 0 ) [ 1 + G ] G .
g HBT ( 2 ) ( τ = 0 ) = g ~ HBT ( 2 ) ( τ = 0 ) 2 R 1 2 + 4 R 1 1 1 + R 1 2 + 2 R 1 1 ,
g HBT ( 2 ) ( τ = 0 ) = g ~ HBT ( 2 ) ( τ = 0 ) [ 1 + G ] 2 G .
g P ( 2 ) ( τ ) = ( T 1 T 1 ) ( T 2 T 2 ) σ T 1 σ T 2 ; g ~ P ( 2 ) ( τ ) = ( I 1 I 1 ) ( I 2 I 2 ) σ I 1 σ I 2 ,
g P ( 2 ) ( τ ) = g ~ P ( 2 ) ( τ ) 1 1 + 1 + 2 R 1 2 R ~ 1 2 R 1 2 1 1 + 1 + 2 R 2 2 R ~ 2 2 R 2 2 ,
g P ( 2 ) ( τ ) = g ~ P ( 2 ) ( τ ) 1 1 + 1 + 2 R 2 R ~ 2 R 2 .
g P ( 2 ) ( τ ) = T 1 T 2 T 1 T 2 C 1 C 2 T 1 T 2 = 1 C 1 C 2 [ g HBT ( 2 ) ( τ ) 1 ] ,
n t ( n ω + Δ n ω , τ ) n t ( n ω , τ ) = Δ n ω τ n t ( n ω , τ ) .
n t ( n ω , τ , w s ) n t ( n ω , τ ) = n s ,
K ( z , x , t , n f , p ) t K ( z , x , t , n f , p ) n ω t , t + τ = 1 n ω τ t = 1 n ω τ K ( z , x , t , n f , p ) K ( z , x , t + τ , n f , p ) .
K ( z , x , t , n f , p ) t K ( z , x , t , n f , p ) n ω , τ = 1 n ω τ t = 1 n ω τ K ( z , x , t , n f , p ) .
g ~ HBT ( 2 ) ( τ ) = I ( z , x , n Ω n ω + t , n f , p ) t I ( z , x , n Ω n ω + t , n f , p ) n ω t , t + τ I ( z , x , n Ω n ω + t , n f , p ) t I ( z , x , n Ω n ω + t , n f , p ) n ω , τ I ( z , x , n Ω n ω + t + τ , n f , p ) t I ( z , x , n Ω n ω + t + τ , n f , p ) n ω , τ .
R = t I ( z , x , n Ω n ω + t , n f , p ) n ω , 0 Q ( z , p ) 1.
K ( z , x , t , n f , p ) s K ( z , x , t , n f , p ) N w s w ^ s = 1 ( N x + 1 ) ( N z + 1 ) n x = N x 2 N x 2 n z = N z 2 N z 2 w ^ s ( n x , n z ) K ( z + n z , x + n x , t , n f , p ) ,
w ^ s ( x , y ) = w s ( x , y ) n z , n x w s ( n z , n x ) .
g ~ HBT ( 2 ) ( τ ) = t s I ( z , x , n Ω n ω + t , n f , p ) N w s w s ^ t I ( z , x , n Ω n ω + t , n f , p ) s I ( z , x , n Ω n ω + t , n f , p ) N w s w s ^ n ω t , t + τ s I ( z , x , n Ω n ω + t , n f , p ) N w s w ^ s t I ( z , x , n Ω n ω + t , n f , p ) s I ( z , x , n Ω n ω + t , n f , p ) N w s w ^ s n ω , τ s I ( z , x , n Ω n ω + t + τ , n f , p ) N w s w ^ s t I ( z , x , n Ω n ω + t + τ , n f , p ) s I ( z , x , n Ω n ω + t + τ , n f , p ) N w s w ^ s n ω , τ .
R = s I ( z , x , n Ω n ω + t , n f , p ) N w s w ^ s t I ( z , x , n Ω n ω + t , n f , p ) s I ( z , x , n Ω n ω + t , n f , p ) N w s w ^ s n ω , 0 Q ( z , p ) s Q ( z , p ) N w s w ^ s 1.
g ~ HBT ( 2 ) ( τ ) = t I ( z , x , n Ω n ω + t , n f , p ) n ω t , t + τ t I ( z , x , n Ω n ω + t , n f , p ) n ω , τ t I ( z , x , n Ω n ω + t + τ , n f , p ) n ω , τ s I ( z , x , n Ω n ω + t , n f , p ) t I ( z , x , n Ω n ω + t , n f , p ) n ω t , t + τ I ( z , x , n Ω n ω + t , n f , p ) t I ( z , x , n Ω n ω + t , n f , p ) n ω , τ I ( z , x , n Ω n ω + t + τ , n f , p ) t I ( z , x , n Ω n ω + t + τ , n f , p ) n ω , τ N w s w ^ s .
