Abstract

We propose a simplification for a robust and easy to implement fast phase unwrapping (FPU) algorithm that is used to solve the phase wrapping problem encountered in various fields of optical imaging and metrology. We show that the number of necessary computations using the algorithm can be reduced compared to its original version. FPU can be easily extended from two to three spatial dimensions. We demonstrate the applicability of the two- and three-dimensional FPU algorithm for Doppler optical coherence tomography (DOCT) in numerical simulations, and in the imaging of a flow phantom and blood flow in the human retina in vivo. We introduce an FPU applicability plot for use as a guide in the selection of the most suitable version of the algorithm depending on the phase noise in the acquired data. This plot uses the circular standard deviation of the wrapped phase distribution as a measure of noise and relates it to the root-mean-square error of the recovered, unwrapped phase.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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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).
    [Crossref] [PubMed]
  2. A. F. Fercher, C. K. Hitzenberger, G. Kamp, and S. Y. Elzaiat, “Measurement of Intraocular Distances by Backscattering Spectral Interferometry,” Opt. Commun. 117(1-2), 43–48 (1995).
    [Crossref]
  3. C. L. Chen and R. K. Wang, “Optical coherence tomography based angiography [Invited],” Biomed. Opt. Express 8(2), 1056–1082 (2017).
    [Crossref] [PubMed]
  4. S. Makita, Y. Hong, M. Yamanari, T. Yatagai, and Y. Yasuno, “Optical coherence angiography,” Opt. Express 14(17), 7821–7840 (2006).
    [Crossref] [PubMed]
  5. R. K. Wang, S. L. Jacques, Z. Ma, S. Hurst, S. R. Hanson, and A. Gruber, “Three dimensional optical angiography,” Opt. Express 15(7), 4083–4097 (2007).
    [Crossref] [PubMed]
  6. J. Tokayer, Y. Jia, A. H. Dhalla, and D. Huang, “Blood flow velocity quantification using split-spectrum amplitude-decorrelation angiography with optical coherence tomography,” Biomed. Opt. Express 4(10), 1909–1924 (2013).
    [Crossref] [PubMed]
  7. 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] [PubMed]
  8. V. J. Srinivasan, S. Sakadzić, I. Gorczynska, S. Ruvinskaya, W. Wu, J. G. Fujimoto, and D. A. Boas, “Quantitative cerebral blood flow with optical coherence tomography,” Opt. Express 18(3), 2477–2494 (2010).
    [Crossref] [PubMed]
  9. 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), 612–629 (2012).
    [Crossref] [PubMed]
  10. W. J. Choi, Y. Li, W. Qin, and R. K. Wang, “Cerebral capillary velocimetry based on temporal OCT speckle contrast,” Biomed. Opt. Express 7(12), 4859–4873 (2016).
    [Crossref] [PubMed]
  11. F. Jaillon, S. Makita, and Y. Yasuno, “Variable velocity range imaging of the choroid with dual-beam optical coherence angiography,” Opt. Express 20(1), 385–396 (2012).
    [Crossref] [PubMed]
  12. M. Szkulmowski, A. Szkulmowska, T. Bajraszewski, A. Kowalczyk, and M. Wojtkowski, “Flow velocity estimation using joint Spectral and Time domain Optical Coherence Tomography,” Opt. Express 16(9), 6008–6025 (2008).
    [Crossref] [PubMed]
  13. Y. K. Tao, A. M. Davis, and J. A. Izatt, “Single-pass volumetric bidirectional blood flow imaging spectral domain optical coherence tomography using a modified Hilbert transform,” Opt. Express 16(16), 12350–12361 (2008).
    [Crossref] [PubMed]
  14. J. Walther and E. Koch, “Relation of joint spectral and time domain optical coherence tomography (jSTdOCT) and phase-resolved Doppler OCT,” Opt. Express 22(19), 23129–23146 (2014).
    [Crossref] [PubMed]
  15. 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]
  16. R. A. Leitgeb, R. M. Werkmeister, C. Blatter, and L. Schmetterer, “Doppler optical coherence tomography,” Prog. Retin. Eye Res. 41, 26–43 (2014).
    [Crossref] [PubMed]
  17. R. Leitgeb, L. Schmetterer, W. Drexler, A. Fercher, R. Zawadzki, and T. Bajraszewski, “Real-time assessment of retinal blood flow with ultrafast acquisition by color Doppler Fourier domain optical coherence tomography,” Opt. Express 11(23), 3116–3121 (2003).
    [Crossref] [PubMed]
  18. B. White, M. Pierce, N. Nassif, B. Cense, B. Park, G. Tearney, B. Bouma, T. Chen, and J. de Boer, “In vivo dynamic human retinal blood flow imaging using ultra-high-speed spectral domain optical coherence tomography,” Opt. Express 11(25), 3490–3497 (2003).
    [Crossref] [PubMed]
  19. B. Vakoc, S. Yun, J. de Boer, G. Tearney, and B. Bouma, “Phase-resolved optical frequency domain imaging,” Opt. Express 13(14), 5483–5493 (2005).
    [Crossref] [PubMed]
  20. A. Szkulmowska, M. Szkulmowski, A. Kowalczyk, and M. Wojtkowski, “Phase-resolved Doppler optical coherence tomography--limitations and improvements,” Opt. Lett. 33(13), 1425–1427 (2008).
    [Crossref] [PubMed]
  21. S. Tozburun, C. Blatter, M. Siddiqui, E. F. J. Meijer, and B. J. Vakoc, “Phase-stable Doppler OCT at 19 MHz using a stretched-pulse mode-locked laser,” Biomed. Opt. Express 9(3), 952–961 (2018).
    [Crossref] [PubMed]
  22. R. K. Wang and L. An, “Doppler optical micro-angiography for volumetric imaging of vascular perfusion in vivo,” Opt. Express 17(11), 8926–8940 (2009).
    [Crossref] [PubMed]
  23. A. Szkulmowska, M. Szkulmowski, D. Szlag, A. Kowalczyk, and M. Wojtkowski, “Three-dimensional quantitative imaging of retinal and choroidal blood flow velocity using joint Spectral and Time domain Optical Coherence Tomography,” Opt. Express 17(13), 10584–10598 (2009).
    [Crossref] [PubMed]
  24. 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(26), 23736–23754 (2009).
    [Crossref] [PubMed]
  25. 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(10), 1219–1223 (2009).
    [Crossref] [PubMed]
  26. V. J. Srinivasan, S. Sakadzić, I. Gorczynska, S. Ruvinskaya, W. Wu, J. G. Fujimoto, and D. A. Boas, “Quantitative cerebral blood flow with optical coherence tomography,” Opt. Express 18(3), 2477–2494 (2010).
    [Crossref] [PubMed]
