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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 22, Iss. 4 — Feb. 24, 2014
  • pp: 4830–4848
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Noise statistics of phase-resolved optical coherence tomography imaging: single-and dual-beam-scan Doppler optical coherence tomography

Shuichi Makita, Franck Jaillon, Israt Jahan, and Yoshiaki Yasuno  »View Author Affiliations


Optics Express, Vol. 22, Issue 4, pp. 4830-4848 (2014)
http://dx.doi.org/10.1364/OE.22.004830


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Abstract

Noise statistics of phase-resolved optical coherence tomography (OCT) imaging are complicated and involve noises of OCT, correlation of signals, and speckles. In this paper, the statistical properties of phase shift between two OCT signals that contain additive random noises and speckle noises are presented. Experimental results obtained with a scattering tissue phantom are in good agreement with theoretical predictions. The performances of the dual-beam method and conventional single-beam method are compared. As expected, phase shift noise in the case of the dual-beam-scan method is less than that for the single-beam method when the transversal sampling step is large.

© 2014 Optical Society of America

1. Introduction

Phase-resolved optical coherence tomography (OCT) is a powerful extension to several functional imaging. For example, cross-sectional flow images are obtained by using the Doppler phase shift caused by the motion of blood cells [1

1. Y. Zhao, Z. Chen, C. Saxer, S. Xiang, J. F. d. Boer, and J. S. Nelson, “Phase-resolved optical coherence tomography and optical Doppler tomography for imaging blood flow in human skin with fast scanning speed and high velocity sensitivity,” Opt. Lett. 25, 114–116 (2000). [CrossRef]

7

7. B. Braaf, K. A. Vermeer, V. A. D. Sicam, E. van Zeeburg, J. C. van Meurs, and J. F. de Boer, “Phase-stabilized optical frequency domain imaging at 1-m for the measurement of blood flow in the human choroid,” Opt. Express 19, 20886–20903 (2011). [CrossRef] [PubMed]

], the cross-sectional biomechanical property can be mapped by detecting local deformation from the phase of OCT [8

8. R. K. Wang, S. Kirkpatrick, and M. Hinds, “Phase-sensitive optical coherence elastography for mapping tissue microstrains in real time,” Appl. Phys. Lett. 90, 164105, 2007). [CrossRef]

10

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

], and the local photothermal effect can be detected [11

11. T. Akkin, D. P. Dav, J.-I. Youn, S. A. Telenkov, H. G. R. III, and T. E. Milner, “Imaging tissue response to electrical and photothermal stimulation with nanometer sensitivity,” Lasers Surg. Med. 33, 219–225 (2003). [CrossRef] [PubMed]

14

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

]. Because these methods are based on the OCT technique [15

15. W. Drexler and J. G. Fujimoto, Optical Coherence Tomography: Technology and Applications (Springer, 2008). [CrossRef]

], they allow three-dimensional high-resolution imaging.

Recently, our and other groups introduced the dual-beam-scan Doppler detection method, where two probing beams are separated along the scanning direction, to increase the sensitivity to motion [19

19. S. Makita, F. Jaillon, M. Yamanari, M. Miura, and Y. Yasuno, “Comprehensive in vivo micro-vascular imaging of the human eye by dual-beam-scan Doppler optical coherence angiography,” Opt. Express 19, 1271–1283 (2011). [CrossRef] [PubMed]

21

21. F. Jaillon, S. Makita, E.-J. Min, B. H. Lee, and Y. Yasuno, “Enhanced imaging of choroidal vasculature by high-penetration and dual-velocity optical coherence angiography,” Biomed. Opt. Express 2, 1147–1158 (2011). [CrossRef] [PubMed]

]. This dual-beam-scan Doppler method measures phase shift between two OCT signals obtained with two probing beams. Because the detection scheme and signal processing of this technique differ from those of conventional phase-resolved OCT, evaluation of its performance is difficult.

In this paper, statistics of phase-resolved OCT imaging with additive, speckle, and decorre-lation noises are addressed. The statistics of generalized phase-resolved OCT are formulated in Section 2. The essential parameter is correlation coefficient between two OCT signals and described with specifications of OCT (Section 2.2). The performances of Doppler OCT with conventional single-beam and dual-beam methods are presented in Section 3 according to the formulation in Section 2. We evaluate the performances of phase-resolved Doppler OCT. Phase-resolved imaging performances are compared between the dual-beam-scan and conventional single-beam methods in phantom tissue experiments (Section 4).

2. Statistics of phase-resolved OCT

Table 1. List of notations

table-icon
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2.1. Statistics of phase shift

The statistics of phase-resolved OCT with additive, speckle, and decorrelation noises are described here. A statistical model with speckle, where the OCT signals vary randomly, is assumed. It is then shown that the effect of additive noise in complex OCT signals can be expressed as a part of decorrelation.

For phase-resolved imaging, the phase shift between two complex OCT signals is used. By considering the two measurements as random variables G1, G2, phase shift is calculated as a phase term of the Hermitian product of two measured OCT signals;
Δϕ=arg[g1*g2],
(1)
where g1 and g2 are respectively realizations of G1 and G2. Here, the measured OCT signals G1 and G2 are the sum of complex signals S1 and S2 and additive noises N1 and N2, respectively.

G1=S1+N1,G2=S2+N2.
(2)

An realization of OCT signal s is the sum of interference signals from scatterers in a coherent detection volume:
s=aRmamexp[i2kc(zmz0)],
(3)
where am is the amplitude of scattered light from the m-th scatterer, and zm is the axial location of the m-th scatterer. aR is the amplitude of the reference light, kc is the central wave number of a broadband light source, and z0 is the depth at zero delay of the interferometer. By assuming that the scatterers’ distribution is random and the density is high compared with resolutions of OCT, the OCT signal is random in space and exhibits fully developed speckle. Hence, S1 and S2 can be considered as zero-mean complex circular Gaussian variables.

