Multiresolution imaging in elastography

The range of strains that can be imaged by any practical elastographic imaging system is inherently limited, and a performance measure is valuable to evaluate these systems from the signal and noise properties of their output images. Such a measure was previously formulated for systems employing cross-correlation based time-delay estimators through the strain filter. While the strain filter predicts the signal-to-noise ratio (SNR(e)) for each tissue strain in the elastogram and provides valuable insights into the nature of image noise, it understated the effects of image resolution (axial resolution, as determined by the cross-correlation window length) on the noise. In this work, the strain filter is modified to study the strain noise at multiple resolutions. The effects of finite window length on signal decorrelation and on the variance of the strain estimator are investigated. Long-duration windows are preferred for improved sensitivity, dynamic range, and SNR(e). However, in this limit the elastogram is degraded due to poor resolution. The results indicate that for nonzero strain, a window length exists at which the variance of strain estimator attains its minima, and consequently the elastographic sensitivity, dynamic range and SNR(e) are strongly affected by the selected window length. Simulation results corroborate the theoretical results, illustrating the presence of a window length where the strain estimation variance is minimized for a given strain value. Multiresolution elastography, where the strain estimate with the highest SNR(e) obtained by processing the pre- and post-compression waveforms at different window lengths is used to generate a composite elastogram and is proposed to improve elastograms. All the objective elastogram parameters (namely: SNR(e), dynamic range, sensitivity and the average elastographic resolution-defined as the cross-correlation window length) are improved with multiresolution elastography when compared to the traditional method of utilizing a single window length to generate the elastogram. Experimental results using a phantom with a hard inclusion illustrates the improvement in elastogram obtained using multiresolution analysis.     

A deconvolution filter for improvement of time-delay estimation in elastography

In elastography, tissue under investigation is compressed, and the resulting strain is estimated from the gradient of displacement estimates. Therefore, it is important to accurately estimate the displacements (time-delay) for good quality elastograms. A principal source of error in time-delay estimation in elastography is the decorrelation of the echo signal due to tissue compression (decorrelation noise). Temporal stretching of the postcompression signals has been shown to reduce the decorrelation noise at small strains. In this article, we present a deconvolution filter that reduces the decorrelation even further when applied in conjunction with signal stretching. The performance of the proposed filter is evaluated using simulated data.     

Estimation and reduction of decorrelation effect due to tissue lateral displacement in elastography

In cross-correlation based elastography, the quality of the strain image is degraded by the distortion of echo waveforms due to tissue axial and lateral displacement. To study the effects of tissue lateral displacement on echo decorrelation, a tissue axial stretching model is developed and a concept called correlation signal-to-noise ratio (CSNR) is introduced to quantify the decorrelation effect due to tissue lateral displacement. A computer simulation based on the tissue stretching model is carried out to study the influence of several important elastographic parameters on echo decorrelation due to tissue lateral displacement. Finally, guided by the CSNR concept, a 2-D spatial comprehensive cross-correlation method is proposed to reduce the decorrelation noise. Results indicate that CSNR can be used as a quality indicator of elastography and the 2-D spatial comprehensive cross-correlation method can effectively reduce the decorrelation noise while slightly decreasing the lateral resolution of the strain image     

Enhancement of echo-signal correlation in elastography using temporal stretching

Echo-signal decorrelation due to tissue compression is a significant source of error in tissue displacement estimates obtained using crosscorrelation. Tissue displacement estimates are used to compute strain values for imaging the elasticity of biological soft tissues. The correlation coefficient between the pre- and post-compression echo rf signals reduces rapidly with signal decorrelation due to increased compression. Miniscule reductions in the value of the correlation coefficient can have a significant impact on the performance of the strain estimator as illustrated by the strain filter. Reducing the rate of signal decorrelation using temporal stretching (which improves the value of the correlation coefficient), significantly improves the performance of the strain filter. The reduction in the rate of signal decorrelation with the subsequent increase in the correlation coefficient using temporal stretching is discussed in this paper. Theoretical, simulation and experimental results quantify the enhancement in the value of the correlation coefficient attained with temporal stretching.     

An adaptive strain estimator for elastography

Elastography is based on the estimation of strain due to applied tissue compression. In conventional elastography, strain is computed from the gradient of the displacement estimates between gated pre- and postcompression echo signals. Gradient-based estimation methods are known to be susceptible to noise. In elastography, in addition to the electronic noise, a principal source of estimation error is the decorrelation of the echo signal as a result of tissue compression (decorrelation noise). Temporal stretching of postcompression signals previously was shown to reduce the decorrelation noise. In this paper, we introduce a novel estimator that uses the stretch factor itself as an estimator of the strain. It uses an iterative algorithm that adaptively maximises the correlation between the pre- and postcompression echo signals by appropriately stretching the latter. We investigate the performance of this adaptive strain estimator using simulated and experimental data. The estimator has exhibited a vastly superior performance compared with the conventional gradient-based estimator.     

