Effects of motion on correlations of pulse-echo ultrasound signals: Applications in delay estimation and aperture coherence

Abstract

The correlation between two pulse-echo ultrasound signals is used to achieve a wide range of ultrasound techniques, such as Doppler imaging and elastography. Prior theoretical descriptions of pulse-echo correlations were restricted to stationary scatterers. Here, a theory for the correlation of moving scatterers is presented. An expression is derived for the correlation of two pulse-echo signals with arbitrary transmit and receive apertures acquired from a medium undergoing bulk motion using the Fresnel approximation. The derivation is shown to coincide with prior derivations in the absence of scatterer motion. The theory was compared against simulations in applications of phase-shift estimation and aperture coherence measurements. The phase-shift estimate and jitter were accurately predicted under axial and transverse motion for focused transmit apertures and for sequential and interleaved synthetic transmit apertures. The theory also accurately predicted how motion affects the correlation coefficient between receive aperture elements for a synthetic transmit aperture. The presented theory provides a framework for analyzing the correlations of arbitrary pulse-echo configurations for applications in which scatterer motion is expected.

I. INTRODUCTION

In pulse-echo ultrasound, an aperture is used to transmit an ultrasonic pulse into a scattering medium, and the reflected pressure wave is received using a second (possibly the same) aperture. The pulse-echo signal is a radiofrequency trace whose amplitude in time corresponds to the echogenicity of the medium in depth. A raster scan of pulse-echo signals is used to reconstruct medical B-mode images, which display the echogenicity of the underlying tissue anatomy. Most medical pulse-echo signals contain speckle, a noise that arises due to diffuse unresolvable scatterers in the medium.^1–4 Although speckle is a noise multiplicative with the echogenicity of tissue, it is used advantageously across a wide range of advanced medical ultrasound techniques that utilize the joint statistics of pairs of pulse-echo signals. In this work, we derive an expression for the complex correlation of two pulse-echo signals as a function of scatterer motion.

The complex correlation of two pulse-echo signals is a second-order statistic of particular interest. For instance, Doppler and speckle-tracking techniques measure the complex correlation between subsequent pulse-echo acquisitions of a moving medium; the displacement between acquisitions is proportional to the complex angle of the correlation,^5–8 or to the time-shift that maximizes its normalized magnitude.⁹ Similarly, phase aberration correction and sound speed estimation techniques measure the complex angle of the correlation between neighboring receive elements to measure the relative time-of-arrival of the echo from a stationary medium.^10–12 The Cramér-Rao lower bound of the displacement estimator variance is a function of the magnitude of the correlation coefficient between the two pulse-echo signals.^1,7,9 Additionally, several adaptive beamforming methods based on spatial coherence have been proposed, such as generalized coherence factor,¹³ filter-delay-multiply-and-sum,¹⁴ phase coherence factor,¹⁵ and short-lag spatial coherence.¹⁶ The latter has been used to differentiate tissue signals from incoherent acoustical and electrical noise according to the correlation coefficients as a function of receive aperture spacing.^16–18

More recently, correlation-based methods have been combined with a retrospective beamforming technique called synthetic aperture.^19,20 Unlike traditional methods where a single tightly-focused pulse-echo signal is swept over the field of view (FOV), synthetic transmit apertures (STAs) are the retrospective coherent sums of multiple broad transmissions. STAs can be composed of different subapertures,¹⁹ virtual sources,^21,22 or steered plane waves,^23,24 and effectively image the medium using the aggregate transmit aperture. STAs provide transmit focusing throughout the entire FOV and enable significantly higher frame rates in excess of 1000 frames per second. The benefits of STAs are a balance between the quality of transmit focusing and the sensitivity to motion artifacts,²⁰ the latter of which can be corrected using correlation-based techniques.^25–27

STAs have augmented a wide range of correlation-based methods. For instance, STAs have enabled the acquisition of very long ensembles at high pulse-repetition frequencies (PRFs) for Doppler²⁸ and vector velocity imaging²⁹ by eliminating the need to raster-scan across the FOV. Long ensemble lengths in conjunction with eigen-based filtering^30,31 have been used to detect extremely slow motion for applications such as functional ultrasound imaging of the brain.³² STAs have also been used for shear wave elastography imaging,³³ intracardiac vector flow,³⁴ aperture coherence imaging,³⁵ phase aberration correction,³⁶ and sound speed estimation.^37,38 Each of these methods utilizes the correlation between two STA pulse-echo signals.

The second-order statistics of pulse-echo speckle signals (e.g., correlation, covariance, coherence) have been studied extensively in the literature. Goodman^1,39 analyzed the statistics of speckle signals in general. Flax et al.⁴⁰ and Wagner et al.⁴ described the correlations of speckle field points and their relationship to the system point spread function. Burckhardt,² Trahey et al.,⁴¹ and Wagner et al.⁴² provided theoretical and experimental measurements of the correlation of translated receive apertures. Mallart and Fink⁴³ provided a derivation for the correlation of two point receivers using an impulsional formulism^44,45 based on the Fresnel approximation.⁴⁶ Walker and Trahey⁴⁷ extended the theory to large receive apertures in the presence of phase aberration.