By considering that the additive noises N1 and N2 are zero-mean complex circular Gaussian variables and independent of each other and the signals S1 and S2, the measured signals G1 and G2 are also zero-mean complex circular Gaussian variables. The statistical properties of the product of two complex zero-mean circular Gaussian variables have been studied in the field of synthetic aperture radar [22

22. L. Jong-Sen, K. Hoppel, S. Mango, and A. Miller, “Intensity and phase statistics of multilook polarimetric and interferometric SAR imagery,” IEEE Trans. Geosci. Remote Sens. 32, 1017–1028 (1994). [CrossRef]

, 23

23. R. J. A. Tough, D. Blacknell, and S. Quegan, “A statistical description of polarimetric and interferometric synthetic aperture radar data,” Proc. R. Soc. Lond. A 449, 567–589 (1995). [CrossRef]

] and it is known that the probability density function (PDF) of the sample phase shift Δϕ (Eq. (1)) can be expressed as
pΔΦ(Δϕ|ρ,Δϕ0)=1ρ22π{β(π2+sin1β)[1β2]3/2+11β2},
(4)
where β = ρ cos(Δϕ − Δϕ0). Additionally,
ρeiΔϕ0=E[G1*G2]E[|G1|2]E[|G2|2],
(5)
where E[·] is the expectation operator. Equation (5) indicates that the parameters ρ and Δϕ0 are the amplitude and phase of the population complex correlation coefficient for the measured signals, respectively. Δϕ0 represents the population phase shift.

The expectation and standard deviation of the sample phase shift Δϕ are described as
E[Δϕ]=ρsinΔϕ01ρ2cos2Δϕ0cos1(ρcosΔϕ0),
(6)
σΔϕ=1ρ21β2[π24πsin1β+(sin1β)2]+π212Li2(ρ2)2,
(7)
where β′ = ρ cosΔϕ0 and Li2 is Euler’s dilogarithm. The estimators of the expectation and standard deviation are the arithmetic mean and sample standard deviation:
Δϕ¯=1Nn=1NΔϕn,
(8)
SΔϕ=1Nn=1N(ΔϕnΔϕ¯)2,
(9)
where N is the number of realizations.

2.2. Correlation coefficient of OCT signals

The correlation coefficient ρ is an essential parameter for defining the statistics of the phase shift. Hence, it determines the performance of phase-resolved OCT. Here, the generalized formulation of correlation coefficient ρ of OCT is described and can be applied to both conventional and dual-beam-scan OCT. The estimations of correlation coefficients are then presented.

The parameter ρ can be described as the following according to the definition of measured signals G1, G2 (Eq. (2));
ρ=ρs(1+SNR11)(1+SNR21),
(10)
where
ρs=|E[S1*S2]|E[|S1|2]E[|S2|2].
(11)
Equation (11) gives the correlation coefficient between two OCT signals S1 and S2 and SNRi = E[|Si|2]/E[|Ni|2] (i=1,2) are the expected signal-to-noise ratios of each measurement. As mentioned in the following section (Section 2.2), ρs is decreased by means of the displacement of the sampling location on tissue, tissue deformation, scattering, and also, in the case of dual-beam-scan OCT, differences in the system properties between two detections. It can be understood that the denominator of Eq. (10) represents the degree of decorrelation caused by additive random noise. For simplicity, here we define a representative of the SNR as
11+ESNR11(1+SNR11)(1+SNR21).
(12)

Note that the previously presented formula for decorrelation noise of Doppler OCT (Eq. (10) in [18

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

]) is identical to Eq. (4) when ρ = α2: i.e., the correlation coefficient depends only on transversal sampling displacement, and Δϕ0 = 0. However, the current model presented here includes the effects of both additive noise and speckle noise. The measured OCT signal G is assumed to be the sum of the varying signal S and noise N. The effect of speckle on phase shift might be accounted for by the varying instantaneous signal-to-noise ratio of each realization |s|2/|n|2.

The population correlation coefficient of OCT signals in Eq. (11) and population phase shift Δϕ0 can be described from the signal cross-correlation coefficient between realizations s1, s2:
ρseiΔϕ0ρS1,S2(t1,t2)=E[s1*(t1)s2(t2)]E[|s1(t1)|2]E[|s2(t2)|2].
(14)

Considering Gaussian beam profiles and a Gaussian coherence function of the light source, PSFs are expressed as
hχ(xL,yL,zL)e2xL2wχ2e2yL2wχ2eΔkχ2zL28e2ikcχ(zLz0)(χ=1,2),
(15)
where wχ is the beam spot radius at 1/e2 of the χ-th channel. Δk is the full width at 1/e2 of a Gaussian spectrum of a light source in unit of wavenumber. Then, the population correlation coefficient of OCT signals ρs and population phase shift Δϕ0 are obtained by substituting Eq. (15) and (13) into Eq. (14):
ρsρη1,η2×2w1w2w12+w222Δk1Δk2Δk12+Δk22×e2x(t1,t2)2+y(t1,t2)2w12+w22eΔk12Δk22Δk12+Δk22z(t1,t2)28×e8(kc1kc2)2Δk12+Δk22×e4kc12Δk22+kc22Δk12Δk12+Δk22D(t2+t12)(t2t1),
(16)
Δϕ02kc1Δk22+kc2Δk12Δk12+Δk22z(t1,t2),
(17)
where ρη1,η2 denotes the correlation coefficient of the scattering process between two channels and (x′(t1, t2), y′(t1, t2), z′(t1, t2)) = r2(t2) − r1(t1) is the displacement of the sampling point between two channels caused by motions of the sample and/or beam scan. D(t) is the diffusivity at time t owing to the diffusion process. Note that the shift between spectra of the two channels kc1kc2 decreases the correlation coefficient.

In the case of solid tissues (no diffusion and no deformation), the correlation coefficient can be defined as:
ρs,SSρη1,η2|ρh1,h2(r(Δt))|,
(18)
where r′t) = r2(t + Δt) − r1(t) and ρh1,h2 is the correlation coefficient of PSFs of the two channels:
ρh1,h2(r)=2w1w2w12+w222Δk1Δk2Δk12+Δk22e2x2+y2w12+w22eΔk12Δk22Δk12+Δk22z28e8(kc1kc2)2Δk12+Δk12e2ikc1Δk22+kc2Δk12Δk12+Δk22z.
(19)

The estimation of the parameter ρ is the sample correlation between realizations g1 and g2:
r=|g1*(t1)g2(t2)¯||g1(t1)|2¯|g2(t2)|2¯.
(20)
Because the sample correlation r is a biased estimation for ρ, a large number of realizations are required for accurate estimation.

