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Experimental Evaluation of the Correlation Interpolation Technique to Measure Regional Tissue Velocity
Abstract
A newly developed correlation interpolation method to measure the regional velocity of moving tissue is evaluated in an experimental setup. Pulsed ultrasound echo signals (center frequency 3.5 MHz) are received from a rotating Agar disk containing scattering particles. When averaging over a depth range of 2.2 mm at a pulse repetition frequency (PRF) of 930 Hz, the standard deviation of the measured displacement between 2 successive pulses was found to be +/- 6 microns. In a second series of experiments, the angular velocity of the disk is estimated from the displacement, as measured simultaneously in two different regions located on separate echo lines (PRF = 465 Hz per line). The exact position of both regions in respect to the center of rotation was found to be irrelevant. The accuracy of the calculated angular velocity was found to be better for large angles between the two lines of observation than for small angles.
... The SinMod method is based on the same principle as has been used to determine motion from ultrasound pulses with limited bandwidth [9], [10]. SinMod is designed to map displacement in a cross section of the heart by analysis of a sequence of images. ...
... Within the passband, frequencies are distinguished by applying two bandpass filters and , being skewed to low and high frequencies, respectively, as (4) Back in the spatial domain through inverse Fourier transform, the pair of bandpass filters renders four images with complex intensities as (5) Power of the sequential images, bandpass filtered with bias for low and high frequencies, respectively, is added, resulting in a low-and a high-frequency power image and (6) The complex cross-power image is obtained by summing products of filtered results of the first image and corresponding complex conjugates of the second image as (7) Interpretation of the images , , and is facilitated by substituting the modeled image signals (3) in (5)- (7). Neglecting cross products with uncorrelated noise, it is found that (8) Using (8), local frequency and local displacement are estimated by (9) Note that local frequency differs from the chosen bandpass center frequency , although should be within the passband around (2). The proposed sine-wave model and related algorithms (1)-(9) are focused on the direct environment of a single pixel . ...
... The images are related to the images (5) by (13) Remarkably, the power images , (6), and the complex power image (7) remain unchanged, when substituting the coarse images (13) for the larger images (5), because complex conjugate multiplication removes the effects of fixed frequency shift completely. A coarsely sampled map of displacement is obtained with (9). Next, the displacement map is scaled to the original image size by resampling. ...
The new SinMod method extracts motion from magnetic resonance imaging (MRI)-tagged (MRIT) image sequences. Image intensity in the environment of each pixel is modeled as a moving sine wavefront. Displacement is estimated at subpixel accuracy. Performance is compared with the harmonic-phase analysis (HARP) method, which is currently the most common method used to detect motion in MRIT images. SinMod can handle line tags, as well as speckle patterns. In artificial images (tag distance six pixels), SinMod detects displacements accurately (error < 0.02 pixels). Effects of noise are suppressed effectively. Sharp transitions in motion at the boundary of an object are smeared out over a width of 0.6 tag distance. For MRIT images of the heart, SinMod appears less sensitive to artifacts, especially later in the cardiac cycle when image quality deteriorates. For each pixel, the quality of the sine-wave model in describing local image intensity is quantified objectively. If local quality is low, artifacts are avoided by averaging motion over a larger environment. Summarizing, SinMod is just as fast as HARP, but it performs better with respect to accuracy of displacement detection, noise reduction, and avoidance of artifacts.
... In Section IV, we compare our concept with those of other groups that also estimate the time shift from signal phases. The performance of our new algorithm is discussed in Section V. We compare our algorithm with correlation techniques and the algorithm presented in [8] and [10] using theoretical analysis and simulations. Among the curvefitting methods, we limit the comparison to the CIM. ...
... which is expected to be a good guess for the centroid frequency for a wide range. To estimate the time shifts at a discrete time sample k, the time shift at the sample k −1 is used as an initial value of the iteration given by (10), to ensure that the correlation function is always evaluated in the vicinity of t = −τ . In elastography, the displacement is a monotonous function of depth. ...
... where the brackets [· · · ] denote rounding toward the nearest integer. Note that in this case, the phase-changing term in (10) has to be changed to e j[τ/T ]ωmT . The second applied method, exact interpolation, is de- ...
In ultrasonic elastography, the exact estimation of temporal displacements between two signals is the key to estimating strain. An algorithm was previously proposed that estimates these displacements using phase differences of the corresponding base-band signals. A major advantage of these algorithms compared with correlation techniques is the computational efficiency. In this paper, an extension of the algorithm is presented that iteratively takes into account the time shifts of the signals to overcome the problems of aliasing and accuracy in the estimation of the phase shift. Thus, it can be proven that the algorithm is equivalent to the search of the maximum of the correlation function. Furthermore, a robust logarithmic compression is proposed that only compresses the envelope of the signal. This compression does not introduce systematic errors and significantly reduces decorrelation noise. The resulting algorithm is a computationally simple and very fast alternative to conventional correlation techniques, and the accuracy of strain images is improved.
... ANNEXES voir la figure 7.17 où sont illustrés l'utilisation d'une parabole [119], d'une gaussienne [120] ou encore d'un cosinus [121,122]. Après ajustement de l'une de ces fonctions paramétriques au voisinage du maximum de la corrélation croisée, la position estimée du maximum de la f parabole (t) = at 2 + bt + c, (7.17) ...
... Ainsi, l'ajustement par une gaussienne de la fonction xcorr{x, y} autour des points Ajustement par un cosinus. Une autre méthode d'interpolation à 3 points est l'utilisation d'un cosinus [121,122] : ...
Ce travail de thèse traite du développement d’une antenne microphonique compacte et d’une chaîne de traitement du signal dédiée, pour la reconnaissance et la localisation angulaire de cibles aériennes. L’approche globale proposée consiste en une détection initiale de cible potentielle, la localisation et le suivi de la cible, et une détection affinée par un filtrage spatial adaptatif informé par la localisation de la cible. Un algorithme original de localisation goniométrique est proposé. Il utilise l’algorithme RANSAC sur des données pression-vitesse large bande [100 Hz - 10 kHz], estimées en temps réel, dans le domaine temporel, par des différences et sommes finies avec des doublets de microphones à espacements inter-microphoniques adaptés à la fréquence. L’extension de la bande passante de l’antenne en hautes fréquences est rendue possible par l’utilisation de différences finies d’ordre élevé, ou de variantes de la méthode PAGE (Phase and Amplitude Gradient Estimation) adaptées à l’antenne développée. L’antenne acoustique compacte ainsi développée utilise 32 microphones MEMS numériques répartis dans le plan horizontal sur une zone de 7.5 centimètres, selon une géométrie d’antenne adaptée aux l’algorithmes de localisation et de filtrage spatial employés. Des essais expérimentaux de localisation et de suivi de trajectoire contrôlée par une sphère de spatialisation dans le domaine ambisonique ont montré une erreur de localisation moyenne de 4 degrés. Une base de données de signatures acoustiques de drones en vol a été créée, avec connaissance de la position du drone par rapport à l’antenne microphonique apportée par des mesures GPS. L’augmentation des données par bruitage artificiel, et la sélection dedescripteurs acoustiques par des algorithmes évolutionnistes, ont permis de détecter un drone inconnu dans un environnement sonore inconnu jusqu’à 200 mètres avec le classifieur JRip. Afin de faciliter la détection et d’en augmenter la portée, l’étape de détection initiale est précédée d’une formation de voies différentielle dans 4 directions principales (nord, sud, est, ouest), et l’étape de détection affinée est précédée d’une formation de voies de Capon informée par la localisation et le suivi de la cible à identifier.
... The cosine fit was used to interpolate three points of the cross correlation function: the maximum, y(0), and the two closest points to the maximum, y(−1) and y(1). Therefore, the equations presented below were used to estimate the time delay [10,37,38]: ...
... The majority of the work found in the literature sought to compare the processing methods using a window length of echo/backscattered signals smaller than 1 μs [10,11,27,31,37,38]. However, in this study, the window length was 20 μs owing to the thickness of the material. ...
Although few studies on ultrasonic materials characterization present the expanded uncertainty, it is very important to quantify the final quality of the result. In addition, many of these studies do not mention the method used to estimate time delay, which is employed to calculate the longitudinal phase velocity (vTS). Therefore, the purpose of this study is to estimate the uncertainties of vTS values obtained by different time-delay estimation methods based on cross-correlation. In addition, computational simulation was used to validate the experimental results. The results of the computational simulation showed that when the sampling frequency is 50 MHz, the bias phase velocity was greater than 1.2 m·s⁻¹ for almost all time-delay estimation methods. Therefore, it is not advisable to use a sampling frequency less than or equal to 50 MHz. In all cases, the expanded uncertainty was below 3.5 m·s⁻¹. The statistical test results indicated a significant difference between the values of the longitudinal phase velocity estimated with a given method, as a function of the sampling frequency and the estimated velocity with each method as a function of the received bandwidth of transducer. The expanded uncertainty for phase velocity is intrinsically dependent on accurate speed of ultrasound estimation in the reference medium. The speed of ultrasound and temperature are correlated, therefore, precise estimation of the reference medium temperature is important to maintain the overall precision of the method employed in this study. The novelty of this work resides in the comparison of different time-delay estimation using measurement uncertainty as parameter. Furthermore, after the detailed step by step assessment of uncertainties for the dissimilar computational methods, one can easily estimate the uncertainty by his own using this paper as guidance.
... Ultrasonic imaging and measurement offers direct interaction of the probing signal with the mechanical properties of the test object. Many measurement systems explore the ultrasound delay time: temperature [7], load measurement [9], food product quality monitoring [3]; thickness [4], nondestructive imaging [6] or biomedical applications [8]; flow rate measurement [5]. Essential procedure carried out in such measurements is the estimation of signal delay or time-of-flight (ToF) [1,2]. ...