  27. A. Bouwens, T. Bolmont, D. Szlag, C. Berclaz, and T. Lasser, “Quantitative cerebral blood flow imaging with extended-focus optical coherence microscopy,” Opt. Lett. 39(1), 37–40 (2014).
    [Crossref] [PubMed]
  28. S. Yazdanfar, C. Yang, M. Sarunic, and J. Izatt, “Frequency estimation precision in Doppler optical coherence tomography using the Cramer-Rao lower bound,” Opt. Express 13(2), 410–416 (2005).
    [Crossref] [PubMed]
  29. B. Park, M. C. Pierce, B. Cense, S. H. Yun, M. Mujat, G. Tearney, B. Bouma, and J. de Boer, “Real-time fiber-based multi-functional spectral-domain optical coherence tomography at 1.3 microm,” Opt. Express 13(11), 3931–3944 (2005).
    [Crossref] [PubMed]
  30. 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]
  31. 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]
  32. H. C. Hendargo, R. P. McNabb, A. H. Dhalla, N. Shepherd, and J. A. Izatt, “Doppler velocity detection limitations in spectrometer-based versus swept-source optical coherence tomography,” Biomed. Opt. Express 2(8), 2175–2188 (2011).
    [Crossref] [PubMed]
  33. 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]
  34. 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]
  35. R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: Two-dimensional phase unwrapping,” Radio Sci. 23(4), 713–720 (1988).
    [Crossref]
  36. S. Chavez, Q. S. Xiang, and L. An, “Understanding phase maps in MRI: a new cutline phase unwrapping method,” IEEE Trans. Med. Imaging 21(8), 966–977 (2002).
    [Crossref] [PubMed]
  37. V. S. Jeught, J. Sijbers, and J. J. Dirckx, “Fast Fourier-Based Phase Unwrapping on the Graphics Processing Unit in Real-Time Imaging Applications,” J. Imaging 1(1), 31–44 (2015).
    [Crossref]
  38. D. C. Ghiglia and M. D. Pritt, Two-dimensional phase unwrapping: theory, algorithms, and software (Wiley, New York, 1998).
  39. 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]
  40. 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]
  41. J. Walther and E. Koch, “Impact of a detector dead time in phase-resolved Doppler analysis using spectral domain optical coherence tomography,” J. Opt. Soc. Am. A 34(2), 241–251 (2017).
    [Crossref] [PubMed]
  42. K. Itoh, “Analysis of the phase unwrapping algorithm,” Appl. Opt. 21(14), 2470 (1982).
    [Crossref] [PubMed]
  43. M. A. Schofield and Y. Zhu, “Fast phase unwrapping algorithm for interferometric applications,” Opt. Lett. 28(14), 1194–1196 (2003).
    [Crossref] [PubMed]
  44. W. Li, A. V. Avram, B. Wu, X. Xiao, and C. Liu, “Integrated Laplacian-based phase unwrapping and background phase removal for quantitative susceptibility mapping,” NMR Biomed. 27(2), 219–227 (2014).
    [Crossref] [PubMed]
  45. H. Bagher-Ebadian, Q. Jiang, and J. R. Ewing, “A modified Fourier-based phase unwrapping algorithm with an application to MRI venography,” J. Magn. Reson. Imaging 27(3), 649–652 (2008).
    [Crossref] [PubMed]
  46. M. Loecher, E. Schrauben, K. M. Johnson, and O. Wieben, “Phase unwrapping in 4D MR flow with a 4D single-step laplacian algorithm,” J. Magn. Reson. Imaging 43(4), 833–842 (2016).
    [Crossref] [PubMed]
  47. E. Barnhill, P. Kennedy, C. L. Johnson, M. Mada, and N. Roberts, “Real-time 4D phase unwrapping applied to magnetic resonance elastography,” Magn. Reson. Med. 73(6), 2321–2331 (2015).
    [Crossref] [PubMed]
  48. T. Latychevskaia, P. Formanek, C. T. Koch, and A. Lubk, “Off-axis and inline electron holography: Experimental comparison,” Ultramicroscopy 110(5), 472–482 (2010).
    [Crossref]
  49. D. Parshall and M. K. Kim, “Digital holographic microscopy with dual-wavelength phase unwrapping,” Appl. Opt. 45(3), 451–459 (2006).
    [Crossref] [PubMed]
  50. H. C. Hendargo, M. Zhao, N. Shepherd, and J. A. Izatt, “Synthetic wavelength based phase unwrapping in spectral domain optical coherence tomography,” Opt. Express 17(7), 5039–5051 (2009).
    [Crossref] [PubMed]
  51. Y. Wang, A. A. Fawzi, O. Tan, X. Zhang, and D. Huang, “Flicker-induced changes in retinal blood flow assessed by Doppler optical coherence tomography,” Biomed. Opt. Express 2(7), 1852–1860 (2011).
    [Crossref] [PubMed]
  52. Y. Wang, D. Huang, Y. Su, and X. S. Yao, “Two-dimensional phase unwrapping in Doppler Fourier domain optical coherence tomography,” Opt. Express 24(23), 26129–26145 (2016).
    [Crossref] [PubMed]
  53. S. Xia, Y. Huang, S. Peng, Y. Wu, and X. Tan, “Robust phase unwrapping for phase images in Fourier domain Doppler optical coherence tomography,” J. Biomed. Opt. 22(3), 36014 (2017).
    [Crossref] [PubMed]
  54. T. E. Gureyev and K. A. Nugent, “Phase retrieval with the transport-of-intensity equation. II. Orthogonal series solution for nonuniform illumination,” J. Opt. Soc. Am. A 13(8), 1670–1682 (1996).
    [Crossref]
  55. E. Pijewska, I. Gorczynska, and M. Szkulmowski, “Matlab implementation of various forms of the Fast Phase Unwrapping algorithm for 2 and 3 dimensional data,” figshare (2019) [retrieved 11 Feb 2019], https://doi.org/10.6084/m9.figshare.7308413 .
  56. H. Sorensen, D. Jones, M. Heideman, and C. Burrus, “Real-valued fast Fourier transform algorithms,” IEEE Trans. Acoust. Speech Signal Process. 35(6), 849–863 (1987).
    [Crossref]
  57. M. Frigo and S. G. Johnson, “benchFFT” (2019), accessed on 15–01–2019, http://www.fftw.org/benchfft/ .
  58. 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(2-1), 022604 (2016).
    [Crossref] [PubMed]
  59. N. I. Fisher, Statistical Analysis of Circular Data (Cambridge University Press, 1995).
  60. D. Ruminski, B. L. Sikorski, D. Bukowska, M. Szkulmowski, K. Krawiec, G. Malukiewicz, L. Bieganowski, and M. Wojtkowski, “OCT angiography by absolute intensity difference applied to normal and diseased human retinas,” Biomed. Opt. Express 6(8), 2738–2754 (2015).
    [Crossref] [PubMed]
  61. World Medical Association, “World Medical Association Declaration of Helsinki: ethical principles for medical research involving human subjects,” JAMA 310(20), 2191–2194 (2013).
    [Crossref] [PubMed]

