Abstract

OCT is a popular cross-sectional microscale imaging modality in medicine and biology. While structural imaging using OCT is a mature technology in many respects, flow and motion estimation using OCT remains an intense area of research. In particular, there is keen interest in maximizing information extraction from the complex-valued OCT signal. Here, we introduce a Bayesian framework into the data workflow in OCT-based velocimetry. We demonstrate that using prior information in this Bayesian framework can significantly improve velocity estimate precision in a correlation-based, model-based framework for Doppler and transverse velocimetry. We show results in calibrated flow phantoms as well as in vivo in a Drosophila melanogaster (fruit fly) heart. Thus, our work improves upon the current approaches in terms of improved information extraction from the complex-valued OCT signal.

© 2015 Optical Society of America

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References

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  1. W. Drexler, M. Liu, A. Kumar, T. Kamali, A. Unterhuber, and R. A. Leitgeb, “Optical coherence tomography today: speed, contrast, and multimodality,” J. Biomed. Opt. 19(7), 071412 (2014).
    [Crossref] [PubMed]
  2. B. J. Berne and R. Pecora, Dynamic Light Scattering (John Wiley & Sons, Inc., New York, 1976).
  3. Y. Imai and K. Tanaka, “Direct velocity sensing of flow distribution based on low-coherence interferometry,” J. Opt. Soc. Am. A 16(8), 2007–2012 (1999).
    [Crossref]
  4. J. Lee, W. Wu, J. Y. Jiang, B. Zhu, and D. A. Boas, “Dynamic light scattering optical coherence tomography,” Opt. Express 20(20), 22262–22277 (2012).
    [Crossref] [PubMed]
  5. V. J. Srinivasan, H. Radhakrishnan, E. H. Lo, E. T. Mandeville, J. Y. Jiang, S. Barry, and A. E. Cable, “OCT methods for capillary velocimetry,” Biomed. Opt. Express 3(3), 612–629 (2012).
    [Crossref] [PubMed]
  6. X. Liu, Y. Huang, J. C. Ramella-Roman, S. A. Mathews, and J. U. Kang, “Quantitative transverse flow measurement using optical coherence tomography speckle decorrelation analysis,” Opt. Lett. 38(5), 805–807 (2013).
    [Crossref] [PubMed]
  7. N. Weiss, T. G. van Leeuwen, and J. Kalkman, “Localized measurement of longitudinal and transverse flow velocities in colloidal suspensions using optical coherence tomography,” Phys. Rev. E. 88(4), 042312 (2013).
    [Crossref] [PubMed]
  8. B. K. Huang and M. A. Choma, “Resolving directional ambiguity in dynamic light scattering-based transverse motion velocimetry in optical coherence tomography,” Opt. Lett. 39(3), 521–524 (2014).
    [Crossref] [PubMed]
  9. B. K. Huang, U. A. Gamm, V. Bhandari, M. K. Khokha, and M. A. Choma, “Three-dimensional, three-vector-component velocimetry of cilia-driven fluid flow using correlation-based approaches in optical coherence tomography,” Biomed. Opt. Express 6(9), 3515–3538 (2015).
    [Crossref] [PubMed]
  10. N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, “Equation of State Calculations by Fast Computing Machines,” J. Chem. Phys. 21(6), 1087–1092 (1953).
    [Crossref]
  11. M. Plummer, “JAGS: A program for analysis of Bayesian graphical models using Gibbs sampling ” in Proceedings of the 3rd International Workshop on Distributed Statistical Computing (DSC 2003), F. K. Hornik, ed. (Technische Universität Wien, Vienna, Austria, 2003).
  12. M. Steyvers, “MATJAGS: a Matlab interface for JAGS,” (2011), http://psiexp.ss.uci.edu/research/programs_data/jags/ .
  13. A. Charnes, E. L. Frome, and P. L. Yu, “The Equivalence of Generalized Least Squares and Maximum Likelihood Estimates in the Exponential Family,” J. Am. Stat. Assoc. 71(353), 169–171 (1976).
    [Crossref]
  14. C. Kasai, K. Namekawa, A. Koyano, and R. Omoto, “Real-Time Two-Dimensional Blood Flow Imaging Using an Autocorrelation Technique,” IEEE Trans. Sonics and Ultrasonics 32(3), 458–464 (1985).
    [Crossref]
  15. V. X. D. Yang, M. L. Gordon, A. Mok, Y. Zhao, Z. Chen, R. S. C. Cobbold, B. C. Wilson, and I. Alex Vitkin, “Improved phase-resolved optical Doppler tomography using the Kasai velocity estimator and histogram segmentation,” Opt. Commun. 208(4-6), 209–214 (2002).
    [Crossref]
  16. S. Geman and C. Graffigne, “Markov Random Field Image Models and Their Applications to Computer Vision,” in International Congress of Mathematicians, A. M. Gleason, ed. (American Mathematical Society, Berkeley, California, 1986), pp. 1496–1517.

