Frequency estimation precision in Doppler optical coherence tomography using the Cramer-Rao lower bound

Siavash Yazdanfar; Changhuei Yang; Marinko V. Sarunic; Joseph A. Izatt

doi:10.1364/OPEX.13.000410

Frequency estimation precision in Doppler optical coherence tomography using the Cramer-Rao lower bound

Siavash Yazdanfar, Changhuei Yang, Marinko V. Sarunic, and Joseph A. Izatt

Optics Express
Vol. 13,
Issue 2,
pp. 410-416
(2005)
•doi: 10.1364/OPEX.13.000410

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Siavash Yazdanfar, Changhuei Yang, Marinko V. Sarunic, and Joseph A. Izatt, "Frequency estimation precision in Doppler optical coherence tomography using the Cramer-Rao lower bound," Opt. Express 13, 410-416 (2005)

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Abstract

Doppler optical coherence tomography (DOCT) is a technique for simultaneous cross-sectional imaging of tissue structure and blood flow. We derive the fundamental uncertainty limits on frequency estimation precision in DOCT using the Cramer-Rao lower bound in the case of additive (e.g., thermal, shot) noise. Experimental results from a mirror and a scattering phantom are used to verify the theoretical limits. Our results demonstrate that the stochastic nature of frequency noise influences the precision of flow imaging, and that the noise model must be selected judiciously in order to estimate the frequency precision.

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Z. P. Chen, T. E. Milner, D. Dave, and J. S. Nelson, “Optical Doppler tomographic imaging of fluid flow velocity in highly scattering media,” Opt. Lett. 22, 64–66 (1997).
[Crossref] [PubMed]
J. A. Izatt, M. D. Kulkarni, S. Yazdanfar, J. K. Barton, and A. J. Welch, “In vivo bidirectional color Doppler flow imaging of picoliter blood volumes using optical coherence tomography,” Opt. Lett. 22, 1439–1441 (1997).
[Crossref]
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, 1178–1181 (1991).
[Crossref] [PubMed]
S. Yazdanfar, A. M. Rollins, and J. A. Izatt, “Imaging and velocimetry of the human retinal circulation using color Doppler optical coherence tomography,” Opt. Lett. 25, 1448–1450 (2000).
[Crossref]
S. Yazdanfar, A. M. Rollins, and J. A. Izatt, “In vivo imaging of human retinal flow dynamics by color Doppler optical coherence tomography,” Arch. Ophthalmol. 121, 235–239 (2003).
[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. Express11, 3116–3121 (2003). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-23-3116
[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. Express11, 3490–3497 (2003). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-25-3490
[Crossref] [PubMed]
Y. Zhao, Z. P. Chen, C. Saxer, S. Xiang, J. F. de Boer, and J. S. Nelson, “Phase-resolved optical coherence tomography and optical Doppler tomography for imaging blood flow in human skin with fast scanning speed and high velocity sensitivity,” Opt. Lett. 25, 114–116 (2000).
[Crossref]
D. Piao, L. L. Otis, N. K. Dutta, and Q. Zhu, “Quantitative assessment of flow velocity-estimation algorithms for optical Doppler tomography imaging,” Appl. Opt. 41, 6118–6127 (2002).
[Crossref] [PubMed]
V. X. D. Yang, M. L. Gordon, S.-J. Tang, N. E. Marcon, G. Gardiner, B. Qi, S. Bisland, E. Seng-Yue, S. Lo, J. Pekar, B. C. Wilson, and I. A. Vitkin, “High speed, wide velocity dynamic range Doppler optical coherence tomography (Part III): in vivo endoscopic imaging of blood flow in the rat and human gastrointestinal tracts,” Opt. Express11, 2416–2424 (2003). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-19-2416
[Crossref] [PubMed]
Y. Imai and K. Tanaka, “Direct velocity sensing of flow distribution based on low-coherence interferometry,” J. Opt. Soc. Am. A 16, 2007–2012 (1999).
[Crossref]
S. Yazdanfar, M. D. Kulkarni, and J. A. Izatt, “High resolution imaging of in vivo cardiac dynamics using color Doppler optical coherence tomography,” Opt. Express1, 424–431 (1997). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-1-13-424
[Crossref] [PubMed]
S. Yazdanfar and J. A. Izatt, “Self-referenced Doppler optical coherence tomography,” Opt. Lett. 27, 2085–2087 (2002).
[Crossref]
H. L. Van Trees, Detection, estimation, and modulation theory, Pt. 1 (John Wiley & Sons, New York, 1968).
L. L. Scharf, Statistical signal processing: Detection, estimation, and time series analysis (Addison-Wesley Publishing Co., Reading, MA, 1991).
J. W. Goodman, Statistical Optics (John Wiley & Sons, Inc., Hoboken, NJ, 1985).
M. Bashkansky and J. Reintjes, “Statistics and reduction of speckle in optical coherence tomography,” Opt. Lett. 25, 545–547 (2000).
[Crossref]
M. D. Kulkarni, T. G. van Leeuwen, S. Yazdanfar, and J. A. Izatt, “Velocity estimation accuracy and frame rate limitations in color Doppler optical coherence tomography,” Opt. Lett. 23, 1057–1059 (1998).
[Crossref]
A. M. Rollins, S. Yazdanfar, J. K. Barton, and J. A. Izatt, “Real-time in vivo color Doppler optical coherence tomography,” J. Biomed. Opt. 7, 123–129 (2002).
[Crossref] [PubMed]
V. X. D. Yang, M. L. Gordon, B. Qi, J. Pekar, S. Lo, E. Seng-Yue, A. Mok, B. C. Wilson, and I. A. Vitkin, “High speed, wide velocity dynamic range Doppler optical coherence tomography (Part I): System design, signal processing, and performance,” Opt. Express11, 794–809 (2003). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-7-794
[Crossref] [PubMed]
D. A. Boas, K. K. Bizheva, and A. M. Siegel, “Using dynamic low-coherence interferometry to image Brownian motion within highly scattering media,” Opt. Lett. 23, 319–321 (1998).
[Crossref]