2018 (1)

2017 (3)

2016 (4)

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(2-1), 022604 (2016).
[Crossref] [PubMed]

M. Loecher, E. Schrauben, K. M. Johnson, and O. Wieben, “Phase unwrapping in 4D MR flow with a 4D single-step laplacian algorithm,” J. Magn. Reson. Imaging 43(4), 833–842 (2016).
[Crossref] [PubMed]

Y. Wang, D. Huang, Y. Su, and X. S. Yao, “Two-dimensional phase unwrapping in Doppler Fourier domain optical coherence tomography,” Opt. Express 24(23), 26129–26145 (2016).
[Crossref] [PubMed]

W. J. Choi, Y. Li, W. Qin, and R. K. Wang, “Cerebral capillary velocimetry based on temporal OCT speckle contrast,” Biomed. Opt. Express 7(12), 4859–4873 (2016).
[Crossref] [PubMed]

2015 (3)

E. Barnhill, P. Kennedy, C. L. Johnson, M. Mada, and N. Roberts, “Real-time 4D phase unwrapping applied to magnetic resonance elastography,” Magn. Reson. Med. 73(6), 2321–2331 (2015).
[Crossref] [PubMed]

V. S. Jeught, J. Sijbers, and J. J. Dirckx, “Fast Fourier-Based Phase Unwrapping on the Graphics Processing Unit in Real-Time Imaging Applications,” J. Imaging 1(1), 31–44 (2015).
[Crossref]

D. Ruminski, B. L. Sikorski, D. Bukowska, M. Szkulmowski, K. Krawiec, G. Malukiewicz, L. Bieganowski, and M. Wojtkowski, “OCT angiography by absolute intensity difference applied to normal and diseased human retinas,” Biomed. Opt. Express 6(8), 2738–2754 (2015).
[Crossref] [PubMed]

2014 (6)

2013 (4)

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]

J. Tokayer, Y. Jia, A. H. Dhalla, and D. Huang, “Blood flow velocity quantification using split-spectrum amplitude-decorrelation angiography with optical coherence tomography,” Biomed. Opt. Express 4(10), 1909–1924 (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]

World Medical Association, “World Medical Association Declaration of Helsinki: ethical principles for medical research involving human subjects,” JAMA 310(20), 2191–2194 (2013).
[Crossref] [PubMed]

2012 (2)

2011 (2)

2010 (4)

T. Latychevskaia, P. Formanek, C. T. Koch, and A. Lubk, “Off-axis and inline electron holography: Experimental comparison,” Ultramicroscopy 110(5), 472–482 (2010).
[Crossref]

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]

V. J. Srinivasan, S. Sakadzić, I. Gorczynska, S. Ruvinskaya, W. Wu, J. G. Fujimoto, and D. A. Boas, “Quantitative cerebral blood flow with optical coherence tomography,” Opt. Express 18(3), 2477–2494 (2010).
[Crossref] [PubMed]

V. J. Srinivasan, S. Sakadzić, I. Gorczynska, S. Ruvinskaya, W. Wu, J. G. Fujimoto, and D. A. Boas, “Quantitative cerebral blood flow with optical coherence tomography,” Opt. Express 18(3), 2477–2494 (2010).
[Crossref] [PubMed]

2009 (7)

R. K. Wang and L. An, “Doppler optical micro-angiography for volumetric imaging of vascular perfusion in vivo,” Opt. Express 17(11), 8926–8940 (2009).
[Crossref] [PubMed]

A. Szkulmowska, M. Szkulmowski, D. Szlag, A. Kowalczyk, and M. Wojtkowski, “Three-dimensional quantitative imaging of retinal and choroidal blood flow velocity using joint Spectral and Time domain Optical Coherence Tomography,” Opt. Express 17(13), 10584–10598 (2009).
[Crossref] [PubMed]

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(26), 23736–23754 (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(10), 1219–1223 (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(6), 814–821 (2009).
[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]

H. C. Hendargo, M. Zhao, N. Shepherd, and J. A. Izatt, “Synthetic wavelength based phase unwrapping in spectral domain optical coherence tomography,” Opt. Express 17(7), 5039–5051 (2009).
[Crossref] [PubMed]

2008 (4)

2007 (1)

2006 (2)

2005 (3)

2004 (1)

2003 (3)

2002 (1)

S. Chavez, Q. S. Xiang, and L. An, “Understanding phase maps in MRI: a new cutline phase unwrapping method,” IEEE Trans. Med. Imaging 21(8), 966–977 (2002).
[Crossref] [PubMed]

1996 (1)

1995 (1)

A. F. Fercher, C. K. Hitzenberger, G. Kamp, and S. Y. Elzaiat, “Measurement of Intraocular Distances by Backscattering Spectral Interferometry,” Opt. Commun. 117(1-2), 43–48 (1995).
[Crossref]

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]

1988 (1)

R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: Two-dimensional phase unwrapping,” Radio Sci. 23(4), 713–720 (1988).
[Crossref]

1987 (1)

H. Sorensen, D. Jones, M. Heideman, and C. Burrus, “Real-valued fast Fourier transform algorithms,” IEEE Trans. Acoust. Speech Signal Process. 35(6), 849–863 (1987).
[Crossref]

1982 (1)

An, L.

R. K. Wang and L. An, “Doppler optical micro-angiography for volumetric imaging of vascular perfusion in vivo,” Opt. Express 17(11), 8926–8940 (2009).
[Crossref] [PubMed]

S. Chavez, Q. S. Xiang, and L. An, “Understanding phase maps in MRI: a new cutline phase unwrapping method,” IEEE Trans. Med. Imaging 21(8), 966–977 (2002).
[Crossref] [PubMed]

Avram, A. V.

W. Li, A. V. Avram, B. Wu, X. Xiao, and C. Liu, “Integrated Laplacian-based phase unwrapping and background phase removal for quantitative susceptibility mapping,” NMR Biomed. 27(2), 219–227 (2014).
[Crossref] [PubMed]

Bagher-Ebadian, H.

H. Bagher-Ebadian, Q. Jiang, and J. R. Ewing, “A modified Fourier-based phase unwrapping algorithm with an application to MRI venography,” J. Magn. Reson. Imaging 27(3), 649–652 (2008).
[Crossref] [PubMed]

Bajraszewski, T.

Barnhill, E.

E. Barnhill, P. Kennedy, C. L. Johnson, M. Mada, and N. Roberts, “Real-time 4D phase unwrapping applied to magnetic resonance elastography,” Magn. Reson. Med. 73(6), 2321–2331 (2015).
[Crossref] [PubMed]

Barry, S.

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(10), 1219–1223 (2009).
[Crossref] [PubMed]

Berclaz, C.

Bieganowski, L.

Blatter, C.

Boas, D. A.

Bolmont, T.

Bouma, B.

Bouma, B. E.

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(2-1), 022604 (2016).
[Crossref] [PubMed]

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] [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(10), 1219–1223 (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(6), 814–821 (2009).
[Crossref] [PubMed]

Bouwens, A.

Bukowska, D.

Burrus, C.

H. Sorensen, D. Jones, M. Heideman, and C. Burrus, “Real-valued fast Fourier transform algorithms,” IEEE Trans. Acoust. Speech Signal Process. 35(6), 849–863 (1987).
[Crossref]

Cable, A. E.

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]

Chavez, S.

S. Chavez, Q. S. Xiang, and L. An, “Understanding phase maps in MRI: a new cutline phase unwrapping method,” IEEE Trans. Med. Imaging 21(8), 966–977 (2002).
[Crossref] [PubMed]

Chen, C. L.

Chen, T.

Choi, W. J.

Cuevas, M.

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]

Davis, A. M.

de Boer, J.

Dhalla, A. H.

Dirckx, J. J.

V. S. Jeught, J. Sijbers, and J. J. Dirckx, “Fast Fourier-Based Phase Unwrapping on the Graphics Processing Unit in Real-Time Imaging Applications,” J. Imaging 1(1), 31–44 (2015).
[Crossref]

Drexler, W.

Elzaiat, S. Y.

A. F. Fercher, C. K. Hitzenberger, G. Kamp, and S. Y. Elzaiat, “Measurement of Intraocular Distances by Backscattering Spectral Interferometry,” Opt. Commun. 117(1-2), 43–48 (1995).
[Crossref]

Ewing, J. R.

H. Bagher-Ebadian, Q. Jiang, and J. R. Ewing, “A modified Fourier-based phase unwrapping algorithm with an application to MRI venography,” J. Magn. Reson. Imaging 27(3), 649–652 (2008).
[Crossref] [PubMed]

Fawzi, A. A.

Fercher, A.

Fercher, A. F.

