The effect of frequency dependent scattering and attenuation on the estimation of blood velocity using ultrasound

A comprehensive theoretical performance comparison of the wideband maximum-likelihood (WMLE) and cross-correlation strategies, previously proposed and evaluated for the estimation of blood velocity using ultrasound is presented. It is based on evaluation of the bias, local and global accuracy, and signal-to-noise ratio (SNR) performance. The results show that the intervening medium does not bias either wideband estimation, due to the effect of tracking the scattering target. The presence of intervening tissue actually improves the global accuracy of both wideband estimators, without a significant change in the local accuracy of either wideband estimator. After the transmission of P pulses, a comparison of the performance of the two strategies shows that the cross-correlation estimator requires P(2 ) correlations to achieve performance similar to that of the WMLE with P operations. In addition, the WMLE can increase the effective SNR in comparison with cross correlation.     

A new time-domain narrowband velocity estimation technique for Doppler ultrasound flow imaging. I. Theory

A significant improvement in blood velocity estimation accuracy can be achieved by simultaneously processing both temporal and spatial information obtained from a sample volume. Use of the spatial information becomes especially important when the temporal resolution is limited. By using a two-dimensional sequence of spatially sampled Doppler signal "snapshots" an improved estimate of the Doppler correlation matrix can be formed. Processing Doppler data in this fashion addresses the range-velocity spread nature of the distributed red blood cell target, leading to a significant reduction in spectral speckle. Principal component spectral analysis of the "snapshot" correlation matrix is shown to lead to a new and robust Doppler mode frequency estimator. By processing only the dominant subspace of the Doppler correlation matrix, the Cramer-Rao bounds on the estimation error of target velocity is significantly reduced in comparison to traditional narrowband blood velocity estimation methods and achieves almost the same local accuracy as a wideband estimator. A time-domain solution is given for the velocity estimate using the root-MUSIC algorithm, which makes the new estimator attractive for real-time implementation.     

A new time-domain narrowband velocity estimation technique for Doppler ultrasound flow imaging. II. Comparative performance assessment

For pt.I see ibid., vol.45, no.4, pp.939-54 (1998). The statistical performance of the new 2-D narrowband time-domain root-MUSIC blood velocity estimator described previously is evaluated using both simulated and flow phantom wideband (50% fractional bandwidth) ultrasonic data. Comparisons are made with the standard 1-D Kasai estimator and two other wideband strategies: the time domain correlator and the wideband point maximum likelihood estimator. A special case of the root-MUSIC, the "spatial" Kasai, is also considered. Simulation and flow phantom results indicate that the root-MUSIC blood velocity estimator displays a superior ability to reconstruct spatial blood velocity information under a wide range of operating conditions. The root-MUSIC mode velocity estimator can be extended to effectively remove the clutter component from the sample volume data. A bimodal velocity estimator is formed by processing the signal subspace spanned by the eigenvectors corresponding to the two largest eigenvalues of the Doppler correlation matrix. To test this scheme, in vivo common carotid flow complex Doppler data was obtained from a commercially available color flow imaging system. Velocity estimates were made using a reduced form of this data corresponding to higher frame rates. The extended root-MUSIC approach was found to produce superior results when compared to both 1- and 2-D Kasai-type estimators that used initialized clutter filters. The results obtained using simulated, flow phantom, and in vivo data suggest that increased sensitivity as well as effective clutter suppression can be achieved using the root-MUSIC technique, and this may be particularly important for wideband high frame rate imaging applications.     

A new wideband spread target maximum likelihood estimator for blood velocity estimation. II. Evaluation of estimator with experimental data

For pt.I see ibid., vol.38, p.1-16, Jan. 1991. The signal models and performance of the estimation strategies described in pt.I are tested with experimental ultrasonic data. The ultrasonic data analyzed verify the theoretical model and predicted performance. The averaged correlation, verified experimentally, confirms that the correlation envelope can be used to estimate the velocity of scatterers and that the shape of the correlation function conveys information regarding the velocity profile within the sample volume. For both the wideband point and range spread estimators, the predicted improvement in velocity resolution and the reduction in height of subsidiary velocity peaks are demonstrated. Through the use of these estimation strategies, information regarding the mean velocity and velocity variation are available for each spatial location within the vessel. This information is presented using a three-dimensional spatial velocity profile display, which appears to offer a number of advantages in the rapid identification of pathology.     