2003 (1)

S. Yazdanfar, A. M. Rollins, and J. A. Izatt, “In vivo imaging of human retinal flow dynamics by color Doppler optical coherence tomography,” Arch. Ophthalmol. 121, 235–239 (2003).
[PubMed]

2002 (3)

D. Piao, L. L. Otis, N. K. Dutta, and Q. Zhu, “Quantitative assessment of flow velocity-estimation algorithms for optical Doppler tomography imaging,” Appl. Opt. 41, 6118–6127 (2002).
[Crossref] [PubMed]

S. Yazdanfar and J. A. Izatt, “Self-referenced Doppler optical coherence tomography,” Opt. Lett. 27, 2085–2087 (2002).
[Crossref]

A. M. Rollins, S. Yazdanfar, J. K. Barton, and J. A. Izatt, “Real-time in vivo color Doppler optical coherence tomography,” J. Biomed. Opt. 7, 123–129 (2002).
[Crossref] [PubMed]

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, 1178–1181 (1991).
[Crossref] [PubMed]

Bajraszewski, T.

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. Express11, 3116–3121 (2003). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-23-3116
[Crossref] [PubMed]

Barton, J. K.

A. M. Rollins, S. Yazdanfar, J. K. Barton, and J. A. Izatt, “Real-time in vivo color Doppler optical coherence tomography,” J. Biomed. Opt. 7, 123–129 (2002).
[Crossref] [PubMed]

Bashkansky, M.

M. Bashkansky and J. Reintjes, “Statistics and reduction of speckle in optical coherence tomography,” Opt. Lett. 25, 545–547 (2000).
[Crossref]

Bisland, S.

V. X. D. Yang, M. L. Gordon, S.-J. Tang, N. E. Marcon, G. Gardiner, B. Qi, S. Bisland, E. Seng-Yue, S. Lo, J. Pekar, B. C. Wilson, and I. A. Vitkin, “High speed, wide velocity dynamic range Doppler optical coherence tomography (Part III): in vivo endoscopic imaging of blood flow in the rat and human gastrointestinal tracts,” Opt. Express11, 2416–2424 (2003). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-19-2416
[Crossref] [PubMed]

Bizheva, K. K.

D. A. Boas, K. K. Bizheva, and A. M. Siegel, “Using dynamic low-coherence interferometry to image Brownian motion within highly scattering media,” Opt. Lett. 23, 319–321 (1998).
[Crossref]

Boas, D. A.

