## Noise statistics of phase-resolved optical coherence tomography imaging: single-and dual-beam-scan Doppler optical coherence tomography |

Optics Express, Vol. 22, Issue 4, pp. 4830-4848 (2014)

http://dx.doi.org/10.1364/OE.22.004830

Acrobat PDF (2056 KB)

### 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

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. 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]

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]

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]

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]

## 2. Statistics of phase-resolved OCT

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]

### 2.1. Statistics of phase shift

**G**

_{1},

**G**

_{2}, phase shift is calculated as a phase term of the Hermitian product of two measured OCT signals; where

*g*

_{1}and

*g*

_{2}are respectively realizations of

**G**

_{1}and

**G**

_{2}. Here, the measured OCT signals

**G**

_{1}and

**G**

_{2}are the sum of complex signals

**S**

_{1}and

**S**

_{2}and additive noises

**N**

_{1}and

**N**

_{2}, respectively.

*s*is the sum of interference signals from scatterers in a coherent detection volume: where

*a*is the amplitude of scattered light from the

_{m}*m*-th scatterer, and

*z*is the axial location of the

_{m}*m*-th scatterer.

*a*is the amplitude of the reference light,

_{R}*k*is the central wave number of a broadband light source, and

_{c}*z*

_{0}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,

**S**

_{1}and

**S**

_{2}can be considered as zero-mean complex circular Gaussian variables.

**N**

_{1}and

**N**

_{2}are zero-mean complex circular Gaussian variables and independent of each other and the signals

**S**

_{1}and

**S**

_{2}, the measured signals

**G**

_{1}and

**G**

_{2}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. 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]

*ϕ*(Eq. (1)) can be expressed as where

*β*=

*ρ*cos(Δ

*ϕ*− Δ

*ϕ*

_{0}). Additionally, where

*E*[·] is the expectation operator. Equation (5) indicates that the parameters

*ρ*and Δ

*ϕ*

_{0}are the amplitude and phase of the population complex correlation coefficient for the measured signals, respectively. Δ

*ϕ*

_{0}represents the population phase shift.

*ϕ*are described as where

*β′*=

*ρ*cosΔ

*ϕ*

_{0}and Li

_{2}is Euler’s dilogarithm. The estimators of the expectation and standard deviation are the arithmetic mean and sample standard deviation: where

*N*is the number of realizations.

### 2.2. Correlation coefficient of OCT signals

*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.

*ρ*can be described as the following according to the definition of measured signals

**G**

_{1},

**G**

_{2}(Eq. (2)); where Equation (11) gives the correlation coefficient between two OCT signals

**S**

_{1}and

**S**

_{2}and

*SNR*=

_{i}*E*[|

**S**

*|*

_{i}^{2}]/

*E*[|

**N**

*|*

_{i}^{2}] (

*i*=1,2) are the expected signal-to-noise ratios of each measurement. As mentioned in the following section (Section 2.2),

*ρ*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

_{s}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]

*ρ*=

*α*

^{2}: i.e., the correlation coefficient depends only on transversal sampling displacement, and Δ

*ϕ*

_{0}= 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}.

*ρ*is defined by referring to previous studies [24

_{s}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]

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]

*h*(

**r**

*) and the complex reflectivity distribution of a sample*

_{L}*η*(

**r**

*).*

_{L}**r**

*= (*

_{L}*x*,

_{L}*y*,

_{L}*z*) is the laboratory coordinate. The coordinate

_{L}**r**consists of a lateral location of a probing beam and a depth of the sample from zero delay of the interferometer.

**r**is a function of time since it will change with beam scanning and motion of objects.

*ϕ*

_{0}can be described from the signal cross-correlation coefficient between realizations

*s*

_{1},

*s*

_{2}:

*w*is the beam spot radius at 1/

_{χ}*e*

^{2}of the

*χ*-th channel. Δ

*k*is the full width at 1/

*e*

^{2}of a Gaussian spectrum of a light source in unit of wavenumber. Then, the population correlation coefficient of OCT signals

*ρ*and population phase shift Δ

_{s}*ϕ*

_{0}are obtained by substituting Eq. (15) and (13) into Eq. (14):

*ρ*

_{η1,η2}denotes the correlation coefficient of the scattering process between two channels and (

*x′*(

*t*

_{1},

*t*

_{2}),

*y′*(

*t*

_{1},

*t*

_{2}),

*z′*(

*t*

_{1},

*t*

_{2})) =

**r**

_{2}(

*t*

_{2}) −

**r**

_{1}(

*t*

_{1}) 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

*k*

_{c}_{1}−

*k*

_{c}_{2}decreases the correlation coefficient.