In each of these descriptions of second-order statistics, the scattering function was assumed to remain stationary between the two pulse echo signals. However, motion is commonly encountered in medical ultrasonography, and in the case of motion estimation, is actually the target of interest. STAs are particularly susceptible to motion because they are acquired over a distributed period of time. There is a need for a motion-sensitive coherence theory in order to improve our understanding of how these techniques perform under scatterer motion.

Below, we present a general formulation for the correlation of two pulse-echo signals from diffuse scatterers undergoing bulk motion in Sec. II. The derivation is based on the Fresnel approximation, and predicts the correlation of backscatter from a given depth for arbitrary transmit and receive apertures. In Sec. III, we examine the theoretical consequences of several common types of motion. In Sec. IV, we demonstrate the predictive power of the theory in sample applications of phase-shift estimation and aperture coherence estimation, illustrating the impact of scatterer motion on correlation estimates.

II. THEORY

The imaging configuration described below is illustrated in Fig. 1. Below, we derive the correlation of two pulse-echo signals from scatterers that have undergone a uniform bulk motion.

FIG. 1.

A pair of pulse-echo configurations is depicted. The first is due to transmit T₁, receive R₁, and scattering function S. The second is due to transmit T₂, receive R₂, and scattering function S with a bulk uniform displacement. In the Fresnel approximation, the apertures and the scatterers are assumed to be at a distance z.

A. Transmitted pressure field

The pressure field transmitted by an aperture T is measured at field point ${\mathit{X}}_s={[x_s,y_s,z_s]}^T$ as a function of temporal frequency f as

$$H({\mathit{X}}_s,f)=\int T({\mathit{X}}_t,f)\frac{e^{j2\pi f{r}_{ts}/c}}{2\pi r_{ts}}\psi ({\mathit{\theta }}_{ts})d{\mathit{X}}_t,$$ (1)

where ${\mathit{X}}_t$ is a point on the transmitting aperture, r_ts is equal to $\left|\right|{\mathit{X}}_s-{\mathit{X}}_t|{|}_2$ where $\left|\right|·|{|}_2$ denotes Euclidean distance, and ψ is an obliquity factor that depends on the angle ${\mathit{\theta }}_{ts}$ between ${\mathit{X}}_t$ and ${\mathit{X}}_s$.^44–46

Assumption 1. Assume that the axial distance $z=|z_s-z_t|$ is much larger than the transverse distance, i.e., $z\gg \Vert {\mathit{x}}_s-{\mathit{x}}_t{\parallel }_2$, where the lower-case boldface ${\mathit{x}}_i={[x_i,y_i]}^T$ denotes the transverse coordinate vector. We can use a first order approximation for amplitude terms

$$r_{ts}\approx z,$$ (2)

and a second order approximation for phase terms

$$r_{ts}\approx z\left(1+\frac{\Vert {\mathit{x}}_s-{\mathit{x}}_t{\Vert }_2^2}{2z^2}\right).$$ (3)

Additionally, Eq. (2) implies $\psi ({\mathit{\theta }}_{ts})\approx 1$. These approximations are collectively referred to as the Fresnel or the “paraxial” approximation.⁴⁶ Applied to Eq. (1), the transmitted pressure field is

$$\begin{array}{c}H({\mathit{x}}_s,z,f)\approx \frac{1}{2\pi z}\int T({\mathit{x}}_t,f)\\ \times \text{exp}\hspace{0.17em}\left[\frac{j2\pi z}{\lambda }\left(1+\frac{{\mathit{x}}_s^T{\mathit{x}}_s-2{\mathit{x}}_s^T{\mathit{x}}_t+{\mathit{x}}_t^T{\mathit{x}}_t}{2z^2}\right)\right]d{\mathit{x}}_t,\end{array}$$ (4)

where $\lambda =c/f$. This approximation is not valid for large steering angles. However, note that the approximation can be applied to steered phased arrays by rotating the coordinate system to align with the direction of wave propagation and by appropriately transforming the apertures, e.g., via projection/propagation onto the rotated z = 0 plane.