2015 (1)

2014 (2)

W. Drexler, M. Liu, A. Kumar, T. Kamali, A. Unterhuber, and R. A. Leitgeb, “Optical coherence tomography today: speed, contrast, and multimodality,” J. Biomed. Opt. 19(7), 071412 (2014).
[Crossref] [PubMed]

B. K. Huang and M. A. Choma, “Resolving directional ambiguity in dynamic light scattering-based transverse motion velocimetry in optical coherence tomography,” Opt. Lett. 39(3), 521–524 (2014).
[Crossref] [PubMed]

2013 (2)

X. Liu, Y. Huang, J. C. Ramella-Roman, S. A. Mathews, and J. U. Kang, “Quantitative transverse flow measurement using optical coherence tomography speckle decorrelation analysis,” Opt. Lett. 38(5), 805–807 (2013).
[Crossref] [PubMed]

N. Weiss, T. G. van Leeuwen, and J. Kalkman, “Localized measurement of longitudinal and transverse flow velocities in colloidal suspensions using optical coherence tomography,” Phys. Rev. E. 88(4), 042312 (2013).
[Crossref] [PubMed]

2012 (2)

2002 (1)

V. X. D. Yang, M. L. Gordon, A. Mok, Y. Zhao, Z. Chen, R. S. C. Cobbold, B. C. Wilson, and I. Alex Vitkin, “Improved phase-resolved optical Doppler tomography using the Kasai velocity estimator and histogram segmentation,” Opt. Commun. 208(4-6), 209–214 (2002).
[Crossref]

1999 (1)

1985 (1)

C. Kasai, K. Namekawa, A. Koyano, and R. Omoto, “Real-Time Two-Dimensional Blood Flow Imaging Using an Autocorrelation Technique,” IEEE Trans. Sonics and Ultrasonics 32(3), 458–464 (1985).
[Crossref]

1976 (1)

A. Charnes, E. L. Frome, and P. L. Yu, “The Equivalence of Generalized Least Squares and Maximum Likelihood Estimates in the Exponential Family,” J. Am. Stat. Assoc. 71(353), 169–171 (1976).
[Crossref]

1953 (1)

N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, “Equation of State Calculations by Fast Computing Machines,” J. Chem. Phys. 21(6), 1087–1092 (1953).
[Crossref]

Alex Vitkin, I.

V. X. D. Yang, M. L. Gordon, A. Mok, Y. Zhao, Z. Chen, R. S. C. Cobbold, B. C. Wilson, and I. Alex Vitkin, “Improved phase-resolved optical Doppler tomography using the Kasai velocity estimator and histogram segmentation,” Opt. Commun. 208(4-6), 209–214 (2002).
[Crossref]

Barry, S.

Bhandari, V.

Boas, D. A.

Cable, A. E.

Charnes, A.