A. F. Fercher, C. K. Hitzenberger, G. Kamp, and S. Y. Elzaiat, “Measurement of Intraocular Distances by Backscattering Spectral Interferometry,” Opt. Commun. 117(1-2), 43–48 (1995).
[Crossref]

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]

Formanek, P.

T. Latychevskaia, P. Formanek, C. T. Koch, and A. Lubk, “Off-axis and inline electron holography: Experimental comparison,” Ultramicroscopy 110(5), 472–482 (2010).
[Crossref]

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(10), 1219–1223 (2009).
[Crossref] [PubMed]

Goldstein, R. M.

R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: Two-dimensional phase unwrapping,” Radio Sci. 23(4), 713–720 (1988).
[Crossref]

Gorczynska, I.

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]

Gruber, A.

Grulkowski, I.

Gureyev, T. E.

Hanson, S. R.

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]

Heideman, M.

H. Sorensen, D. Jones, M. Heideman, and C. Burrus, “Real-valued fast Fourier transform algorithms,” IEEE Trans. Acoust. Speech Signal Process. 35(6), 849–863 (1987).
[Crossref]

Hendargo, H. C.

Hitzenberger, C. K.

A. F. Fercher, C. K. Hitzenberger, G. Kamp, and S. Y. Elzaiat, “Measurement of Intraocular Distances by Backscattering Spectral Interferometry,” Opt. Commun. 117(1-2), 43–48 (1995).
[Crossref]

Hong, Y.

Huang, D.

Huang, Y.

S. Xia, Y. Huang, S. Peng, Y. Wu, and X. Tan, “Robust phase unwrapping for phase images in Fourier domain Doppler optical coherence tomography,” J. Biomed. Opt. 22(3), 36014 (2017).
[Crossref] [PubMed]

Hurst, S.

Itoh, K.

Izatt, J.

Izatt, J. A.

Jacques, S. L.

Jahan, I.

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(10), 1219–1223 (2009).
[Crossref] [PubMed]

Jeught, V. S.

V. S. Jeught, J. Sijbers, and J. J. Dirckx, “Fast Fourier-Based Phase Unwrapping on the Graphics Processing Unit in Real-Time Imaging Applications,” J. Imaging 1(1), 31–44 (2015).
[Crossref]

Jia, Y.

Jiang, J. Y.

Jiang, Q.

H. Bagher-Ebadian, Q. Jiang, and J. R. Ewing, “A modified Fourier-based phase unwrapping algorithm with an application to MRI venography,” J. Magn. Reson. Imaging 27(3), 649–652 (2008).
[Crossref] [PubMed]

Johnson, C. L.

E. Barnhill, P. Kennedy, C. L. Johnson, M. Mada, and N. Roberts, “Real-time 4D phase unwrapping applied to magnetic resonance elastography,” Magn. Reson. Med. 73(6), 2321–2331 (2015).
[Crossref] [PubMed]

Johnson, K. M.

M. Loecher, E. Schrauben, K. M. Johnson, and O. Wieben, “Phase unwrapping in 4D MR flow with a 4D single-step laplacian algorithm,” J. Magn. Reson. Imaging 43(4), 833–842 (2016).
[Crossref] [PubMed]

Jones, D.

H. Sorensen, D. Jones, M. Heideman, and C. Burrus, “Real-valued fast Fourier transform algorithms,” IEEE Trans. Acoust. Speech Signal Process. 35(6), 849–863 (1987).
[Crossref]

Kamp, G.

A. F. Fercher, C. K. Hitzenberger, G. Kamp, and S. Y. Elzaiat, “Measurement of Intraocular Distances by Backscattering Spectral Interferometry,” Opt. Commun. 117(1-2), 43–48 (1995).
[Crossref]

Kennedy, P.

E. Barnhill, P. Kennedy, C. L. Johnson, M. Mada, and N. Roberts, “Real-time 4D phase unwrapping applied to magnetic resonance elastography,” Magn. Reson. Med. 73(6), 2321–2331 (2015).
[Crossref] [PubMed]

Kim, M. K.

Koch, C. T.

T. Latychevskaia, P. Formanek, C. T. Koch, and A. Lubk, “Off-axis and inline electron holography: Experimental comparison,” Ultramicroscopy 110(5), 472–482 (2010).
[Crossref]

Koch, E.

J. Walther and E. Koch, “Impact of a detector dead time in phase-resolved Doppler analysis using spectral domain optical coherence tomography,” J. Opt. Soc. Am. A 34(2), 241–251 (2017).
[Crossref] [PubMed]

J. Walther and E. Koch, “Relation of joint spectral and time domain optical coherence tomography (jSTdOCT) and phase-resolved Doppler OCT,” Opt. Express 22(19), 23129–23146 (2014).
[Crossref] [PubMed]

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]

Kowalczyk, A.

Krawiec, K.

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]

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(10), 1219–1223 (2009).
[Crossref] [PubMed]

Lasser, T.

Latychevskaia, T.

T. Latychevskaia, P. Formanek, C. T. Koch, and A. Lubk, “Off-axis and inline electron holography: Experimental comparison,” Ultramicroscopy 110(5), 472–482 (2010).
[Crossref]

Leitgeb, R.

Leitgeb, R. A.

Li, W.

W. Li, A. V. Avram, B. Wu, X. Xiao, and C. Liu, “Integrated Laplacian-based phase unwrapping and background phase removal for quantitative susceptibility mapping,” NMR Biomed. 27(2), 219–227 (2014).
[Crossref] [PubMed]

Li, Y.

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]

Liu, C.

W. Li, A. V. Avram, B. Wu, X. Xiao, and C. Liu, “Integrated Laplacian-based phase unwrapping and background phase removal for quantitative susceptibility mapping,” NMR Biomed. 27(2), 219–227 (2014).
[Crossref] [PubMed]

Lo, E. H.

Loecher, M.

M. Loecher, E. Schrauben, K. M. Johnson, and O. Wieben, “Phase unwrapping in 4D MR flow with a 4D single-step laplacian algorithm,” J. Magn. Reson. Imaging 43(4), 833–842 (2016).
[Crossref] [PubMed]

Lubk, A.

T. Latychevskaia, P. Formanek, C. T. Koch, and A. Lubk, “Off-axis and inline electron holography: Experimental comparison,” Ultramicroscopy 110(5), 472–482 (2010).
[Crossref]

Ma, Z.

Mada, M.

E. Barnhill, P. Kennedy, C. L. Johnson, M. Mada, and N. Roberts, “Real-time 4D phase unwrapping applied to magnetic resonance elastography,” Magn. Reson. Med. 73(6), 2321–2331 (2015).
[Crossref] [PubMed]

Makita, S.

Malukiewicz, G.

Mandeville, E. T.

McNabb, R. P.

Meijer, E. F. J.

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.

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(10), 1219–1223 (2009).
[Crossref] [PubMed]

Nassif, N.

Nugent, K. A.

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(10), 1219–1223 (2009).
[Crossref] [PubMed]

Park, B.

Parshall, D.

Peng, S.

S. Xia, Y. Huang, S. Peng, Y. Wu, and X. Tan, “Robust phase unwrapping for phase images in Fourier domain Doppler optical coherence tomography,” J. Biomed. Opt. 22(3), 36014 (2017).
[Crossref] [PubMed]

Pierce, M.

Pierce, M. C.

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

Qin, W.

Radhakrishnan, H.

Roberts, N.

E. Barnhill, P. Kennedy, C. L. Johnson, M. Mada, and N. Roberts, “Real-time 4D phase unwrapping applied to magnetic resonance elastography,” Magn. Reson. Med. 73(6), 2321–2331 (2015).
[Crossref] [PubMed]

Ruminski, D.