... ToF values obtained using (4), (5), (6) and (7). Bias error due to neighbor reflection was obtained using (8). ...
Performance comparison of conventional and spread spectrum signals in Time of Flight estimation is given. Ultrasonic measurement under multiple reflections condition is analyzed. It was indicated that if two reflections are in close proximity the neighboring signal induces energy leak into its opponent signal. Due to such situation ToF estimate is obtained with bias error. Narrow signals like single rectangular pulse, should suffer less from the aforementioned phenomena. But use of spread spectrum signals is preferred thanks to their compressibility. It was hypothesized that such long signals will have worse bias error due to neighbor reflection. Goal of the investigation was to compare the performance in multiple reflections environment in Time of Flight estimation for classical signals and spread spectrum signals. Investigation revealed that spread spectrum signals have better performance in a sense of bias error caused by neighboring reflection.
... ToF is exploited for navigation in air and water (13,14) ; surface profiling (15) ; gas or liquid flow velocity measurement (16,17) . It is used in biology for blood coagulation (18) , glucose level measurement (19) ; tissue wall thickness (20) , skeletal diagnostics (21) , internal temperature or elastography (22,23) . Food industry explores ToF for sugar content; fruit ripeness, meat or milk products quality management (24,25) . ...
... In ideal case sinc interpolating function should be used for the ToF estimation between the samples. But usually in real time application processing speed is required so several truncated interpolation techniques exist (22,39,(47)(48)(49)(50)59) where discrete peak position m ( Figure 4) is located first and then the interpolation function is used to estimate the subsample delay. ...
Estimation of the time-of-flight is the most recently used procedure in material thickness measurement, ultrasound velocity estimation, mechanical load and stiffness evaluation, curing or melting monitoring, temperature measurement etc.
A review of the pulse time of flight estimation techniques is given. Simple techniques like thresholding, zero crossing or phase shift are discussed. For high accuracy measurements more advanced techniques are needed to exploit whole information contained in signal. Advanced techniques use maximum likelihood criteria for time position estimation. Likeness between the reference (transmitted) signal and received signal is analyzed here. A comparison of the correlation maximum, difference L1 norm minimum and difference L2 norm minimum techniques is given. Processing in digital domain introduces additional errors. Augmentation of the noise component of the Cramer-Rao equation accounting the analog-to-digital conversion influence (quantization and clock jitter) is presented. Estimate of the time of flight in digital domain is discrete. Subsample interpolated value can be obtained using sinc function, but usually truncated version is used. Bias errors introduced by parabolic, cosine and Gaussian interpolation are analyzed and compared to frequency domain interpolator. Application of the findings discussed above for ultrasonic system engineering is given.
... Estimation of signal propagation time, the time-of-flight (ToF) is quite a recent procedure in measurements [1,2]. Many measurement systems use the ToF of the ultrasonic signal: food processing [3], thickness [4], flow [5], imaging [6], temperature meters [7], biomedical [8] or load measurement [9]. ...
... If matched filter is applied in digital space, then conversion into digital domain will introduce aliasing errors, quantization errors, jittering errors. Then, N 0 in (8) has to be modified to account for digital space [21]: ...
Investigation of the errors of Time of Flight (ToF) estimate of ultrasonic signals in case of multiple reflections environment is presented. Multiple signals environment induces neighboring signal energy leak into analyzed signal. This in turn introduces some errors in ToF estimate. Narrow signals like single pulse, should posses lower influence that CW burst or chirp signals. Then high energy signals (burst or chirp) will always be jeopardized in case of multiple reflections. New class of SS signals, trains of pulses of arbitrary pulse width and position (APWP), is expected to have properties similar to chirp (wide, controllable bandwidth) and the single pulse (low correlation sidelobes) signals. Goal of the investigation was to compare the multi-reflections performance in Time of Flight estimation for classical signals (rectangular pulse, step and CW burst) and spread spectrum signals (chirp and APWP). Signals from fixed delay line were recorded, aligned in time and averaged to produce noise-free reference signal which was used to simulate the plate thickness measurement and liquid velocity estimation with variable spacing of signals in time. Simulation and experimental investigation results are presented and possible problem solutions suggested.
... The straightforward procedure to determine the discrete time delay between two signals is by computing a pattern matching function, as for example the cross-correlation function, and determine its maximum (or minimum depending on the algorithm) [10]- [12]. Methods based on: adaptive filters [13], Fast Fourier transform [14], Hilbert transform correlation [15], combinations among autocorrelation and crosscorrelation functions [16], analog quadrature detectors [17], and many others have been proposed. ...
... The simplest and mostly used method for continuos time delay estimation consists on the computation of the correlation function, by fitting a parabola around its peak, and finally by determining the zero of the derivative of the fitted parabola [10], [11]. Other types of fitting functions were proposed, such as cosine [12]. It is referred that as these methods are based on a pattern-matching function that is sampled at the original frequency, they usually suffer from relatively high bias and variance [1]. ...
In this paper we propose a new algorithm for sub-sample time-delay estimation between two discrete-time signals. The new algorithm operates just over the discrete-time signals without the need for interpolation or fitting. The proposed method is compared with a method based on spline fitting, and that is referred to outperform other methods over a broader set of conditions (Viola&Walker, 2005). The new approach, besides presenting a low computational cost (70% less), seemed to outperforms the spline approach in situations of high noise levels (typically signal-to noise ratios smaller than 20dB).
... The easiest and most widely used method consists of fitting a parabola around the peak of the pattern-matching curve [15], [16], [31]. DeJong et al. [32], [33] have proposed instead the use of a cosine function to interpolate the pattern-matching function. Their results show that this method performs better than parabolic fitting, but that it exhibits aliasing for velocities exceeding the Nyquist limit [33], [34]. ...
... Cosine curves have been used extensively in the past to interpolate the discrete "correlation" function and thus estimate subsample displacements [32], [33]. Given the largest sample of the "correlation" function, y(0), and its two neighbors y(−1) and y(1), the estimated subsample shift is given by the following expression [31]- [33]: ...
Time delay estimation (TDE) lies at the heart of signal processing algorithms in a broad range of application areas, including communications, coherent imaging, speech processing, and acoustics. In medical ultrasound for example, TDE is used in blood flow estimation, tissue motion measurement, tissue elasticity estimation, phase aberration correction, and a number of other algorithms. Because of its central significance, TDE accuracy, precision, and computational cost are of critical importance. Furthermore, because TDE is typically performed on sampled signals-and delay estimates are usually desired over a continuous domain-time delay estimator performance should be considered in conjunction with associated interpolation. In this paper we present a new time-delay estimator that directly determines continuous time-delay estimates from sampled data. The technique forms a spline-based, piecewise continuous representation of the reference signal then solves for the minimum of the sum squared error between the reference and the delayed signals to determine their relative time delay. Computer simulation results clearly show that the proposed algorithm significantly outperforms other algorithms in terms of jitter and bias over a broad range of conditions. We also describe a modified version of the algorithm that includes companding with only a minor increase in computational cost.
... The most widely used interpolation method is the parabolic-fit that is simple, but its estimation bias is high when the sampling rate to center frequency ratio (f s /f 0 ) is low (in the order of 4)[6],[9], and[11]. In addition to the parabolic-fit, the cosine-fit[7],[8], and[9]and the reconstructive interpolation methods[9]also are used. The cosine-fit interpolation can be used at f s /f 0 = 4 with high estimation accuracy; but as mentioned in[7], it has velocity aliasing for velocities exceeding the Nyquist limit. ...
... The noise depends on the SNR of the signal and the length of the finite observation time. According to the criteria of maximizing expected signal peak at τ relative to the background noise, the resulting optimum filter[8]in terms of signal-and-noise spectral density is: ...
... Cross-correlation estimates the time delay based on the maximum correlation of the emitted and received signals in full waveform. It is widely utilized in applications with high noise and high accuracy requirements [30][31][32][33][34][35][36][37]. An advantage of cross-correlation method compared with peak detection is that it can easily apply interpolation methods to improve the accuracy of the discrete signals. ...
Background light noise is one of the major challenges in the design of Light Detection and Ranging (LiDAR) systems. In this paper, we build a single-beam LiDAR module to investigate the effect of light intensity on the accuracy/precision and success rate of measurements in environments with strong background noises. The proposed LiDAR system includes the laser signal emitter and receiver system, the signal processing embedded platform, and the computer for remote control. In this study, two well-known time-of-flight (ToF) estimation methods, which are peak detection and cross-correlation (CC), were applied and compared. In the meanwhile, we exploited the cross-correlation technique combined with the reduced parabolic interpolation (CCP) algorithm to improve the accuracy and precision of the LiDAR system, with the analog-to-digital converter (ADC) having a limited resolution of 125 mega samples per second (Msps). The results show that the CC and CCP methods achieved a higher success rate than the peak method, which is 12.3% in the case of applying emitted pulses 10 µs/frame and 8.6% with 20 µs/frame. In addition, the CCP method has the highest accuracy/precision in the three methods reaching 7.4 cm/10 cm and has a significant improvement over the ADC’s resolution of 1.2 m. This work shows our contribution in building a LiDAR system with low cost and high performance, accuracy, and precision.
... Some of them are described in Refs. [20][21][22][23][24], in which they are mainly used to recognize the echo signals or motion tracking. In this paper, we investigated four interpolation methods to refine the RBS and therefore improve the accuracy to estimate the shift of the RBS, including sinc-interpolation, centroid, cosine interpolation, and parabolic interpolation. ...
