Abstract

A variety of promising approaches for quantitative flow velocity measurement in OCT have been proposed in recent years. The question is: Which method gets the most precise flow velocity out of the interference signals detected. We have compared the promising joint spectral and time domain optical coherence tomography (jSTdOCT) and the commonly used phase-resolved Doppler OCT (DOCT) and describe the link between these two proven methods for OCT in the Fourier domain (FD OCT). First, we show that jSTdOCT can be significantly improved by calculating the center of gravity via an unbiased complex algorithm instead of detecting the maximum intensity signal of the broadened Doppler frequency spectrum. Secondly, we introduce a unified mathematical description for DOCT and jSTdOCT that differs only in one exponent and call it enhjSTdOCT. Third, we present that enhjSTdOCT has the potential to significantly reduce the noise of the velocity measurement by choosing an exponent depending on the transverse sample velocity component and the signal-to-noise ratio. EnhjSTdOCT is verified numerically and experimentally to find the optimal parameters for maximal velocity noise reduction.

© 2014 Optical Society of America

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  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 J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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  18. S. H. Yun, G. Tearney, J. De Boer, and B. Bouma, “Motion artifacts in optical coherence tomography with frequency-domain ranging,” Opt. Express 12(13), 2977–2998 (2004).
    [Crossref] [PubMed]
  19. J. Walther, A. Krüger, M. Cuevas, and E. Koch, “Effects of axial, transverse, and oblique sample motion in FD OCT in systems with global or rolling shutter line detector,” J. Opt. Soc. Am. A 25(11), 2791–2802 (2008).
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  23. M. Szkulmowski, I. Grulkowski, D. Szlag, A. Szkulmowska, A. Kowalczyk, and M. Wojtkowski, “Flow velocity estimation by complex ambiguity free joint Spectral and Time domain Optical Coherence Tomography,” Opt. Express 17(16), 14281–14297 (2009).
    [Crossref] [PubMed]
  24. A. Bouwens, D. Szlag, M. Szkulmowski, T. Bolmont, M. Wojtkowski, and T. Lasser, “Quantitative lateral and axial flow imaging with optical coherence microscopy and tomography,” Opt. Express 21(15), 17711–17729 (2013).
    [Crossref] [PubMed]
  25. H. Wehbe, M. Ruggeri, S. Jiao, G. Gregori, C. A. Puliafito, and W. Zhao, “Automatic retinal blood flow calculation using spectral domain optical coherence tomography,” Opt. Express 15(23), 15193–15206 (2007).
    [Crossref] [PubMed]
  26. E. Koch, J. Walther, and M. Cuevas, “Limits of Fourier domain Doppler-OCT at high velocities,” Sens. Actuators A Phys. 156(1), 8–13 (2009).
    [Crossref]
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  30. P. Cimalla, J. Walther, M. Mehner, M. Cuevas, and E. Koch, “Simultaneous dual-band optical coherence tomography in the spectral domain for high resolution in vivo imaging,” Opt. Express 17(22), 19486–19500 (2009).
    [Crossref] [PubMed]

2014 (1)

2013 (3)

A. C. Chan, E. Y. Lam, and V. J. Srinivasan, “Comparison of Kasai autocorrelation and maximum likelihood estimators for Doppler optical coherence tomography,” IEEE Trans. Med. Imaging 32(6), 1033–1042 (2013).
[Crossref] [PubMed]

A. Bouwens, D. Szlag, M. Szkulmowski, T. Bolmont, M. Wojtkowski, and T. Lasser, “Quantitative lateral and axial flow imaging with optical coherence microscopy and tomography,” Opt. Express 21(15), 17711–17729 (2013).
[Crossref] [PubMed]

J. Walther and E. Koch, “Velocity noise reduction by using enhanced joint spectral and time domain optical coherence tomography,” Proc. SPIE 8802,” Optical Coherence Tomography and Coherence Techniques VI, 88020K (2013).

2011 (1)

J. Walther and E. Koch, “Enhanced joint spectral and time domain optical coherence tomography for quantitative flow velocity measurement,” Proc. SPIE 8091,” Optical Coherence Tomography and Coherence Techniques V, 80910L (2011).