According to Eq. (10), the correlation coefficient of the OCT signal ρ̂s can be estimated as
ρ^s=(1+ESNR^1)r=|g1*(t1)g2(t2)¯|[|g1(t1)|2¯|n1|2¯][|g2(t2)|2¯|n2|2¯],
(21)
where ESNR^ is the estimation of the representative SNR (Eq. (12)):
1+ESNR^1=|g1(t1)|2¯|g2(t2)|2¯[|g1(t1)|2¯|n1|2¯][|g2(t2)|2¯|n2|2¯].
(22)
The mean power of noise |nχ|2¯ will be estimated from the noise floor level of the OCT system without any tissue.

2.3. Maximum likelihood estimation of phase shift

The mean of phase shift Δϕ (Eq. (8)) is a biased estimator for Δϕ0. According to the expectation (Eq. (6)), the mean estimator results in large offset of the estimation from the population parameter Δϕ0 when it is close to the boundaries of phase measurement range [27

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

]. The maximum likelihood estimation (MLE) of parameter Δϕ0 will be used for better estimation. The MLE of population phase shift Δϕ0 with ν independent realizations g1(κ) and g2(κ) (κ = 1, ...,ν) is [23

23. R. J. A. Tough, D. Blacknell, and S. Quegan, “A statistical description of polarimetric and interferometric synthetic aperture radar data,” Proc. R. Soc. Lond. A 449, 567–589 (1995). [CrossRef]

]
Δϕ^0=arg[κ=1νg1(κ)*g2(κ)].
(23)
The PDF of the MLE for phase shift becomes [22

22. L. Jong-Sen, K. Hoppel, S. Mango, and A. Miller, “Intensity and phase statistics of multilook polarimetric and interferometric SAR imagery,” IEEE Trans. Geosci. Remote Sens. 32, 1017–1028 (1994). [CrossRef]

, 23

23. R. J. A. Tough, D. Blacknell, and S. Quegan, “A statistical description of polarimetric and interferometric synthetic aperture radar data,” Proc. R. Soc. Lond. A 449, 567–589 (1995). [CrossRef]

]
pΔΦ^0(Δϕ^0|ρ,Δϕ0)=Γ(ν+12)(1ρ2)νρcos(Δϕ^0Δϕ0)2πΓ(ν)[1ρ2cos2(Δϕ^0Δϕ0)]ν+1/2+(1ρ2)ν2πF12(ν,1;12;ρ2cos2(Δϕ^0Δϕ0)),
(24)
where 2F1 is Gauss hypergeometric function.

2.4. Practical estimators

The MLE of phase shift Δϕ̂0 has been shown as Eq. (23). However, in the real case, it is almost impossible to acquire several independent samples for a single location.

Frequently, spatial averaging around the point of interest in an image is applied [4

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

, 28

28. V. X. Yang, M. L. Gordon, A. Mok, Y. Zhao, Z. Chen, R. S. Cobbold, B. C. Wilson, and I. A. Vitkin, “Improved phase-resolved optical Doppler tomography using Kasai velocity estimator and histogram segmentation,” Opt. Commun. 208, 209–214 (2002). [CrossRef]

]. The extension of the MLE to a spatial moving average is
Δϕ^0=arg[iIjJlLg1*(x1+i,y1+j,z1+l)g2(x1+i,y1+j,z1+l)],
(26)
where (I, J, L) is the size of the three-dimensional averaging window. In the averaging window, a tissue should be homogeneous and statistical parameters constant; i.e., the temporal changes between two signals and detection conditions should be equivalent. That means that motion of the sample and scanning speed of the probing beam should be constant and deformation of objects is equivalent inside the window. The problem is that the realizations, the Hermitian products within a moving window, are not independent of each other. The detection regions of each realization partially overlap owing to the spatial extent of the PSF. Hence, the number of independent realizations is not equal to the number of realizations within the window νIJL.

To estimate the moments of the estimated phase shift and the sample correlation coefficient, the effective number of independent samples (ENIS) within an averaging window should be known. Taking the analogy of synthetic aperture radar [29

29. A. Moreira, “Improved multilook techniques applied to SAR and SCANSAR imagery,” IEEE Trans. Geosci. Remote Sens. 29, 529–534, 1991). [CrossRef]

], the ENIS can be defined using the cross-correlation coefficient between Hermitian products as
ENIS=I1+2i=1I1IlIρg22(iΔx,0,0)J1+2j=1J1JlJρg22(0,jΔy,0)×L1+2l=1L1LlLρg22(0,0,lΔz),
(27)
where (Δx, Δy, Δz) is the spatial separation between neighboring pixels in the image along each direction. ρg2 (iΔx, 0, 0) is the correlation coefficient between two Hermitian products with the displacement of i image pixels in the x-direction;
ρg2(iΔx,0,0)=|E[G1*(x1)G2(x1)G1(x1+i)G2*(x1+i)]E[G1*(x1)G2(x1)]E[G1(x1+i)G2*(x1+i)]|E[|G1*(x1)G2(x1)E[G1*(x1)G2(x1)]|2]E[|G1*(x1+i)G2(x1+i)E[G1*(x1+i)G2(x1+i)]|2].
(28)

In the case of solid tissues, the correlation coefficient can be described by expanding the fourth-order moment in Eq. (28) [30

30. J. S. Bendat and A. G. Piersol, Random Data: Analysis and Measurement Procedures (John Wiley and Sons, 2010). [CrossRef]

] and using Eq. (18):
ρg2,SS(iΔx,jΔy,lΔz)=1(1+ESNR1)2|ρh1,h1*(iΔx,jΔy,lΔz)ρh2,h2(iΔx,jΔy,lΔz)|,
(29)
ρh1,h1 and ρh2,h2 are the auto-correlation coefficients of each channel;
|ρh1,h1*(iΔx,jΔy,lΔz)ρh2,h2(iΔx,jΔy,lΔz)|=ew12+w22w12w22[(iΔx)2+(jΔy)2]e(lΔz)216(Δk12+Δk22).
(30)

3. Performance of flow imaging with phase-resolved OCT

Phase-resolved imaging is a common method for cross-sectional flow imaging by OCT. The phase shift between OCT signals at different time points is caused by axial movements of samples, and expressed as
Δϕflow=2nkcVΔtcosθ,
(31)
where symbols are defined as follows: V, the velocity of moving tissue; θ, Doppler angle; Δt, the time delay between the two time points; and n, the refractive index of the sample.