A. Charnes, E. L. Frome, and P. L. Yu, “The Equivalence of Generalized Least Squares and Maximum Likelihood Estimates in the Exponential Family,” J. Am. Stat. Assoc. 71(353), 169–171 (1976).
[Crossref]

Chen, Z.

V. X. D. Yang, M. L. Gordon, A. Mok, Y. Zhao, Z. Chen, R. S. C. Cobbold, B. C. Wilson, and I. Alex Vitkin, “Improved phase-resolved optical Doppler tomography using the Kasai velocity estimator and histogram segmentation,” Opt. Commun. 208(4-6), 209–214 (2002).
[Crossref]

Choma, M. A.

Cobbold, R. S. C.

V. X. D. Yang, M. L. Gordon, A. Mok, Y. Zhao, Z. Chen, R. S. C. Cobbold, B. C. Wilson, and I. Alex Vitkin, “Improved phase-resolved optical Doppler tomography using the Kasai velocity estimator and histogram segmentation,” Opt. Commun. 208(4-6), 209–214 (2002).
[Crossref]

Drexler, W.

W. Drexler, M. Liu, A. Kumar, T. Kamali, A. Unterhuber, and R. A. Leitgeb, “Optical coherence tomography today: speed, contrast, and multimodality,” J. Biomed. Opt. 19(7), 071412 (2014).
[Crossref] [PubMed]

Frome, E. L.

A. Charnes, E. L. Frome, and P. L. Yu, “The Equivalence of Generalized Least Squares and Maximum Likelihood Estimates in the Exponential Family,” J. Am. Stat. Assoc. 71(353), 169–171 (1976).
[Crossref]

Gamm, U. A.

Gordon, M. L.

V. X. D. Yang, M. L. Gordon, A. Mok, Y. Zhao, Z. Chen, R. S. C. Cobbold, B. C. Wilson, and I. Alex Vitkin, “Improved phase-resolved optical Doppler tomography using the Kasai velocity estimator and histogram segmentation,” Opt. Commun. 208(4-6), 209–214 (2002).
[Crossref]

Huang, B. K.

Huang, Y.

Imai, Y.

Jiang, J. Y.

Kalkman, J.

N. Weiss, T. G. van Leeuwen, and J. Kalkman, “Localized measurement of longitudinal and transverse flow velocities in colloidal suspensions using optical coherence tomography,” Phys. Rev. E. 88(4), 042312 (2013).
[Crossref] [PubMed]

Kamali, T.

W. Drexler, M. Liu, A. Kumar, T. Kamali, A. Unterhuber, and R. A. Leitgeb, “Optical coherence tomography today: speed, contrast, and multimodality,” J. Biomed. Opt. 19(7), 071412 (2014).
[Crossref] [PubMed]

Kang, J. U.

Kasai, C.

C. Kasai, K. Namekawa, A. Koyano, and R. Omoto, “Real-Time Two-Dimensional Blood Flow Imaging Using an Autocorrelation Technique,” IEEE Trans. Sonics and Ultrasonics 32(3), 458–464 (1985).
[Crossref]

Khokha, M. K.

Koyano, A.

C. Kasai, K. Namekawa, A. Koyano, and R. Omoto, “Real-Time Two-Dimensional Blood Flow Imaging Using an Autocorrelation Technique,” IEEE Trans. Sonics and Ultrasonics 32(3), 458–464 (1985).
[Crossref]

Kumar, A.

W. Drexler, M. Liu, A. Kumar, T. Kamali, A. Unterhuber, and R. A. Leitgeb, “Optical coherence tomography today: speed, contrast, and multimodality,” J. Biomed. Opt. 19(7), 071412 (2014).
[Crossref] [PubMed]

Lee, J.

Leitgeb, R. A.

W. Drexler, M. Liu, A. Kumar, T. Kamali, A. Unterhuber, and R. A. Leitgeb, “Optical coherence tomography today: speed, contrast, and multimodality,” J. Biomed. Opt. 19(7), 071412 (2014).
[Crossref] [PubMed]

Liu, M.