**r′**(Δ

*t*) =

**r**

_{2}(

*t*+ Δ

*t*) −

**r**

_{1}(

*t*) and

*ρ*

_{h1,h2}is the correlation coefficient of PSFs of the two channels:

*ρ*is the sample correlation between realizations

*g*

_{1}and

*g*

_{2}: Because the sample correlation

*r*is a biased estimation for

*ρ*, a large number of realizations are required for accurate estimation.

### 2.3. Maximum likelihood estimation of phase shift

*ϕ*(Eq. (8)) is a biased estimator for Δ

*ϕ*

_{0}. According to the expectation (Eq. (6)), the mean estimator results in large offset of the estimation from the population parameter Δ

*ϕ*

_{0}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]

*ϕ*

_{0}will be used for better estimation. The MLE of population phase shift Δ

*ϕ*

_{0}with

*ν*independent realizations

*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]

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]

**449**, 567–589 (1995). [CrossRef]

_{2}

*F*

_{1}is Gauss hypergeometric function.

*σ*

_{Δϕ̂0}is characterized by first- and second-order moments

*E*[Δ

*ϕ̂*

_{0}],

*E*[Δ

*ϕ̂*

_{0}

^{2}] as The moments of Δ

*ϕ̂*

_{0}can be numerically calculated using Eq. (24). However, the calculation cost is high. To reduce the computation time, approximations of the moments have been found. The moments of Δ

*ϕ̂*

_{0}can be expressed by the summation of an infinite series as shown in Appendix A. The decrement of the higher-order term from the previous term in the series is from about 10 to more than 90 %. Hence, asymptotic expressions of expectation, variance, and other statistics of Δ

*ϕ̂*

_{0}can be obtained by taking the first several ten terms of the series. Summing up to the ∼ 30-th order provides a good approximation. The only exception is the case when

*ν*→ ∞ or

*ρ*→ 1, where the summation does not asymptotically converge to the real value. However, it is a rare case in real experiments and can thus be ignored.

### 2.4. Practical estimators

*ϕ̂*

_{0}has been shown as Eq. (23). However, in the real case, it is almost impossible to acquire several independent samples for a single location.

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

*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

*ν*≠

*IJL*.

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

*x*, Δ

*y*, Δ

*z*) is the spatial separation between neighboring pixels in the image along each direction.

*ρ*

_{g2}(

*i*Δ

*x*, 0, 0) is the correlation coefficient between two Hermitian products with the displacement of

*i*image pixels in the

*x*-direction;

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

*ρ*

_{h1,h1}and

*ρ*

_{h2,h2}are the auto-correlation coefficients of each channel;

## 3. Performance of flow imaging with phase-resolved OCT

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]

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]

*V*, the velocity of moving tissue;

*θ*, Doppler angle; Δ

*t*, the time delay between the two time points; and

*n*, the refractive index of the sample.

*σ*

_{ΔϕSS}indicates a standard deviation of the spatial distribution of the phase shift for the surrounding solid tissue.

*K*= 1/2

*nk*

_{c}cos

*θ*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

*h*

_{1}=

*h*

_{2}and

*η*

_{1}=

*η*

_{2}. Under this condition, the signal correlation coefficient with a static tissue is obtained from Eqs. (18) and (19) as 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]

*ϕ*

_{0}= 0, the phase shift noise in a static tissue is obtained as where

*δx*= Δ

*x/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

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]

*ρ*

_{η1,η2}≡

*ρ*

_{Pol.}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,

## 4. Evaluation of phase shift noise

### 4.1. Experimental setup and method

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]

*μ*m using optical simulation software (ZEMAX, Radiant Zemax, LLC, Redmond, WA). The axial resolution was measured to be about 9.5

*μ*m (full-width at half maximum, 6 dB width) in air. Theoretically, this corresponds to

*δx*from 0.1 to 2.

