A near-optimal estimation framework for velocity magnitude and direction in two-beam Doppler systems

Doppler ultrasound systems are widely used in medical imaging and biomedical applications to measure movement and fluid velocity. Although, accurate estimation of velocity magnitude and flow direction can be quite difficult when the Doppler frequency shift is unknown and contaminated by measurement noise. This paper introduces an estimate, approximate minimum variance unbiased estimate, for obtaining velocity measurements using Doppler ultrasound with two beams, when the Doppler frequency is unknown. A statistical model of Doppler measurements that are non-linear has been first used to develop a sufficient statistics based estimate, after which a second-order approximation of the likelihood function is incorporated to obtain asymptotically efficient parameter estimates for the beam velocities. The resulting beam velocity estimates have therefore been used to estimate the velocity magnitude and blood flow direction. Analytical closed-form expressions for the estimation of variances are extracted through higher-order error propagation analysis, and is used for estimating theoretical performance of the estimators. The analytical variances were measured against the Cramer–Rao lower bound (CRLB) across different signal to noise ratios, beam angles, and ensembles. Simulations demonstrated that the performance of the proposed estimator exhibiting a near-optimal asymptotic efficiency which is near to the CRLB, thus corresponding to near-optimal asymptotic efficiency. Analyzing the data showed that the value of beam intersection angle, doppler frequency uncertainty, and measurement of noise affected the accuracy of both magnitudes and directions of velocity. The presented framework provides theoretical approach for developing reliable ultrasound Doppler velocity estimation systems in situations where the Doppler frequency is uncertain.