D. A. Boas, K. K. Bizheva, and A. M. Siegel, “Using dynamic low-coherence interferometry to image Brownian motion within highly scattering media,” Opt. Lett. 23, 319–321 (1998).
[Crossref]

Bouma, B. E.

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. Express11, 3490–3497 (2003). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-25-3490
[Crossref] [PubMed]

Cense, B.

Chang, W.

Chen, T. C.

Chen, Z. P.

Z. P. Chen, T. E. Milner, D. Dave, and J. S. Nelson, “Optical Doppler tomographic imaging of fluid flow velocity in highly scattering media,” Opt. Lett. 22, 64–66 (1997).
[Crossref] [PubMed]

Dave, D.

Z. P. Chen, T. E. Milner, D. Dave, and J. S. Nelson, “Optical Doppler tomographic imaging of fluid flow velocity in highly scattering media,” Opt. Lett. 22, 64–66 (1997).
[Crossref] [PubMed]

de Boer, J. F.

Drexler, W.

Dutta, N. K.

Fercher, A. F.

Flotte, T.

Fujimoto, J. G.

Gardiner, G.

Goodman, J. W.

J. W. Goodman, Statistical Optics (John Wiley & Sons, Inc., Hoboken, NJ, 1985).

Gordon, M. L.

V. X. D. Yang, M. L. Gordon, B. Qi, J. Pekar, S. Lo, E. Seng-Yue, A. Mok, B. C. Wilson, and I. A. Vitkin, “High speed, wide velocity dynamic range Doppler optical coherence tomography (Part I): System design, signal processing, and performance,” Opt. Express11, 794–809 (2003). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-7-794
[Crossref] [PubMed]

Gregory, K.

Hee, M. R.

Huang, D.

Imai, Y.

Y. Imai and K. Tanaka, “Direct velocity sensing of flow distribution based on low-coherence interferometry,” J. Opt. Soc. Am. A 16, 2007–2012 (1999).
[Crossref]

Izatt, J. A.

S. Yazdanfar, A. M. Rollins, and J. A. Izatt, “In vivo imaging of human retinal flow dynamics by color Doppler optical coherence tomography,” Arch. Ophthalmol. 121, 235–239 (2003).
[PubMed]

A. M. Rollins, S. Yazdanfar, J. K. Barton, and J. A. Izatt, “Real-time in vivo color Doppler optical coherence tomography,” J. Biomed. Opt. 7, 123–129 (2002).
[Crossref] [PubMed]

S. Yazdanfar and J. A. Izatt, “Self-referenced Doppler optical coherence tomography,” Opt. Lett. 27, 2085–2087 (2002).
[Crossref]

S. Yazdanfar, M. D. Kulkarni, and J. A. Izatt, “High resolution imaging of in vivo cardiac dynamics using color Doppler optical coherence tomography,” Opt. Express1, 424–431 (1997). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-1-13-424
[Crossref] [PubMed]

Kulkarni, M. D.

Leitgeb, R. A.

Lin, C. P.

Lo, S.

Marcon, N. E.

Milner, T. E.

Z. P. Chen, T. E. Milner, D. Dave, and J. S. Nelson, “Optical Doppler tomographic imaging of fluid flow velocity in highly scattering media,” Opt. Lett. 22, 64–66 (1997).
[Crossref] [PubMed]

Mok, A.

Nassif, N.

Nelson, J. S.

Z. P. Chen, T. E. Milner, D. Dave, and J. S. Nelson, “Optical Doppler tomographic imaging of fluid flow velocity in highly scattering media,” Opt. Lett. 22, 64–66 (1997).
[Crossref] [PubMed]

Otis, L. L.

Park, B. H.

Pekar, J.

Piao, D.

Pierce, M. C.

Puliafito, C. A.

Qi, B.

Reintjes, J.

M. Bashkansky and J. Reintjes, “Statistics and reduction of speckle in optical coherence tomography,” Opt. Lett. 25, 545–547 (2000).
[Crossref]

Rollins, A. M.

S. Yazdanfar, A. M. Rollins, and J. A. Izatt, “In vivo imaging of human retinal flow dynamics by color Doppler optical coherence tomography,” Arch. Ophthalmol. 121, 235–239 (2003).
[PubMed]

A. M. Rollins, S. Yazdanfar, J. K. Barton, and J. A. Izatt, “Real-time in vivo color Doppler optical coherence tomography,” J. Biomed. Opt. 7, 123–129 (2002).
[Crossref] [PubMed]

Saxer, C.