The quadratic term ${\mathit{x}}_t^T{\mathit{x}}_t$ can be incorporated as a parabolic focusing term into the transmit aperture definition as $T({\mathit{x}}_t,f)e^{j\pi ({\mathit{x}}_t^T{\mathit{x}}_t)/(\lambda z)}\to T({\mathit{x}}_t,f)$, allowing the expression to be rewritten as

$$\begin{array}{c}H({\mathit{x}}_s,z,f)\approx \frac{1}{2\pi z}\text{exp}\hspace{0.17em}\left[\frac{j2\pi z}{\lambda }\right]\\ \hspace{1em}\times \int T({\mathit{x}}_t,f)\hspace{0.17em}\text{exp}\hspace{0.17em}\left[\frac{j\pi }{\lambda z}\left({\mathit{x}}_s^T{\mathit{x}}_s-2{\mathit{x}}_s^T{\mathit{x}}_t\right)\right]d{\mathit{x}}_t\end{array}.$$ (5)

In the case that the transmit aperture cannot be focused, Eq. (5) is equivalent to applying the stronger Fraunhofer or “far-field” approximation, where the quadratic term is ignored (i.e., ${\mathit{x}}_t^T{\mathit{x}}_t\approx 0$).⁴⁶ Below, we omit f and z as parameters for readability.

B. Backscattered pressure field

Consider a field of scatterers with a backscattering function $S({\mathit{X}}_s)$. When insonified by pressure field $H({\mathit{X}}_s)$, the backscattered pressure received by a receive aperture R with coordinates ${\mathit{X}}_r$ is

$$\begin{array}{c}P(T,S,R)=\int H({\mathit{X}}_s)S({\mathit{X}}_s)\int R({\mathit{X}}_r)\frac{e^{j2\pi r_{sr}/\lambda }}{2\pi r_{sr}}d{\mathit{X}}_{s}d{\mathit{X}}_r\end{array}.$$ (6)

Let us assume that R has the same z coordinate as T. Plugging in Eq. (5) and applying the same approximations to the receive aperture sensitivity function, the total backscattered signal is approximated as

$$\begin{array}{c}P(T,S,R)\approx \frac{e^{j4\pi z/\lambda }}{{(2\pi z)}^2}\iiint T({\mathit{x}}_t)S({\mathit{x}}_s)R({\mathit{x}}_r)\\ \hspace{1em}\times \hspace{0.17em}\text{exp}\hspace{0.17em}\left[\frac{j2\pi }{\lambda z}\left({\mathit{x}}_s^T({\mathit{x}}_s-{\mathit{x}}_t-{\mathit{x}}_r)\right)\right]d{\mathit{x}}_{r}d{\mathit{x}}_{s}d{\mathit{x}}_t.\end{array}$$ (7)

C. Correlation of two backscattered signals

Consider two backscattered signals obtained with arbitrary transmit and receive apertures and scatterers, denoted $P(T_1,S_1,R_1)$ and $P(T_2,S_2,R_2)$. Their correlation is given as

$${\Gamma }_{12}=\mathbf{E}\left[P(T_1,S_1,R_1)P^{*}(T_2,S_2,R_2)\right]$$ (8)

$$\begin{array}{c}\approx \mathbf{E}[\frac{e^{j4\pi (z_1-z_2)/\lambda }}{{(2\pi )}^4{z}_1^2{z}_2^2}\iint T_1({\mathit{x}}_{t_1})T_2^{*}({\mathit{x}}_{t_2})\\ \hspace{1em}\times \iint S_1({\mathit{x}}_{s_1})S_2^{*}({\mathit{x}}_{s_2})\iint R_1({\mathit{x}}_{r_1})R_2^{*}({\mathit{x}}_{r_2})\\ \hspace{1em}\hspace{0.17em}\times \text{exp}\hspace{0.17em}\left[j2\pi {\mathit{x}}_{s_1}^T({\mathit{x}}_{s_1}-{\mathit{x}}_{t_1}-{\mathit{x}}_{r_1})/(z_1\lambda )\right]\\ \hspace{1em}\hspace{0.17em}\times \text{exp}\hspace{0.17em}{\left[j2\pi {\mathit{x}}_{s_2}^T({\mathit{x}}_{s_2}-{\mathit{x}}_{t_2}-{\mathit{x}}_{r_2})/(z_2\lambda )\right]}^{*}\\ \hspace{1em}\times d{\mathit{x}}_{r_1}d{\mathit{x}}_{r_2}d{\mathit{x}}_{s_1}d{\mathit{x}}_{s_2}d{\mathit{x}}_{t_1}d{\mathit{x}}_{t_2}],\end{array}$$ (9)

where $*$ denotes complex conjugation. From here, further assumptions are necessary to proceed with the analysis.

D. Correlation under bulk motion of diffuse scatterers

Assumption 2. Consider the case of spatially incoherent diffuse scatterers undergoing a constant and uniform bulk motion ${\mathit{X}}_d={[x_d,y_d,z_d]}^T$, i.e.,

$$\mathbf{E}\left[S_1({\mathit{X}}_1)S_2^{*}({\mathit{X}}_2)\right]=|S_0{|}^2\delta ({\mathit{X}}_1-{\mathit{X}}_2+{\mathit{X}}_d).$$ (10)

When S₁ and S₂ are the only random components in Eq. (9), the expectation can be taken inside the integral. Note in Eq. (9) that the scattering functions are weighted by the transmit pressure and receive sensitivity fields, meaning that assumption 2 only needs to be satisfied within the focal spot defined by the transmit and receive apertures.