We use a spectrum interpolation technique to improve the distributed strain measurement accuracy in a Rayleigh-scatter-based optical frequency domain reflectometry sensing system. We demonstrate that strain accuracy is not limited by the “uncertainty principle” that exists in the time–frequency analysis. Different interpolation methods are investigated and used to improve the accuracy of peak position of the cross-correlation and, therefore, improve the accuracy of the strain. Interpolation implemented by padding zeros on one side of the windowed data in the spatial domain, before the inverse fast Fourier transform, is found to have the best accuracy. Using this method, the strain accuracy and resolution are both improved without decreasing the spatial resolution. The strain of 3 μϵ within the spatial resolution of 1 cm at the position of 21.4 m is distinguished, and the measurement uncertainty is 3.3 μϵ.
... Secondly, the image enhancement obtained by video processing is limited by the lack of phase information in the demodulated signals. As the phase shift in the radio-frequency (RF) signals is a sensitive index for motion detection(Bohs et al. 1991;de Jong et al. 1991;Ferrara et al. 1994), development of improved image processing methods is therefore directed towards processing of the original RF echo data. To evaluate the feasibility of RF signal processing techniques, the temporal characteristics of the blood scattering during IVUS imaging needs to be investigated.Oronningsaeter et al. (1995) have studied the effects of Doppler shift and frequency broadening respectively caused by the axial and lateral movements of blood particles in simulated data. ...
Intravascular ultrasound (rvUS) is a new imaging mOdality providing real-time, crosssectional,
high-resolution images of the arterial lumen and vessel wall. In contrast to
conventional x-ray angiography that only displays silhouette views of the vessel lumen,
IVUS imaging permits visualization of lesion morphology and accurate measurements
of arterial cross-sectional dimensions in patients. These unique capabilities have led to
many important clinical applications including quantitative assessment of the severity,
restenosis, progression of atherosclerosis, selection and guidance of catheterbased
therapeutic procedures and short- and long-term evaluation of the outcome of an
intravascular intervention.
Like the progress of other medial imaging modalities, the advent of IVUS techniques
has brought in new challenges in the field of signal and image processing. Quantitative
analysis of IVUS images requires the identification of arterial structures such as the
lumen and plaque within an image. Manual contour tracing is well known to be time
consuming and subjective. Development of an automated contour detection method
may improve the reproducibility of quantitative IVUS and avoid a tedious manual
procedure. Computerized three-dimensional (3D) reconstruction of an IVUS image
series may extend the tomographic data to a more powerful volumetric assessment of
the vessel segment. Obviously, this could not be achieved without the advance of 3D
image processing techniques. Furthermore, it is demonstrated that processing of the
original radio frequency (RF) echo signals provides an efficient means to improve the
IVUS image quality as well as a new approach to extract volumetric flow information.
The goals of the studies reported in this thesis are therefore directed toward
development of video image and RF signal processing techniques for image
enhancement, automated contour detection, D reconstruction and flow imaging.
In this chapter several IVUS scanning mechanisms and some background information
about ultrasonic imaging are briefly introduced. The principles of different video-based
contour detection approaches and examples of contour detection in echocardiograms
are discussed. Subsequently, applications of RF analysis in IVUS images are reviewed,
followed by the scope of this thesis in the final part.
... features. Correlation-based processing has mainly been used to estimate tissue motion from ultrasound imaging in a number of studies [69][70][71][72][73][74][75][76][77][78]. The autocorrelation function of an image is an assessment of the amount of regularity as well as the fineness/coarseness of the texture present in the image. ...
Speckle has been widely considered a noisy feature in ultrasound images, thus it is intended to be suppressed and eliminated. On the other hand, speckle can be studied as a signal modeled by various statistical distributions or by analyzing its intensity with spatial relations in image space that characterize its nature, and hence, the nature of the underlying tissue. This knowledge can then be used in order to classify the different speckle regions into anatomical structures. In fact, speckle characterization in echocardiography and other ultrasonic images is important for motion tracking, tissue characterization, image segmentation, registration, and other medical applications for diagnosis, therapy planning and decision making. In this paper, we review and discuss various speckle characterization methods, which are often applied to confirm the speckle nature of the elements.
... In ideal case sinc function should be used for the ToF estimation between the samples. But usually in real time application processing speed is required so several truncated interpolation techniques exist [20,21,22]. All these techniques locate the peak position m (Fig.1) using discrete signal and then use some interpolation function to estimate the subsample delay. ...
This work presents an investigation of the bias error introduced in time of flight estimation realized by subsample interpolation in digital domain. The time of flight estimation is accomplished based on the evaluation of the peak position of the cross correlation function. In order to cope with the discrete nature of the cross-correlation function, subsample estimation exploits three time domain interpolation techniques: parabolic, cosine, Gaussian and frequency domain interpolation using phase angle. An empirical equation relating the maximum value of the bias error to sampling frequency and signal parameters (center frequency and envelope bandwidth) has been derived. It is found that the maximum value of the bias error is in inverse cubic relation to sampling frequency and in quadratic relation envelope bandwidth for cosine interpolation. The maximum value of the bias error is in inverse cubic relation to sampling frequency and in quadratic relation to center frequency and envelope bandwidth for parabolic interpolation. The coefficients related to the approximation technique are given. Results can be applied for bias errors estimation or correction when fast subsample interpolation is used and application of phase domain interpolation is unacceptable due to processing speed limitations. The equations for minimum required sampling frequency are derived by balancing the interpolation error against Cramer-Rao lower bound.
... We have tested three different crosscorrelation-based algorithms: (1) cosine fitting, (2) seismogram interpolation, and (3) cross-correlation interpolation. Cosine curve fitting (method 1) has been used extensively in the past in interpolating the discrete correlation function to estimate subsample delay time (de Jong et al., 1990de Jong et al., , 1991 Cespedes et al., 1995). In algorithm (2) we calculate the Fourier transform of a time series, pad the frequency spectrum with zeros, and calculate the inverse Fourier transform. ...
We have conducted a series of cross-well experiments to continuously measure in situ temporal variations in seismic velocity at two test sites: building 64 (B64) and Richmond Field Station (RFS) of the Lawrence Berkeley National Laboratory in California. A piezoelectric source was used to generate highly repeat- able signals, and a string of 24 hydrophones was used to record the signals. The B64 experiment was conducted utilizing two boreholes 17 m deep and 3 m apart for � 160 h. At RFS, we collected a 36-day continuous record in a cross-borehole facility using two 70-m-deep holes separated by 30 m. With signal enhancement techniques we were able to achieve a precision of � 6.0 nsec and � 10 nsec in delay-time esti- mation from stacking of 1-hr records during the � 7- and � 35-day observation pe- riods at the B64 and RFS sites, which correspond to 3 and 0.5 ppm of their travel times, respectively. Delay time measured at B64 has a variation of � 2 lsec in the 160-hr period and shows a strong and positive correlation with the barometric pres- sure change at the site. At RFS, after removal of a linear trend, we find a delay-time variation of � 2.5 lsec, which exhibits a significant negative correlation with baro- metric pressure. We attribute the observed correlations to stress sensitivity of seismic velocity known from laboratory studies. The positive and negative sign observed in the correlation is likely related to the expected near- and far-field effects of this stress dependence in a poroelastic medium. The stress sensitivity is estimated to be � 106/ Pa and � 107/Pa at the B64 and RFS site, respectively.
... Since the underlying correlation function is not known, any adopted interpolation function could lead to biased estimates for the sub-sample displacement. In the literature, the grid slopes algorithm [44] has shown to be relatively unbiased in comparison with the use of other conventional interpolation functions such as cosine [45], cubic spline [44] and parabolic [46] functions. ...
We report a summary of recent developments and current status of our team's efforts to image and quantify in vivo nonlinear strain and tissue mechanical properties. Our work is guided by a focus on applications to cancer diagnosis and treatment using clinical ultrasound imaging and quasi-static tissue deformations. We review our recent developments in displacement estimation from ultrasound image sequences. We discuss cross correlation approaches, regularized optimization approaches, guided search methods, multiscale methods, and hybrid methods. Current implementations can return results of high accuracy in both axial and lateral directions at several frames per second. We compare several strain estimators. Again we see a benefit from a regularized optimization approach. We then discuss both direct and iterative methods to reconstruct tissue mechanical property distributions from measured strain and displacement fields. We review the formulation, discretization, and algorithmic considerations that come into play when attempting to infer linear and nonlinear elastic properties from strain and displacement measurements. Finally we illustrate our progress with example applications in breast disease diagnosis and tumor ablation monitoring. Our current status shows that we have demonstrated quantitative determination of nonlinear parameters in phantoms and in vivo, in the context of 2D models and data. We look forward to incorporating 3D data from 2D transducer arrays to noninvasively create calibrated 3D quantitative maps of nonlinear elastic properties of breast tissues in vivo.
... This window length was chosen because it was expected to give a good trade off between resolution and robustness. In order to detect subsample displacements in the axial direction, cosine interpolation [6] of R(r, r 0 , t) was performed by fitting a curve through the maximum/minimum and its neighboring points. This could not be done in the case of sum of absolute differences. ...
Techniques have been described in the literature to enable two-dimensional strain rate estimation. They are based on two-dimensional velocity estimation. One of the problems to obtain robust lateral strain rate estimates is the fact that lateral velocity estimates are intrinsically noisier than axial ones. The aim of this study was to find the optimal estimator for tracking of the radio frequency patterns both axially and laterally. Theoretical properties for the following estimators were investigated using simulations: cross correlation, normalized cross correlation, sum of absolute differences and sum of squared differences. Two-dimensional velocity estimation was not feasible using cross correlation. However, normalized cross correlation, sum of absolute differences and sum of squared differences showed accurate axial and lateral results.