2010 (1)

J. Walther, G. Mueller, H. Morawietz, and E. Koch, “Signal power decrease due to fringe washout as an extension of the limited Doppler flow measurement range in spectral domain optical coherence tomography,” J. Biomed. Opt. 15(4), 041511 (2010).
[Crossref] [PubMed]

2009 (6)

2008 (4)

2007 (2)

2005 (3)

2004 (1)

2003 (5)

2000 (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 J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

1985 (1)

C. Kasai, K. Namekawa, A. Koyano, and R. Omoto, “Real-Time Two-Dimensional Blood-Flow Imaging Using An Auto-Correlation Technique,” IEEE Trans. Sonics Ultrason. 32, 458–464 (1985).

Bachmann, A. H.

Bajraszewski, T.

Blatter, C.

Bolmont, T.

Bouma, B.

Bouma, B. E.

Bouwens, A.

Büttner, L.

Cense, B.

Chan, A. C.

A. C. Chan, E. Y. Lam, and V. J. Srinivasan, “Comparison of Kasai autocorrelation and maximum likelihood estimators for Doppler optical coherence tomography,” IEEE Trans. Med. Imaging 32(6), 1033–1042 (2013).
[Crossref] [PubMed]

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 J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Chen, T. C.

Chen, Z.

Cimalla, P.

Cuevas, M.

Czarske, J.

De Boer, J.

De Boer, J. F.

Drexler, W.

Fercher, A.

Fercher, A. F.

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 J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Fujimoto, J. 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 J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Gordon, M.

Gregori, G.

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 J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Grulkowski, I.

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 J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Hitzenberger, C.

Huang, D.

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 J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Izatt, J.

Jahan, I.

Jaillon, F.

Jiao, S.

Kasai, C.

C. Kasai, K. Namekawa, A. Koyano, and R. Omoto, “Real-Time Two-Dimensional Blood-Flow Imaging Using An Auto-Correlation Technique,” IEEE Trans. Sonics Ultrason. 32, 458–464 (1985).

Koch, E.

J. Walther and E. Koch, “Velocity noise reduction by using enhanced joint spectral and time domain optical coherence tomography,” Proc. SPIE 8802,” Optical Coherence Tomography and Coherence Techniques VI, 88020K (2013).

J. Walther and E. Koch, “Enhanced joint spectral and time domain optical coherence tomography for quantitative flow velocity measurement,” Proc. SPIE 8091,” Optical Coherence Tomography and Coherence Techniques V, 80910L (2011).

J. Walther, G. Mueller, H. Morawietz, and E. Koch, “Signal power decrease due to fringe washout as an extension of the limited Doppler flow measurement range in spectral domain optical coherence tomography,” J. Biomed. Opt. 15(4), 041511 (2010).
[Crossref] [PubMed]

E. Koch, J. Walther, and M. Cuevas, “Limits of Fourier domain Doppler-OCT at high velocities,” Sens. Actuators A Phys. 156(1), 8–13 (2009).
[Crossref]

P. Cimalla, J. Walther, M. Mehner, M. Cuevas, and E. Koch, “Simultaneous dual-band optical coherence tomography in the spectral domain for high resolution in vivo imaging,” Opt. Express 17(22), 19486–19500 (2009).
[Crossref] [PubMed]

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(22), 19698–19713 (2009).
[Crossref] [PubMed]

J. Walther, A. Krüger, M. Cuevas, and E. Koch, “Effects of axial, transverse, and oblique sample motion in FD OCT in systems with global or rolling shutter line detector,” J. Opt. Soc. Am. A 25(11), 2791–2802 (2008).
[Crossref] [PubMed]

Kowalczyk, A.

Koyano, A.

C. Kasai, K. Namekawa, A. Koyano, and R. Omoto, “Real-Time Two-Dimensional Blood-Flow Imaging Using An Auto-Correlation Technique,” IEEE Trans. Sonics Ultrason. 32, 458–464 (1985).

Krüger, A.

Lam, E. Y.

A. C. Chan, E. Y. Lam, and V. J. Srinivasan, “Comparison of Kasai autocorrelation and maximum likelihood estimators for Doppler optical coherence tomography,” IEEE Trans. Med. Imaging 32(6), 1033–1042 (2013).
[Crossref] [PubMed]

Lasser, T.

Leitgeb, R.

Leitgeb, R. A.

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 J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Lo, S.

Makita, S.

Mehner, M.

Mok, A.

Morawietz, H.

J. Walther, G. Mueller, H. Morawietz, and E. Koch, “Signal power decrease due to fringe washout as an extension of the limited Doppler flow measurement range in spectral domain optical coherence tomography,” J. Biomed. Opt. 15(4), 041511 (2010).
[Crossref] [PubMed]

Mueller, G.