W. Drexler, M. Liu, A. Kumar, T. Kamali, A. Unterhuber, and R. A. Leitgeb, “Optical coherence tomography today: speed, contrast, and multimodality,” J. Biomed. Opt. 19(7), 071412 (2014).
[Crossref] [PubMed]

Liu, X.

Lo, E. H.

Mandeville, E. T.

Mathews, S. A.

Metropolis, N.

N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, “Equation of State Calculations by Fast Computing Machines,” J. Chem. Phys. 21(6), 1087–1092 (1953).
[Crossref]

Mok, A.

V. X. D. Yang, M. L. Gordon, A. Mok, Y. Zhao, Z. Chen, R. S. C. Cobbold, B. C. Wilson, and I. Alex Vitkin, “Improved phase-resolved optical Doppler tomography using the Kasai velocity estimator and histogram segmentation,” Opt. Commun. 208(4-6), 209–214 (2002).
[Crossref]

Namekawa, K.

C. Kasai, K. Namekawa, A. Koyano, and R. Omoto, “Real-Time Two-Dimensional Blood Flow Imaging Using an Autocorrelation Technique,” IEEE Trans. Sonics and Ultrasonics 32(3), 458–464 (1985).
[Crossref]

Omoto, R.

C. Kasai, K. Namekawa, A. Koyano, and R. Omoto, “Real-Time Two-Dimensional Blood Flow Imaging Using an Autocorrelation Technique,” IEEE Trans. Sonics and Ultrasonics 32(3), 458–464 (1985).
[Crossref]

Radhakrishnan, H.

Ramella-Roman, J. C.

Rosenbluth, A. W.

N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, “Equation of State Calculations by Fast Computing Machines,” J. Chem. Phys. 21(6), 1087–1092 (1953).
[Crossref]

Rosenbluth, M. N.

N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, “Equation of State Calculations by Fast Computing Machines,” J. Chem. Phys. 21(6), 1087–1092 (1953).
[Crossref]

Srinivasan, V. J.

Tanaka, K.

Teller, A. H.

N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, “Equation of State Calculations by Fast Computing Machines,” J. Chem. Phys. 21(6), 1087–1092 (1953).
[Crossref]

Teller, E.

N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, “Equation of State Calculations by Fast Computing Machines,” J. Chem. Phys. 21(6), 1087–1092 (1953).
[Crossref]

Unterhuber, A.

W. Drexler, M. Liu, A. Kumar, T. Kamali, A. Unterhuber, and R. A. Leitgeb, “Optical coherence tomography today: speed, contrast, and multimodality,” J. Biomed. Opt. 19(7), 071412 (2014).
[Crossref] [PubMed]

van Leeuwen, T. G.

N. Weiss, T. G. van Leeuwen, and J. Kalkman, “Localized measurement of longitudinal and transverse flow velocities in colloidal suspensions using optical coherence tomography,” Phys. Rev. E. 88(4), 042312 (2013).
[Crossref] [PubMed]

Weiss, N.

N. Weiss, T. G. van Leeuwen, and J. Kalkman, “Localized measurement of longitudinal and transverse flow velocities in colloidal suspensions using optical coherence tomography,” Phys. Rev. E. 88(4), 042312 (2013).
[Crossref] [PubMed]

Wilson, B. C.

V. X. D. Yang, M. L. Gordon, A. Mok, Y. Zhao, Z. Chen, R. S. C. Cobbold, B. C. Wilson, and I. Alex Vitkin, “Improved phase-resolved optical Doppler tomography using the Kasai velocity estimator and histogram segmentation,” Opt. Commun. 208(4-6), 209–214 (2002).
[Crossref]

Wu, W.

Yang, V. X. D.