Spectral correlation in ultrasonic pulse echo signal processing

The effects of using spectral correlation in a maximum-likelihood estimator (MLE) for backscattered energy corresponding to coherent reflectors embedded in media of microstructure scatterers is considered. The spectral autocorrelation (SAC) function is analyzed for various scatterer configurations based on the regularity of the interspacing distance between scatterers. It is shown that increased regularity gives rise to significant spectral correlation, whereas uniform distribution of scatters throughout a resolution cell results in no significant correlation between spectral components. This implies that when a true uniform distribution for the effective scatterers exists, the power spectral density (PSD) is sufficient to characterize their echoes. However, as the microstructure scatterer distribution becomes more regular, SAC terms become more significant. MLE results for 15 A-scans from stainless steel specimens with three different grain sizes indicate an average 6-dB signal-to-noise ratio (SNR) improvement in the coherent scatterer (flat-bottom hole) echo intensities for estimators using the SAC characterization as opposed to the PSD characterization.     

A new estimator for vector velocity estimation

A new estimator for determining the two-dimensional velocity vector using a pulsed ultrasound field is derived. The estimator uses a transversely modulated ultrasound field for probing the moving medium under investigation. A modified autocorrelation approach is used in the velocity estimation. The new estimator automatically compensates for the axial velocity when determining the transverse velocity. The estimation is optimized by using a lag different from one in the estimation process, and noise artifacts are reduced by averaging RF samples. Further, compensation for the axial velocity can be introduced, and the velocity estimation is done at a fixed depth in tissue to reduce the influence of a spatial velocity spread. Examples for different velocity vectors and field conditions are shown using both simple and more complex field simulations. A relative accuracy of 10.1% is obtained for the transverse velocity estimates for a parabolic velocity profile for flow transverse to the ultrasound beam and a SNR of 20 dB using 20 pulse-echo lines. The overall bias in the estimates was -4.3%     

Estimation of mean frequency and variance of ultrasonic Doppler signal by using second-order autoregressive model

In order to estimate the mean frequency and variance of the diagnostic ultrasound Doppler signal in the presence of clutter noise, a new estimator using a second-order autoregressive (AR) model, called the AR estimator, is proposed. The sampled signal that contains information of both the Doppler signal and clutter is described by the second-order AR model with two poles. The mean frequency and variance of a unidirectional Doppler signal can be estimated, respectively, from the phase and the magnitude of the pole, with larger phase between the two poles. If the clutter is not completely rejected, all conventional estimators, including the autocorrelation (AC) estimator, result in erroneous estimations for the mean frequency and variance of the Doppler signal, whereas the AR estimator gives an accurate estimation. In the absence of clutter, however, the performance of both the AC and AR estimators are similar. If the blood flows in both directions in a sample volume and the clutter is rejected to the extent that it no longer obscures the Doppler signal, the proposed method can estimate simultaneously the mean frequencies and variances of both the forward and reverse blood flows. The performance of the proposed AR estimator was compared with that of the AC estimator by computer simulations and experiments, and it was found that when the number of available sampled data is small, the AR estimator does not require the use of a clutter filter, which simplifies Doppler signal detection.     

H.L. Van Trees (ed.) and K.L. Bell (ed.), Bayesian Bounds for parameter estimation and nonlinear filtering/tracking, Wiley & Sons Ltd, 2007 [Book review]

Parameter and random-process estimation play a central role in signal processing. A common requirement in the estimation problems is to predict and/or evaluate the error performance. This task is often complicated because the system and measurement models are typically non-linear with some inherent uncertainty. The error performance prediction is supposed to be theoretical - its goal is to establish the best achievable limit and is to be carried out even before one develops a suitable estimator. The error performance evaluation is usually done by Monte Carlo simulations once a candidate estimator has been built. Evaluation serves to assess the estimator quality by comparing its actual error with the theoretical prediction. The fundamental question in algorithm design is "Can we do better?" In this context, it is of paramount importance to find a theoretical bound on the error performance and compare it against the performance of various sub-optimal candidate estimators.     

A maximum likelihood estimator for linear and nonlinear systems-apractical application of estimation techniques in measurement problems

A method is presented for estimating the parameters of linear systems and nonlinear systems. The linear systems are modeled by their transfer function, while the nonlinear systems are described by a Volterra series. The estimator belongs to the class of maximum-likelihood estimators. During the estimation process, the Cramer-Rao lower bound on the covariance matrix of the estimates is derived     

Estimating mean scatterer spacing with the frequency-smoothedspectral autocorrelation function

The quasiperiodicity of regularly spaced scatterers results in characteristic patterns in the spectra of backscattered ultrasonic signals from which the mean scatterer spacing can be estimated. The mean spacing has been considered for classifying certain biological tissue. This paper addresses the problem of estimating the mean scatterer spacing from backscattered ultrasound signals using the frequency-smoothed spectral autocorrelation (SAC) function. The SAC function exploits characteristic differences between the phase spectrum of the resolvable quasiperiodic scatterers and the unresolvable uniformly distributed (diffuse) scatterers to improve estimator performance over other estimators that operate directly on the magnitude spectrum. Mean scatterer spacing estimates are compared for the frequency-smoothed SAC function and a cepstral technique using an AR model. Simulation results indicate that SAC-based estimates converge more reliably over smaller amounts of data than cepstrum-based estimates. An example of computing an estimate from liver tissue scans is also presented for the SAC function and the AR cepstrum     