A fundamental limit on delay estimation using partially correlatedspeckle signals

Delay estimation is used in ultrasonic imaging to estimate blood or soft tissue motion, to measure echo arrival time differences for phase aberration correction, and to estimate displacement for tissue elasticity measurements. In each of these applications delay estimation is performed using speckle signals which are at least partially decorrelated relative to one another. Delay estimates which utilize such data are subject to large errors known as false peaks and smaller magnitude errors known as jitter. While false peaks can sometimes be removed through nonlinear processing, jitter errors place a fundamental limit on the performance of delay estimation techniques. The authors apply the Cramer-Rao Lower Bound to derive an analytical expression which predicts the magnitude of jitter errors incurred when estimating delays using radio frequency (RF) data from speckle targets. The analytical expression presented includes the effects of signal decorrelation due to physical processes, corruption by electronic noise, and a number of other factors. Simulation results are presented which show that the performance of the normalized cross correlation algorithm closely matches theoretical predictions. These results indicate that for poor signal to noise ratios (0 dB) a small improvement in signal to noise ratio can dramatically reduce jitter magnitude. At high signal to noise ratios (30 dB) small amounts of signal decorrelation can significantly increase the magnitude of jitter errors     

Noise reduction for ultrasonic elastography using transmit-side frequency compounding: a preliminary study

Ultrasonic elastography is an imaging technique providing information about the relative stiffness of biological tissues. In general, elastography suffers from noise artifacts, which degrade lesion detectability and increase the likelihood of misdiagnosis. This paper proposes a method called transmit- side frequency compounding for elastography (TSFC). Beamforming is modified to transmit frames with N alternating center frequencies. Pairs of frames with the same center frequency are used to calculate sub-elastograms that are then averaged to produce one compounded elastogram. Simulation results based on an uniformly elastic tissue model demonstrate the decorrelation among sub-elastograms and the improvement in elastographic signal-to-noise ratio (SNRe) achieved by compounding sub-elastograms. An elastic phantom experiment further validates the noise reduction obtained by the proposed technique.     

Three-dimensional tissue motion and its effect on image noise inelastography

In elastography both high correlation coefficient between pre- and post-compression RF signals and high applied strain are required to achieve the best quality in elastograms. Because the elastogram is computed using a 1-D cross-correlation technique applied to a 1-D ultrasound signal, it is assumed that tissue motion occurs only within the axis of compression (axis of the acoustic wave propagation), or at least that the scatterers remain within the acoustic beam during tissue motion. In practice, soft tissues are incompressible and, therefore, the lateral and elevational (out-of-plane) tissue strains are 50% of the applied strain. Therefore, tissue scatterers may move across the beam due to the applied compression. In this paper we address the degradation of the elastographic quality due to the lateral and elevational motion of the scatterers in uniformly elastic media. A full 3-D model predicting the correlation coefficient as measured using 1-D cross-correlations is proposed. It is shown that the signal-to-noise ratio in elastograms (SNR_e) is nonstationary, and that it depends on the beamwidth and on the applied strain. In order to achieve a higher stationary SNR_e, it is proposed to confine the tissue in the lateral direction. Phantom experiments are used to corroborate the theoretical developments     

Optimizing multicompression approaches to elasticity imaging

Breast lesion visibility in static strain imaging ultimately is noise limited. When correlation and related techniques are applied to estimate local displacements between two echo frames recorded before and after a small deformation, target contrast increases linearly with the amount of deformation applied. However, above some deformation threshold, decorrelation noise increases more than contrast such that lesion visibility is severely reduced. Multicompression methods avoid this problem by accumulating displacements from many small deformations to provide the same net increase in lesion contrast as one large deformation but with minimal decorrelation noise. Unfortunately, multicompression approaches accumulate echo noise (electronic and sampling) with each deformation step as contrast builds so that lesion visibility can be reduced again if the applied deformation increment is too small. This paper uses signal models and analysis techniques to develop multicompression strategies that minimize strain image noise. The analysis predicts that displacement variance is minimal in elastically homogeneous media when the applied strain increment is 0.0035. Predictions are verified experimentally with gelatin phantoms. For in vivo breast imaging, a strain increment as low as 0.0015 is recommended for minimum noise because of the greater elastic heterogeneity of breast tissue.     