The sensitivity of flow imaging is defined by the minimum detectable flow in images. This minimum detectable flow can be defined as the velocity corresponding to the random variation of the phase shift for surrounding solid tissue.
vmin=KσΔϕSSΔt,
(32)
where σΔϕSS indicates a standard deviation of the spatial distribution of the phase shift for the surrounding solid tissue. K = 1/2nkc cosθ is a factor depending on the tissue and system features. Equation (32) clearly shows that longer time delay and smaller phase shift noise increase the sensitivity of flow imaging. To compare the phase-resolved flow imaging performances of conventional Doppler OCT and dual-beam-scan OCT, phase noise in each method is defined in the following sections.

3.1. Conventional phase-resolved Doppler OCT

Conventional Doppler OCT uses a single probe beam and single detection channel, and applies auto-correlation processing to obtain the phase shift. In this case, h1 = h2 and η1 = η2. Under this condition, the signal correlation coefficient with a static tissue is obtained from Eqs. (18) and (19) as
ρs,SS(SB)=exb2+yb2w2,
(33)
where (x′b, y′b) is the transversal displacement of a probing beam between two measurements. In the case of inter-line Doppler [2

2. 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. d. Boer, “In vivo dynamic human retinal blood flow imaging using ultra-high-speed spectral domain optical Doppler tomography,” Opt. Express 11, 3490–3497 (2003). [CrossRef] [PubMed]

], xb2+yb2Δx, which is the transversal sampling step between adjacent axial lines.

By substituting Eq. (33) into Eqs. (10) and (29) and using Eqs. (7), (25), and (27) with Δϕ0 = 0, the phase shift noise in a static tissue is obtained as
σΔϕSS(SB)={σΔϕ|ρ=eδx21+1/ESNR(SB)(IJL=1)σΔϕ^0|ρ=eδx21+1/ESNR(SB),ν=ENISI1,J,L(IJL>=2),
(34)
where δx = Δx/w is the fractional sampling step between two adjacent axial lines. Since a single-line-shifted image is used to calculate phase shifts, the window size may be reduced by 1 to maintain the same spatial resolution. Note that the ESNR also affects the ENIS as shown by Eq. (29).

3.2. Dual-beam-scan Doppler OCT

The polarization-multiplexing dual-beam-scan Doppler method detects two OCT signals using different polarization states at the same location of the static tissue [19

19. S. Makita, F. Jaillon, M. Yamanari, M. Miura, and Y. Yasuno, “Comprehensive in vivo micro-vascular imaging of the human eye by dual-beam-scan Doppler optical coherence angiography,” Opt. Express 19, 1271–1283 (2011). [CrossRef] [PubMed]

]. Hence, the signal correlation coefficient with a static tissue can be obtained from Eqs. (18) and (19) as
ρs,SS(DB)=ρPol.2w1w2w12+w222Δk1Δk2Δk12+Δk22e8(kc1kc2)2Δk12+Δk22,
(35)
where ρη1,η2ρPol. is the correlation coefficient between the scattering process with two different polarization states of probing beams. It is shown that an increasing difference in the PSFs of the two channels decreases the signal correlation coefficient. The same light source, identical performances of detectors, and the same optical setup for two channels are required to maximize the performance of the dual-beam method. In the ideal case, ρs,SS(DB)=ρPol..

By substituting Eq. (35) into Eqs. (10) and (29) and using Eqs. (7), (25), and (27) with Δϕ0 = 0, phase shift noise in a static tissue can be described.

σΔϕSS(DB)={σΔϕ|ρ=ρs,SS(DB)1+1/ESNR(DB)(IJL=1)σΔϕ^0|ρ=ρs,SS(DB)1+1/ESNR(DB),ν=ENISI,J,L(IJL>=2).
(36)

4. Evaluation of phase shift noise

To validate and demonstrate the phase-resolved OCT analysis, an experiment using static tissue was conducted. The behaviors of the phase shift noise in phase-resolved OCT and dual-beam-scan OCT are compared.

4.1. Experimental setup and method

The scattering phantom was made by fixing 1 % soybean oil lipid emulsion (Intralipos®20%, Otsuka Pharmaceutical Factory Inc., Japan) with 10 % porcine gelatin (G2500, Sigma-Aldrich Corp., St. Louis, MO).

Phase-resolved OCT imaging was performed with 256 axial lines/frame and different fractional sampling steps δx from 0.1 to 2.

The conventional single-beam Doppler OCT system can use power of two beams into single probe beam. To emulate the single-beam Doppler method using this DB-OCT system, the two detected OCT signals are summed after a bulk motion correction as
g(xi,zl)=gH(xi,zl)+gV(xi,zl)exp[iΔϕch(xi)],
(37)
where gH and gV are measured OCT signals from the two polarization channels of DB-OCT, and Δϕch(xi)=arg[lgH*(xi,zl)gV(xi,zl)] is the phase difference between two channels at each line estimated by taking the argument of summed Hermitian products along the axial direction [33

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

]. Hence, the ESNR of the single-beam method is theoretically twice that of the dual-beam method; ESNR(SB) = 2ESNR(SB). Two signals g1 and g2 are assigned as g1gH(xi, zl), g2gV(xi, zl) in the case of the dual-beam method and g1g(xi, zl), g2g(xi+1, zl) in the case of the single-beam method using Eq. (37). The sample phase differences of the dual-beam and single-beam methods are calculated and analyzed.

4.2. Results

Figure 1 is a cross-sectional OCT image of the scattering phantom. The phase-resolved phantom images with several fractional sampling steps are shown in Fig. 2. A part of an image with a constant image depth was assigned as a region of interest (ROI) for analysis as indicated by a yellow box in Fig. 1 (256 lines × 10 pixels). A set of 100 B-scans are acquired and each statistics are measured every B-scan. The final measurements of statistics are averages of 100 realizations.

Fig. 1 A cross-sectional OCT image of the tissue phantom. The yellow box indicates the ROI of phase shift analysis.
Fig. 2 Phase-resolved images with dual-beam-scan (right column; a, c, e) and emulated conventional phase-resolved (left column; b, d, f) methods. The fractional sampling step was set to be (a), (b) 0.84, (c), (d) 0.43, (e), (f) and 0.1.