Phase-Shifting Interferometry and Maximum-Likelihood Estimation Theory. II. A Generalized Solution

A novel means of quantitatively assessing the performance of a phase-shifting interferometer is further investigated. We show how maximum-likelihood estimation theory can be used to estimate the surface profile from the general case of M noisy, phase-shifted measurements. Monte Carlo experiments show that the maximum-likelihood estimator is unbiased and efficient, achieving the theoretical Cramér-Rao lower bound on the variance of the error. We then use Monte Carlo experiments to compare the performance of the maximum-likelihood estimator with that of two conventional algorithms.     

Single-ensemble-based eigen-processing methods for color flow imaging--Part II. The matrix pencil estimator

Parametric spectral estimators can potentially be used to obtain flow estimates directly from raw slow-time ensembles whose clutter has not been suppressed. We present a new eigen-based parametric flow estimation method called the matrix pencil, whose principles are based on a matrix form under the same name. The presented method models the slow-time signal as a sum of dominant complex sinusoids in the slow-time ensemble, and it computes the principal Doppler frequencies by using a generalized eigen-value problem-formulation and matrix rank reduction principles. Both fixed rank (rank-one, rank-two) and adaptive-rank matrix pencil flow estimators are proposed, and their potential applicability to color flow signal processing is discussed. For the adaptive-rank estimator, the nominal rank was defined as the minimum eigen-structure rank that yields principal frequency estimates with a spread greater than a prescribed bandwidth. In our initial performance evaluation, the fixed-rank matrix pencil estimators were applied to raw color flow data (transmit frequency: 5 MHz; pulse repetition period: 0.175 ms; ensemble size: 14) acquired from a steady flow phantom (70 cm/s at centerline) that was surrounded by rigid-tissue-mimicking material. These fixed-rank estimators produced velocity maps that are well correlated with the theoretical flow profile (correlation coefficient: 0.964 to 0.975). To facilitate further evaluation, the matrix pencil estimators were applied to synthetic slow-time data (transmit frequency: 5 MHz; pulse repetition period: 1.0 ms; ensemble size: 10) modeling flow scenarios without and with tissue motion (up to 1 cm/s). The bias and root-mean-squared error of the estimators were computed as a function of blood-signal-to-noise ratio and blood velocity. The matrix pencil flow estimators showed that they are comparatively less biased than most of the existing frequency-based flow estimators like the lagone autocorrelator.     

Optimal velocity estimation in ultrasound color flow imaging in presence of clutter

In color flow imaging (CFI), the rejection of tissue clutter signal is treated separately from blood velocity estimation by high-pass filtering the received Doppler signal. The complete suppression of clutter is then difficult to achieve without affecting the subsequent velocity estimates. In this work, a different approach to velocity estimation is investigated, based on a statistical model of the signal from both clutter and blood. An analytic expression for the Cramer-Rao lower bound (CRLB) is developed, and used to determine the existence of an efficient maximum likelihood estimator (MLE) of blood velocity in CFI when assuming full knowledge of the clutter statistics. We further simulate and compare the performance of the MLE to that of the autocorrelation method (ACM) using finite-impulse response (FIR) and polynomial regression clutter filters. Two signal scenarios are simulated, representing a central and peripheral vessel. Simulations showed that, by including 3-9 (independent) spatial points, the MLE variance approached the CRLB in both scenarios. The ACM was approximately unbiased only for the central scenario in the clutter filter pass band, then with a variance of up to four times the CRLB. The ACM suffered from a severe bias in the filter transition region, and a significant performance gain was achieved here using the MLE. For practical use, the clutter properties must be estimated. We finally replaced the known clutter statistics with an estimate obtained from low-rank approximations of the received sample correlation matrix. Used in the model-based framework, this method came close to the performance of the MLE, and it may be an important step toward a practical model-based estimator, including tissue clutter with optimal performance.     

Clutter rejection filters in color flow imaging: a theoretical approach

A general class of linear clutter rejection filters is described, covering the commonly used filter types including FIR/IIR filters with linear initialization, as well as regression filters, where the clutter component is estimated by least square curve fitting. The filter can be described by a complex valued matrix, and a frequency response is defined. However, in contrast to a time invariant filter, the general linear filter may create frequency components which are not present in the input signal. This produces bias in the velocity and velocity spread estimates. It is shown that the clutter filter effect on the autocorrelation estimates can be described by a frequency domain transfer function, but unlike time invariant filters, the transfer function is different for each temporal lag of the autocorrelation function. Using a two dimensional (axial and temporal dimension) model of the received signal, the bias in velocity and velocity spread is quantified, both for the autocorrelation algorithm and the time shift cross-correlation estimator. Theoretical expressions, as well as numerical examples are given.     