V. X. D. Yang, M. L. Gordon, A. Mok, Y. Zhao, Z. Chen, R. S. C. Cobbold, B. C. Wilson, and I. Alex Vitkin, “Improved phase-resolved optical Doppler tomography using the Kasai velocity estimator and histogram segmentation,” Opt. Commun. 208(4-6), 209–214 (2002).
[Crossref]

Yu, P. L.

A. Charnes, E. L. Frome, and P. L. Yu, “The Equivalence of Generalized Least Squares and Maximum Likelihood Estimates in the Exponential Family,” J. Am. Stat. Assoc. 71(353), 169–171 (1976).
[Crossref]

Zhao, Y.

V. X. D. Yang, M. L. Gordon, A. Mok, Y. Zhao, Z. Chen, R. S. C. Cobbold, B. C. Wilson, and I. Alex Vitkin, “Improved phase-resolved optical Doppler tomography using the Kasai velocity estimator and histogram segmentation,” Opt. Commun. 208(4-6), 209–214 (2002).
[Crossref]

Zhu, B.

Biomed. Opt. Express (2)

IEEE Trans. Sonics and Ultrasonics (1)

C. Kasai, K. Namekawa, A. Koyano, and R. Omoto, “Real-Time Two-Dimensional Blood Flow Imaging Using an Autocorrelation Technique,” IEEE Trans. Sonics and Ultrasonics 32(3), 458–464 (1985).
[Crossref]

J. Am. Stat. Assoc. (1)

A. Charnes, E. L. Frome, and P. L. Yu, “The Equivalence of Generalized Least Squares and Maximum Likelihood Estimates in the Exponential Family,” J. Am. Stat. Assoc. 71(353), 169–171 (1976).
[Crossref]

J. Biomed. Opt. (1)

W. Drexler, M. Liu, A. Kumar, T. Kamali, A. Unterhuber, and R. A. Leitgeb, “Optical coherence tomography today: speed, contrast, and multimodality,” J. Biomed. Opt. 19(7), 071412 (2014).
[Crossref] [PubMed]

J. Chem. Phys. (1)

N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, “Equation of State Calculations by Fast Computing Machines,” J. Chem. Phys. 21(6), 1087–1092 (1953).
[Crossref]

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

Opt. Commun. (1)

V. X. D. Yang, M. L. Gordon, A. Mok, Y. Zhao, Z. Chen, R. S. C. Cobbold, B. C. Wilson, and I. Alex Vitkin, “Improved phase-resolved optical Doppler tomography using the Kasai velocity estimator and histogram segmentation,” Opt. Commun. 208(4-6), 209–214 (2002).
[Crossref]

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. E. (1)

N. Weiss, T. G. van Leeuwen, and J. Kalkman, “Localized measurement of longitudinal and transverse flow velocities in colloidal suspensions using optical coherence tomography,” Phys. Rev. E. 88(4), 042312 (2013).
[Crossref] [PubMed]

Other (4)

S. Geman and C. Graffigne, “Markov Random Field Image Models and Their Applications to Computer Vision,” in International Congress of Mathematicians, A. M. Gleason, ed. (American Mathematical Society, Berkeley, California, 1986), pp. 1496–1517.

B. J. Berne and R. Pecora, Dynamic Light Scattering (John Wiley & Sons, Inc., New York, 1976).

M. Plummer, “JAGS: A program for analysis of Bayesian graphical models using Gibbs sampling ” in Proceedings of the 3rd International Workshop on Distributed Statistical Computing (DSC 2003), F. K. Hornik, ed. (Technische Universität Wien, Vienna, Austria, 2003).