τ 1 = 1 τ c 1 / log ( g c ) .
τ ^ 1 ( z , x , y ) = max { 0 , Re { τ 2 ( z , x , y ) τ ¯ 2 ( y ) } } ,
g ( 1 ) ( τ ) = S 1 S 2 | S 1 | 2 | S 2 | 2 ; g ~ ( 1 ) ( τ ) = F 1 F 2 | F 1 | 2 | F 2 | 2 .
| F | 2 = | S | 2 + | N | 2 + S N + S N ,
| F 1 | 2 | F 2 | 2 = | S 1 | 2 + | N 1 | 2 ) ( | S 2 | 2 + | N 2 | 2 = | S 1 | 2 | S 2 | 2 ( 1 + | N 1 | 2 | S 1 | 2 ) ( 1 + | N 2 | 2 | S 2 | 2 ) .
F 1 F 2 = ( S 1 + N 1 ) ( S 2 + N 2 ) = S 1 S 2 + S 1 N 2 + N 1 S 2 + N 1 N 2 = S 1 S 2 .
g ~ ( 1 ) ( τ ) = S 1 S 2 | S 1 | 2 | S 2 | 2 1 ( 1 + | N 1 | 2 | S 1 | 2 ) ( 1 + | N 2 | 2 | S 2 | 2 ) , = g ( 1 ) ( τ ) 1 ( 1 + | N 1 | 2 | S 1 | 2 ) ( 1 + | N 2 | 2 | S 2 | 2 ) .
I = | S + N | 2 = | S | 2 + | N | 2 + 2 Re { S N } ,
I = | S | 2 + | N | 2 + 2 Re { S N } , = T + Q ,
2 Re { S N } = S N + S N = S N + S N = 0.
g DLS ( 2 ) ( τ ) = T 1 T 2 T 1 2 ; g ~ DLS ( 2 ) ( τ ) = I 1 I 2 I 1 2 .
I 1 2 = ( T 1 + Q 1 ) ( T 1 + Q 1 ) = T 1 2 + Q 1 2 + 2 T 1 Q 1 ,
I 1 I 2 = | S 1 | 2 | S 2 | 2 + | S 1 | 2 | N 2 | 2 + 2 | S 1 | 2 Re { S 2 N 2 } + | N 1 | 2 | S 2 | 2 + | N 1 | 2 | N 2 | 2 + 2 | N 1 | 2 Re { S 2 N 2 } + 2 Re { S 1 N 1 } | S 2 | 2 + 2 Re { S 1 N 1 } | N 2 | 2 + 4 Re { S 1 N 1 } Re { S 2 N 2 } . I 1 I 2 = | S 1 | 2 | S 2 | 2 + | S 1 | 2 | N 2 | 2 + 2 | S 1 | 2 Re { S 2 N 2 } + | N 1 | 2 | S 2 | 2 + | N 1 | 2 | N 2 | 2 + 2 | N 1 | 2 Re { S 2 N 2 } + 2 Re { S 1 N 1 } | S 2 | 2 + 2 Re { S 1 N 1 } | N 2 | 2 + 4 Re { S 1 N 1 } Re { S 2 N 2 } .
2 | S 1 | 2 Re { S 2 N 2 } = | S 1 | 2 ( S 2 N 2 + S 2 N 2 ) = | S 1 | 2 S 2 N 2 + | S 1 | 2 S 2 N 2 = 0 ,
I 1 I 2 = | S 1 | 2 | S 2 | 2 + | S 1 | 2 | N 2 | 2 + | N 1 | 2 | S 2 | 2 + | N 1 | 2 | N 2 | 2 = T 1 T 2 + T 1 Q 2 + Q 1 T 2 + Q 1 Q 2 .
g ~ DLS ( 2 ) ( τ > 0 ) = T 1 T 2 + T 1 Q 2 + Q 1 T 2 + Q 1 Q 2 T 1 2 + Q 1 2 + 2 T 1 Q 1 = g DLS ( 2 ) ( τ > 0 ) + Q 2 T 1 + T 2 Q 1 T 1 2 + Q 1 Q 2 T 1 2 1 + Q 1 2 T 1 2 + 2 Q 1 T 1 .