Assumption 3. Assume that the axial displacement z_d is small relative to z such that $z_d/(z+z_d)\approx 0$. The correlation can then be simplified by integrating over the delta function in Eq. (10) (see the Appendix for details),

$$\begin{array}{c}{\Gamma }_{12}({\mathit{x}}_d,z_d)\approx \frac{|S_0{|}^2}{{(2\pi z)}^4}{e}^{-j4\pi z_d/\lambda }\int e^{-j2\pi {\mathit{x}}_d^T({\mathit{x}}_d/(z\lambda )+2\mathit{u})}\\ \times \widetilde{T_1}(\mathit{u})\widetilde{R_1}(\mathit{u}){\left[\widetilde{T_2}\left(\mathit{u}+\frac{{\mathit{x}}_d}{\lambda z}\right)\widetilde{R_2}\left(\mathit{u}+\frac{{\mathit{x}}_d}{\lambda z}\right)\right]}^{*}\hspace{0.17em}d\mathit{u},\end{array}$$ (11)

where $\tilde{X}(\mathit{u})$ denotes the spatial two-dimensional Fourier transform of $X(\mathit{x})$ at spatial frequency vector u.

Equation (11) gives the monochromatic approximation of the correlation of two pulse-echo signals acquired from moving diffuse scatterers for arbitrary transmit and receive apertures for wavelength λ. The monochromatic expression can be extended to broadband (BB) apertures by incorporating the aperture frequency response and integrating over the frequency. In the case that all four apertures have the same response F(f),

$${\Gamma }_{12}^{\text{broad}}({\mathit{x}}_d,z_d)=\int {\left|F(f)\right|}^4{\Gamma }_{12}({\mathit{x}}_d,z_d)\hspace{0.17em}df,$$ (12)

where ${\Gamma }_{12}$ is implicitly a function of f.

III. COMMON SCENARIOS

We illustrate the theoretical consequences of Eq. (11) in a few common scenarios of scatterer motion. For simplicity, we consider the monochromatic correlation ${\Gamma }_{12}$.

A. Stationary scatterer case

In the case of no scatterer motion (i.e., ${\mathit{x}}_d=\mathbf{0}$ and z_d = 0), the scatterer model reduces to

$$\mathbf{E}\left[S_1({\mathit{X}}_1)S_2^{*}({\mathit{X}}_2)\right]=|S_0{|}^2\delta ({\mathit{X}}_1-{\mathit{X}}_2).$$ (13)

The correlation is then simplified as

$${\Gamma }_{12}(0,0)\approx A^2\int \widetilde{T_1}(\mathit{u})\widetilde{R_1}(\mathit{u}){\left[\widetilde{T_2}(\mathit{u})\widetilde{R_2}(\mathit{u})\right]}^{*}d\mathit{u},$$ (14)

where $A=\left|S_0\right|/{(2\pi z)}^2$.

B. Translating aperture case

In the case that the scatterers are stationary and the transmit and receive apertures are identical up to a translational offset, i.e., $T({\mathit{x}}_t)=T_1({\mathit{x}}_t)=T_2({\mathit{x}}_t+{\Delta }_t)$ and $R({\mathit{x}}_r)=R_1({\mathit{x}}_r)=R_2({\mathit{x}}_r+{\Delta }_r)$, the correlation reduces to

$${\Gamma }_{12}(\mathbf{0},0)\approx A^2\int {\left|\tilde{T}(\mathit{u})\right|}^2{\left|\tilde{R}(\mathit{u})\right|}^2{e}^{-j2\pi {\mathit{u}}^T({\Delta }_t+{\Delta }_r)}d\mathit{u}.$$ (15)

This is equivalent to the statement that the correlation between two backscattered signals is equal to the cross-correlation of the autocorrelations of the transmit and receive apertures, evaluated at lag ${\Delta }_t+{\Delta }_r$, which coincides with previous derivations for the correlation given stationary incoherent sources.^43,47 This model applies to techniques that rely on coherence measurements between aperture elements, such as phase aberration correction, sound speed estimation, and aperture coherence imaging.