Ruvinskaya, S.

Sakadzic, S.

Sarunic, M.

Schmetterer, L.

Schofield, M. A.

Schrauben, E.

M. Loecher, E. Schrauben, K. M. Johnson, and O. Wieben, “Phase unwrapping in 4D MR flow with a 4D single-step laplacian algorithm,” J. Magn. Reson. Imaging 43(4), 833–842 (2016).
[Crossref] [PubMed]

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]

Shepherd, N.

Siddiqui, M.

Sijbers, J.

V. S. Jeught, J. Sijbers, and J. J. Dirckx, “Fast Fourier-Based Phase Unwrapping on the Graphics Processing Unit in Real-Time Imaging Applications,” J. Imaging 1(1), 31–44 (2015).
[Crossref]

Sikorski, B. L.

Sorensen, H.

H. Sorensen, D. Jones, M. Heideman, and C. Burrus, “Real-valued fast Fourier transform algorithms,” IEEE Trans. Acoust. Speech Signal Process. 35(6), 849–863 (1987).
[Crossref]

Srinivasan, V. J.

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]

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(10), 1219–1223 (2009).
[Crossref] [PubMed]

Su, Y.

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.

D. Ruminski, B. L. Sikorski, D. Bukowska, M. Szkulmowski, K. Krawiec, G. Malukiewicz, L. Bieganowski, and M. Wojtkowski, “OCT angiography by absolute intensity difference applied to normal and diseased human retinas,” Biomed. Opt. Express 6(8), 2738–2754 (2015).
[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]

A. Szkulmowska, M. Szkulmowski, D. Szlag, A. Kowalczyk, and M. Wojtkowski, “Three-dimensional quantitative imaging of retinal and choroidal blood flow velocity using joint Spectral and Time domain Optical Coherence Tomography,” Opt. Express 17(13), 10584–10598 (2009).
[Crossref] [PubMed]

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(26), 23736–23754 (2009).
[Crossref] [PubMed]

A. Szkulmowska, M. Szkulmowski, A. Kowalczyk, and M. Wojtkowski, “Phase-resolved Doppler optical coherence tomography--limitations and improvements,” Opt. Lett. 33(13), 1425–1427 (2008).
[Crossref] [PubMed]

M. Szkulmowski, A. Szkulmowska, T. Bajraszewski, A. Kowalczyk, and M. Wojtkowski, “Flow velocity estimation using joint Spectral and Time domain Optical Coherence Tomography,” Opt. Express 16(9), 6008–6025 (2008).
[Crossref] [PubMed]

Szlag, D.

Tan, O.

Tan, X.

S. Xia, Y. Huang, S. Peng, Y. Wu, and X. Tan, “Robust phase unwrapping for phase images in Fourier domain Doppler optical coherence tomography,” J. Biomed. Opt. 22(3), 36014 (2017).
[Crossref] [PubMed]

Tao, Y. K.

Tearney, G.

Tearney, G. J.

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(10), 1219–1223 (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(6), 814–821 (2009).
[Crossref] [PubMed]

Tokayer, J.

Tozburun, S.

Tyrrell, J. 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(10), 1219–1223 (2009).
[Crossref] [PubMed]

Uribe-Patarroyo, N.

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(2-1), 022604 (2016).
[Crossref] [PubMed]

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] [PubMed]

Vakoc, B.

Vakoc, B. J.

S. Tozburun, C. Blatter, M. Siddiqui, E. F. J. Meijer, and B. J. Vakoc, “Phase-stable Doppler OCT at 19 MHz using a stretched-pulse mode-locked laser,” Biomed. Opt. Express 9(3), 952–961 (2018).
[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(10), 1219–1223 (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(6), 814–821 (2009).
[Crossref] [PubMed]

Villiger, M.

Walther, J.

J. Walther and E. Koch, “Impact of a detector dead time in phase-resolved Doppler analysis using spectral domain optical coherence tomography,” J. Opt. Soc. Am. A 34(2), 241–251 (2017).
[Crossref] [PubMed]

J. Walther and E. Koch, “Relation of joint spectral and time domain optical coherence tomography (jSTdOCT) and phase-resolved Doppler OCT,” Opt. Express 22(19), 23129–23146 (2014).
[Crossref] [PubMed]

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]

Wang, R. K.

Wang, Y.

Werkmeister, R. M.

R. A. Leitgeb, R. M. Werkmeister, C. Blatter, and L. Schmetterer, “Doppler optical coherence tomography,” Prog. Retin. Eye Res. 41, 26–43 (2014).
[Crossref] [PubMed]

Werner, C. L.

R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: Two-dimensional phase unwrapping,” Radio Sci. 23(4), 713–720 (1988).
[Crossref]

White, B.

Wieben, O.

M. Loecher, E. Schrauben, K. M. Johnson, and O. Wieben, “Phase unwrapping in 4D MR flow with a 4D single-step laplacian algorithm,” J. Magn. Reson. Imaging 43(4), 833–842 (2016).
[Crossref] [PubMed]

Wojtkowski, M.

D. Ruminski, B. L. Sikorski, D. Bukowska, M. Szkulmowski, K. Krawiec, G. Malukiewicz, L. Bieganowski, and M. Wojtkowski, “OCT angiography by absolute intensity difference applied to normal and diseased human retinas,” Biomed. Opt. Express 6(8), 2738–2754 (2015).
[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]

A. Szkulmowska, M. Szkulmowski, D. Szlag, A. Kowalczyk, and M. Wojtkowski, “Three-dimensional quantitative imaging of retinal and choroidal blood flow velocity using joint Spectral and Time domain Optical Coherence Tomography,” Opt. Express 17(13), 10584–10598 (2009).
[Crossref] [PubMed]

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(26), 23736–23754 (2009).
[Crossref] [PubMed]

A. Szkulmowska, M. Szkulmowski, A. Kowalczyk, and M. Wojtkowski, “Phase-resolved Doppler optical coherence tomography--limitations and improvements,” Opt. Lett. 33(13), 1425–1427 (2008).
[Crossref] [PubMed]

M. Szkulmowski, A. Szkulmowska, T. Bajraszewski, A. Kowalczyk, and M. Wojtkowski, “Flow velocity estimation using joint Spectral and Time domain Optical Coherence Tomography,” Opt. Express 16(9), 6008–6025 (2008).
[Crossref] [PubMed]

Wu, B.

W. Li, A. V. Avram, B. Wu, X. Xiao, and C. Liu, “Integrated Laplacian-based phase unwrapping and background phase removal for quantitative susceptibility mapping,” NMR Biomed. 27(2), 219–227 (2014).
[Crossref] [PubMed]

Wu, W.

Wu, Y.

S. Xia, Y. Huang, S. Peng, Y. Wu, and X. Tan, “Robust phase unwrapping for phase images in Fourier domain Doppler optical coherence tomography,” J. Biomed. Opt. 22(3), 36014 (2017).
[Crossref] [PubMed]

Xia, S.

S. Xia, Y. Huang, S. Peng, Y. Wu, and X. Tan, “Robust phase unwrapping for phase images in Fourier domain Doppler optical coherence tomography,” J. Biomed. Opt. 22(3), 36014 (2017).
[Crossref] [PubMed]

Xiang, Q. S.

S. Chavez, Q. S. Xiang, and L. An, “Understanding phase maps in MRI: a new cutline phase unwrapping method,” IEEE Trans. Med. Imaging 21(8), 966–977 (2002).
[Crossref] [PubMed]

Xiao, X.