As expected from Eq. (7), the phase shift noise can be characterized by the correlation co-efficient of measured signals ρ. In Fig. 3, sample standard deviations of sample phase shift SΔϕ of dual-beam and emulated single-beam methods are plotted against the sample correlation r obtained using Eqs. (20) and (9). The solid curve is the line calculated with Eq. (7) at Δϕ0 = 0. Experimental and theoretical results are in good agreement.

Fig. 3 Scatter plot of phase shift noise vs correlation coefficient. The solid curve shows the population standard deviation of phase shift (Eq. (7)).

To compare the dual-beam and single-beam methods, phase shift noise SΔϕ is plotted against the fractional sampling step δx in Fig. 4. Each curve represents expected phase shift noise (standard deviation of the phase shift, Eq. (7)) for the dual-beam and single-beam methods. The correlation coefficient ρPol. in the dual-beam method was estimated to be 0.91 by averaging the estimation ρPol.,n=(1+1/ESNR^n(DB))rn of each n-th measurement, where we assume ρs(DB)ρPol. and use Eq. (21). ESNR^(DB) is the sample representative SNR of the dual-beam method calculated using Eq. (22) as approx. 11 dB. The population representative SNR ESNR is set to 11 dB for the dual-beam method and 14 dB for the single-beam method. As expected, phase shift noise is almost constant for all fractional sampling steps in the dual-beam method, because two signals are obtained at the same position on the sample no matter the magnitude of the fractional sampling step. The phase shift noise is significantly small compared with that for the single-beam method at large δx. The transitional point of the fractional sampling step where the magnitudes of phase shift noise become identical between single-beam and dual-beam methods is
δxc=ln[ρPol.ESNR(DB)ESNR(SB)ESNR(SB)+1ESNR(DB)+1]=ln[ρPol.ESNR(SB)+1ESNR(SB)+2].
(38)
If the fractional step is larger than this δx, the dual-beam method provides superior performance in terms of phase noise compared with the conventional single-beam method.

Fig. 4 Phase shift noise vs fractional sampling step δx.

This is plotted as Fig. 5 for ρPol. = 0.91. When the fractional sampling step is larger than δxc, the dual-beam method exhibits less phase shift noise. When δx < δxc, the single-beam method is better. And δxc is larger as ESNR decreases. These characteristics can be easily understood as follows. In the case of smaller δxc and lower ESNR, phase shift noise caused by additive random noise is dominant. Since the single-beam method exhibits a larger SNR by a factor of 2, the phase shift noise of the single-beam method is less than that of the dual-beam method.

Fig. 5 The transitional point δxc (Eq. (38)) is plotted. In the upper region, the dual-beam method exhibits less phase shift noise than that of the single-beam method.

In Fig. 4, the predicted phase shift noise is greater than the experimental results at large δx in the case of the single-beam method. This would be explained by elongation of the beam profile [34

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

]. A broadened beam profile increases the signal correlation coefficient ρs and decreases the phase shift noise.

The phase shift noise against the ESNR is shown in Fig. 6. To virtually change the ESNR, complex circular Gaussian noise is numerically generated and added to complex OCT data. The phase shift noise decreases as the ESNR increases. However, the phase shift noise approaches an asymptotic value.

Fig. 6 Phase shift noises vs ESNR.

In the high-ESNR regime, decorrelation phase shift noise is dominant. The equivalent representative SNR of a signal correlation coefficient ESNRρs can be described by equating ρs=1/(1+ESNRρs1) as
ESNRρs=ρs1ρs.
(39)
When the ESNR is larger than this ESNRρs, the ESNR is no longer a dominant limitation of phase noise but the signal correlation ρs is. In the case of the current dual-beam system, the ESNRρs |ρs=ρPol. ≈ 10.6 dB and phase shift noise approaches σΔϕ|ρ=ρPol. ≈ 0.63 radians in the high-ESNR regime.

4.2.1. Averaged phase shift noise

In practical applications, spatial complex averaging (Eq. (26)) is used to enhance the contrast of phase-resolved images. The performances with averaging in conventional and dual-beam-scan phase-resolved flow imaging are compared in this section.

First, estimations of the correlation coefficient of the Hermitian product ρg2 are evaluated because ρg2 is essential to the estimation of the effective number of independent samples ENIS (Eq. (27)). The Hermitian product g1*g2 was calculated for the single B-scan image and the spatial autocorrelation was obtained in the ROI according to the spatial displacement steps Δx and Δz, which correspond to the spatial lengths according to the single pixel. Figure 7 shows the profiles of estimated ρg2,SS. The horizontal axis of each plot is normalized by the beam spot radius, w = 16.5 μm, and the axial resolution defined as half width at e−2 of axial PSF, ζ=4/Δk=9.5μm/2log2=8.1μm. The solid curves in each plot show the expected profiles from Eqs. (29) by substituting ρs,SS(DB)=ρPol. and ρs,SS(SB)=eδx2. Here, ρPol. and the ESNR were calculated from data obtained in the experiment. They show that the experimental data and estimation using Eq. (29) are in good agreement.

Fig. 7 Profiles of the correlation coefficients of the Hermitian products with displacement along (a) the lateral direction and (b) the axial direction. In each figure, the horizontal axis is fractional displacement iδx = iΔx/w and lδz = lΔz/ζ, where i and l are the displacements in the number of pixels along lateral and axial directions, respectively. δx = 0.22 and δz = 0.52. Solid curves are the expected correlation coefficients from Eq. (29).

The suppression of phase shift noise by the moving average is shown in Fig. 8. The effective number of independent samples within the window ENIS is calculated using Eq. (27). Solid curves show the approximate phase shift noise numerically simulated using Eqs. (25), (43), and (44) by summing series up to the 50-th order. The experimental results and numerical estimations are in good agreement for large δx.

Fig. 8 Phase shift noise vs effective number of independent samples.

When the fractional sampling step is very small, (i.e., δx < 0.2), measured results with lateral averaging deviate from predicted values. Perhaps under this condition, the OCT signals do not significantly differ between the two axial lines. The phase shift estimation σΔϕ̂0 does not obey Eq. (24). When correlation coefficient ρs is close to 1, phase shift Δϕ0 is constant. In addition, if δx is small, the Hermitian products extracted along the lateral direction can be considered as a sum of a constant phasor and a random phasor. If this assumption is valid, the phase shift noise will decrease by the square root of the number of averaged realizations. In fact, the noise suppression ratio under this condition is close to 1N, where N is the number of sampling points in the lateral averaging window.