C. Estimation of purely axial scatterer motion

Consider the case of purely axial motion, i.e., ${\mathit{x}}_d=\mathbf{0}$ and $z_d\ne 0$. The correlation simplifies to

$${\Gamma }_{12}(\mathbf{0},z_d)\approx {\Gamma }_{12}(\mathbf{0},0)\hspace{0.17em}{e}^{-j4\pi z_d/\lambda }.$$ (16)

Note that axial motion effectively contributes a complex phase rotation to the correlation for stationary scatterers. For cases where ${\Gamma }_{12}(\mathbf{0},0)$ has no inherent complex phase (i.e., is real-valued), the displacement can be estimated using the phase-shift of the measured correlation ${\widehat{\Gamma }}_{12}$,

$${\widehat{z}}_d=\frac{\lambda }{4\pi }\hspace{0.17em}\text{arg}\left[{\widehat{\Gamma }}_{12}\right].$$ (17)

In the case where T₁ = T₂ and R₁ = R₂, the phase-shift estimator is sometimes called an “autocorrelator.” Phase-shift estimators are commonly employed in speckle tracking methods such as Color Doppler and elastography.

D. Estimation of jitter due to speckle decorrelation

Decorrelation of the underlying speckle fundamentally degrades the ability to estimate the true phase-shift between two backscattered signals. The amount of degradation depends on the correlation coefficient magnitude μ,^1,6,9 which is defined as

$$\mu =\left|\frac{{\Gamma }_{12}({\mathit{x}}_d,z_d)}{\sqrt{{\Gamma }_{11}(\mathbf{0},0){\Gamma }_{22}(\mathbf{0},0)}}\right|.$$ (18)

Decorrelation (i.e., $\mu <1$) introduces jitter, causing the measurable phase-shift to deviate from the true phase-shift. The variance of the phase-shift error $\Delta \theta$ between two arbitrary speckle signals with correlation μ is¹

$${\sigma }_{\Delta \theta }^2=\frac{{\pi }^2}{3}-\pi \arcsin (\mu )+{(\arcsin (\mu ))}^2-\frac{1}{2}{\text{Li}}_2({\mu }^2),$$ (19)

where ${\text{Li}}_2$ is the dilogarithm function. The variance is zero when μ = 1 and grows rapidly as μ is reduced. Figure 2 plots the phase-shift estimator jitter as a function of μ for an example case of an 8 MHz signal with sound speed of 1540 m/s.

FIG. 2.

(Color online) The expected jitter of a phase-shift estimate is plotted as a function of the magnitude of the correlation coefficient between the two signals. For an 8 MHz signal and sound speed of 1540 m/s, the maximum detectable phase shift is 48 μm. The expected jitter is approximately 8 μm when $\mu =0.95$.

In the monochromatic case for T₁ = T₂ and R₁ = R₂, Eqs. (16) and (19) predict that pure axial motion only rotates the complex phase (which has no effect on μ) and therefore does not contribute to estimator jitter.

E. Impact of transverse scatterer motion

In Eq. (11), transverse scatterer motion (i.e., ${\mathit{x}}_d\ne \mathbf{0}$) induces the equivalent of both a transverse shift and modulation to one of the transmit-receive aperture pairs. Perfect signal correlation (i.e., μ = 1) can be achieved only when, for all u,

$$\widetilde{T_1}(\mathit{u})\widetilde{R_1}(\mathit{u})=\widetilde{T_2}\left(\mathit{u}+\frac{{\mathit{x}}_d}{\lambda z}\right)\widetilde{R_2}\left(\mathit{u}+\frac{{\mathit{x}}_d}{\lambda z}\right)e^{j4\pi {\mathit{u}}^T{\mathit{x}}_d}.$$ (20)

This equality is not satisfied in typical ultrasound aperture configurations (e.g., rectangular apertures), in which case $\mu <1$. Thus transverse scatterer motion generally introduces jitter into axial phase-shift estimation.

F. Correlation of synthetic transmit apertures

In aperture synthesis, multiple acquisitions are coherently summed to retrospectively form an aggregate transmit aperture. Like traditional apertures, STAs can be used in correlation measurements. However, each STA component may be acquired from a different scattering function due to scatterer motion, which could cause decorrelation and increased jitter in STA-based phase-shift estimation.

Consider a STA synthesized from N transmit-receive aperture pairs $(T_1,R_1),\dots ,(T_N,R_N)$. For two consecutive STA acquisitions, denote each component scattering function as $S_1,\dots ,S_{2N}$. The correlation of two STAs is the sum of the cross-correlations of each component,

$$\begin{array}{c}{\Gamma }_{{\Sigma }_1{\Sigma }_2}=\mathbf{E}\left[\sum _{p=1}^{N}P(T_p,S_p,R)\sum _{q=1}^N{P}^{*}(T_q,S_{N+q},R)\right]\\ =\sum _{p=1}^N\sum _{q=1}^N\mathbf{E}\left[P(T_p,S_p,R)P^{*}(T_q,S_{N+q},R)\right].\end{array}$$ (21)