W. Li, A. V. Avram, B. Wu, X. Xiao, and C. Liu, “Integrated Laplacian-based phase unwrapping and background phase removal for quantitative susceptibility mapping,” NMR Biomed. 27(2), 219–227 (2014).
[Crossref] [PubMed]

Yamanari, M.

Yang, C.

Yao, X. S.

Yasuno, Y.

Yatagai, T.

Yazdanfar, S.

Yun, S.

Yun, S. H.

Zawadzki, R.

Zebker, H. A.

R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: Two-dimensional phase unwrapping,” Radio Sci. 23(4), 713–720 (1988).
[Crossref]

Zhang, X.

Zhao, M.

Zhu, Y.

Appl. Opt. (2)

Biomed. Opt. Express (8)

Y. Wang, A. A. Fawzi, O. Tan, X. Zhang, and D. Huang, “Flicker-induced changes in retinal blood flow assessed by Doppler optical coherence tomography,” Biomed. Opt. Express 2(7), 1852–1860 (2011).
[Crossref] [PubMed]

D. Ruminski, B. L. Sikorski, D. Bukowska, M. Szkulmowski, K. Krawiec, G. Malukiewicz, L. Bieganowski, and M. Wojtkowski, “OCT angiography by absolute intensity difference applied to normal and diseased human retinas,” Biomed. Opt. Express 6(8), 2738–2754 (2015).
[Crossref] [PubMed]

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), 612–629 (2012).
[Crossref] [PubMed]

W. J. Choi, Y. Li, W. Qin, and R. K. Wang, “Cerebral capillary velocimetry based on temporal OCT speckle contrast,” Biomed. Opt. Express 7(12), 4859–4873 (2016).
[Crossref] [PubMed]

C. L. Chen and R. K. Wang, “Optical coherence tomography based angiography [Invited],” Biomed. Opt. Express 8(2), 1056–1082 (2017).
[Crossref] [PubMed]

S. Tozburun, C. Blatter, M. Siddiqui, E. F. J. Meijer, and B. J. Vakoc, “Phase-stable Doppler OCT at 19 MHz using a stretched-pulse mode-locked laser,” Biomed. Opt. Express 9(3), 952–961 (2018).
[Crossref] [PubMed]

J. Tokayer, Y. Jia, A. H. Dhalla, and D. Huang, “Blood flow velocity quantification using split-spectrum amplitude-decorrelation angiography with optical coherence tomography,” Biomed. Opt. Express 4(10), 1909–1924 (2013).
[Crossref] [PubMed]

H. C. Hendargo, R. P. McNabb, A. H. Dhalla, N. Shepherd, and J. A. Izatt, “Doppler velocity detection limitations in spectrometer-based versus swept-source optical coherence tomography,” Biomed. Opt. Express 2(8), 2175–2188 (2011).
[Crossref] [PubMed]

IEEE Trans. Acoust. Speech Signal Process. (1)

H. Sorensen, D. Jones, M. Heideman, and C. Burrus, “Real-valued fast Fourier transform algorithms,” IEEE Trans. Acoust. Speech Signal Process. 35(6), 849–863 (1987).
[Crossref]

IEEE Trans. Med. Imaging (3)

S. Chavez, Q. S. Xiang, and L. An, “Understanding phase maps in MRI: a new cutline phase unwrapping method,” IEEE Trans. Med. Imaging 21(8), 966–977 (2002).
[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]

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]

J. Biomed. Opt. (2)

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]

S. Xia, Y. Huang, S. Peng, Y. Wu, and X. Tan, “Robust phase unwrapping for phase images in Fourier domain Doppler optical coherence tomography,” J. Biomed. Opt. 22(3), 36014 (2017).
[Crossref] [PubMed]

J. Imaging (1)

V. S. Jeught, J. Sijbers, and J. J. Dirckx, “Fast Fourier-Based Phase Unwrapping on the Graphics Processing Unit in Real-Time Imaging Applications,” J. Imaging 1(1), 31–44 (2015).
[Crossref]

J. Magn. Reson. Imaging (2)

H. Bagher-Ebadian, Q. Jiang, and J. R. Ewing, “A modified Fourier-based phase unwrapping algorithm with an application to MRI venography,” J. Magn. Reson. Imaging 27(3), 649–652 (2008).
[Crossref] [PubMed]

M. Loecher, E. Schrauben, K. M. Johnson, and O. Wieben, “Phase unwrapping in 4D MR flow with a 4D single-step laplacian algorithm,” J. Magn. Reson. Imaging 43(4), 833–842 (2016).
[Crossref] [PubMed]

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

JAMA (1)

World Medical Association, “World Medical Association Declaration of Helsinki: ethical principles for medical research involving human subjects,” JAMA 310(20), 2191–2194 (2013).
[Crossref] [PubMed]

Magn. Reson. Med. (1)

E. Barnhill, P. Kennedy, C. L. Johnson, M. Mada, and N. Roberts, “Real-time 4D phase unwrapping applied to magnetic resonance elastography,” Magn. Reson. Med. 73(6), 2321–2331 (2015).
[Crossref] [PubMed]

Nat. Med. (1)

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(10), 1219–1223 (2009).
[Crossref] [PubMed]

NMR Biomed. (1)

W. Li, A. V. Avram, B. Wu, X. Xiao, and C. Liu, “Integrated Laplacian-based phase unwrapping and background phase removal for quantitative susceptibility mapping,” NMR Biomed. 27(2), 219–227 (2014).
[Crossref] [PubMed]

Opt. Commun. (1)

A. F. Fercher, C. K. Hitzenberger, G. Kamp, and S. Y. Elzaiat, “Measurement of Intraocular Distances by Backscattering Spectral Interferometry,” Opt. Commun. 117(1-2), 43–48 (1995).
[Crossref]

Opt. Express (22)

S. Makita, Y. Hong, M. Yamanari, T. Yatagai, and Y. Yasuno, “Optical coherence angiography,” Opt. Express 14(17), 7821–7840 (2006).
[Crossref] [PubMed]

R. K. Wang, S. L. Jacques, Z. Ma, S. Hurst, S. R. Hanson, and A. Gruber, “Three dimensional optical angiography,” Opt. Express 15(7), 4083–4097 (2007).
[Crossref] [PubMed]

R. Leitgeb, L. Schmetterer, W. Drexler, A. Fercher, R. Zawadzki, and T. Bajraszewski, “Real-time assessment of retinal blood flow with ultrafast acquisition by color Doppler Fourier domain optical coherence tomography,” Opt. Express 11(23), 3116–3121 (2003).
[Crossref] [PubMed]

B. White, M. Pierce, N. Nassif, B. Cense, B. Park, G. Tearney, B. Bouma, T. Chen, and J. de Boer, “In vivo dynamic human retinal blood flow imaging using ultra-high-speed spectral domain optical coherence tomography,” Opt. Express 11(25), 3490–3497 (2003).
[Crossref] [PubMed]

B. Vakoc, S. Yun, J. de Boer, G. Tearney, and B. Bouma, “Phase-resolved optical frequency domain imaging,” Opt. Express 13(14), 5483–5493 (2005).
[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(1), 385–396 (2012).
[Crossref] [PubMed]

M. Szkulmowski, A. Szkulmowska, T. Bajraszewski, A. Kowalczyk, and M. Wojtkowski, “Flow velocity estimation using joint Spectral and Time domain Optical Coherence Tomography,” Opt. Express 16(9), 6008–6025 (2008).
[Crossref] [PubMed]