In order to compare dual-beam and single-beam methods, phase shift noise with lateral moving average was calculated where the window size is up to the optical resolution. The window sizes for each method are as follows.
I(DB)={2δx2δx1,1otherwise
(40)
I(SB)={2δx12δx2,1otherwise.
(41)
These phase shift noises and corresponding ENISs calculated from Eqs. (27), (40), and (41) are plotted in Fig. 9. Since the window size must be an integers (Eqs. (40) and (41)), the population standard deviation of the MLE of phase shift σΔϕ̂0 and estimated effective number ENIS exhibit discontinuous values along δx as shown by solid curves in Fig. 9. The transitional point of the fractional sampling step is nearly the same as that without averaging. However, the phase shift noise of the dual-beam method at small fractional sampling step is reduced and approaches that of the single-beam method.

Fig. 9 Phase shift noise vs fractional sampling step with averaging using a constant window size.

5. Discussions

The essential factor that explains the phase shift noise in phase-resolved OCT is the correlation coefficient of measured OCT signals. The phase shift noise relying on the SNR can be treated as the decorrelation of measured signals caused by additive noise. Hence, the phase shift noises derived from additive noise and structural decorrelation are unified. The presented statistical model accounts for the spatial variation of the instantaneous SNR; i.e., speckle. The introduced statistics well describe the imaging performance of phase-resolved OCT.

In this study, we compared the dual-beam Doppler method and the inter-line single-beam Doppler method. When the transversal sampling is coarse, the phase shift noise of the dual-beam method is less than that of the conventional Doppler method as expected. This indicates that there is a great advantage in the case of systems with high spatial resolution. The imaging speed and/or imaging range can be increased by increasing the transversal sampling step as a level of phase shift noise is low.

Recently, Doppler methods with dedicated scanning protocols have been employed to increase the time delay and increase the flow sensitivity [35

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

37

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

]. With high-dense transversal sampling, it is predicted that the single-beam method will surpass the dual-beam method. However, repeatability of a beam scanning mechanism and/or sample fluctuation perhaps limit the advantage [37

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

]. As shown in Fig. 3, a small reduction of the correlation coefficient will result in a rapid increase of the phase shift noise. On the other hand, the dual-beam-scan method can be used with a simple raster scanning protocol.

For vasculature imaging in optical coherence angiography, squared Doppler phase shifts are calculated to contrast vessels [4

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

]. The response to flow can be defined by the second moment of the phase shift estimation Eϕ̂2]. Since the lateral motion of samples reduces the correlation coefficient between OCT signals at different time points and hence increases Eϕ̂2], the squared Doppler phase shift imaging is expected to be sensitive to not only axial motion but also lateral movement.

6. Conclusion

The statistical properties of phase-resolved OCT imaging were described. The investigated statistics of phase-resolved OCT were validated by evaluating phase shift noise measured with a static tissue phantom. Flow imaging performances of dual-beam-scan phase-resolved Doppler OCT and the conventional single-beam method were compared and discussed using the presented statistics. The dual-beam method exhibited lower phase shift noise for coarse transversal sampling than the single-beam method. The presented statistics of phase-resolved OCT are useful in investigating, comparing, and designing phase-resolved OCT systems.

Appendix A. Moments of the maximum likelihood estimate of phase shift

From the moment generating function, the n-th order moment of the sample phase shift is obtained as
E[Δϕ^0n]=(1)n/2πncos(nπ2)n+1+2l=1{(1)lρlΓ(l2+1)Γ(l2+ν)F12(l2,l2ν+1;l+1;ρ2)πΓ(ν)Γ(l+1)×[nsnsinh(πs)[scos(lΔϕ0)lsin(lΔϕ0)]l2+s2]|s0}.
(42)
The first and second moments are then obtained by setting n as 1 and 2 in Eq. (42):
E[Δϕ^0]=l=12(ρ)lsin(lΔϕ0)Γ(l2+1)Γ(l2+ν)F12[l2,l2ν+1;l+1;ρ2]lΓ(ν)Γ(l+1),
(43)
E[Δϕ^02]=π23+l=14(ρ)lcos(lΔϕ0)Γ(l2+1)Γ(l2+ν)F12[l2,l2ν+1;l+1;ρ2]l2Γ(ν)Γ(l+1).
(44)

References and links

1.

Y. Zhao, Z. Chen, C. Saxer, S. Xiang, J. F. d. Boer, and J. S. Nelson, “Phase-resolved optical coherence tomography and optical Doppler tomography for imaging blood flow in human skin with fast scanning speed and high velocity sensitivity,” Opt. Lett. 25, 114–116 (2000). [CrossRef]

2.

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. d. Boer, “In vivo dynamic human retinal blood flow imaging using ultra-high-speed spectral domain optical Doppler tomography,” Opt. Express 11, 3490–3497 (2003). [CrossRef] [PubMed]

3.

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, 3116–3121 (2003). [CrossRef] [PubMed]

4.

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

5.

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

6.

D. Y. Kim, J. Fingler, J. S. Werner, D. M. Schwartz, S. E. Fraser, and R. J. Zawadzki, “In vivo volumetric imaging of human retinal circulation with phase-variance optical coherence tomography,” Biomed. Opt. Express 2, 1504–1513 (2011). [CrossRef] [PubMed]

7.

B. Braaf, K. A. Vermeer, V. A. D. Sicam, E. van Zeeburg, J. C. van Meurs, and J. F. de Boer, “Phase-stabilized optical frequency domain imaging at 1-m for the measurement of blood flow in the human choroid,” Opt. Express 19, 20886–20903 (2011). [CrossRef] [PubMed]

8.

R. K. Wang, S. Kirkpatrick, and M. Hinds, “Phase-sensitive optical coherence elastography for mapping tissue microstrains in real time,” Appl. Phys. Lett. 90, 164105, 2007). [CrossRef]

9.

S. G. Adie, X. Liang, B. F. Kennedy, R. John, D. D. Sampson, and S. A. Boppart, “Spectroscopic optical coherence elastography,” Opt. Express 18, 25519–25534 (2010). [CrossRef] [PubMed]

10.

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

11.

T. Akkin, D. P. Dav, J.-I. Youn, S. A. Telenkov, H. G. R. III, and T. E. Milner, “Imaging tissue response to electrical and photothermal stimulation with nanometer sensitivity,” Lasers Surg. Med. 33, 219–225 (2003). [CrossRef] [PubMed]

12.