Scharf, L. L.

L. L. Scharf, Statistical signal processing: Detection, estimation, and time series analysis (Addison-Wesley Publishing Co., Reading, MA, 1991).

Schmetterer, L.

Schuman, J. S.

Seng-Yue, E.

Siegel, A. M.

D. A. Boas, K. K. Bizheva, and A. M. Siegel, “Using dynamic low-coherence interferometry to image Brownian motion within highly scattering media,” Opt. Lett. 23, 319–321 (1998).
[Crossref]

Stinson, W. G.

Swanson, E. A.

Tanaka, K.

Y. Imai and K. Tanaka, “Direct velocity sensing of flow distribution based on low-coherence interferometry,” J. Opt. Soc. Am. A 16, 2007–2012 (1999).
[Crossref]

Tang, S.-J.

Tearney, G. J.

van Leeuwen, T. G.

Van Trees, H. L.

H. L. Van Trees, Detection, estimation, and modulation theory, Pt. 1 (John Wiley & Sons, New York, 1968).

Vitkin, I. A.

Welch, A. J.

White, B. R.

Wilson, B. C.

Xiang, S.

Yang, V. X. D.

Yazdanfar, S.

S. Yazdanfar, A. M. Rollins, and J. A. Izatt, “In vivo imaging of human retinal flow dynamics by color Doppler optical coherence tomography,” Arch. Ophthalmol. 121, 235–239 (2003).
[PubMed]

A. M. Rollins, S. Yazdanfar, J. K. Barton, and J. A. Izatt, “Real-time in vivo color Doppler optical coherence tomography,” J. Biomed. Opt. 7, 123–129 (2002).
[Crossref] [PubMed]

S. Yazdanfar and J. A. Izatt, “Self-referenced Doppler optical coherence tomography,” Opt. Lett. 27, 2085–2087 (2002).
[Crossref]

Zawadzki, R. J.

Zhao, Y.

Zhu, Q.

Appl. Opt. (1)

Arch. Ophthalmol. (1)

S. Yazdanfar, A. M. Rollins, and J. A. Izatt, “In vivo imaging of human retinal flow dynamics by color Doppler optical coherence tomography,” Arch. Ophthalmol. 121, 235–239 (2003).
[PubMed]

J. Biomed. Opt. (1)

A. M. Rollins, S. Yazdanfar, J. K. Barton, and J. A. Izatt, “Real-time in vivo color Doppler optical coherence tomography,” J. Biomed. Opt. 7, 123–129 (2002).
[Crossref] [PubMed]

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

Y. Imai and K. Tanaka, “Direct velocity sensing of flow distribution based on low-coherence interferometry,” J. Opt. Soc. Am. A 16, 2007–2012 (1999).
[Crossref]

Opt. Lett. (8)

M. Bashkansky and J. Reintjes, “Statistics and reduction of speckle in optical coherence tomography,” Opt. Lett. 25, 545–547 (2000).
[Crossref]

S. Yazdanfar and J. A. Izatt, “Self-referenced Doppler optical coherence tomography,” Opt. Lett. 27, 2085–2087 (2002).
[Crossref]

Z. P. Chen, T. E. Milner, D. Dave, and J. S. Nelson, “Optical Doppler tomographic imaging of fluid flow velocity in highly scattering media,” Opt. Lett. 22, 64–66 (1997).
[Crossref] [PubMed]

D. A. Boas, K. K. Bizheva, and A. M. Siegel, “Using dynamic low-coherence interferometry to image Brownian motion within highly scattering media,” Opt. Lett. 23, 319–321 (1998).
[Crossref]

Science (1)

Other (8)

H. L. Van Trees, Detection, estimation, and modulation theory, Pt. 1 (John Wiley & Sons, New York, 1968).

L. L. Scharf, Statistical signal processing: Detection, estimation, and time series analysis (Addison-Wesley Publishing Co., Reading, MA, 1991).

J. W. Goodman, Statistical Optics (John Wiley & Sons, Inc., Hoboken, NJ, 1985).

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Fig. 1.