Y. K. Tao, A. M. Davis, and J. A. Izatt, “Single-pass volumetric bidirectional blood flow imaging spectral domain optical coherence tomography using a modified Hilbert transform,” Opt. Express 16(16), 12350–12361 (2008).
[Crossref] [PubMed]

J. Walther and E. Koch, “Relation of joint spectral and time domain optical coherence tomography (jSTdOCT) and phase-resolved Doppler OCT,” Opt. Express 22(19), 23129–23146 (2014).
[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]

V. J. Srinivasan, S. Sakadzić, I. Gorczynska, S. Ruvinskaya, W. Wu, J. G. Fujimoto, and D. A. Boas, “Quantitative cerebral blood flow with optical coherence tomography,” Opt. Express 18(3), 2477–2494 (2010).
[Crossref] [PubMed]

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] [PubMed]

V. J. Srinivasan, S. Sakadzić, I. Gorczynska, S. Ruvinskaya, W. Wu, J. G. Fujimoto, and D. A. Boas, “Quantitative cerebral blood flow with optical coherence tomography,” Opt. Express 18(3), 2477–2494 (2010).
[Crossref] [PubMed]

R. K. Wang and L. An, “Doppler optical micro-angiography for volumetric imaging of vascular perfusion in vivo,” Opt. Express 17(11), 8926–8940 (2009).
[Crossref] [PubMed]

A. Szkulmowska, M. Szkulmowski, D. Szlag, A. Kowalczyk, and M. Wojtkowski, “Three-dimensional quantitative imaging of retinal and choroidal blood flow velocity using joint Spectral and Time domain Optical Coherence Tomography,” Opt. Express 17(13), 10584–10598 (2009).
[Crossref] [PubMed]

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(26), 23736–23754 (2009).
[Crossref] [PubMed]

S. Yazdanfar, C. Yang, M. Sarunic, and J. Izatt, “Frequency estimation precision in Doppler optical coherence tomography using the Cramer-Rao lower bound,” Opt. Express 13(2), 410–416 (2005).
[Crossref] [PubMed]

B. Park, M. C. Pierce, B. Cense, S. H. Yun, M. Mujat, G. Tearney, B. Bouma, and J. de Boer, “Real-time fiber-based multi-functional spectral-domain optical coherence tomography at 1.3 microm,” Opt. Express 13(11), 3931–3944 (2005).
[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]

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]

Y. Wang, D. Huang, Y. Su, and X. S. Yao, “Two-dimensional phase unwrapping in Doppler Fourier domain optical coherence tomography,” Opt. Express 24(23), 26129–26145 (2016).
[Crossref] [PubMed]

H. C. Hendargo, M. Zhao, N. Shepherd, and J. A. Izatt, “Synthetic wavelength based phase unwrapping in spectral domain optical coherence tomography,” Opt. Express 17(7), 5039–5051 (2009).
[Crossref] [PubMed]

Opt. Lett. (3)

Phys. Rev. E (1)

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(2-1), 022604 (2016).
[Crossref] [PubMed]

Prog. Retin. Eye Res. (1)

R. A. Leitgeb, R. M. Werkmeister, C. Blatter, and L. Schmetterer, “Doppler optical coherence tomography,” Prog. Retin. Eye Res. 41, 26–43 (2014).
[Crossref] [PubMed]

Radio Sci. (1)

R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: Two-dimensional phase unwrapping,” Radio Sci. 23(4), 713–720 (1988).
[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 J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Sens. Actuators A Phys. (1)

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]

Ultramicroscopy (1)

T. Latychevskaia, P. Formanek, C. T. Koch, and A. Lubk, “Off-axis and inline electron holography: Experimental comparison,” Ultramicroscopy 110(5), 472–482 (2010).
[Crossref]

Other (4)

D. C. Ghiglia and M. D. Pritt, Two-dimensional phase unwrapping: theory, algorithms, and software (Wiley, New York, 1998).

N. I. Fisher, Statistical Analysis of Circular Data (Cambridge University Press, 1995).

E. Pijewska, I. Gorczynska, and M. Szkulmowski, “Matlab implementation of various forms of the Fast Phase Unwrapping algorithm for 2 and 3 dimensional data,” figshare (2019) [retrieved 11 Feb 2019], https://doi.org/10.6084/m9.figshare.7308413 .

M. Frigo and S. G. Johnson, “benchFFT” (2019), accessed on 15–01–2019, http://www.fftw.org/benchfft/ .

Supplementary Material (5)

NameDescription
» Code 1       Matlab implementation of various forms of the Fast Phase Unwrapping algorithm for 2 and 3 dimensional data.
» Visualization 1       2D and 3D fast phase unwrapping results in a set of simulated 3D Doppler OCT data with increasing noise level. Top: a plot of phase unwrapping error-metric (RMSE (psi)) vs. phase noise-metric (scirc (fi)). Dots represent mean RMSE (psi) values comput
» Visualization 2       Phase unwrapping in the flow phantom. Top: plot of the circular standard deviation values (scirc) in subsequent B-scans. The values were calculated in the region of interest indicated by yellow rectangles in wrapped phase images. Blue line – linear f
» Visualization 3       Comparison of 2D and 3D complex FPU method in imaging of the human retina in vivo with the eye intentionally misaligned to decrease the imaging sensitivity. Colors in the images represent DOCT phase values: red – blood flow direction against the inco
» Visualization 4       Comparison of 2D and 3D complex FPU method in imaging of the human retina in vivo with the eye intentionally misaligned to decrease the imaging sensitivity. Colors in the images represent DOCT phase values: red – blood flow direction against the inco

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Figures (6)