I 1 2 = | S 1 | 2 | S 1 | 2 + 2 | S 1 | 2 | N 1 | 2 + | N 1 | 2 | N 1 | 2 + 4 Re 2 { S 1 N 1 } ,
4 Re 2 { S 1 N 1 } = S 1 2 N 1 2 + S 1 2 N 1 2 + 2 | S 1 | 2 | N 1 | 2 = 2 Re { S 1 2 N 1 2 } + 2 | S 1 | 2 | N 1 | 2 = 2 | S 1 | 2 | N 1 | 2 ,
I 1 2 = | S 1 | 2 | S 1 | 2 + 4 | S 1 | 2 | N 1 | 2 + | N 1 | 2 | N 1 | 2 , = T 1 2 + 2 Q 1 2 + 4 T 1 Q 1
g DLS ( 2 ) ( τ = 0 ) = T 1 2 + 2 Q 1 2 + 4 T 1 Q 1 T 1 2 + Q 1 2 + 2 T 1 Q 1 = g DLS ( 2 ) ( τ = 0 ) + 2 Q 1 2 T 1 2 + 4 Q 1 T 1 1 + Q 1 2 T 1 2 + 2 Q 1 T 1 .
g HBT ( 2 ) ( τ ) = T 1 T 2 T 1 T 2 ; g ~ HBT ( 2 ) ( τ ) = I 1 I 2 I 1 I 2 .
I 1 I 2 = ( T 1 + Q 1 ) ( T 2 + Q 2 ) = T 1 T 2 + T 1 Q 2 + Q 1 T 2 + Q 1 Q 2 ,
g ~ HBT ( 2 ) ( τ > 0 ) = T 1 T 2 + T 1 Q 2 + Q 1 T 2 + Q 1 Q 2 T 1 T 2 + T 1 Q 2 + Q 1 T 2 + Q 1 Q 2 g ~ HBT ( 2 ) ( τ > 0 ) = g HBT ( 2 ) ( τ > 0 ) + Q 2 T 2 + Q 1 T 1 + Q 1 Q 2 T 1 T 2 1 + Q 2 T 2 + Q 1 T 1 + Q 1 Q 2 T 1 T 2 .
g HBT ( 2 ) ( τ = 0 ) = T 1 2 + 2 Q 1 2 + 4 T 1 Q 1 T 1 2 + Q 1 2 + 2 T 1 Q 1 g ~ HBT ( 2 ) ( τ = 0 ) = g HBT ( 2 ) ( τ = 0 ) + 2 Q 1 2 T 1 2 + 4 Q 1 T 1 1 + Q 1 2 T 1 2 + 2 Q 1 T 1 .
g P ( 2 ) ( τ ) = ( T 1 T 1 ) ( T 2 T 2 ) σ T 1 σ T 2 ; g ~ P ( 2 ) ( τ ) = ( I 1 I 1 ) ( I 2 I 2 ) σ I 1 σ I 2 .
σ I 1 =   I 1 2 I 1 2   T 1 2 + 2 Q 1 2 + 4 T 1 Q 1 ( T 1 2 + Q 1 2 + 2 T 1 Q 1 ) =   T 1 2 T 1 2 + Q 1 2 + 2 T 1 Q 1 .
( I 1 I 1 ) ( I 2 I 2 ) =   I 1 I 2 I 1 I 2 =   T 1 T 2 + T 1 Q 2 + Q 1 T 2 + Q 1 Q 2   ( T 1 T 2 + T 1 Q 2 + Q 1 T 2 + Q 1 Q 2 ) =   T 1 T 2 T 1 T 2 .
g ~ P ( 2 ) ( τ ) =   T 1 T 2 T 1 T 2 T 1 2 T 1 2 Q 1 2 + 2 T 1 Q 1 T 2 2 T 2 2 Q 2 2 + 2 T 2 Q 2 =   T 1 T 2 T 1 T 2 T 1 2 T 1 2 T 2 2 T 2 2 1 1 + Q 1 2 + 2 T 1 Q 1 T 1 2 T 1 2 1 1 + Q 2 2 + 2 T 2 Q 2 T 2 2 T 2 2 =   g P ( 2 ) ( τ ) 1 1 + 1 + 2 T 1 Q 1 2 T 1 2 Q 1 T 1 2 Q 1 2 1 1 + 1 + 2 T 2 Q 2 2 T 2 2 Q 2 T 2 2 Q 2 2 .
