Statistical properties of phase-decorrelation in phase-resolved Doppler optical coherence tomography

Abstract

Decorrelation noise limits the ability of phase-resolved Doppler optical coherence tomography systems to detect smaller vessels exhibiting slower flow velocities, which limits the utility of the technique in many clinical and biological settings. An understanding of the statistical properties of decorrelation noise can aid in the optimal design of these systems and guide the development of noise mitigating strategies. In this work, the statistical properties of decorrelation noise are derived from the underlying statistics of the coherent imaging system and validated through comparison with empirical results and Monte Carlo modeling. Expressions for the noise distribution and the noise variance as a function of relevant imaging system parameters are given, and the implications of these results on both system and algorithm design are discussed.

Fig. 1

The beam profiles and defining parameters (δx, 2ω_o) of the displaced Gaussian beams resulting in a single pair of phase measurements, and a single phase difference measurement. The parameter 2ωo defines the 1/e² (power) beam diameter at its focus.

Fig. 2

(a) Each of the beams A and B yield a complex reflectivity (for a given depth) represented by the phasors Γ_A and Γ_B, respectively. The signal of interest is the phase difference between these phasors, θ. (b) The correlation between the phasors Γ_A and Γ_B is described explicitly by expressing the phasor Γ_B as a linear combination of Γ_A and a third independent phasor Γ_C, with the coefficients of this linear combination being determined by the fractional displacement of the two measuring beams, (δx/ω_o). (c) Using a rotated coordinate frame in which the X’ axis is aligned to the phasor Γ_A. equates the phase difference, θ, with the angle of the phasor ${\mathrm{\Gamma }}_B={\alpha }^2\mid {\mathrm{\Gamma }}_A\mid +\sqrt{1-{\alpha }^4}{\mathrm{\Gamma }}_C$.

Fig. 3

The decorrelation noise distribution is shown for three fractional displacements, (δx/ω_o).

Fig. 4

The experimental and numerical validating methods are illustrated. Decorrelation noise was measured by imaging a homogenous scattering phantom using a Doppler optical frequency domain imaging (OFDI) system. Corresponding intensity (a) and phase difference (b) images over a reduced ROI are shown for a fractional displacement of 0.156. Note that the colormap on the Doppler image is truncated from its full range of [−π, π] to [−π/2, π/2] to increase contrast. (c) The generation of decorrelation noise by numerical simulation is illustrated. The beam profile (E²) given by the intensity of the Gaussian imaging is multiplied by an array of randomly generated complex reflectivities (R) with and without an offset of Δ and summed to give correlated complex reflectivities Γ_A and Γ_B respectively. The fractional displacement is given by the ratio of Δ to the beam width.

Fig. 5

(a) The derived analytic expression for the decorrelation noise distribution is compared with measured noise distributions from a PR-DOCT system and simulated noise distributions from a Monte Carlo model for (a) fractional displacement (δx/ω_o) = 0.078 and (b) (δx/ω_o) = 0.156. The analytic expression displays excellent agreement with both the measured and simulated distributions over the full angular range.

Fig. 6

The fractional displacement and signal-to-noise ratio (SNR) are plotted versus their resulting RMS phase noise. Using these relations, imaging systems can be designed to achieve the highest imaging speed (resulting from the highest fractional displacements) while limiting decorrelation noise levels at or below the level resulting from an anticipated SNR level. As an example, SNR levels of 15 dB to 30 dB require limiting fractional displacements to less than 0.064 and 0.009 respectively to maintain near SNR limited Doppler performance.

Fig. 7

The performance of mean, median, and weighted mean filters applied to decorrelation noise is plotted for a series of sample counts (n = 3, 10, 20, 30, and 40). The estimator errors were calculated as the standard deviation of that estimator applied to 25000 independent sets (of sample count n). These sets were generated using the Monte Carlo model of decorrelation noise. For the weighted mean, weights were given by |Γ_A|². Both the median and weighted mean posses outlier rejection properties that reduce the impact of the large power in the tails of the decorrelation noise distribution function. The error for the median and weighted mean estimators as a fraction of the error of the mean estimator is given as the label for each point in their respective curves. The fractional displacement (δx/ω_o) for this calculation was 0.12 corresponding to an unfiltered RMS noise level of 0.31 rad. Inset: the estimator error for the weighted mean filter and sample count n = 20 is shown as a function of the weighting exponent, k. Optimal filtering is achieved near k = 2.