M. Steyvers, “MATJAGS: a Matlab interface for JAGS,” (2011), http://psiexp.ss.uci.edu/research/programs_data/jags/ .

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Figures (9)

Fig. 1
Fig. 1

(a) Model of the time-varying, complex-valued OCT signal i(r). Each particle in an ensemble of M identical, uniformly moving, randomly distributed particles contributes to i(r). The contribution is weighted by the point-spread function (psf), which is complex-valued in the z-axis and real-valued in the x- and y-axes. Typically, psf(y) = psf(x). For clarity, only a subset of the M scatterers are shown. The scatterers also undergo diffusion with a root mean squared displacement given by a diffusivity parameter D. (b) iz(t) is the axial response, that is, the complex-valued OCT signal along the z-axis. As shown in Eq. (2), autocorrelation of the complex-valued signal yields a complex-valued fringe as well as an amplitude envelope. Here, the ★operator indicates correlation. The frequency of the fringe is given by the Doppler shift imparted by the axial component of the moving scatterers. The amplitude envelope reproduces the shape of the point-spread function envelope. The width of the envelope is modulated by the axial speed (vz), that is, the magnitude of the axial velocity. Assuming psf(y) = psf(x), the real-valued response along the x- and y-axes is likewise an amplitude envelope with a width modulated by the total in-plane speed (vx2 + vy2)1/2. (c) In the case of purely diffusive motion of monodisperse scatterers in the axial direction, the magnitude of the autocorrelation of the complex-valued signal is an exponential decay. The characteristic decay time is inversely proportional to the particle diffusivity. For short periods of time (shown here), the signal may wander in a local neighborhood. Over time, the signal fills out speckle statistics in the complex plane.

Fig. 2
Fig. 2

Left panel: The measured velocity is the difference between flow velocity in the object being imaged (vflow) and the velocity of the bean scanner that defects the imaging beam (vscan). Varying vscan breaks a symmetry that is otherwise present in DLS-OCT. Symmetry is broken because vmeas is different when sign of vscan is flipped. Moreover, the velocity component along the x-axis (or y-axis) can be estimated by exploiting the fact that vmeas is minimized when vflow = vscan. Right panel: Value of the autocorrelation of the complex-valued OCT signal as a function of time lag (τ) and axial location (z). Data is from a calibrated flow phantom described in Section 4.1. Here, the direction of vscan defines the x-axis, and vflow is nominally parallel to the x-axis. Decorrelation times are longer (i.e. rates of decorrelation are slower) as vscan approaches vflow. Note that the fringe frequency (Doppler shift) varies slowly with scan bias speed, indicating a small beam scanning-induced Doppler shift. By analyzing the total Doppler shift as a function of scan bias velocity, we estimate that the scanner-induced Doppler shift equivalent to −16.4 μm/s per 1 mm/s of scan velocity.

Fig. 3
Fig. 3

Axial velocity (vz) estimation using the time-varying signals acquired at a scan bias velocity of −1.7 mm/s. (a) Representative B-scan at a single scan bias velocity. (b) Kasai estimate of the axial velocity, (c) Bayesian analysis with an uninformative prior and (d) an adaptive hyperprior. (e) A sample uncertainty comparison at the center of the channel (blue = uninformative prior, green = adaptive hyperprior, red = Kasai). Here, the posterior distribution of the axial velocity P(vz|data) is defined in Eqs. (4) and (7).

Fig. 4
Fig. 4

Doppler flow velocity profiles generated using the Kasai Doppler estimator and by centroiding the posterior probability density functions for the uninformative hyperprior (UHP) and adaptive hyperprior (AHP) estimators. The R2 values (minimum velocity values) for Kasai, UHP, and AHP parabolic fits are 0.989 (−0.221 mm/s), 0.991 (−0.225 mm/s), and 0.995 (−0.224 mm/s), respectively.

Fig. 5
Fig. 5

Bayesian estimates of two-component flow velocity vectors: lateral flow velocity (vx) and axial velocity (vz). They were reconstructed using either an uninformative prior (i.e. very broad prior probability P(θ)) or an adaptive hyperprior (i.e. prior probability is defined by a neighboring posterior distribution). Using a larger sample size and incorporating neighboring information improves the precision. We define precision by the width of the posterior probability density function. Each row of subfigures uses the same color bar. The posterior distribution of the lateral velocity P(vx|data) and of the axial velocity P(vz|data) is defined in Eqs. (4) and (7).