J. Walther, G. Mueller, H. Morawietz, and E. Koch, “Signal power decrease due to fringe washout as an extension of the limited Doppler flow measurement range in spectral domain optical coherence tomography,” J. Biomed. Opt. 15(4), 041511 (2010).
[Crossref] [PubMed]

Mujat, M.

Namekawa, K.

C. Kasai, K. Namekawa, A. Koyano, and R. Omoto, “Real-Time Two-Dimensional Blood-Flow Imaging Using An Auto-Correlation Technique,” IEEE Trans. Sonics Ultrason. 32, 458–464 (1985).

Nassif, N.

Nelson, J. S.

Omoto, R.

C. Kasai, K. Namekawa, A. Koyano, and R. Omoto, “Real-Time Two-Dimensional Blood-Flow Imaging Using An Auto-Correlation Technique,” IEEE Trans. Sonics Ultrason. 32, 458–464 (1985).

Park, B. H.

Pekar, J.

Pfister, T.

Pierce, M. C.

Puliafito, C. A.

H. Wehbe, M. Ruggeri, S. Jiao, G. Gregori, C. A. Puliafito, and W. Zhao, “Automatic retinal blood flow calculation using spectral domain optical coherence tomography,” Opt. Express 15(23), 15193–15206 (2007).
[Crossref] [PubMed]

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 J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Qi, B.

Ruggeri, M.

Sarunic, M.

Saxer, C.

Schmetterer, L.

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 J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Seng-Yue, E.

Srinivasan, V. J.

A. C. Chan, E. Y. Lam, and V. J. Srinivasan, “Comparison of Kasai autocorrelation and maximum likelihood estimators for Doppler optical coherence tomography,” IEEE Trans. Med. Imaging 32(6), 1033–1042 (2013).
[Crossref] [PubMed]

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 J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

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 J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Szkulmowska, A.

Szkulmowski, M.

Szlag, D.

Tearney, G.

Tearney, G. J.

Vakoc, B.

Vakoc, B. J.

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(6), 814–821 (2009).
[Crossref] [PubMed]

Villiger, M. L.

Vitkin, I.

Walther, J.

J. Walther and E. Koch, “Velocity noise reduction by using enhanced joint spectral and time domain optical coherence tomography,” Proc. SPIE 8802,” Optical Coherence Tomography and Coherence Techniques VI, 88020K (2013).

J. Walther and E. Koch, “Enhanced joint spectral and time domain optical coherence tomography for quantitative flow velocity measurement,” Proc. SPIE 8091,” Optical Coherence Tomography and Coherence Techniques V, 80910L (2011).

J. Walther, G. Mueller, H. Morawietz, and E. Koch, “Signal power decrease due to fringe washout as an extension of the limited Doppler flow measurement range in spectral domain optical coherence tomography,” J. Biomed. Opt. 15(4), 041511 (2010).
[Crossref] [PubMed]

E. Koch, J. Walther, and M. Cuevas, “Limits of Fourier domain Doppler-OCT at high velocities,” Sens. Actuators A Phys. 156(1), 8–13 (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(22), 19698–19713 (2009).
[Crossref] [PubMed]

P. Cimalla, J. Walther, M. Mehner, M. Cuevas, and E. Koch, “Simultaneous dual-band optical coherence tomography in the spectral domain for high resolution in vivo imaging,” Opt. Express 17(22), 19486–19500 (2009).
[Crossref] [PubMed]

J. Walther, A. Krüger, M. Cuevas, and E. Koch, “Effects of axial, transverse, and oblique sample motion in FD OCT in systems with global or rolling shutter line detector,” J. Opt. Soc. Am. A 25(11), 2791–2802 (2008).
[Crossref] [PubMed]

Wehbe, H.

White, B. R.

Wilson, B.

Wojtkowski, M.

Xiang, S.

Yang, C.

Yang, V.

Yasuno, Y.

Yazdanfar, S.

Yun, S.

Yun, S. H.

Zawadzki, R. J.

Zhao, W.

Zhao, Y.