I T = | S p 1 + N p 1 | 2 + | S p 2 + N p 2 | 2 = | α | 2 | S | 2 + | N p 1 | 2 + 2 Re { α S N p 1 } + | β | 2 | S | 2 + | N p 2 | 2 + 2 Re { β S N p 2 } = | S | 2 + | N p 1 | 2 + | N p 2 | 2 + 2 Re { α S N p 1 } + 2 Re { β S N p 2 } ,
g HBT ( 2 ) ( τ ) = T 1 T 2 T 1 T 2 ; g ~ HBT ( 2 ) ( τ ) = I T 1 I T 2 I T 1 I T 2 ,
I T 1 I T 2 = | S 1 | 2 | S 2 | 2 + | S 1 | 2 | N p 1 , 2 | 2 + | S 1 | 2 | N p 2 , 2 | 2 + 2 | S 1 | 2 Re { α S 2 N p 1 , 2 } + 2 | S 1 | 2 Re { β S 2 N p 2 , 2 } + | N p 1 , 1 | 2 | S 2 | 2 + | N p 1 , 1 | 2 | N p 1 , 2 | 2 + | N p 1 , 1 | 2 | N p 2 , 2 | 2 + 2 | N p 1 , 1 | 2 Re { α S 2 N p 1 , 2 } + 2 | N p 1 , 1 | 2 Re { β S 2 N p 2 , 2 } + | N p 2 , 1 | 2 | S 2 | 2 + | N p 2 , 1 | 2 | N p 1 , 2 | 2 + | N p 2 , 1 | 2 | N p 2 , 2 | 2 + 2 | N p 2 , 1 | 2 Re { α S 2 N p 1 , 2 } + 2 | N p 2 , 1 | 2 Re { β S 2 N p 2 , 2 } + 2 | S 2 | 2 Re { α S 1 N p 1 , 1 } + | N p 1 , 2 | 2 Re { α S 1 N p 1 , 1 } + | N p 2 , 2 | 2 Re { α S 1 N p 1 , 1 } + 4 Re { α S 1 N p 1 , 1 } Re { α S 2 N p 1 , 2 } + 4 Re { α S 1 N p 1 , 1 } Re { β S 2 N p 2 , 2 } + 2 | S 2 | 2 Re { β S 1 N p 2 , 1 } + | N p 1 , 2 | 2 Re { β S 1 N p 2 , 1 } + | N p 2 , 2 | 2 Re { β S 1 N p 2 , 1 } + 4 Re { β S 1 N p 2 , 1 } Re { α S 2 N p 1 , 2 } + 4 Re { β S 1 N p 2 , 1 } Re { β S 2 N p 2 , 2 } ,
I T 1 I T 2 =   | S 1 | 2 | S 2 | 2 + | S 1 | 2 | N p 1 , 2 | 2 + | S 1 | 2 | N p 2 , 2 | 2 + | N p 1 , 1 | 2 | S 2 | 2 + | N p 1 , 1 | 2 | N p 1 , 2 | 2 + | N p 1 , 1 | 2 | N p 2 , 2 | 2 + | N p 2 , 1 | 2 | S 2 | 2 + | N p 2 , 1 | 2 | N p 1 , 2 | 2 + | N p 2 , 1 | 2 | N p 2 , 2 | 2 =   T 1 T 2 + T 1 Q p 1 , 2 + T 1 Q p 2 , 2 + Q p 1 , 1 T 2 + Q p 1 , 1 Q p 1 , 2 + Q p 1 , 1 Q p 2 , 2 + Q p 2 , 1 T 2 + Q p 2 , 1 Q p 1 , 2 2 + Q p 2 , 1 Q p 2 , 2 .