... Both, the C W B and MSE have been normalized to the cycle duration of a 7.5 MHz pulse. The time shift was estimated using Phase Root Seeking.Fig. 3 shows the same comparison using Correlation Interpolation [3] for time delay estimation. ...
With the Phase-Root-Seeking algorithm a new and very fast
algorithm for time delay estimation was recently introduced, permitting
the estimation of strain images in real-time. In this paper the accuracy
of the real-time strain imaging concept is investigated by the most
rigorous approach, i.e. a comparison to the theoretical accuracy limit:
the Cramer-Rao-Lower-Bound. Simulations show, that this limit can be
approximately reached by carefully selecting window length and window
shift for time delay estimations and using a generalized least square
estimator for the estimation of strain from time delays
... One of the few papers to address lateral interpolation is [9], in which the grid slopes algorithm is shown to be relatively unbiased in comparison with other techniques . More prone to bias are cosine [3, 5, 6], cubic spline [9] and parabolic interpolation [2, 3, 7]. Fourier reconstruction [3] has been shown to work well in the axial direction, provided the signal is bandlimited and the sampling frequency above the Nyquist limit. ...
Freehand 3D ultrasound can be acquired without a position sensor by deducing the elevational probe motion from the interframe speckle decorrelation. However, a freehand scan involves lateral and axial, as well as elevational, probe motion. The lateral sampling is determined by the A-line separation and is relatively sparse: lateral motion tracking therefore requires subsample interpolation. In this paper, we investigate the resilience of lateral interpolation techniques to simultaneous lateral and elevational probe motion. We propose a novel interpolation strategy and, through a series of in vitro experiments, compare its performance with that of established alternatives. The new technique is shown to be superior, limiting interpolation errors to around 5% of the length of the freehand reconstruction. (E-mail: [email protected]
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Three-dimensional (3D) ultrasound elastography can provide 3D tissue stiffness information that may be used during clinical diagnoses. In the framework of strain elastography, motion tracking plays an important role. In this study, an improved 3D region-growing motion tracking (RGMT) algorithm based on a concept of exterior boundary points was developed. In principle, the proposed method first determines displacement at some seed points by strictly checking the local correlation and continuity in the neighborhood of those seeds. Subsequent displacement estimation is then conducted from these initial seeds to obtain displacements associated with other locations. This RGMT algorithm is designed to use more known information -including displacements and correlation values of all known-displacement neighboring points -to estimate the displacement of an unknown-displacement point, whereas previous RGMT methods employed information from only one such point. The algorithm was tested on 3D ultrasound volumetric data acquired from a simulation, a tissue-mimicking phantom and 5 human subjects. Motion compensated cross-correlations (MCCC), strain contrast, and displacement Laplacian values (representing smoothness of an estimated displacement field) were calculated and used to evaluate merits of the proposed RGMT method. Compared to a previously published RGMT method, results show that the proposed RGMT method can provide smaller displacement errors, smoother displacements and improve strain contrast while maintaining reasonably high MCCC values, indicating good motion tracking quality. The proposed method is also computationally more efficient. In summary, our preliminary results demonstrated that the proposed RGMT algorithm is capable of obtaining high-quality 3D strain elastographic data using modified clinical equipment.
This chapter provides an overview of the methods that are used for ultrasound elastography, which utilizes tissue displacements as fundamental information to infer mechanical properties of biological soft tissues. In signal processing, cross‐correlation is a basic concept and has been commonly used for pattern recognition in various applications, including time‐delay estimation in elastography. Constitutive equations of materials based on continuum mechanics describe the macroscopic stress response to external mechanical stimuli. Most contemporary motion‐tracking techniques are based on two important ingredients, a coarse‐to‐fine search scheme and injection of biases, to overcome large tracking errors. To use the motion regularization, speckle tracking can be modeled as an optimization problem using an energy function combining the correlation coefficient with speckle motion continuity. While region‐based methods are commonly used, optical flow estimation can also be done by assuming certain forms of motion smoothness through nonparametric motion models.
Various relationships are suggested in the literature to qualify the elastic behavior of (a segment of) an artery as a function of measurable parameters.
Due to their low-power levels, GPS signals are very susceptible to interference from either intentional or unintentional sources. This vulnerability is further aggravated by the increased reliance of the civilian infrastructure on GPS for time synchronization. This brings about the pressing need for detection and localisation of the interferers in real-time to prevent disturbance to everyday operation of this infrastructure. This paper details the upgrades to one such system, GNSS Environmental Monitoring System (GEMS) which consists of a number of sensor stations connected to a central processing unit, to achieve real-time interference localisation. The paper describes the GEMS environment and details and compares performance of various time and frequency domain Time Difference of Arrival (TDOA) estimation approaches.
Ultrasound Elastography (UE) is a non-invasive technique to image the elasticity distribution of the soft tissues. In UE two ultrasound RF signals obtained before and after applying a compressing load on the tissue are processed for local time delay estimation (TDE) between them. To increase the accuracy of the TDE, it is necessary to use a sub-sample TDE method. In this paper a new TDE method based on the SSD-spline TDE is presented which not only directly produces continuous time delay estimates from the sampled data but also provides a higher rate of accuracy in final elastograms by compensating for the decorrelation of the post-compression RF signals due to the applied compression. Moreover, a two-stage procedure is proposed to reduce the processing time. The experimental results show the proposed algorithm outperforms the golden TDE method of ldquoSSD with cubic spline fitting sub-sample TDErdquo technique in terms of the SNRe and CNRe.
Traditional cross-correlation considers situations where two functions or data sets are linked by a constant shift either in time or space. Correlation provides estimates of such shifts even in the presence of considerable noise corruption. This makes the technique valuable in applications like sonar, displacement or velocity determination and pattern recognition. When regions are decomposed into patches in applications such as Particle Image Velocimetry it also allows estimates to be made of whole displacement/flow fields. The fundamental problem with traditional correlation is that patch size and hence statistical reliability must be compromised with resolution. This article develops a natural generalization of cross-correlation which removes the need for such compromises by replacing the constant shift with a function of time or space. This permits correlation to be applied globally to a whole domain retaining any long-range coherences present and dramatically improves statistical reliability by using all the data present in the domain for each estimate.
This paper is concerned with the implementation of a new image processing technique called Generalised Cross-Correlation designed to detect a two-dimensional displacement field between consecutive pictures of some object. The particular problem considered here is the determination of a flow field from experimental tank data. The method consists of maximizing a Generalised Cross-Correlation (GC-C) function in order to compute the parameters in a global mathematical description of the displacement field. This is in complete contrast to traditional approaches such Cellular Correlation, where an array of constant vectors are estimated using spatially local data. The focus here is on the case where the displacements are too large to employ perturbation techniques and thus numerical optimization is required to calculate the parameters in the flow/ displacement model used. A combination of a Genetic Algorithm (GA) and a Hillclimbing Method (HC) is applied to recover the global maximum of the GC-C function in the presence of many large local maxima. The paper also briefly reviews the basic background of GC-C and develops those GA issues relevant for the implementation of the method.
A recent generalisation of cross-correlation (GC-C) makes it possible to model transformations between data such as pairs
of images by sets of parameterised functions as opposed to constant shifts, rotations etc. as employed in conventional cross-correlation.
Typical applications of GC-C are in areas such as particle image velocimetry (PIV), or two-dimensional or three-dimensional
surface strain field determinations. As the flow, strain etc. descriptions developed by GC-C are global or zonal, the parameters
required are estimated using all or a large fraction of the information in the images typically used to provide the basic
data in such techniques. This is in complete contrast to traditional cross-correlation methods used in PIV, where the image
domains are segmented into small sub-regions and a constant shift, rotation etc. is determined separately in each local cell.
Such local cellular methods inevitably introduce a compromise between spatial resolution and the statistical confidence that
can be placed in the estimates of the shifts, rotations etc. GC-C removes the need for such compromises. This paper examines
the application of the small perturbation form of GC-C to real experimental data sets with special emphasis on showing the
effects of the analytical approximations employed in the perturbation scheme. In particular, the key issue of the effects
of the bandwidth of the images used are explored and a very simple procedure is described for checking that optimal results
are being obtained.
In this paper we present a new time delay estimator that directly determines continuous time delay estimates from sampled data. The technique forms a spline-based piecewise continuous representation of the reference signal and then solves for the minimum of the sum squared error between the reference and the delayed signals to determine their relative time delay. The mathematical formulation of the proposed tracking algorithm makes it particularly well suited for application in tissue elasticity estimation, since it can easily include companding. We have performed a series of simulations using ultrasound data to test the performance of this algorithm and compare it to currently accepted delay estimators implementing a variety of sub-sample interpolation methods. Simulation results show that this algorithm significantly outperforms other algorithms in terms of jitter and bias over a broad range of conditions.
Quantitative elasticity imaging is investigated for a wide range
of gel-based, tissue equivalent phantoms and excised tissue specimens.
To do this, strain imaging methods have been developed for very large
surface deformations. Images produced by this approach exhibit a high
signal to noise ratio (SNR). Results are presented demonstrating both
the accuracy and sensitivity of the method for imaging internal strain
in models of kidney pathology. In particular, strain images have been
used to detect renal scar in a rabbit model of chronic nephritis before
any detectable change in kidney function. Finally, reconstruction of the
elastic modulus based on these high SNR strain images has been tested.