Assuming a constant uniform motion of ${\mathit{X}}_d$, i.e.,

$$\mathbf{E}\left[S_p({\mathit{X}}_1)S_q^{*}({\mathit{X}}_2)\right]\approx |S_0{|}^2\delta ({\mathit{X}}_1-{\mathit{X}}_2+(q-p){\mathit{X}}_d),$$ (22)

the correlation between two STA signals is expressed as

$${\Gamma }_{{\Sigma }_1{\Sigma }_2}=\sum _{p=1}^N\sum _{q=1}^N{\Gamma }_{pq}\left((N+q-p){\mathit{x}}_d,(N+q-p)z_d\right).$$ (23)

For this model of scatterer motion, the autocorrelation of either synthetic signal is the same,

$${\Gamma }_{{\Sigma }_1{\Sigma }_1}={\Gamma }_{{\Sigma }_2{\Sigma }_2}=\sum _{p=1}^N\sum _{q=1}^N{\Gamma }_{pq}\left((q-p){\mathit{x}}_d,(q-p)z_d\right).$$ (24)

Consequently, the correlation coefficient magnitude is simplified as

$$\mu =\left|\frac{{\Gamma }_{{\Sigma }_1{\Sigma }_2}}{\sqrt{{\Gamma }_{{\Sigma }_1{\Sigma }_1}{\Gamma }_{{\Sigma }_2{\Sigma }_2}}}\right|=\left|\frac{{\Gamma }_{{\Sigma }_1{\Sigma }_2}}{{\Gamma }_{{\Sigma }_1{\Sigma }_1}}\right|.$$ (25)

IV. NUMERICAL EXPERIMENTS

We demonstrate the presented theory in three applications of pulse-echo correlation with scatterer motion: (1) phase-shift estimation with focused apertures; (2) phase-shift estimation with plane wave STA using sequential or interleaved⁴⁸ sequences; and (3) correlation coefficient measurements as a function of receive aperture spacing with STAs. The first two applications are examples of so-called autocorrelators in the sense that the same transmit and receive apertures are used for the two correlated signals; the third is an example of a cross-correlator, where different receive apertures are used. Each application was evaluated using numerical integration of Eqs. (11) and (12) and compared against matched pulse-echo simulations.

A. Numerical and simulation methods

The correlation ${\Gamma }_{12}$ and the correlation coefficient magnitude μ were evaluated numerically using the matlab Symbolic Toolbox (The Mathworks, Natick, MA) with parameters for a Verasonics L12–3v transducer (128 elements, 200 μm pitch, c = 1540 m/s, z = 50 mm). The transducer was modeled with a center frequency of f = 8 MHz. For x within the domain of the aperture T, parabolic and plane wave transmit focusing was applied as

$$T^{\text{par}}(\mathit{x})=1,$$ (26)

$$T^{\text{pln}}(\mathit{x})=\text{exp}\hspace{0.17em}\left[\frac{j2\pi }{\lambda }\left(\mathit{x}\hspace{0.17em}\text{sin}\hspace{0.17em}\alpha -\frac{{\mathit{x}}^T\mathit{x}}{2z}\right)\right],$$ (27)

where α denotes the steering angle. [Note that these definitions take into account the parabolic focusing term $e^{j\pi {\mathit{x}}^T\mathit{x}/(\lambda z)}$ that was incorporated into the derivations of Eqs. (11) and (12).] Elevation displacement was not considered. The expected jitter was predicted using Eq. (19) based on the predicted value of μ. Monochromatic [narrowband (NB)] analysis was performed using Eq. (11), whereas BB analysis used Eq. (12) assuming a Gaussian frequency response with a fractional bandwidth of B = 60% about center frequency $f_0=8$ MHz, i.e.,

$$F(f)=\text{exp}\hspace{0.17em}\left[-\pi {\left(\frac{f-f_0}{B{f}_0}\right)}^2\right].$$ (28)

Paired simulations were executed using identical imaging parameters with Field II Pro,^49–51 a C program that simulates the spatial impulse response of ultrasound transducers using linear systems theory. A small field of scatterers with a density of 30 scatterers per resolution cell was placed at a distance of z = 50 mm unless otherwise noted. The (x, y, z) extent of scatterers was roughly $(8\lambda ,13\lambda ,4\lambda )$ to fully encompass the point spread function at the focal depth. Broadband simulations used a transducer fractional bandwidth of 60%, whereas NB simulations used a fractional bandwidth of 1%. A 2-cycle sinusoid excitation pulse was employed. For each experiment, 128 independent scatterer realizations were simulated to obtain the mean and standard deviation results. In the simulations, the transmit aperture was focused using the true geometric distance (i.e., without the Fresnel approximation),

$$T^{\text{geo}}(\mathit{x})=\text{exp}\hspace{0.17em}\left[\frac{j2\pi }{\lambda }\left(\sqrt{{\mathit{x}}^T\mathit{x}+z^2}-\frac{{\mathit{x}}^T\mathit{x}}{2z}\right)\right].$$ (29)

B. Experiment 1: Focused transmit autocorrelator

A focused transmit aperture was used to acquire two consecutive pulse-echo signals with scatterer motion. Two cases were considered:

• (1)Lateral displacement ranging from x_d = 0 to 100 μm; fixed axial displacement of z_d = 10 μm.
• (2)Axial displacement ranging from z_d = 0 to 100 μm; fixed lateral displacement of x_d = 10 μm.