Fig. 1
Fig. 1 Graphical representation of the data flow in the fast phase unwrapping (FPU) algorithm. The wrapped phase distribution, φ (r), is the input data. It is used to calculate the phase estimate ψest (r). The phase correcting function Q(r) is computed as the difference between the phase estimate and the input phase scaled by 2π and rounded to the nearest integer numbers, i.e., it is an integer map of 2π phase wraps. The phase wrap map is multiplied by 2π and added to the input phase to yield the output unwrapped phase distribution ψ (r).
Fig. 2
Fig. 2 Result from the applications of 2D and 3D fast phase unwrapping in a set of simulated 3D Doppler OCT data at increasing noise levels. (a) A plot of phase unwrapping error metrics (RMSE (ψ)) vs. phase noise metrics (σcirc (φ)). Dots represent mean RMSE (ψ) values computed from data sets comprising 100 data points each. Dashed lines indicate the standard deviations of the mean RMSE [2D FPU algorithm (green) and 3D FPU algorithm (black) outcomes]. The blue plot shows a linear dependence of the RMSE (ψ) on σcirc (φ) in the absence of phase unwrapping errors in ψ (r), i.e. RMSE (ψ) is only affected by the noise level. The deviations of the 2D and 3D FPU RMSE plots from this reference line indicate an increasing probability of phase unwrapping errors in ψ (r). The inset presents an example of phase-wrapped input data φ (r). Yellow rectangle denote the region-of-interest used to calculate σcirc. (b) Example cross-sectional images of the unwrapped phase ψ (r) extracted from four selected 3D data sets. The values of the circular standard deviation σcirc (φ) in the input data are listed in the top left corners. The top row of images shows the results of the 2D FPU method, and the bottom row demonstrates the outcomes of the 3D FPU algorithm. The Doppler OCT images were intentionally not thresholded for noise elimination to demonstrate the influences of the 4FT FPU algorithms on the noise areas. The results of the algorithms for all the data points in the plots on the top are presented in Visualization 1.
Fig. 3
Fig. 3 Phase unwrapping using the 4FT FPU algorithm for multiple phase wraps in four sets of data at different maximum values of unwrapped phases ψmax. (a, b) Plots of phase unwrapping error metric (RMSE (ψ)) as a function of phase noise metric (σcirc (φ)) for 3D Doppler OCT data simulated at increasing noise levels. (a) 2D 4FT FPU and (b) 3D 4FT FPU. Dots represent the mean RMSE (ψ) values computed from data sets comprising 100 data points each. Dashed lines indicate the standard deviations of the mean RMSE. Colors represent different maximum values of unwrapped phases ψmax. The blue solid line shows a linear dependence of the RMSE (ψ) on σcirc (φ) in the absence of phase unwrapping errors in ψ (r), i.e., RMSE (ψ) is only affected by the noise level. The deviation of the 2D and 3D FPU RMSE plots from this reference line indicates increasing probability of phase unwrapping errors in ψ (r). As the number of phase wraps increases, the sensitivities of the algorithms to noise also increase. (c) An example of phase-wrapped input data φ (r) for σcirc. = 0.48 rad. 2D: Phase unwrapped using 2D 4FT FPU, and 3D: phase unwrapped using 3D 4FT FPU. Black rectangle denote the region-of-interest used to calculate σcirc.
Fig. 4
Fig. 4 Phase unwrapping in the flow phantom. (a) Plot of the circular standard deviation values (σcirc) in subsequent B-scans. The values were calculated based on the regions-of-interest indicated by the yellow rectangles in the wrapped phase images in (b); black line: linear fit to the data points. (b) A0: cross-sectional structural OCT images of the glass tube with milk as the circulating fluid at four locations in the 3D data set; φ: wrapped phase; 2D: phase unwrapped with the 2D 4FT FPU method, arrows B and C indicate the phase unwrapping artefacts introduced in the areas of noise. 3D: outcomes of the 3D 4FT FPU method, arrow A: error-free phase unwrapping of the flow, arrow D: erroneous phase unwrapping (phase has remained wrapped). The colors in the images represent Doppler OCT phase values (red: blood flow direction against the incoming beam of light, and blue: blood flow along the light propagation direction). Increasing color intensities indicate increasing axial flow velocity values as indicated by the color bar. The thresholding of the Doppler OCT images to remove the noise was intentionally avoided to demonstrate how the 4FT FPU algorithms influenced the noise areas. Yellow rectangle denote the region-of-interest used to calculate σcirc. A movie showing all the frames from the data set is presented in Visualization 2.
Fig. 5
Fig. 5 Comparison of 2D and 3D 4FT FPU methods in the imaging of the flow phantom with the use of multiple phase wraps. (a) Cross-sectional structural OCT image of the glass tube with milk as the circulating fluid. (b) Doppler OCT image with multiple phase wraps within the capillary lumen. Black rectangle denote the region-of-interest used to calculate σcirc. (c) Outcome of the 2D 4FT FPU method. (d) Results of the 3D 4FT FPU method.
Fig. 6
Fig. 6 Comparison of the 2D and 3D 4FT FPU methods in the imaging of the human retina in vivo in the vicinity of the head of the optic nerve. (e-h) The eye was intentionally misaligned to decrease the imaging sensitivity as compared to (a-d). (a, e) Structural OCT images of the eye fundus. Arrows A, B, C, and D, indicate large and visible vessels. (b, f) Doppler OCT image with phase wraps within the vessels. (c, g) Results following the use of the 2D 4FT FPU method. (d, h) Results following the use of the 3D 4FT FPU method. Arrows E and F point to small vessels which are not apparent in the structural image but are detected with Doppler OCT. Colors in the images represent the Doppler OCT phase values (red: blood flow direction against the incoming beam of light, blue: flow direction along the light propagation direction). Increasing color intensity indicates increasing axial flow velocity values as indicated by the color bar. The thresholding of the Doppler OCT images to remove the noise was intentionally avoided to demonstrate how the 4FT FPU algorithms influenced the noise areas. Yellow rectangles denote the regions-of-interest used to calculate the signal-to-noise ratio and σcirc. Movies showing all the frames from both data sets are presented in Visualization 3 (top) and Visualization 4 (bottom).

Tables (2)

Tables Icon

Table 1 Comparison of fast phase unwrapping (FPU) algorithms

Tables Icon

Table 2 Imaging protocols applied for the experiments with the use of the flow phantom and human eye.

Equations (23)

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ψ( r )=φ( r )+2πQ( r ),
Q ( r )= ( 2π ) 1 ( ψ est ( r )φ( r ) ).
Q ( r )= ( 2π ) 1 2 ( 2 ψ est ( r ) 2 φ( r ) ).
P( r )=exp( j ψ est ( r ) )=exp( jφ( r ) ),
2 P( r )=( P( r )j ψ est ( r ) )=P( r ) ( ψ est ( r ) ) 2 +jP( r ) 2 ψ est ( r ),
2 ψ est ( r )=Im{ P ( r ) 1 2 P( r ) }.
2 ψ est ( r )=Im{ exp( jφ( r ) ) 2 exp( jφ( r ) ) }= =cosφ( r ) 2 sinφ( r )sinφ( r ) 2 cosφ( r ).
ψ est ( r )= 2 [ cosφ( r ) 2 sinφ( r ) ] 2 [ sinφ( r ) 2 cosφ( r ) ].
2 g( r )= ( 2π ) 2 F 1 { K 2 F{ g( r ) } } 2 g( r )= ( 2π ) 2 F 1 { K 2 F{ g( r ) } },
ψ est ( r )= F 1 { K 2 F{ cosφ( r ) F 1 { K 2 F{ sinφ( r ) } } } }+ F 1 { K 2 F{ sinφ( r ) F 1 { K 2 F{ cosφ( r ) } } } }.
ψ est ( r )= 2 [ cosφ( r ) 2 sinφ( r )sinφ( r ) 2 cosφ( r ) ],
ψ est ( r )= F 1 { K 2 F{ cosφ( r ) F 1 { K 2 F{ sinφ( r ) } }sinφ( r ) F 1 { K 2 F{ cosφ( r ) } } } }.
Q( r )=round( ( 2π ) 1 ( ψ est ( r )φ( r ) ) )=round( Q ( r ) ).
ψ est ( r )= F 1 { K 2 F{ Im{ P ( r ) 1 F 1 { K 2 F{ P( r ) } } } } }.
A( r,t )=| A( r,t ) |exp( jφ(r,t) ),
φ D ( r,Δt )=Im{ ln( A( r,t ) A * ( r,t+Δt ) ) }=2 k z n v z ( r )Δt,
Δ t max = π 2 k z n v zmax ,
P( r,Δt )= A( r,t ) A * ( r,t+Δt ) | A( r,t ) A * ( r,t+Δt ) | =exp( j φ D (r,Δt) ).
v z (r)={ v max ( v max / R 2 ) | r | 2 for | r |<R (inside the capillary) 0 for | r |R (outside the capillary)
A( r,t )= A s ( r,t )+ A n ( r,t )=| A( r,t ) |exp[iφ(r,t)]+ A n ( r,t ),
A( r,t+Δt )= A s ( r,t+Δt )+ A n ( r,t+Δt )= =| A( r,t ) |exp[iφ(r,t)+i φ D (r,Δt)]+ A n ( r,t+Δt ).
σ circ ( φ( r,t ) )= 2ln( 1/ | rS p( φ( r,t ) )exp( iφ( r,t ) ) | ) ,
RMSE( ψ )= M 1 r ( ψ(r) φ D ref (r) ) 2 ,