S. A. Telenkov, D. P. Dave, S. Sethuraman, T. Akkin, and T. E. Milner, “Differential phase optical coherence probe for depth-resolved detection of photothermal response in tissue,” Phys. Med. Biol. 49, 111–119 (2004). [CrossRef] [PubMed]

13.

D. C. Adler, S.-W. Huang, R. Huber, and J. G. Fujimoto, “Photothermal detection of gold nanoparticles using phase-sensitive optical coherence tomography,” Opt. Express 16, 4376–4393 (2008). [CrossRef] [PubMed]

14.

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

15.

W. Drexler and J. G. Fujimoto, Optical Coherence Tomography: Technology and Applications (Springer, 2008). [CrossRef]

16.

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, 410–416 (2005). [CrossRef] [PubMed]

17.

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

18.

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

19.

S. Makita, F. Jaillon, M. Yamanari, M. Miura, and Y. Yasuno, “Comprehensive in vivo micro-vascular imaging of the human eye by dual-beam-scan Doppler optical coherence angiography,” Opt. Express 19, 1271–1283 (2011). [CrossRef] [PubMed]

20.

S. Zotter, M. Pircher, T. Torzicky, M. Bonesi, E. Gtzinger, R. A. Leitgeb, and C. K. Hitzenberger, “Visualization of microvasculature by dual-beam phase-resolved Doppler optical coherence tomography,” Opt. Express 19, 1217–1227 (2011). [CrossRef] [PubMed]

21.

F. Jaillon, S. Makita, E.-J. Min, B. H. Lee, and Y. Yasuno, “Enhanced imaging of choroidal vasculature by high-penetration and dual-velocity optical coherence angiography,” Biomed. Opt. Express 2, 1147–1158 (2011). [CrossRef] [PubMed]

22.

L. Jong-Sen, K. Hoppel, S. Mango, and A. Miller, “Intensity and phase statistics of multilook polarimetric and interferometric SAR imagery,” IEEE Trans. Geosci. Remote Sens. 32, 1017–1028 (1994). [CrossRef]

23.

R. J. A. Tough, D. Blacknell, and S. Quegan, “A statistical description of polarimetric and interferometric synthetic aperture radar data,” Proc. R. Soc. Lond. A 449, 567–589 (1995). [CrossRef]

24.

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

25.

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

26.

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

27.

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

28.

V. X. Yang, M. L. Gordon, A. Mok, Y. Zhao, Z. Chen, R. S. Cobbold, B. C. Wilson, and I. A. Vitkin, “Improved phase-resolved optical Doppler tomography using Kasai velocity estimator and histogram segmentation,” Opt. Commun. 208, 209–214 (2002). [CrossRef]

29.

A. Moreira, “Improved multilook techniques applied to SAR and SCANSAR imagery,” IEEE Trans. Geosci. Remote Sens. 29, 529–534, 1991). [CrossRef]

30.

J. S. Bendat and A. G. Piersol, Random Data: Analysis and Measurement Procedures (John Wiley and Sons, 2010). [CrossRef]

31.

F. Jaillon, S. Makita, and Y. Yasuno, “Variable velocity range imaging of the choroid with dual-beam optical coherence angiography,” Opt. Express 20, 385–396 (2012). [CrossRef] [PubMed]

32.

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

33.

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

34.

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

35.

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

36.

L. An, J. Qin, and R. K. Wang, “Ultrahigh sensitive optical microangiography for in vivo imaging of microcirculations within human skin tissue beds,” Opt. Express 18, 8220–8228 (2010). [CrossRef] [PubMed]

37.

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

OCIS Codes
(110.4500) Imaging systems : Optical coherence tomography
(120.5050) Instrumentation, measurement, and metrology : Phase measurement
(170.3340) Medical optics and biotechnology : Laser Doppler velocimetry
(170.3880) Medical optics and biotechnology : Medical and biological imaging

ToC Category:
Fourier Optics and Signal Processing

History
Original Manuscript: January 10, 2014
Manuscript Accepted: February 10, 2014
Published: February 21, 2014

Virtual Issues
Vol. 9, Iss. 4 Virtual Journal for Biomedical Optics

Citation
Shuichi Makita, Franck Jaillon, Israt Jahan, and Yoshiaki Yasuno, "Noise statistics of phase-resolved optical coherence tomography imaging: single-and dual-beam-scan Doppler optical coherence tomography," Opt. Express 22, 4830-4848 (2014)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-4-4830