Appl. Opt. (1)

IEEE Trans. Med. Imaging (2)

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(6), 814–821 (2009).
[Crossref] [PubMed]

A. C. Chan, E. Y. Lam, and V. J. Srinivasan, “Comparison of Kasai autocorrelation and maximum likelihood estimators for Doppler optical coherence tomography,” IEEE Trans. Med. Imaging 32(6), 1033–1042 (2013).
[Crossref] [PubMed]

IEEE Trans. Sonics Ultrason. (1)

C. Kasai, K. Namekawa, A. Koyano, and R. Omoto, “Real-Time Two-Dimensional Blood-Flow Imaging Using An Auto-Correlation Technique,” IEEE Trans. Sonics Ultrason. 32, 458–464 (1985).

J. Biomed. Opt. (1)

J. Walther, G. Mueller, H. Morawietz, and E. Koch, “Signal power decrease due to fringe washout as an extension of the limited Doppler flow measurement range in spectral domain optical coherence tomography,” J. Biomed. Opt. 15(4), 041511 (2010).
[Crossref] [PubMed]

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

Opt. Express (17)

V. Yang, M. Gordon, B. Qi, J. Pekar, S. Lo, E. Seng-Yue, A. Mok, B. Wilson, and I. Vitkin, “High speed, wide velocity dynamic range Doppler optical coherence tomography (Part I): System design, signal processing, and performance,” Opt. Express 11(7), 794–809 (2003).
[Crossref] [PubMed]

R. Leitgeb, C. Hitzenberger, and A. Fercher, “Performance of Fourier domain vs. time domain optical coherence tomography,” Opt. Express 11(8), 889–894 (2003).
[Crossref] [PubMed]

M. Szkulmowski, I. Grulkowski, D. Szlag, A. Szkulmowska, A. Kowalczyk, and M. Wojtkowski, “Flow velocity estimation by complex ambiguity free joint Spectral and Time domain Optical Coherence Tomography,” Opt. Express 17(16), 14281–14297 (2009).
[Crossref] [PubMed]

P. Cimalla, J. Walther, M. Mehner, M. Cuevas, and E. Koch, “Simultaneous dual-band optical coherence tomography in the spectral domain for high resolution in vivo imaging,” Opt. Express 17(22), 19486–19500 (2009).
[Crossref] [PubMed]

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(22), 19698–19713 (2009).
[Crossref] [PubMed]

A. Bouwens, D. Szlag, M. Szkulmowski, T. Bolmont, M. Wojtkowski, and T. Lasser, “Quantitative lateral and axial flow imaging with optical coherence microscopy and tomography,” Opt. Express 21(15), 17711–17729 (2013).
[Crossref] [PubMed]

S. Makita, F. Jaillon, I. Jahan, and Y. Yasuno, “Noise statistics of phase-resolved optical coherence tomography imaging: single-and dual-beam-scan Doppler optical coherence tomography,” Opt. Express 22(4), 4830–4848 (2014).
[Crossref] [PubMed]

R. A. Leitgeb, L. Schmetterer, W. Drexler, A. F. Fercher, R. J. 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(23), 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. de Boer, “In vivo dynamic human retinal blood flow imaging using ultra-high-speed spectral domain optical Doppler tomography,” Opt. Express 11(25), 3490–3497 (2003).
[Crossref] [PubMed]

S. H. Yun, G. Tearney, J. De Boer, and B. Bouma, “Motion artifacts in optical coherence tomography with frequency-domain ranging,” Opt. Express 12(13), 2977–2998 (2004).
[Crossref] [PubMed]

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Proc. SPIE 8091,” Optical Coherence Tomography and Coherence Techniques (1)

J. Walther and E. Koch, “Enhanced joint spectral and time domain optical coherence tomography for quantitative flow velocity measurement,” Proc. SPIE 8091,” Optical Coherence Tomography and Coherence Techniques V, 80910L (2011).

Proc. SPIE 8802,” Optical Coherence Tomography and Coherence Techniques (1)

J. Walther and E. Koch, “Velocity noise reduction by using enhanced joint spectral and time domain optical coherence tomography,” Proc. SPIE 8802,” Optical Coherence Tomography and Coherence Techniques VI, 88020K (2013).