Results on gel-based phantoms indicate that a method based on stress
continuity can be used to clearly identify bounded inclusions in an
otherwise infinite, homogeneous medium
Intravascular ultrasound imaging systems are effective in
identifying and characterizing occlusive arterial disease. In order to
investigate changes in vessel wall elasticity due to atherosclerosis, a
real-time, high-frequency intravascular ultrasound imaging system has
been developed for the study of excised artery sections in vitro. The
system consists of a miniature ultrasound probe constructed from a
42-MHz lead zirconate titanate (PZT) transducers, a high-frequency scan
converter, and a computer-controlled flow system. By correlating vessel
wall displacement, determined from the ultrasound data, with
intraluminal pressure variations recorded with a catheter-tip pressure
transducer, the elastic properties of the vessel wall can be assessed.
Dynamic vessel wall motion along one radial view, assessed using
one-dimensional cross-correlation, corresponds well with the
simultaneously measured intraluminal pressure waveform. Preliminary
studies using a two-dimensional speckle tracking algorithm show
potential for determination of regional variation in vessel elasticity
We review the principles of elastography and some of its accomplishments to date in the area of breast imaging in vivo. We present a literature review of a variety of methods for the estimation of tissue elasticity that have been reported in the literature in the past 15 years, and of data on the elastic properties of soft tissues. This is followed by a description of elastography and its relationship to the theory of elasticity. The principles underlying the elastography imaging technique, and methods for time delay estimations and their tradeoffs are discussed. We then present results of computer simulations, phantom and tissue studies in vitro, and some breast imaging studies in vivo. After a discussion of the origin and appearance of some important elastographic artifacts, we conclude with some general observations and conclusions.
One of the major contributions to the improvement of spectral Doppler and colour flow imaging instruments has been the development of advanced signal-processing techniques made possible by increasing computing power. Model-based or parametric spectral estimators, time-frequency transforms, station-arising algorithms and spectral width correction techniques have been investigated as possible improvements on the FFT-based estimators currently used for real-time spectral estimation of Doppler signals. In colour flow imaging some improvement on velocity estimation accuracy has been achieved by the use of new algorithms but at the expense of increased computational complexity compared with the conventional autocorrelation method. Polynomial filters have been demonstrated to have some advantages over IIR filters for stationary echo cancellation. Several methods of velocity vector estimation to overcome the problem of angle dependence have been studied, including 2D feature tracking, two and three beam approaches and the use of spectral width in addition to mean frequency. 3D data acquisition and display and Doppler power imaging have also been investigated. The use of harmonic imaging, using the second harmonic generated by encapsulated bubble contrast media, seems promising particularly for imaging slow flow. Parallel image data acquisition using non-sequential scanning or broad beam transmission, followed by simultaneous reception along a number of beams, has been studied to speed up 'real-time' imaging.
Thesis (Ph. D.)--Massachusetts Institute of Technology, Whitaker College of Health Sciences and Technology, 1996. Includes bibliographical references (p. 135-150). by Robert William Stadler. Ph.D.
In elastography, several methods for 2-D strain imaging have been introduced, based on both raw frequency (RF) data and speckle-tracking. Although the precision and lesion detectability of axial strain imaging in terms of elastographic signal-to-noise ratio (SNRe) and elastographic contrast-to-noise ratio (CNRe) have been reported extensively, analysis of lateral precision is still lacking. In this paper, the performance of different 2-D correlation RF- and envelope-based strain estimation methods was evaluated using simulation data and phantom experiments. Besides window size and interpolation methods for subsample displacement estimation, the influence of recorrelation techniques was examined. Precision and contrast of the measured displacements and strains were assessed using the difference between modeled and measured displacements, SNRe and CNRe. In general, a 2-D coarse-to-fine displacement estimation method is favored, using envelope data for window sizes exceeding the theoretical upper bound for strain estimation. Using 2-D windows of RF data resulted in better displacement estimates for both the axial and lateral direction than 1-D RF-based or envelope-based techniques. Obtaining subsample lateral displacement estimates by fitting a predefined shape through the cross-correlation function (CCF) yielded results similar to those obtained with up-sampling of RF data in the lateral direction. Using a CCF model was favored because of the decreased computation time. Local aligning and stretching of the windows (recorrelation) resulted in an increase of 2-17 and 6-7 dB in SNRe for axial and lateral strain estimates, respectively, over a range of strains (0.5 to 5.0%). For a simulated inhomogeneous phantom (2.0% applied strain), the measured axial and lateral SNRes were 29.2 and 20.2 dB, whereas the CNRes were 50.2 dB and 31.5 dB, respectively. For the experimental data, lower SNRe (axial: 28.5 dB; lateral: 17.5 dB) and CNRe (axial: 39.3 dB; lateral: 31 dB) were found. In conclusion, a coarse-to-fine approach is favored using RF data on a fine scale. The use of 2D parabolic interpolation is favored to obtain subsample displacement estimates. Recorrelation techniques, such as local aligning and stretching, increase SNRe and CNRe in both directions. (E-mail: [email protected]
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Spectral tissue strain (STS) is a new technique for measuring and imaging tissue strain from a set of images using intravascular ultrasound. The technique is based on the Fourier scaling property and uses the chirp z-transform (CZT) to estimate strain within the vessel walls. Some preliminary results, both in vitro and in vivo, are described. A novel display technique has also been developed for encoding radial strain and displaying the resulting colour map as an overlay on the original image.
One limitation encountered using high frequency intravascular ultrasound (IVUS) is the echogenicity of blood, which increases dramatically at frequencies of 20-40 MHz. Because of the higher velocity of moving blood particles, the echo pattern of flowing blood shows more variations in time than that of the wall. To investigate the time-varying characteristics of the blood scattering measurements were performed on the radiofrequency (RF) data collected in vivo from five pig experiments. After positioning the echo catheter inside the iliac artery, an M-mode sequence of 30 RF traces was acquired at a high pulse repetition rate (5 kHz). The RF correlation time was measured on the regions of blood and the arterial wall. Two processing techniques, temporal averaging and correlation, were tested for suppression of the blood echo intensity. The correlation time Tc measured in the blood region was approximately 1 ms, which was shorter than that measured in the wall region (Tc > 6 ms). The correlation values calculated in a small window showed a large variation in the blood region while the wall region produced a constant high output. After processing eight consecutive RF traces (delta T = 200 microseconds), the temporal averaging method results in a 50% intensity reduction in the blood region. Using the correlation output as a weighting function, the blood echo intensity can be further reduced to only 10% of its original value. Application of the RF correlation processing to a cross-sectional image data demonstrates the feasibility of this technique to remove most of the blood echoes and enhance the image contrast of the luminal interface.
Measurements of endothelium-dependent vasoreactivity and arterial compliance are important metrics of vascular pathophysiology which may be used for the development and evaluation of therapeutic methods. The technique of ultrasonic echo tracking is applicable to measurements of endothelium-dependent vasoreactivity and arterial compliance. To evaluate the application of echo tracking to these measurements, we constructed a system based upon analog-to-digital conversion and storage of the radio frequency (RF) ultrasound signals. Off-line analysis of the RF data with various echo-tracking algorithms demonstrated two potential sources of error: tracking drift and RF transition regions. The tracking drift resulted from the slow accumulation of tracking error. The RF transition regions were associated with disparate motions of neighboring reflectors or the insonation of a new series of tissue layers. As a result of these sources of error, the application of echo tracking to endothelium-dependent vasoreactivity measurements is unlikely to outperform duplex ultrasound methods. The application of echo tracking to arterial compliance measurements via the arterial pressure/diameter relationship may produce variable results due to RF transition regions. Finally, the application of echo tracking to arterial compliance measurements via the pulse wave velocity is relatively insensitive to these sources of error because the pulse-wave velocity measurement depends upon the timing of the peak arterial distension, not on the absolute value of the distension.
A technique is described for measuring the local hardness of the vessel wall and atheroma using intravascular ultrasound. Strain images were constructed using the relative local displacements, which are estimated from the time shifts between gated echo signals acquired at two levels of intravascular pressure. Time shifts were estimated using one-dimensional correlation with bandlimited interpolation around the peak. Tissue-mimicking phantoms with typical morphology and hardness topology of some atherosclerotic vessels were constructed. Hard and soft regions could be distinguished on the strain image, independently of their contrast in echogenicity. Thus, the potential of ultrasonic hardness imaging to provide information that may be unavailable from the echogram alone was demonstrated. The strain images of the homogeneous and layered phantoms showed some artifacts that need to be corrected for, to obtain images of the modulus of elasticity. For in vitro and in vivo experiments, the spatial resolution of the technique needs to be improved. Furthermore, two-dimensional correlation techniques may be necessary in case of nonradial expansion and an off-centre catheter position.
In this study several time delay estimation (TDE) methods were investigated for estimation of time shifts of less than 10 ns at a frequency of 30 MHz. Using simulated and experimental echosignals we investigated the performance of five methods: two phase related methods (phase shift and phase difference method); two correlation methods (cross-correlation and correlation interpolation method); and a demodulation method. The results showed that the correlation interpolation method is by far the most accurate for all time delays. With this method, estimation errors of about 200 ps are achievable with an signal-to-noise ratio (SNR) of 40 dB (f0 = 30 MHz, bandwidth = 20 MHz) for time shifts of up to 10 ns.