Fig. 6
Fig. 6

Comparison of adaptive hyperprior estimation when the estimation process begins on the left-hand side and moves right (left to right; green colormap) and when it begins on the right-hand side and moves left (right to left; blue colormap). The flow velocity profiles are similar in either case. Data are from the ndata = 1, nbias = 8 (second column in Fig. 5). The “merge” images are RGB color images in which the green channel is the left-to-right data and the blue channel is the right-to-left data. Cyan-appearing pixels in the “merge” images indicate a high degree of overlap between the left-to-right and right-to-left profiles. The left-to-right profile has a slight rightward shift and, likewise, the right-to-left profile has a slight leftward shift.

Fig. 7
Fig. 7

Bayesian estimates of DLS parameters using 1 sample and the 4 fastest scan bias velocities. The first column consists of results from an uninformative prior; the second, uninformative priors on all parameters except the lateral beam waist; and the third, adaptive hyperprior. Only the second column assumes a fixed beam waist of 5 μm.

Fig. 8
Fig. 8

Bayesian estimates of DLS parameters using 1 sample and the 4 slowest scan bias velocities.

Fig. 9
Fig. 9

Bayesian heart wall velocimetry in a Drosophila melanogaster pre-pupa. All horizontal axes are identical (time). (a) M-mode image of D. melanogaster heart. (b) Heart wall velocity trace (black) and 95% CI for Bayesian analysis using a broad uninformative prior. Note that CI plot is on a log scale. (c) Heart wall velocity trace (black) and 95% CI for Bayesian analysis using an adaptive hyperprior. (d) Reduction in uncertainty when using an adaptive hyperprior compared to a broad uninformative prior. The vertical axis is truncated at RU = 0. The uninformative prior occasionally outperformed the adaptive hyperprior (i.e. RU<0). These datapoints are highlighted in red.

Tables (1)

Tables Icon

Table 1 Improvements with fewer scan biases

Equations (17)

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

i( r,t )= m=1 M psf ( r )δ( r -[ r r m ( t ) ] )d r
r m ( t )= r m ( 0 )+vt
G(r,τ)=Rexp( 2jk v 2 τ )exp( v x 2 + v y 2 τ 2 w xy 2 v z 2 τ 2 w z 2 )exp( 4 k 2 D| τ | )
G(r, v x scan ,τ)=Rexp( 2jk v 2 τ )exp( ( ( v x v x,scan ) 2 +C ) τ 2 w xy 2 )
P( θ| data )= P( data|θ )P( θ ) P( data )
P( data )= P( data|θ ) P( θ )dθ
P( θ n | data )= θ n c P( θ| data ) d θ n c
θ= θ n θ n c
P r o ( data|θ )= τ, v x,scan 1 2π σ exp( | G r o ( τ, v x,scan )dat a r o ( τ, v x,scan ) | 2 2 σ 2 )
θ={ R, v x, w xy ,C, σ 2 }
P( θ curr , θ neigh | dat a curr )= P( dat a curr | θ curr )P( θ curr | θ neigh )P( θ neigh ) P( dat a curr )
P( dat a curr )= P( dat a curr | θ curr ) P( θ curr | θ neigh )P( θ neigh )d θ curr d θ neigh
P( θ curr,n | dat a curr )= P ( dat a curr , θ curr | dat a curr )d θ curr,n c | d θ neigh
θ curr = θ curr,n θ curr,n c
ω dop = 1 τ lag tan 1 Im{G(r, τ lag )} Re{G(r, τ lag )}
RU= C I 95% UHP C I 95% AHP C I 95% UHP ×100%,
v ¯ z = v z P( v z |data ) d v z

Metrics