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[Crossref]

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Figures (8)

Fig. 1
Fig. 1 One way to apply the 2D FT in joint spectral and time domain OCT (jSTdOCT) as used in this manuscript; (a) 2D interferogram consisting of 704 interference spectra of the detected 1% Intralipid flow within a 315 µm glass capillary imaged at a Doppler angle β of 81.7°. (b) Grayscale structural image (M-scan with 704 A-scans) containing the amplitude information of the capillary center with an Intralipid flow rate of 1.3 ml/h. (c) Grayscale image of the Doppler frequency shift fD distribution in depth z extracted from 704 complex-valued A-scans.
Fig. 2
Fig. 2 (a) Normalized amplitude of the Doppler frequency spectrum caused by a single obliquely moving particle passing the sample beam center in the middle of the detection period with K measurements. Simulated data (K = 32) in blue without and in red with zero padding by using 480 elements. (b) Same diagram for two scatterers of equal amplitude separated by 10 time steps with a phase difference of 2/3π. The signal of both scatterers declines at the border of the interval. (c) Doppler frequency spectrum of a single scatterer in the middle of the sample beam at the beginning of the time window with K samples. (d) Linear frequency spectrum of a large number (100) of point scatterers with random phases and amplitudes passing the beam at different points of time. The sample velocities are in all simulations set to vx’ = 0.1 and vz’ = 0.1. In all cases the maximum amplitude after zero padding was normalized to 1. The green vertical line shows the set axial velocity, which coincides with the result of DOCT in (a) and (c) and exhibits small deviations to the result of DOCT (b) vx’ = 0.089 and in (d) vx’ = 0.088. Only for (a) and (c), the maximum of fD’ after zero padding is identical to the set axial velocity component.
Fig. 3
Fig. 3 Graphic of the Doppler frequency spectrum transferred on the circle (a) and onto the complex plane (b) by drawing each component of the power spectrum as a function of exp[i2πm/K]. The components from the power spectrum (vertical) are the weights for the complex values on the circle. The numerically simulated Doppler frequency spectrum of multiple reflecting particles passing the sample beam, shown in Fig. 2(d), is used. The blue line indicates the angle Φ calculated by Eq. (8).
Fig. 4
Fig. 4 (a) Standard deviation of the normalized axial velocity σ(vz’) calculated as a function of the exponent Ex in Eq. (15) for a transverse velocity component of vx’ = 0.1. The diagram shows an optimal value for Ex = 1. σ(vz’) for Ex = 2 is only larger by 17%, while the limiting value for large exponents is higher by a factor of more than 2.5. The position of Ex = 1 and Ex = 2 are marked by vertical black lines. (b) σ(vz’) as a function of the transverse velocity vx’ for Ex = 1 (green graph), Ex = 2 (corresp. to DOCT, red graph) and Ex = 107 (corresp. to jSTdOCT, blue graph). Ex = 1 is optimal over the total range of transverse velocities. At high transverse velocities, the results with different exponents converge because all values approach the standard deviation for an isotropic distribution of velocities in the range of ± 0.5 which is about 0.289.
Fig. 5
Fig. 5 Standard deviation of the axial velocity σ(vz’) calculated as a function of the transverse velocity vx’ and the reciprocal of the signal-to-noise ratio of the measured OCT signal 1/SNR. For a better visualization, σ(vz’) is negated. Seven specific exponents were used for the calculation of the respective σ(vz’), each presented by a single colored 3D plane. The result of jSTdOCT with Ex = ∞ corresponds to the red data and the one of DOCT with Ex = 2 to the cyan. Ex = 0.5 (yellow), Ex = 1 (green), Ex = 2 (cyan), Ex = 4 (blue), Ex = 8 (violet), Ex = 16 (magenta) and Ex = ∞ (red). For better comprehensibility, the parts (a) to (d) of Fig. 5 show different perspectives of the same 3D plot with identical mathematical content. Furthermore, the relative noise of DOCT and jSTdOCT compared to the enhjSTdOCT with the optimized exponent is shown in (e) and (f), respectively. The red color signals optimal or nearly optimal performance while other colors show higher phase noise. Note the different scale and color scale in (e) and (f).
Fig. 6
Fig. 6 Stepwise processing of the 1% Intralipid flow measurement with an Doppler angle β of 81.7° and an absolute flow velocity of 9.2 mm/s corresponding to vx’ = 0.11 at the capillary center. There, M-scans consisting of 704 A-scans were acquired with fA-scan = 12 kHz at the glass capillary cross-section achieving a measured SNR of 14.9 dB at the capillary center. Backscattering information (a) and overlayed Doppler flow information (b) by usual FD OCT and DOCT processing, respectively, is presented. The depth-resolved Doppler frequency spectra (c) are determined by K = 704 complex A-scans applied to the 2. FT of the enhjSTdOCT algorithm. Doppler flow profiles are calculated as a function of the depth position for four different exponents (d). The direction of time (a and b), frequency (c and e) and depth (a, b, c and e) are indicated. To calculate the standard deviation σ(fD’), 22 input sequences of K = 32 were defined to be Fourier transformed in time (e,f). The exponent resulting in the lowest velocity noise can be read from the black dot in the 3D surface plot (g) being Ex = 1 and Ex = 2, respectively, for vx’ = 0.11 and 1/SNR = 0.18. The standard deviation σ(fD’) as a function of the exponent Ex for vx’ = 0.11 confirm the numerically simulated optimal exponent for the capillary center (h) (red measurement, black theoretical curve). The positions of Ex = 1 and Ex = 2 are marked by vertical lines.
Fig. 7
Fig. 7 Results of the measurement series for presenting the dependency of the optimal exponent Ex on the transverse velocity component and the SNR. Image series show the intensity (a) and flow information (b) of the Intralipid flow at the capillary center A-scan for flow transverse velocities vx’ ranging from 0.04 to 0.2. Corresponding diagrams showing the Doppler frequency shift fD’ and the velocity noise σ(fD’) are presented in (c) and (d). The optimal exponent for the capillary center can be read from image parts (e) and (f).
Fig. 8
Fig. 8 OCT intensity images (a) and broadened Doppler frequency shifts (b) of the representative measurement with vx’ = 0.09 (cp. Fig. 7), where Gaussian white noise is added to Γj(z) to generate incrementally lower backscattering signals of the same measurement. Exemplarily, flow profiles are presented for the original data with SNR = 15.3 (c) in comparison to the attenuated one with SNR = 3.5 (d) for four different exponents (Ex = 1, Ex = 2, Ex = 8, Ex → ∞). As seen, (c) offers an optimal exponent of Ex = 1 and (d) of Ex = 8. The diagrams in (e) and (f) present σ(fD’) as a function of the exponent Ex and the signal quality 1/SNR for exponents ranging from Ex = 0.5 to Ex → ∞ and 1/SNR from 0.17 to 0.67. The theoretical 3D diagram shows the estimated optimal exponents for the flow measurement with different signal amplitudes (marked by specific symbols) due to additive Gaussian white noise in the post processing.