In the first case, the monochromatic theory was compared against BB simulations (60% bandwidth). (The BB theory and NB simulations are omitted for visual clarity; transducer bandwidth was observed to have minimal effect on sensitivity to lateral motion.) In the second case, monochromatic theory was compared against NB simulations (1% bandwidth), and BB theory against BB simulations. For reference, a 100 μm displacement observed at 10 kHz PRF corresponds to 1 m/s velocity.

Figure 3 plots the axial displacement estimates and jitter as a function of lateral and axial scatterer motions. Lateral scatterer motion did not bias the estimates but increased the jitter in both the theory and the simulations. Notably, the monochromatic theory showed good agreement with the BB simulations. Axial scatterer motion was accurately predicted and measured by both the NB and BB theories and simulations, with an aliasing artifact near $z_d\approx 48$ μm, which corresponds to π radians for 1540 m/s sound speed at 8 MHz. Axial scatterer motion was found to increase the jitter only in the BB cases, whereas the NB apertures had no effect on jitter.

FIG. 3.

(Color online) Experiment 1: Focused transmits were used to measure axial displacements (left column) and jitter (right column) in the presence of lateral (top row) and axial (bottom row) scatterer motion. The theoretical results in Eqs. (11) and (12) were compared against NB and BB simulations. (a) Displacement estimates with lateral motion. (b) Estimator jitter with lateral motion. (c) Displacement estimates with axial motion. (d) Estimator jitter with axial motion.

The results of this numerical experiment suggest that decorrelation is likely caused by the re-weighting of scatterers passing through the focus of the imaging system. Axial motion through the tight axial focus of BB apertures caused decorrelation, whereas the same motion through the long and uniform focus of NB apertures had no effect. Similarly, lateral scatterer motion across the lateral focus of the transmit-receive beampattern (e.g., sinc-squared profile for rectangular apertures) caused decorrelation. These results indicate that a monochromatic analysis is insufficient for characterizing jitter under axial motion; by contrast the lateral effect appears to be independent of the bandwidth of the transducer, as shown by the good agreement between monochromatic theory and BB simulations. These findings are generally consistent with previous observations where tighter focusing and greater displacement magnitudes resulted in higher jitter in acoustic radiation force impulse (ARFI) elastography.^52,53

C. Experiment 2: STA autocorrelator

The effects of scatterer motion were investigated for two types of STA autocorrelator sequences:

• •Sequential: $[T_1^{(1)},\dots ,T_N^{(1)},T_1^{(2)},\dots ,T_N^{(2)}]$;
• •Interleaved: $[T_1^{(1)},T_1^{(2)},\dots ,T_N^{(1)},T_N^{(2)}]$,

where $T_n^{(m)}$ denotes the nth transmit of the mth frame. The interleaved configuration has recently been shown to increase the maximum detectable phase-shift estimate N-fold for an N-pulse STA by reducing the effective PRF between the two STAs,^48,54 and can be used to address aliasing artifacts like the one observed in Fig. 3(c). Plane wave STAs were formed using N = 3 angles steered at −3°, 0°, and 3° with either the sequential or interleaved sequence. Two consecutive frames were acquired. The scatterer displacement between frames was selected to match those of experiment 1; that is, a 10 μm displacement between frames was achieved using a 3.3 μm displacement between pulses. Here, only the more applicable BB theory and simulations were considered.

Figure 4 shows close agreement between the theoretical predictions and simulation measurements of axial motion. The results also match the BB results from experiment 1 (Fig. 3), both in trends and in absolute values. The sequential and interleaved STAs had similar performance in estimating the displacements, with lateral and axial motions causing increased jitter in both. Notably, interleaved STA eliminated the aliasing artifact caused by fast axial motion, but exhibited more jitter due to lateral motion than sequential STA.

FIG. 4.

(Color online) Experiment 2: Sequential and interleaved STAs were used to measure axial displacements (left column) and jitter (right column) in the presence of lateral (top row) and axial (bottom row) scatterer motion. The BB theoretical result is compared against BB simulations. Interleaved STA eliminates the aliasing artifact at the cost of increased jitter. (a) Displacement estimates with lateral motion. (b) Estimator jitter with lateral motion. (c) Displacement estimates with axial motion. (d) Estimator jitter with axial motion.

These results indicate that STAs perform similarly to ordinary focused transmits for the same effective displacements (i.e., per-frame for STAs versus per-pulse for focused transmits). Interestingly, intra-STA scatterer motion was not a significant source of jitter, an observation consistent with the evident widespread success of STA Doppler. Despite being composed of multiple distinct acquisitions, the results indicate that the correlation coefficient between two STAs remained similar to those of focused transmits, possibly due to the uniform nature of the motion and its presence in both signals.