Echo decorrelation from displacement gradients in elasticity and velocity estimation

Several ultrasonic techniques for the estimation of blood velocity, tissue motion and elasticity are based on the estimation of displacement through echo time-delay analysis. A common assumption is that tissue displacement is constant within a short observation time used for time delay estimation (TDE). The precision of TDE is mainly limited by noise sources corrupting the echo signals. In addition to electronic and quantization noise, a substantial source of TDE error is the decorrelation of echo signals because of displacement gradients within the observation time. The authors present a theoretical model that describes the mean changes of the crosscorrelation function as a function of observation time and displacement gradient. The gradient is assumed to be small and uniform within the observation time; the decorrelation introduced by the lateral and elevational displacement components is assumed to be small compared with the decorrelation caused by the axial component. The decorrelation model predicts that the expected value of the crosscorrelation function is a low-pass filtered version of the autocorrelation function (i.e., the crosscorrelation obtained without gradients). The filter is a function of the axial gradient and the observation time. This theoretical finding is corroborated experimentally. Limitations imposed by decorrelation in displacement estimation and potential uses of decorrelation in medical ultrasound are discussed.     

Frequency-domain-based strain estimation and high-frame-rate imaging for quasi-static elastography

In freehand elastography, quasi-static tissue compression is applied through the ultrasound probe, and the corresponding axial strain is estimated by calculating the time shift between consecutive echo signals. This calculation typically suffers from a poor signal-to-noise ratio or from the decorrelation between consecutive echoes resulting from an erroneous axial motion impressed by the operator. This paper shows that the quality of elastograms can be improved through the integration of two distinct techniques in the strain estimation procedure. The first technique evaluates the displacement of the tissue by analyzing the phases of the echo signal spectra acquired during compression. The second technique increases the displacement estimation robustness by averaging multiple displacement estimations in a high-frame-rate imaging system, while maintaining the typical elastogram frame-rate. The experimental results, obtained with the Ultrasound Advanced Open Platform (ULA-OP) and a cyst phantom, demonstrate that each of the proposed methods can independently improve the quality of elastograms, and that further improvements are possible through their combination.     

Complex principal components for robust motion estimation

Bias and variance errors in motion estimation result from electronic noise, decorrelation, aliasing, and inherent algorithm limitations. Unlike most error sources, decorrelation is coherent over time and has the same power spectrum as the signal. Thus, reducing decorrelation is impossible through frequency domain filtering or simple averaging and must be achieved through other methods. In this paper, we present a novel motion estimator, termed the principal component displacement estimator (PCDE), which takes advantage of the signal separation capabilities of principal component analysis (PCA) to reject decorrelation and noise. Furthermore, PCDE only requires the computation of a single principal component, enabling computational speed that is on the same order of magnitude or faster than the commonly used Loupas algorithm. Unlike prior PCA strategies, PCDE uses complex data to generate motion estimates using only a single principal component. The use of complex echo data is critical because it allows for separation of signal components based on motion, which is revealed through phase changes of the complex principal components. PCDE operates on the assumption that the signal component of interest is also the most energetic component in an ensemble of echo data. This assumption holds in most clinical ultrasound environments. However, in environments where electronic noise SNR is less than 0 dB or in blood flow data for which the wall signal dominates the signal from blood flow, the calculation of more than one PC is required to obtain the signal of interest. We simulated synthetic ultrasound data to assess the performance of PCDE over a wide range of imaging conditions and in the presence of decorrelation and additive noise. Under typical ultrasonic elasticity imaging conditions (0.98 signal correlation, 25 dB SNR, 1 sample shift), PCDE decreased estimation bias by more than 10% and standard deviation by more than 30% compared with the Loupas method and normalized cross-correlation with cosine fitting (NC CF). More modest gains were observed relative to spline-based time delay estimation (sTDE). PCDE was also tested on experimental elastography data. Compressions of approximately 1.5% were applied to a CIRS elastography phantom with embedded 10.4-mm-diameter lesions that had moduli contrasts of -9.2, -5.9, and 12.0 dB. The standard deviation of displacement estimates was reduced by at least 67% in homogeneous regions at 35 to 40 mm in depth with respect to estimates produced by Loupas, NC CF, and sTDE. Greater improvements in CNR and displacement standard deviation were observed at larger depths where speckle decorrelation and other noise sources were more significant.     