Conventional pulsed ultrasound systems are only able to detect motion along the ultrasound beam (i.e., axial motion). If the angle between the actual motion direction and the ultrasound beam is known, then the magnitude of the actual motion can be derived. This technique can be applied for laminar blood-flow measurements in straight vessels, but for tissue motion it is inadequate because the local tissue motion direction is unknown and may be position-dependent. Assessment of both the axial motion and the lateral motion (i.e., in the direction perpendicular to the ultrasound beam) makes angle-independent assessment of the magnitude of the actual motion feasible. Information about the axial and lateral motion is available in a set of radiofrequency (RF) signals obtained along the same line of observation (M-mode). The experiments described in the present paper show that axial and lateral motion can be estimated from the shape of the envelope of the 2-D (spatial and temporal) correlation function of analytic M-mode RF signals. Furthermore, it is demonstrated that the shape is also affected by the Band width of the received RF signals, signal-to-noise ratio, and local amplitude and phase characteristics of the ultrasound beam.
Delay estimation is used in ultrasonic imaging to estimate blood flow, determine phase aberration corrections, and to calculate elastographic images. Several algorithms have been developed to determine these delays. The accuracy of these methods depends in differing ways on noise, bandwidth, and delay range. In most cases relevant to delay estimation in ultrasonics, a subsample estimate of the delay is required. We introduce two new delay algorithms that use cubic polynomial splines to continuously represent the delay. These algorithms are compared to conventional delay estimators, such as normalized cross correlation and autocorrelation, and to another spline-based method. We present simulations that compare the algorithms' performance for varying amounts of noise, delay, and bandwidth. The proposed algorithms have better performance, in terms of bias and jitter, in a realistic ultrasonic imaging environment. The computational requirements of the new algorithms also are considered.
Ultrasound-based mechanical strain imaging systems utilize signals from conventional diagnostic ultrasound systems to image tissue elasticity contrast that provides new diagnostically valuable information. Previous works (Hall et al 2003 Ultrasound Med. Biol. 29 427, Zhu and Hall 2002 Ultrason. Imaging 24 161) demonstrated that uniaxial deformation with minimal elevation motion is preferred for breast strain imaging and real-time strain image feedback to operators is important to accomplish this goal. The work reported here enhances the real-time speckle tracking algorithm with two significant modifications. One fundamental change is that the proposed algorithm is a column-based algorithm (a column is defined by a line of data parallel to the ultrasound beam direction, i.e. an A-line), as opposed to a row-based algorithm (a row is defined by a line of data perpendicular to the ultrasound beam direction). Then, displacement estimates from its adjacent columns provide good guidance for motion tracking in a significantly reduced search region to reduce computational cost. Consequently, the process of displacement estimation can be naturally split into at least two separated tasks, computed in parallel, propagating outward from the center of the region of interest (ROI). The proposed algorithm has been implemented and optimized in a Windows system as a stand-alone ANSI C++ program. Results of preliminary tests, using numerical and tissue-mimicking phantoms, and in vivo tissue data, suggest that high contrast strain images can be consistently obtained with frame rates (10 frames s(-1)) that exceed our previous methods.
In this paper we present a highly accurate, spline-based time delay estimator (TDE) that directly determines subsample time delay estimates from sampled data. The algorithm uses cubic splines to produce a continuous time representation of a reference signal and then computes an analytical matching function between this reference and a delayed signal. The location of the minima of this function yields estimates of the time delay. We also present a more computationally efficient formulation of this algorithm, which is based on FIR filtering to determine the cubic spline coefficients, polynomial approximation to determine the time delay estimates and adaptive search of the delay estimate. The proposed algorithm is particularly useful in applications such as blood flow estimation and tissue elasticity estimation since it can include companding with little additional computational cost.
This paper introduces a new velocity estimator, referred to as the 2D autocorrelator, which differs from conventional Doppler techniques in two respects: the derivation of axial velocity values by evaluating the Doppler equation using explicit estimates of both the mean Doppler and the mean RF frequency at each range gate location; and, the 2D nature (depth samples versus pulse transmissions) of processing within the range gate. The estimator's output can be calculated by evaluating the 2D autocorrelation function of the demodulated (baseband) backscattered echoes at two lags. A full derivation and mathematical description of the estimator is presented, based on the framework of the 2D Fourier transform. The same framework is adopted to analyze two other established velocity estimators (the conventional 1D autocorrelator and the crosscorrelator) in a unifying manner, and theoretical arguments as well as experimental results are used to highlight the common aspects of all three estimators. In addition, a thorough performance evaluation is carried out by means of extensive simulations, which document the effect of a number of factors (velocity spread, range gate length, ensemble length, noise level, transmitted bandwidth) and provide an insight into the optimum parameters and trade-offs associated with individual algorithms. Overall, the 2D autocorrelator is shown to offer the best performance in the context of the specific simulation conditions considered here. Its superiority over the crosscorrelator is restricted to cases of low signal-to-noise ratios. However, the 2D autocorrelator always outperforms the conventional 1D autocorrelator by a significant margin. These comparisons, when linked to the computational requirements of the proposed estimator, suggest that it combines the generally higher performance of 2D broadband time-domain techniques with the relatively modest complexity of 1D narrowband phase-domain velocity estimators.< >
Previous ultrasound speckle tracking methods have been extended, permitting measurement of internal displacement and strain fields over a wide dynamic range of tissue motion. The markedly increased dynamic range of this approach should lead to enhanced contrast resolution in strain and elasticity images. Results of experiments on gelatin-based, tissue equivalent phantoms show the capabilities of the method.< >
A real-time ultrasound time-domain correlation (UTDC) blood flowmeter has been developed. Real-time performance has been achieved through the implementation of a custom-designed high-speed residue-number system (RNS) hardware correlator. The flowmeter is interfaced to a commercial ultrasound imager and can produce one-dimensional velocity versus range graphs at a rate of three per second. It has been validated in a blood flow phantom under a variety of conditions along with in vivo measurements in the human carotid artery. The theory of the time-domain correlation technique, design and implementation of flowmeter hardware, and the important correlation parameters which affect the performance of the flowmeter are described.< >
The Doppler technique has traditionally been the method used to extract motion information from ultrasonic echoes reflected by moving tissues. The Doppler technique has been around for a long time, and has been extensively reviewed and analyzed in the literature. Recently, time-domain methodologies for estimating tissue motion have gained in popularity. Time-domain methods have advantages over Doppler methods in many applications, and as of yet have not been comprehensively reviewed. An overview of time-domain techniques that have appeared in the literature over the past few years is presented. Their potential advantages over Doppler are examined, and the individual techniques are compared.< >
The thickness of the left ventricular free wall and internal chamber diameter were continuously measured by pairs of ultrasonic crystals together with left ventricular pressure in normal conscious dogs. During the resting state, wall thickness decreased abruptly with the onset of atrial contraction from 10.5 mm to an average end-diastolic valueof9.8 mm. In contrast to most previous studies, there was no change in wall thickness during isovolumic systole, and with ejection the wall thickened by 31.3 percent of end-diastolic wall thickness. Atrial pacing, phenylephrine, isoproterenol and propranolol produced significant changes in chamber size with reciprocal changes in wall thickness. In addition, changes in the extent and velocity of left ventricular chamber shortening in the minor equator were associated with comparable reciprocal changes in the extent and velocity of free wall thickening (correlation coefficients 0.97 to 0.99). During acute coronary occlusion, progressive reductions in the extent and velocity of regional wall shortening with partial ischemia were associated with comparable changes in systolic wall thickening characteristics (r = 0.96 and 0.95), and holosystolic elongation in fully ischemic areas was associated with holosystolic wall thinning. During chronic pressure overload, despite wall thickening, the relation between chamber shortening and wall thickening were retained and direct computation of dynamic wall stress variations was possible. These measurements allowed precise definition of the dynamics of the left ventricular wall during normal and abnormal cardiac states. The demonstration that in the absence of regional dysfunction analysis of wall thickness in a single region of ventricular free wall can be used to describe myocardial and overall left ventricular function, as well as regional function in the presence of ischemia, constitutes a new approach to the assessment of cardiac function that has potential for echocardiographic applications.
Correlation interpolation is introduced as a method to determine the displacement of moving biological tissue on the basis of a sequence of ultrasonic echo signals. The echo signal is sampled along the echo depth with approximately 4 samples per average high frequency period. Sampling in time occurs with the pulse repetition frequency. The necessary information is extracted from a crosscorrelation function between successive signals, which is modelled using four parameters. The parameters are estimated from five calculated correlation sums and the shift with maximum correlation is determined. In contrast to existing techniques, the performance of this method is determined mainly by the number of samples used, while the ratio of the number of samples in depth and time is irrelevant. Using 64 samples at a signal-to-noise power ratio of 10, the standard deviation of the error in the determination of the shift in depth is 0.08 sampling intervals. As in many other methods, the width of the aliasing interval equals the mean frequency period.
Signal averaging of echocardiographic frames acquired over multiple cardiac cycles has been used to improve image quality. However, misalignment of frames from respiration, transducer or patient movement, and irregular cardiac contraction affects the quality of the resultant averaged image. A motion detection system has been developed using inter-frame subtraction and statistical pattern recognition techniques. Automatic selection of frames exhibiting significant motion in a canine cardiac model compare favorably to manual selection by cardiologists (phi = 0.94) in a test set of 103 images. This method, combined with signal averaging, has resulted in an improvement in image quality.
Four time-domain oriented, real-time frequency estimators, based on the detection of phase, zero-crossings, instantaneous frequency or autocorrelation, were simulated on a digital computer and subjected to computer generated Doppler signals, enabling the investigation of the influence of spectral shape, filtering, frequency shift, noise and quantitation. Three estimators, the autocorrelator as well as the instantaneous frequency detector and the autocorrelator, both with extended frequency range, appeared to be very accurate. They exhibit a bias in the estimator output of less than 2 percent over a wide frequency range, the former up to nearly the Nyquist frequency, the latter two beyond, even for skew spectra and under poor signal conditions regarding bandwidth and noise.