Estimation receiver operating characteristic curve and ideal observers for combined detection/estimation tasks

The localization receiver operating characteristic (LROC) curve is a standard method to quantify performance for the task of detecting and locating a signal. This curve is generalized to arbitrary detection/estimation tasks to give the estimation ROC (EROC) curve. For a two-alternative forced-choice study, where the observer must decide which of a pair of images has the signal and then estimate parameters pertaining to the signal, it is shown that the average value of the utility on those image pairs where the observer chooses the correct image is an estimate of the area under the EROC curve (AEROC). The ideal LROC observer is generalized to the ideal EROC observer, whose EROC curve lies above those of all other observers for the given detection/estimation task. When the utility function is nonnegative, the ideal EROC observer is shown to share many mathematical properties with the ideal observer for the pure detection task. When the utility function is concave, the ideal EROC observer makes use of the posterior mean estimator. Other estimators that arise as special cases include maximum a posteriori estimators and maximum-likelihood estimators.     

Squared envelope-based SNR estimation

Two novel squared envelope (SQE)-based and nondata-aided (NDA) signal-to-noise ratio (SNR) estimators over slow fading channel are proposed, which are robust to carrier offsets and can be applied to both nonconstant modulus (NCM) and constant modulus (CM) constellations. First, the moment-based first-and-second moments (M1M2) estimator is proposed because of its simple computation. The NDA approach has some difficulties when it is applied to NCM constellations, mainly due to self-noise introduced by the variability in the amplitude of transmitted symbols. Therefore, an expectation-maximization (EM) estimator, to solve the problem, instead of using the maximum-likelihood estimation, which is too complex for practical implementation, is proposed. For comparison purposes, the normalized Cramér–Rao lower bound is numerically evaluated as the performance benchmark for those estimators. Simulation results have verified that the two novel SQE and NDA SNR estimators are better at estimating the corresponding SNR range for different constellations. Besides, two hybrid schemes combining M1M2 with EM estimators are suggested to reduce deficiencies caused by these methods when they are employed independently to estimate the SNR.     

Estimating 2-D vector velocities using multidimensional spectrum analysis

Wilson (1991) presented an ultrasonic wideband estimator for axial blood flow velocity estimation through the use of the 2-D Fourier transform. It was shown how a single velocity component was concentrated along a line in the 2-D Fourier space, where the slope was given by the axial velocity. Later, it was shown that this approach could also be used for finding the lateral velocity component by also including a lateral sampling. A single velocity component would then be concentrated along a plane in the 3-D Fourier space, tilted according to the 2 velocity components. This paper presents 2 new velocity estimators for finding both the axial and lateral velocity components. The estimators essentially search for the plane in the 3- D Fourier space, where the integrated power spectrum is largest. The first uses the 3-D Fourier transform to find the power spectrum, while the second uses a minimum variance approach. Based on this plane, the axial and lateral velocity components are estimated. Several phantom measurements, for flow-to-depth angles of 60, 75, and 90 degrees, were performed. Multiple parallel lines were beamformed simultaneously, and 2 different receive apodization schemes were tried. The 2 estimators were then applied to the data. The axial velocity component was estimated with an average standard deviation below 2.8% of the peak velocity, while the average standard deviation of the lateral velocity estimates was between 2.0% and 16.4%. The 2 estimators were also tested on in vivo data from a transverse scan of the common carotid artery, showing the potential of the vector velocity estimation method under in vivo conditions.     

数字采样测量噪声误差的估计和分析

针对高频数字采样中的附加噪声和时间抖动误差估计问题，给出了高频数字采样测量噪声误差的最大似然估计方法。在获得了相应的数学模型后，对最小二乘估计和最大似然估计方法进行了比较。仿真结果表明，对于此类误差最大似然估计比最小二乘估计有效性更好。另外最大似然估计的一个优点是用于不确定度计算的克拉美一罗下限值更易于获得。     

Degree of polarization estimation in the presence of nonuniform illumination and additive Gaussian noise

Within the general framework of active imaging we address the degree of polarization (DOP) estimation in the presence of additive Gaussian detector noise. We first study the performance of standard DOP estimators and propose a method to increase estimation precision using physically relevant a priori information. We then consider the realistic case of nonuniform illumination distribution. We derive the Cramer-Rao lower bound and determine a profile likelihood-based estimator. We demonstrate the efficiency of this new estimator and compare its performance with other standard estimators as a function of the degree of nonuniformity of the illumination.