Equations (15)

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Γ j ( z ) = A j ( z ) exp [ i ϕ j ( z ) ] ,
Γ j + 1 ( z ) Γ j * ( z ) = | A j+1 ( z ) A j ( z ) | exp [ i ( φ j + 1 ( z ) φ j ( z ) ) ] ,
Δ ϕ ( z ) = 2 π f D ( z ) f A s c a n = 4 π n Δ z λ 0 = 2 n k Δ z
Δ ϕ ( z ) ¯ = arg { j = 1 K 1 Γ j + 1 ( z ) Γ j * ( z ) }
v z ( z ) = Δ ϕ ( z ) λ 0 f A s c a n 4 π n = λ 0 f D ( z ) 2 n
v z max = ± λ 0 f A s c a n 4 n
f D ' ( z ) = arg max { F T [ Γ j ( z ) ] } / K
f D ' ( z ) = v z ' ( z ) = 1 2 π Φ ( z ) Φ ( z ) = arg ( m = Κ / 2 + 1 Κ / 2 exp [ i 2 π m / Κ ] | F T ( Γ j ( z ) ) | m 2 )
Φ ( z ) = arg [ F T 1 ( | F T ( Γ j ( z ) ) | 2 ) 1 ]
Φ ( z ) = arg [ F T 1 ( F T ( Γ j ( z ) ) F T ( Γ j ( z ) ) * ) 1 ]
Φ ( z ) = arg [ F T 1 ( F T ( Γ j ( z ) ) F T ( Γ j * ( z ) ) ) 1 ]
Φ ( z ) = arg [ F T 1 ( F T ( Γ j ( z ) Γ j * ( z ) ) ) 1 ]
Φ ( z ) = arg [ ( Γ j ( z ) Γ j * ( z ) ) 1 ]
Φ ( z ) = arg [ j = 1 K 1 Γ j + 1 ( z ) Γ j * ( z ) ] = Δ ϕ ( z ) ¯
Φ ( z ) = lim E x [ arg ( m = - K / 2 + 1 K / 2 exp [ i 2 π m / K ] | F T ( Γ j ( z ) ) | m E x ) ]

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