Wavelet transform-based strain estimator for elastography

A new signal processing algorithm based on a wavelet transform (WT) is proposed for instantaneous strain estimation in acoustic elastography. The proposed estimator locally weighs ultrasonic echo signals acquired before tissue compression by a Gaussian window function and uses the resulting waveform as a mother wavelet to calculate the WT of the postcompression signal. From the location of the WT peak, strain is estimated in the time-frequency domain. Because of the additive noise in signals and the discrete sampling, errors are commonly made in estimating the strain. Statistics of these errors are analyzed theoretically to evaluate the performance of the proposed estimator. The strain estimates are found to be unbiased, but error variances depend on the signal properties (echo signal-to-noise ratio and bandwidth), signal processing parameter (time-bandwidth product), and the applied strain. The results are compared with those obtained from the conventional strain estimator based on time-delay estimates. The proposed estimator is shown to offer strain estimates with greater precision and potentially higher spatial resolution, dynamic range, and sensitivity at the expense of increased computation time.     

2-D companding for noise reduction in strain imaging

Companding is a signal preprocessing technique for improving the precision of correlation-based time delay measurements. In strain imaging, companding is applied to warp 2-D or 3-D ultrasonic echo fields to improve coherence between data acquired before and after compression. It minimizes decorrelation errors, which are the dominant source of strain image noise. The word refers to a spatially variable signal scaling that compresses and expands waveforms acquired in an ultrasonic scan plane or volume. Temporal stretching by the applied strain is a single-scale (global), 1-D companding process that has been used successfully to reduce strain noise. This paper describes a two-scale (global and local), 2-D companding technique that is based on a sum-absolute-difference (SAD) algorithm for blood velocity estimation. Several experiments are presented that demonstrate improvements in target visibility for strain imaging. The results show that, if tissue motion can be confined to the scan plane of a linear array transducer, displacement variance can be reduced two orders of magnitude using 2-D local companding relative to temporal stretching.     

The combined effect of signal decorrelation and random noise on the variance of time delay estimation

Tissue motion and elasticity imaging techniques commonly use time delay estimation (TDE) for the assessment of tissue displacement. The performance of these techniques is limited because the signals are corrupted by various factors including electronic noise, quantization, and speckle decorrelation. Speckle decorrelation is caused by changes in the coherent interference among scatterers when the tissue moves relative to the ultrasound beam. In time delay estimation, the effect of noise is usually addressed through the signal-to-noise ratio (SNR) term. Decorrelation, often a significant source of error in medical ultrasound, is commonly described in terms of the correlation coefficient. A relationship between the correlation coefficient and the SNR was previously derived in the literature, for identical signals corrupted by uncorrelated random noise. In this paper, we derive the relationship between the peak of the correlation coefficient function and the SNR for two jointly stationary signals when a delay is present between the signals. Recently, an expression for the Cramer-Rao lower bound (CRLB) has been derived in the literature for partially decorrelated signals in terms of the SNR and the correlation coefficient. Since the applicability of the CRLB is determined not only by the SNR, but also by the correlation coefficient, it is important to unify the expression for the CRLB for partially correlated signals. In this paper, we derive an expression for the CRLB in term of an equivalent SNR converted from the correlation coefficient using an SNR-p relationship, and show this expression to be equivalent to the expression for CRLB. We also corroborate the validity of the SNR-p expression with a simulation. Using this formulation, correlation measurements can be converted to SNR to obtain a composite SNR. The use of this composite SNR in lieu of those in the CRLB expression in the literature allows the extension of the literature results to the solution of the common TDE problems that involve signal decorrelation.     

Testing the limitations of 2-D companding for strain imaging using phantoms

Companding may be used as a technique for generating low-noise strain images. It involves warping radio-frequency echo fields in two dimensions and at several spatial scales to minimize decorrelation errors in correlation-based displacement estimates. For the appropriate experimental conditions, companding increases the sensitivity and dynamic range of strain images without degrading contrast or spatial resolution significantly. In this paper, we examine the conditions that limit the effectiveness of 2-D local companding through a series of experiments using phantoms with tissue-like acoustic and elasticity properties. We found that strain noise remained relatively unchanged as the applied compression increased to 5% of the phantom height, while target contrast increased in proportion to the compression. Controlling the image noise at high compressions improves target visibility over the broad range induced in elastically heterogeneous media, such as biological tissues. Compressions greater than 5% introduce large strains and complex motions that reduce the effectiveness of companding. Control of boundary conditions and ultrasonic data sampling rates is critical for a successful implementation of our algorithms.     