We have previously reported initial clinical results of a novel blood velocity imaging technique utilizing a two-dimensional correlation search applied to consecutively acquired echoes. In this paper, we describe both the physical principles underlying this technique and test tank experiments which define its performance under a variety of conditions. The results indicate that, unlike Doppler flow imaging systems, this technique defines the flow velocity vector in two dimensions and is not subject to aliasing.
The spiral orientation of left ventricular (LV) fibers suggests that twisting about the ventricular long axis of the apex with respect to the base, i.e., torsional deformation, may be characteristic of LV contraction. To demonstrate this twisting motion, 10 orthotopic human cardiac allograft recipients were studied with biplane cineradiography of tantalum helices implanted within the LV midwall at 12 specific sites. Counterclockwise twisting about the LV long axis (as reviewed from apex to base) accompanied ventricular ejection in all patients. Torsional deformation angles, measured relative to a reference minor axis at the base, were substantially smaller in the anteroapical wall, as compared with counterparts in the apical third of the inferior and lateral walls (anterior = 13.3 +/- 6.0 degrees, inferior = 18.7 +/- 6.3 degrees, and lateral = 23.4 +/- 10.7 degrees). Torsional angles at the midventricular level were roughly half as much and exhibited similar regional variabilities (anterior = 7.6 +/- 3.3 degrees, inferior = 9.0 +/- 3.3 degrees, lateral = 10.7 +/- 5.2 degrees, and septal = 8.8 +/- 3.8 degrees). Comparison of control beats and the initial beat after abrupt cessation of rapid atrial pacing (126 +/- 10 beats/min) with return to the control heart rate (96 +/- 9 beats/min) permitted the mild positive inotropic effect of tachycardia to be assessed at similar levels of ventricular load. Torsional deformation of the anteroapical and inferoapical sites increased significantly (p less than 0.05) over control values to 15.6 +/- 7.5 degrees and 21.2 +/- 5.5 degrees, respectively. In contrast, torsional deformation of the lateral wall was essentially unchanged. These data provide direct evidence for torsional deformation of the left ventricle in humans, demonstrate that torsion of the LV chamber is nonuniform, and suggest a dependence of LV torsion upon contractile strength that is attenuated in the lateral wall.
A method is described for quantifying tissue movement in vivo from the computation of correlation coefficient between pairs of A-scans with appropriate time separation. The method yields quantifiable and repeatable secondary patterns of soft tissue movement in response to primary cardiac movement in a given subject, shows consistently different results as between normal livers and a variety of abdominal tumours, and is sensitive to either progress or therapeutically-induced regression of malignant disease. While the results reported here have been obtained using somewhat simple and crude equipment, the method is well suited to implementation on a commercial real-time scanner.
Echocardiographic measurements of minor axis and wall thickness and calculations from these two measurements of left ventricular end-diastolic volume and mass were performed in 24 patients and compared with angiocardiographic measurements of the same variables in corresponding patients. The echo-measured left ventricular end-diastolic chamber dimension (Dd) correlated closely with the angiographic minor axis in the AP plane (correlation coefficient 0.87 and se ± 0.45 cm) and with the minor axis from the lateral film (r = 0.91, se ±0.39 cm). Similar correlations were found between measurements by these methods of wall thickness (r = 0.89, se ±1.3 mm), of end-diastolic volume (r = 0.94, se ±30.6 cc), and of left ventricular mass (r = 0.88, se ±49.19 g). The reproducibility of this method was established by independent recordings and measurements of echo Polaroid films by two observers. The percent systolic wall thickening, as determined by echocardiography, identified subjects with ejection fractions greater or less than 0.50. Echocardiography offers a reliable and reproducible method for measuring left ventricular wall thickness and mass. Finally, ultrasound may provide an accurate method for measuring systolic wall thickening in man.
A new type of pulse-Doppler is described and results illustrating in vivo operation presented. The system uses a phase detection principle to produce an output proportional to the velocity of moving structures continously as a function of depth into the tissue, similar to an A-mode display. It employs only a single processing channel consisting of double delay line canceller and delay line phase-detector. Operation is equivalent to a parallel processing multiple-gate Doppler having a very large number of gates. This relatively simple system has promise in making true Doppler imaging available with B-mode imaging systems as well as for investigating velocity profiles and diameter of peripheral arteries.
We examined in conscious dogs the effects of reductions in myocardial blood flow (MBF) in three different layers across the wall on regional myocardial contractile function in the ischemic zone, measured as systolic wall thickening (%WT). In 16 dogs, %WT was measured with sonomicrometry and MBF was determined with microspheres (10- to 12-microns diam) during coronary stenosis of the left circumflex coronary artery. The stenoses were categorized into six groups by the effect on %WT (each group representing progressive 20% decrements in %WT from control), and individual and pooled regression analyses were performed on data from six of the dogs having multiple data points to evaluate the shape (linear or curvilinear) of the relationships between MBF and changes in %WT. Transmural contractile function was highly sensitive to acute reductions in MBF, especially reductions in the subendocardium. The shape of the normalized subendocardial MBF-%WT relation was mildly curvilinear by regression analysis (quadratic equation, gamma = -0.75x2 + 2.15x -0.39, r2 = 0.92). Likewise, mean transmural and midmyocardial MBF correlated well and closely with changes in %WT. Subepicardial MBF, however, correlated poorly with changes in %WT, there being no change in subepicardial MBF until %WT had been reduced more than 50%.
To eliminate the need for intramyocardial transducers in measuring regional left ventricular (LV) function we have developed a pulsed Doppler technique utilizing a single epicardial transducer. Wall thickening is evaluated by digitally integrating the velocity of myocardial layers passing through the sample volume located at a selected depth. Thickening fraction (TF) can then be estimated by dividing the systolic excursion by the sample volume depth. The Doppler method was compared with the transit-time method in three acute dogs by placing the 4-mm-diameter epicardial Doppler transducer over a 2-mm-diameter endocardial crystal tunneled through the LV wall. With the sample volume set to 1 mm less than the minimum LV thickness, simultaneous measurements of TF by the Doppler and transit-time methods showed good agreement (r = 0.95) during control, ischemia, volume overload, shock, and anoxia. In 28 chronically instrumented piglets signals were obtained for longer periods with Doppler transducers than with transit-time segment-length crystals. We conclude that the Doppler technique provides a high-fidelity wall thickening waveform and a good estimate of TF with minimal disturbance to the ventricle and that the technique is suitable for use in both acute and chronically instrumented animals.
Torsion of the left ventricle (LV) is associated with rotation of the apex with respect to the base around the long axis of the LV. A mathematical model of LV mechanics, which relates torsion to transmural distribution of fibre shortening, was evaluated with two-dimensional echocardiography in nine anaesthetised closed-chest dogs. Torsion was calculated as the difference between the angles of rotation (radians) of echo-derived transverse cross-section projections of the LV obtained at the mitral valve and low papillary level, divided by the axial distance between these projections measured in a long-axis cross-section, and multiplied by the outer radius in a mid-papillary transverse projection of the LV. A shortening to torsion ratio (STR) was defined as the ratio of inner wall shortening to torsion occurring during ejection. In a series of 11 measurements, each based on frame-to-frame analysis of 15 cardiac cycles, STR was found to be 2.31 +/- 0.23 rad-1 (mean +/- SD), whereas the mathematical model predicted a STR value of 2.4 rad-1 over a wide range of preload, afterload and contractility levels. We conclude that two-dimensional echocardiography validates the presence of torsion in the normal canine left ventricle, as predicted by the model of left ventricular mechanics.
Motion observed in real-time ultrasound images may be very misleading. The speckle pattern in images of soft tissue is an artefact and it may move in a way which bears no simple relationship to the motion of the tissues. The physical origin of this anomaly is described.
The instantaneous and continuous thickness of the left ventricular wall was measured in ten dogs with a specially designed transducer. The thickness change during the "isovolumic" phase of systole was 11%. The average thickness increased an additional 10% during the ejection period of systole. These average values increased to 20% and 15% respectively with the administration of norepinephrine (approximately 0.001 mg/kg per minute).
Transverse and longitudinal epicardial are strains were compared with thickness strain of the myocardium. Assuming the myocardium is incompressible, reasonable agreement was found in the three strains, which lends support to the measurements that were made.
It is concluded that changes in the thickness of the myocardial wall during the cardiac cycle may be important in some considerations of the heart's performance.
The temporal correlation properties of ultrasonic echoes from contracting myocardium are investigated theoretically. Myocar- dium is modeled as a dense suspension of moving particles, with time- varying scattering amplitudes. Single scattering is assumed. Echoes from different particles are added coherently. The variance of particle velocity is proposed as an indicator of contractile performance. It is shown how this parameter may be extracted from the average corre- lation of pairs of echoes arising from separate ultrasound pulses. An experiment is performed in which a manufactured target with known motion characteristics is interrogated using a commercial medical ul- trasound scanner. The experimental results are found to be in good agreement with theory. Finally, practical considerations for perform- ing and interpreting these measurements in human myocardium are discussed. EVERAL INVESTIGATIONS have demonstrated that it may be possible to evaluate cardiac contractile per- formance accurately with ultrasound. One technique in- volves the measurement of the frequency average of the intensity of ultrasonic echoes emanating from myocar- dium. Echo intensity has been found to vary throughout the cardiac cycle in normal canine myocardium, with the maximum level occurring at end-diastole and the mini- mum at end-systole ( l), (2). In addition, the magnitude of this effect has a regional dependence throughout the heart, with areas of greater contractile activity exhibiting greater amplitudes of variation (3). The extent of the cyclic variation of echo intensity has also been observed to drop significantly in canine myocardium making a tran- sition from normal to ischemic 141. Thus the magnitude of cyclic variation may be interpreted as a measure of contractile performance. A second technique entails quantitative analysis of video images of the heart produced by ultrasound scan-
A discrete-time Doppler system is described which features: (a) simultaneous processing of the velocity information for 128 data points or more; (b) sampled time domain frequency analysis; and (c) basic fast response elements for next generation ″real time″ two dimensional flow imaging scanners. The frequency detector measures the instantaneous phase of the echo signal by a dedicated high speed hardware processor. The phase increment between two subsequent transmit/receive cycles is a direct measure of the instantaneous frequency. This parameter is then used to calculate the mean and variance of the detected velocity. A computer simulation study, generating time signals off a given spectrum, has been used to verify the performance of the new processor, such as linearity, adequacy of dynamic range, transient response and noise sensitivity. Experimental, as well as clinical results obtained with a prototype instrument are presented.