Fiber-optic, time-domain Michelson interferometer used to measure the frequency precision of Doppler OCT. BPF, band-pass filter; A/D, analog-to-digital converter; I, Q, in-phase and quadrature signal components after demodulation.

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Fig. 2.

Algorithm for calculating frequency precision and signal-to-noise ratio (SNR). The OCT reflectance image of Liposyn is shown on the left, with the arrow indicating the direction of the incident beam. The standard deviation of the noise is calculated in the region of interest (ROI) located outside of the sample. At each axial position, the average signal value across 100 scans is measured and divided by the noise to measure local SNR. At the same depth, the variance of the Doppler frequency (right) is measured and compared to the theoretical frequency precision.

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Fig. 3.

Experimental and theoretical frequency standard deviation (SD) versus SNR for A) additive noise only and B) both additive and multiplicative (speckle) noise. The observation time was different for the two experiments, resulting in different theoretical limits. The inset of A), magnifying lower SNR values on a log-log scale, suggests that the autocorrelation algorithm used to estimate frequency is a maximum likelihood estimator. CRLB, Cramer-Rao lower bound based on the additive model of Eqs. (2) and (5).

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Equations (10)

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

s (t) = A (t) \cos [2 π (f_{r} - f_{s}) t + β (t)],

s (t) = A (t) \cos [2 π f_{s} t + β (t)], 0 \leq t \leq t_{0}

f_{s}^{min} = \frac{1}{t_{0}}

Var [\hat{α} (\overset{⃑}{R}) - α] \geq {(E {{[\frac{\partial \ln p (\overset{⃑}{R} ∣ α)}{\partial α}]}^{2}})}^{- 1}

r (t) = s (t, α) + n (t)

σ_{\hat{α}}^{2} \geq {(\frac{1}{N_{0}} \int_{0}^{t_{0}} {[\frac{\partial s (t, α)}{\partial α}]}^{2} d t)}^{- 1}

σ_{\hat{f}}^{2} \geq {(\frac{2 π^{2}}{3} \frac{A^{2}}{N_{0}} t_{0}^{2})}^{- 1}

f_{s}^{min} \approx σ \propto \frac{1}{t_{0} \sqrt{SNR}}

R (τ) = ∣ R (τ) ∣ \exp [- i ϕ (τ)] \equiv 〈 s^{*} (t) s (t + τ) 〉

Ω = 16 {(π ⁄ λ_{0})}^{2} D_{T} [rad ⁄ s]

Abstract

References

2003 (1)

2002 (3)

2000 (3)

1999 (1)

1998 (2)

1997 (2)

1991 (1)

Bajraszewski, T.

Barton, J. K.

Bashkansky, M.

Bisland, S.

Bizheva, K. K.

Boas, D. A.

Bouma, B. E.

Cense, B.

Chang, W.

Chen, T. C.

Chen, Z. P.

Dave, D.

de Boer, J. F.

Drexler, W.

Dutta, N. K.

Fercher, A. F.

Flotte, T.

Fujimoto, J. G.

Gardiner, G.

Goodman, J. W.

Gordon, M. L.

Gregory, K.

Hee, M. R.

Huang, D.

Imai, Y.

Izatt, J. A.

Kulkarni, M. D.

Leitgeb, R. A.

Lin, C. P.

Lo, S.

Marcon, N. E.

Milner, T. E.

Mok, A.

Nassif, N.

Nelson, J. S.

Otis, L. L.

Park, B. H.

Pekar, J.

Piao, D.

Pierce, M. C.

Puliafito, C. A.

Qi, B.

Reintjes, J.

Rollins, A. M.

Saxer, C.

Scharf, L. L.

Schmetterer, L.

Schuman, J. S.

Seng-Yue, E.

Siegel, A. M.

Stinson, W. G.

Swanson, E. A.

Tanaka, K.

Tang, S.-J.

Tearney, G. J.

van Leeuwen, T. G.

Van Trees, H. L.

Vitkin, I. A.

Welch, A. J.

White, B. R.

Wilson, B. C.

Xiang, S.

Yang, V. X. D.

Yazdanfar, S.

Zawadzki, R. J.

Zhao, Y.

Zhu, Q.

Appl. Opt. (1)

Arch. Ophthalmol. (1)

J. Biomed. Opt. (1)

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