Performance of minimum-mean-square-error filters for spatially nonoverlapping target and input-scene noise

Using computer simulations, we investigate the performance of a minimum-mean-square-error filter for input-scene noise that is spatially nonoverlapping (disjoint) with a target for a limited set of images. Different input-scene-noise statistics are used to test the filter performance. We show that in the presence of spatially nonoverlapping target and input-scene noise, the output of the minimummean- square-error filter has a well-defined correlation peak, small sidelobes, and a high peak-to-correlationenergy ratio compared with other widely used filters such as the classical matched filter, the phase-only filter, and the inverse filter. We also test the robustness of the minimum-mean-square-error filter to errors in noise statistics used in the filter design. We show that, for the images tested here, the performance of the minimum-mean-square-error filter is not sensitive to errors in noise statistics and the filter can detect the target even if a considerable error exists. The discrimination capability and the illumination sensitivity of the minimum-mean-square-error filter are also tested.     

High-speed dynamic speckle interferometry: phase errors due to intensity, velocity, and speckle decorrelation

The recently developed technique of high-speed phase-shifting speckle interferometry combined with temporal phase unwrapping allows dynamic displacement fields to be measured, even for objects containing global discontinuities such as cracks or boundaries. However, when local speckle averaging is included, small phase errors introduced at each time step are accumulated along the time axis, yielding total phase values that depend strongly on the speckle rereference rate. We present an analysis of the errors introduced in the phase evaluation by three sources: intensity errors, velocity errors, and speckle decorrelation. These errors are analyzed when they act both independently and together, for the most commonly used phase-shifting algorithms, with computer-generated speckle patterns. It is shown that, in a controlled out-of-plane geometry, errors in the unwrapped phase map that are due to speckle decorrelation rise as the time between rereferencing events is increased, whereas those due to intensity and velocity errors are reduced. It is also shown that speckle decorrelation errors are typically more important than the intensity and velocity errors. These results provide guidance as to the optimal speckle rereferencing rate in practical applications of the technique.     

Noise properties of a quadrature phase tracker for interferometricoptical fiber sensors

The noise performance of an electronic quadrature phase-detection system for interferometric optical fiber sensors is presented. Three noise sources are discussed in this work, namely, synchronous detection-circuit noise, phase-perturbation noise; and additive amplitude noise. We determined the output signal-to-noise ratio (SNR) experimentally as a function of input phase power for each of the three noise sources. For uncorrelated synchronous detection-circuit noise the output SNR increases monotonically with input phase power. For correlated noise the output SNR has distinct peaks due to noise cancellation. System performance is limited by uncorrelated detection-circuit noise which exhibits a threshold behavior in output SNR at a phase shift of 25 mrad/Hz^½. The phase noise has a more conventional behavior in the sense that SNR gain occurs only at the expense of dynamic performance. Uncorrelated amplitude noise also displays noise cancellation at certain discrete values of input phase, as is the case for correlated synchronous detection-circuit noise. System insensitivity to correlated light-source amplitude noise is evident from the fact that the output SNR is more than 30 dB higher than the input SNR     

Variational Bayesian Based IMM Robust GPS Navigation Filter

This paper investigates the navigational performance of Global Positioning System (GPS) using the variational Bayesian (VB) based robust filter with interacting multiple model (IMM) adaptation as the navigation processor. The performance of the state estimation for GPS navigation processing using the family of Kalman filter (KF) may be degraded due to the fact that in practical situations the statistics of measurement noise might change. In the proposed algorithm, the adaptivity is achieved by estimating the time-varying noise covariance matrices based on VB learning using the probabilistic approach, where in each update step, both the system state and time-varying measurement noise were recognized as random variables to be estimated. The estimation is iterated recursively at each time to approximate the real joint posterior distribution of state using the VB learning. One of the two major classical adaptive Kalman filter (AKF) approaches that have been proposed for tuning the noise covariance matrices is the multiple model adaptive estimate (MMAE). The IMM algorithm uses two or more filters to process in parallel, where each filter corresponds to a different dynamic or measurement model. The robust Huber's M-estimation-based extended Kalman filter (HEKF) algorithm integrates both merits of the Huber M-estimation methodology and EKF. The robustness is enhanced by modifying the filter update based on Huber's M-estimation method in the filtering framework. The proposed algorithm, referred to as the interactive multi-model based variational Bayesian HEKF (IMM-VBHEKF), provides an effective way for effectively handling the errors with time-varying and outlying property of non-Gaussian interference errors, such as the multipath effect. Illustrative examples are given to demonstrate the navigation performance enhancement in terms of adaptivity and robustness at the expense of acceptable additional execution time.