*g*

_{H}and

*g*

_{V}are measured OCT signals from the two polarization channels of DB-OCT, and

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]

*ESNR*of the single-beam method is theoretically twice that of the dual-beam method;

*ESNR*

^{(SB)}= 2

*ESNR*

^{(SB)}. Two signals

*g*

_{1}and

*g*

_{2}are assigned as

*g*

_{1}≜

*g*

_{H}(

*x*,

_{i}*z*)

_{l}*, g*

_{2}≜

*g*

_{V}(

*x*,

_{i}*z*) in the case of the dual-beam method and

_{l}*g*

_{1}≜

*g*(

*x*,

_{i}*z*)

_{l}*, g*

_{2}≜

*g*(

*x*

_{i}_{+1},

*z*) 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.

_{l}### 4.2. Results

*ρ*. In Fig. 3, sample standard deviations of sample phase shift

*S*

_{Δ}

*of dual-beam and emulated single-beam methods are plotted against the sample correlation*

_{ϕ}*r*obtained using Eqs. (20) and (9). The solid curve is the line calculated with Eq. (7) at Δ

*ϕ*

_{0}= 0. Experimental and theoretical results are in good agreement.

*S*

_{Δϕ}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

*ρ*

_{Pol.}in the dual-beam method was estimated to be 0.91 by averaging the estimation

*n*-th measurement, where we assume

*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 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.

*ρ*

_{Pol.}= 0.91. When the fractional sampling step is larger than

*δx*

_{c}, the dual-beam method exhibits less phase shift noise. When

*δx*<

*δx*

_{c}, the single-beam method is better. And

*δx*

_{c}is larger as

*ESNR*decreases. These characteristics can be easily understood as follows. In the case of smaller

*δx*

_{c}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.

*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]

*ρ*and decreases the phase shift noise.

_{s}*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.

*ESNR*regime, decorrelation phase shift noise is dominant. The equivalent representative SNR of a signal correlation coefficient

*ESNR*

_{ρs}can be described by equating

*ESNR*is larger than this

*ESNR*

_{ρs}, the

*ESNR*is no longer a dominant limitation of phase noise but the signal correlation

*ρ*is. In the case of the current dual-beam system, the

_{s}*ESNR*

_{ρs}|

_{ρs=ρPol.}≈ 10.6 dB and phase shift noise approaches

*σ*

_{Δ}

*|*

_{ϕ}_{ρ=ρPol.}≈ 0.63 radians in the high-ESNR regime.

#### 4.2.1. Averaged phase shift noise

*ρ*

_{g2}are evaluated because

*ρ*

_{g2}is essential to the estimation of the effective number of independent samples

*ENIS*(Eq. (27)). The Hermitian product

*x*and Δ

*z*, which correspond to the spatial lengths according to the single pixel. Figure 7 shows the profiles of estimated

*ρ*

_{g2,SS}. The horizontal axis of each plot is normalized by the beam spot radius,

*w*= 16.5

*μ*m, and the axial resolution defined as half width at

*e*

^{−2}of axial PSF,

*ρ*

_{Pol.}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.

*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*.

*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

*σ*

_{Δϕ̂0}does not obey Eq. (24). When correlation coefficient

*ρ*is close to 1, phase shift Δ

_{s}*ϕ*

_{0}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

*N*is the number of sampling points in the lateral averaging window.

*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

*σ*

_{Δϕ̂0}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.

## 5. Discussions

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]

*ϕ*

_{0}and

*ρ*. The further alterations for the presented study according to the previous work will provide a statistical analysis tool of phase-resolved OCT that is more accurate.

_{s}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. 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]

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]

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

*E*[Δ

*ϕ̂*

^{2}]. Since the lateral motion of samples reduces the correlation coefficient between OCT signals at different time points and hence increases

*E*[Δ

*ϕ̂*

^{2}], the squared Doppler phase shift imaging is expected to be sensitive to not only axial motion but also lateral movement.