I T =   | S | 2 + | N p 1 | 2 + | N p 2 | 2 + 2 Re { α S N p 1 } + 2 Re { β S N p 2 } =   T + Q p 1 + Q p 2
I T 1 I T 2 =   T 1 T 2 + T 1 Q p 1 , 2 + T 1 Q p 2 , 2 + Q p 1 , 1 T 2 + Q p 1 , 1 Q p 1 , 2 + Q p 1 , 1 Q p 2 , 2 + Q p 2 , 1 T 2 + Q p 2 , 1 Q p 1 , 2 + Q p 2 , 1 Q p 2 , 2 .
g ~ HBT ( 2 ) ( τ > 0 ) = g HBT ( 2 ) ( τ > 0 ) + R T 1 1 + R T 2 1 + R T 12 1 + R T 1 1 + R T 2 1 + R T 12 ,
R T j = T j Q p 1 , j + Q p 2 , j = T j Q j
R T 12 = ( R p 11 , 1 + R p 12 , 1 + R p 21 , 1 + R p 22 , 1 ) ( R p 11 , 2 + R p 12 , 2 + R p 21 , 2 + R p 22 , 2 ) ( R p 11 , 1 + R p 21 , 2 ) ( R p 11 , 2 + R p 21 , 2 ) + ( R p 12 , 2 + R p 22 , 2 ) + ( R p 12 , 1 + R p 21 , 1 ) ,
R p j k , i = T p j , i Q p k , i
g ~ HBT ( 2 ) ( τ > 0 ) = g HBT ( 2 ) ( τ > 0 ) + 2 R T 1 + ( R p 11 + R p 12 + R p 21 + R p 22 ) 2 ( R p 11 + R p 21 ) 2 ( R p 12 + R p 22 ) 2 1 + 2 R T 1 + ( R p 11 + R p 12 + R p 21 + R p 22 ) 2 ( R p 11 + R p 21 ) 2 ( R p 12 + R p 22 ) 2
g HBT ( 2 ) ( τ > 0 ) = g ~ HBT ( 2 ) ( τ > 0 ) [ 1 + G T 12 ] G T 12 .
G T 12 = 2 R T 1 + ( R p 11 + R p 12 + R p 21 + R p 22 ) 2 ( R p 11 + R p 21 ) 2 ( R p 12 + R p 22 ) 2 .
I T 1 2 =   | S 1 | 4 + 2 | S 1 | 2 | N p 1 , 1 | 2 + 2 | S 1 | 2 | N p 2 , 1 | 2 + | N p 1 , 1 | 4 + | N p 2 , 1 | 4 + 2 | N p 1 , 1 | 2 | N p 2 , 1 | 2 + 4 Re 2 { α S 1 N p 1 , 1 } + 4 Re 2 { β S 1 N p 2 , 1 } =   T 1 2 + Q p 1 , 1 2 + Q p 2 , 1 2 + 2 Q p 1 , 1 Q p 2 , 1 + 4 T 1 Q p 1 , 1 + 4 T 1 Q p 2 , 1 =   T 1 2 + 2 Q p 1 , 1 2 + 2 Q p 2 , 1 2 + 2 Q p 1 , 1 Q p 2 , 1 + 4 T 1 Q 1 =   T 1 2 + 2 ( Q p 1 , 1 + Q p 2 , 1 ) 2 2 Q p 1 , 1 Q p 2 , 1 + 4 T 1 Q 1 ,
I T 1 2 =   T 1 2 + 2 T 1 Q p 2 , 1 + Q p 1 , 1 2 + Q p 2 , 1 2 + 2 Q p 1 , 1 Q p 2 , 1 + 2 T 1 Q p 1 , 1 =   T 1 2 + Q p 1 , 1 2 + Q p 2 , 1 2 + 2 Q p 1 , 1 Q p 2 , 1 + 2 T 1 Q 1 =   T 1 2 + ( Q p 1 , 1 Q p 2 , 1 ) 2 + 2 T 1 Q 1 .
g ~ HBT ( 2 ) ( τ = 0 ) = g HBT ( 2 ) ( τ = 0 ) + 4 R T 1 + 2 R T 2 2 R p 1 1 + 2 R T 1 + R T 2 ,
R p =   T 1 Q p 1 , 1 T 1 Q p 2 , 1 = T p 1 + T p 2 Q p 1 , 1 T p 1 + T p 2 Q p 2 , 1 = ( R p 11 + R p 11 ) ( R p 12 + R p 22 ) ,
g HBT ( 2 ) ( τ = 0 ) = g ~ HBT ( 2 ) ( τ = 0 ) [ 1 + G T 11 ] 2 G T 11 + 2 R p 1 ,
G T 11 = 1 + 2 R T R T 2 .