Cross correlation transit-time measuring instrumentation can be
accurate to a few parts per million depending upon the sensing
technology. When ultrasonic sensors are used to measure fluid flow the
accuracy is primarily determined by the fluid properties and flow
profile. In multiphase mixtures, very large changes of acoustic
attenuation can occur and this originally provided the impetus to
produce the ultrasonic cross-correlation flowmeter. In its basic form
the instrument measures the flow velocity by determining the transit
time of changes of acoustic impedance, due to flow disturbances, between
two parallel ultrasonic beams spaced a known distance apart. Since only
the transit time is measured between two fixed marker beams, the
measurement is largely unaffected by wide variations in the speed of
sound in the fluid and the fluid properties. The correlation signal
processor used in all trials reported in this paper was a commercially
available instrument specifically designed for accurate transit time
determination and velocity tracking and is based on a multichannel
design system. The measurable transit time range was from 0.5 ms to 400
ms corresponding to a velocity range of 0.1 m/s to 80 m/s for a sensor
spacing of 40 mm. This instrument can also process phase and amplitude
modulation data simultaneously so that fluids varying from single phase
to multiphase can be metered
A new method is presented for computing the local myocardial
deformation from two-dimensional (2-D) echocardiograms. It is based on
the relationship between echocardiographic speckle motion and the
underlying tissue motion. The computer algorithm follows an optical flow
methodology and computes the 2-D interframe linear velocity field of the
speckle pattern inside a myocardial region of interest. Each regional
velocity (motion) field is then decomposed into meaningful components
representing the local myocardial translation, rotation (twist), and
biaxial deformation (contraction and thickening). The diagnostic
potential of the method is illustrated with a clinical example. The
deformation values are in close agreement with manually calculated
values obtained from wall thickness and heart circumference measurements
from end-diastole to end-systole
A new blood flow imaging system is described that com-bines a conventional pulsed Doppler device and a newly developed autocorrelator. In the system blood flow within a given cross section of a live organ is displayed in real time. The direction of blood flow and its variance are expressed by means of a difference in color and its hue, respectively. Experiments were conducted with a mechanical and an electrical scanner using phantoms, and good agreement with the theory was obtained. Studies on clinical significance have also been carried out for normal and diseased hearts, and successful results have been found. Copyright © 1985 by The Institute of Electrical and Electronics Engineers, Inc.
A digital implementation of the multigate Doppler blood flowmeter is described. The observation range of 10 cm is divided into 128 depth channels, thus offering the unique possibility of simultaneously recording the blood vessel geometry as well as the blood flow topography, where topography is defined as a two-dimensional map of blood flow velocity along the axis of a blood vessel. The apparatus consists of three major sections: a) Transmitter/receiver and mixer which produce a range phase signal containing the velocity information. b) The range phase signal is sampled, digitized, and stored into 128 memory ceDs for further processing. A feedback circuit significantly reduces the required resolution of the analog to digital converter and at the same time suppresses unwanted low-frequency signals. c) A frequency discriminator yields a voltage proportional to the velocity in each channel. The system can detect SOvelocity profiles per second, sufficient to characterize the pulsatile flow in arteries. The main features of the novel instrument are minimal hardware due to sequential signal processing, ease of operation, and a variety of display modes, including two-dimensional (2D) flow maps. The results, obtained in laboratory tests under simulated conditions as weD as in transcutaneous applications, agree weD with those reported by other investigators using conventional equipment.
An ultrasonic human-blood-flow velocity profile measurement method using time-domain correlation of consecutive echo pairs has been developed. The time shift between a pair of range gated echoes is estimated by searching for the shift that results in the maximum correlation. The time shift indicates the distance a group of scatterers has moved, from which flow velocity is estimated. The basis for the computer simulations and error analyses of the scheme includes a band-passed white Gaussian noise signal model for an echo from a scattering medium, the estimate of flow velocity from both a single scatterer and multiple scatterers, and a derived precision estimation. The error analysis via computer simulation includes an evaluation of errors associated with the correlation method. For a uniform flow velocity profile, beamwidth modulation represents the greatest error source. However, for a nonuniform flow velocity profile, the jitter caused by a small flow velocity gradient can exceed the other error sources. A detailed computer simulation evaluated the interdependencies of window length, beam width, vessel diameter, and viewing angle on the estimation of flow velocity.< >
A novel ultrasonic volumetric flow measurement method using time-domain correlation of consecutive pairs of echoes has been developed. An ultrasonic data acquisition system determined the time shift between a pair of range gated echoes by searching for the time shift with the maximum correlation between the RF sampled waveforms. Experiments with a 5-MHz transducer indicate that the standard deviation of the estimate of steady fluid velocity through 6-mm-diameter tubes is less than 10% of the mean. Experimentally, Sephadex (G-50; 20-80 mu m dia.) particles in water and fresh porcine blood have been used as ultrasound scattering fluids. Two-dimensional (2-D) flow velocity can be estimated by slowly sweeping the ultrasonic beam across the blood vessel phantom. Volumetric flow through the vessel is estimated by integrating the 2-D flow velocity field and then is compared to hydrodynamic flow measurements to assess the overall experimental accuracy of the time-domain method. Flow rates from 50-500 ml/min have been estimated with an accuracy better than 10% under the idealized characteristics used in this study, which include straight circular thin-walled tubes, laminar axially-symmetric steady flow, and no intervening tissues.< >
In two-dimensional echocardiography the study of the motion of the heart, especially of the left ventricle, is of central interest. Although the clinician can see this motion on real-time two-dimensional echocardiograms, its quantification can still be greatly improved. This paper presents and tests a computer method to quantify the motion of the heart from digitized image sequences. This method computes on every point of an image the two-dimensional velocity vector which characterizes its motion from this image to the next. This approach has the following advantages: 1) border recognition algorithms are no longer needed, 2) motion is not restricted to its radial component, and 3) motion information is available on every point of the image.
A new image processing method is developed to process two-dimensional ultrasound B-scan echo images which makes possible the delineation of the displacement of cardiac boundaries for the precise assessment of infarcted/ischemic regions. An economic image processing system is designed for the purpose. An algorithm to compute the direction and amplitude of the displacement vector at any point in the cardiac image, given two frames of the image at two different time instants, is developed. The performance of the algorithm is examined with synthesized images. Displacement amplitudes along the boundaries of the left ventricle are plotted and compared for normal and diseased conditions.
A new technique for determining the Doppler frequency shift in a phase-coherent pulsed Doppler system is presented. In the new approach, the Doppler frequency shift is given directly in the time domain in terms of the measured I and Q components of the measured Doppler signal. The algorithm is based on an expression for the instantaneous rate of change of phase which separates rapidly varying from slowly varying terms. It permits noise smoothing in each term separately. Since the technique relies solely on signal processing in the time domain, it is significantly simpler to implement than the classic Fourier transform approach. In addition, the algorithm can be shown to give rigorously accurate values for instantaneous frequency and outperform the Fourier transform approach in poor signal-to-noise environments. Experimental results are presented which confirm the superiority of the new domain technique.
The instantaneous systolic relationship between the left ventricular wall thickness and cavity radius by echocardiography
- P Vignon
- G Pelle
- J Ohayon
- P Brun
Vignon, P., Pelle, G., Ohayon, J., and Brun, P., The instantaneous systolic
relationship between the left ventricular wall thickness and cavity radius by
echocardiography, in Proc. Cardiovascular Dynamics and Models, pp. 264-275 (Les Editions INSERM, Paris, 1988).
Analysis of heart motion from two-dimensional echocardiograms by velocity field decomposition, in Computers in Cardiology
- G E Mailloux
- F Langlois
- P Y Simard
- M Bertrand
Mailloux, G.E., Langlois, F., Simard, P.Y., and Bertrand, M., Analysis of heart
motion from two-dimensional echocardiograms by velocity field decomposition, in Computers in Cardiology (IEEE Computer Society Press, Leuven,
Belgium, 1987).
Flow velocity profile via time-domain correlation: error analysis and computer simulation
- S G Foster
- P M Embree
- W D Brien
Foster, S.G., Embree, P.M., and O'Brien Jr., W.D., Flow velocity profile via
time-domain correlation: error analysis and computer simulation, /EEE Trans.
Ultrason. Ferro. Freq. Control 37, 164-l 75 (1990).
Volumetric blood flow via time-domain correlation: experimental verification
- P M Embree
- W D Brien
Embree, P.M., and O'Brien, W.D., Volumetric blood flow via time-domain
correlation: experimental verification,
IEEE Trans. Ultrason. Ferro. Freq.
Control 37, 176-l 89 (1990).