## 6. Conclusion

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

*n*-th order moment of the sample phase shift is obtained as

*n*as 1 and 2 in Eq. (42):

## 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. |

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 |

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 |

4. | S. Makita, Y. Hong, M. Yamanari, T. Yatagai, and Y. Yasuno, “Optical coherence angiography,” Opt. Express |

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. |

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 |

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 |

8. | R. K. Wang, S. Kirkpatrick, and M. Hinds, “Phase-sensitive optical coherence elastography for mapping tissue microstrains in real time,” Appl. Phys. Lett. |

9. | S. G. Adie, X. Liang, B. F. Kennedy, R. John, D. D. Sampson, and S. A. Boppart, “Spectroscopic optical coherence elastography,” Opt. Express |

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 |

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. |

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. |

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 |

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 |

15. | W. Drexler and J. G. Fujimoto, |

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 |

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 |

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 |

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 |

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 |

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 |

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. |

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 |

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 |

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 |

26. | J. Lee, W. Wu, J. Y. Jiang, B. Zhu, and D. A. Boas, “Dynamic light scattering optical coherence tomography,” Opt. Express |

27. | A. Szkulmowska, M. Szkulmowski, A. Kowalczyk, and M. Wojtkowski, “Phase-resolved Doppler optical coherence tomographylimitations and improvements,” Opt. Lett. |

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. |

29. | A. Moreira, “Improved multilook techniques applied to SAR and SCANSAR imagery,” IEEE Trans. Geosci. Remote Sens. |

30. | J. S. Bendat and A. G. Piersol, |

31. | F. Jaillon, S. Makita, and Y. Yasuno, “Variable velocity range imaging of the choroid with dual-beam optical coherence angiography,” Opt. Express |

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 |

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 |

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 |

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 |

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 |

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 |

**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

Sort: Year | Journal | Reset

### References

- 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]
- 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]
- 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]
- S. Makita, Y. Hong, M. Yamanari, T. Yatagai, Y. Yasuno, “Optical coherence angiography,” Opt. Express 14, 7821–7840 (2006). [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, 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- W. Drexler, J. G. Fujimoto, Optical Coherence Tomography: Technology and Applications (Springer, 2008). [CrossRef]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- A. Szkulmowska, M. Szkulmowski, A. Kowalczyk, M. Wojtkowski, “Phase-resolved Doppler optical coherence tomographylimitations and improvements,” Opt. Lett. 33, 1425–1427 (2008). [CrossRef] [PubMed]
- 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]
- A. Moreira, “Improved multilook techniques applied to SAR and SCANSAR imagery,” IEEE Trans. Geosci. Remote Sens. 29, 529–534, 1991). [CrossRef]
- J. S. Bendat, A. G. Piersol, Random Data: Analysis and Measurement Procedures (John Wiley and Sons, 2010). [CrossRef]
- F. Jaillon, S. Makita, Y. Yasuno, “Variable velocity range imaging of the choroid with dual-beam optical coherence angiography,” Opt. Express 20, 385–396 (2012). [CrossRef] [PubMed]
- S. Makita, F. Jaillon, M. Yamanari, Y. Yasuno, “Dual-beam-scan Doppler optical coherence angiography for birefringence-artifact-free vasculature imaging,” Opt. Express 20, 2681–2692 (2012). [CrossRef] [PubMed]
- K. Kurokawa, K. Sasaki, S. Makita, Y.-J. Hong, 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]
- S. H. Yun, G. J. Tearney, J. F. d. Boer, B. E. Bouma, “Motion artifacts in optical coherence tomography with frequency-domain ranging,” Opt. Express 12, 2977–2998 (2004). [CrossRef] [PubMed]
- I. Grulkowski, I. Gorczynska, M. Szkulmowski, D. Szlag, A. Szkulmowska, R. A. Leitgeb, A. Kowalczyk, M. Wojtkowski, “Scanning protocols dedicated to smart velocity ranging in spectral OCT,” Opt. Express 17, 23736–23754 (2009). [CrossRef]
- L. An, J. Qin, 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]
- B. Braaf, K. A. Vermeer, K. V. Vienola, 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]

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