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References

  1. Y. Zhao, Z. Chen, C. Saxer, S. Xiang, J. F. d. Boer, 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]
  2. B. R. White, M. C. Pierce, N. Nassif, B. Cense, B. H. Park, G. J. Tearney, B. E. Bouma, T. C. Chen, J. F. d. Boer, “In vivo dynamic human retinal blood flow imaging using ultra-high-speed spectral domain optical Doppler tomography,” Opt. Express 11, 3490–3497 (2003). [CrossRef] [PubMed]
  3. R. Leitgeb, L. Schmetterer, W. Drexler, A. Fercher, R. Zawadzki, T. Bajraszewski, “Real-time assessment of retinal blood flow with ultrafast acquisition by color Doppler Fourier domain optical coherence tomography,” Opt. Express 11, 3116–3121 (2003). [CrossRef] [PubMed]
  4. S. Makita, Y. Hong, M. Yamanari, T. Yatagai, Y. Yasuno, “Optical coherence angiography,” Opt. Express 14, 7821–7840 (2006). [CrossRef] [PubMed]
  5. 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, B. E. Bouma, “Three-dimensional microscopy of the tumor microenvironment in vivo using optical frequency domain imaging,” Nat. Med. 15, 1219–1223 (2009). [CrossRef] [PubMed]
  6. D. Y. Kim, J. Fingler, J. S. Werner, D. M. Schwartz, S. E. Fraser, R. J. Zawadzki, “In vivo volumetric imaging of human retinal circulation with phase-variance optical coherence tomography,” Biomed. Opt. Express 2, 1504–1513 (2011). [CrossRef] [PubMed]
  7. B. Braaf, K. A. Vermeer, V. A. D. Sicam, E. van Zeeburg, J. C. van Meurs, J. F. de Boer, “Phase-stabilized optical frequency domain imaging at 1-m for the measurement of blood flow in the human choroid,” Opt. Express 19, 20886–20903 (2011). [CrossRef] [PubMed]
  8. R. K. Wang, S. Kirkpatrick, M. Hinds, “Phase-sensitive optical coherence elastography for mapping tissue microstrains in real time,” Appl. Phys. Lett. 90, 164105, 2007). [CrossRef]
  9. S. G. Adie, X. Liang, B. F. Kennedy, R. John, D. D. Sampson, S. A. Boppart, “Spectroscopic optical coherence elastography,” Opt. Express 18, 25519–25534 (2010). [CrossRef] [PubMed]
  10. B. F. Kennedy, S. H. Koh, R. A. McLaughlin, K. M. Kennedy, P. R. T. Munro, D. D. Sampson, “Strain estimation in phase-sensitive optical coherence elastography,” Biomed. Opt. Express 3, 1865–1879 (2012). [CrossRef] [PubMed]
  11. T. Akkin, D. P. Dav, J.-I. Youn, S. A. Telenkov, H. G. R., T. E. Milner, “Imaging tissue response to electrical and photothermal stimulation with nanometer sensitivity,” Lasers Surg. Med. 33, 219–225 (2003). [CrossRef] [PubMed]
  12. S. A. Telenkov, D. P. Dave, S. Sethuraman, T. Akkin, T. E. Milner, “Differential phase optical coherence probe for depth-resolved detection of photothermal response in tissue,” Phys. Med. Biol. 49, 111–119 (2004). [CrossRef] [PubMed]
  13. D. C. Adler, S.-W. Huang, R. Huber, J. G. Fujimoto, “Photothermal detection of gold nanoparticles using phase-sensitive optical coherence tomography,” Opt. Express 16, 4376–4393 (2008). [CrossRef] [PubMed]
  14. H. H. Mller, L. Ptaszynski, K. Schlott, C. Debbeler, M. Bever, S. Koinzer, R. Birngruber, R. Brinkmann, G. Httmann, “Imaging thermal expansion and retinal tissue changes during photocoagulation by high speed OCT,” Biomed. Opt. Express 3, 1025–1046 (2012). [CrossRef]
  15. W. Drexler, J. G. Fujimoto, Optical Coherence Tomography: Technology and Applications (Springer, 2008). [CrossRef]
  16. S. Yazdanfar, C. Yang, M. Sarunic, J. Izatt, “Frequency estimation precision in Doppler optical coherence tomography using the Cramer-Rao lower bound,” Opt. Express 13, 410–416 (2005). [CrossRef] [PubMed]
  17. B. H. Park, M. C. Pierce, B. Cense, S.-H. Yun, M. Mujat, G. J. Tearney, B. E. Bouma, J. F. d. Boer, “Real-time fiber-based multi-functional spectral-domain optical coherence tomography at 1.3 μm,” Opt. Express 13, 3931–3944 (2005). [CrossRef] [PubMed]
  18. B. J. Vakoc, G. J. Tearney, B. E. Bouma, “Statistical properties of phase-decorrelation in phase-resolved Doppler optical coherence tomography,” IEEE Trans. Med. Imaging 28, 814–821 (2009). [CrossRef] [PubMed]
  19. S. Makita, F. Jaillon, M. Yamanari, M. Miura, Y. Yasuno, “Comprehensive in vivo micro-vascular imaging of the human eye by dual-beam-scan Doppler optical coherence angiography,” Opt. Express 19, 1271–1283 (2011). [CrossRef] [PubMed]
  20. S. Zotter, M. Pircher, T. Torzicky, M. Bonesi, E. Gtzinger, R. A. Leitgeb, C. K. Hitzenberger, “Visualization of microvasculature by dual-beam phase-resolved Doppler optical coherence tomography,” Opt. Express 19, 1217–1227 (2011). [CrossRef] [PubMed]
  21. F. Jaillon, S. Makita, E.-J. Min, B. H. Lee, Y. Yasuno, “Enhanced imaging of choroidal vasculature by high-penetration and dual-velocity optical coherence angiography,” Biomed. Opt. Express 2, 1147–1158 (2011). [CrossRef] [PubMed]
  22. L. Jong-Sen, K. Hoppel, S. Mango, A. Miller, “Intensity and phase statistics of multilook polarimetric and interferometric SAR imagery,” IEEE Trans. Geosci. Remote Sens. 32, 1017–1028 (1994). [CrossRef]
  23. R. J. A. Tough, D. Blacknell, S. Quegan, “A statistical description of polarimetric and interferometric synthetic aperture radar data,” Proc. R. Soc. Lond. A 449, 567–589 (1995). [CrossRef]
  24. J. Walther, E. Koch, “Transverse motion as a source of noise and reduced correlation of the Doppler phase shift in spectral domain OCT,” Opt. Express 17, 19698–19713 (2009). [CrossRef] [PubMed]
  25. V. J. Srinivasan, S. Sakadi, I. Gorczynska, S. Ruvinskaya, W. Wu, J. G. Fujimoto, D. A. Boas, “Quantitative cerebral blood flow with optical coherence tomography,” Opt. Express 18, 2477–2494 (2010). [CrossRef] [PubMed]
  26. J. Lee, W. Wu, J. Y. Jiang, B. Zhu, D. A. Boas, “Dynamic light scattering optical coherence tomography,” Opt. Express 20, 22262–22277 (2012). [CrossRef] [PubMed]
  27. A. Szkulmowska, M. Szkulmowski, A. Kowalczyk, M. Wojtkowski, “Phase-resolved Doppler optical coherence tomographylimitations and improvements,” Opt. Lett. 33, 1425–1427 (2008). [CrossRef] [PubMed]
  28. V. X. Yang, M. L. Gordon, A. Mok, Y. Zhao, Z. Chen, R. S. Cobbold, B. C. Wilson, I. A. Vitkin, “Improved phase-resolved optical Doppler tomography using Kasai velocity estimator and histogram segmentation,” Opt. Commun. 208, 209–214 (2002). [CrossRef]
  29. A. Moreira, “Improved multilook techniques applied to SAR and SCANSAR imagery,” IEEE Trans. Geosci. Remote Sens. 29, 529–534, 1991). [CrossRef]
  30. J. S. Bendat, A. G. Piersol, Random Data: Analysis and Measurement Procedures (John Wiley and Sons, 2010). [CrossRef]
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