D. Experiment 3: Receive aperture coherence with STA

In this experiment, the correlation coefficient was measured as a function of receive aperture spacing for a single STA. The STA was constructed using four non-overlapping transmit sub-apertures of 32 elements each. Scatterer motion was present between each transmission, and the resulting pulse-echo signals were summed coherently to form the STA. The receive apertures were selected to be single elements, separated by a lag $0\le {\Delta }_r\le 127d$, where the element pitch was d = 0.2 mm. Lateral displacements of 0, 0.1, 1, and 10 mm per STA (i.e., per every 4 transmissions) and axial displacements of 0, 0.01, 0.1, and 1 mm per STA were considered. Lateral and axial motions were considered independent of one another. All analysis was performed with BB transducers.

Figure 5 plots the correlation coefficients as a function of receive aperture spacing for lateral and axial scatterer motion. Close agreement was observed between theoretical predictions and simulation measurements, including a piecewise behavior that was observed at multiples of the sub-aperture size of 32 elements. Small displacements less than 0.01 mm axially and 0.1 mm laterally showed small deviations from the predictions for stationary scatterers, which corresponds roughly to a triangle function for rectangular apertures.^43,47 The robustness to minor intra-STA scatterer motion observed here is similar to that observed with STA Doppler in experiment 2, and is likely because both received signals observe the same intra-STA motion. Larger displacements degraded the correlation coefficient; such large decorrelations can likely be avoided in practice by using a sufficiently high PRF between STA components. Axial displacements caused greater decorrelation than equivalent transverse motions. A potential explanation for this disparity is that single-element receive apertures result in worse lateral resolution and thus exhibit a greater tolerance for motion across the widened lateral focus.

FIG. 5.

(Color online) Experiment 3: The theoretically predicted and simulation-measured correlation coefficients are plotted as a function of receive aperture spacing for lateral (top) and axial (bottom) scatterer motion. Large motions in either direction resulted in decorrelation throughout. (a) Lateral motion and (b) axial motion.

E. Limitations of the theory

The presented theory utilizes the Fresnel approximation to simplify the analysis. The range of validity of the Fresnel and Fraunhofer approximations are well-studied.^46,55 We utilized a brief study (omitted here) to select an appropriate depth for analysis, and discovered that a focal depth of z = 5 cm or more was necessary to obtain reasonable match between theory and simulation. This depth corresponds to an f-number of approximately 2. However, many ultrasound imaging configurations utilize f-numbers smaller than this, possibly invalidating the Fresnel approximation (and possibly the small-displacement assumption) and leaving open-ended the utility of such a theoretical analysis.

The bulk uniform motion assumption is also commonly violated in pulse-echo imaging. For instance, blood flow is often modeled as having a parabolic displacement profile, and ARFI is modeled as producing a Gaussian displacement profile. In the latter case, rapid changes in the displacement profile within the focal beam (referred to as “shearing”) have been linked to increased jitter.^52,53 Furthermore, real physiological motions are generally pulsatile and time-varying. The presented theory also does not consider other common decorrelating phenomena such as electronic noise from the system and acoustical noise from multiple reflections and strong off-axis scatterers, nor does it examine other implicit assumptions, such as a homogeneous speed of sound.

V. CONCLUSION

We have presented a theory for the correlation of motion in pulse-echo ultrasound. Equation (11) predicts the complex correlation between two pulse-echo ultrasound speckle signals, each due to an arbitrary transmit and receive aperture, with scattering functions that are simple translations of one another. The derivation relies on three primary assumptions: (1) the Fresnel approximation ($|z_s-z_t|\gg \parallel {\mathit{x}}_s-{\mathit{x}}_t{\parallel }_2$); (2) bulk uniform motion of diffuse scatterers [Eq. (10)]; and (3) small axial displacements [$z_d/(z+z_d)\approx 0$]. The resulting expression describes correlation as a function of axial and transverse motion, and simplifies to previously derived formulations of correlation in the absence of motion.

The theory was then compared against simulations in three numerical experiments: focused transmit phase-shift estimation (Fig. 3), STA phase-shift estimation (Fig. 4), and receive aperture coherence with STAs (Fig. 5). A key theoretical result of this paper was that transverse scatterer motion causes decorrelation for typical ultrasound imaging aperture configurations, whereas axial scatterer motion causes decorrelation when using BB apertures. The theory successfully predicted that interleaved STAs increase the aliasing limit of phase-shift estimators at the cost of a slight increase in jitter. The theory also confirmed that intra-STA scatterer motion does not cause decorrelation when both STAs contain the identical motion, supporting the current widespread use of ultrafast STA Doppler techniques. Finally, the theory was used to predict receive-aperture correlation coefficients with STAs, which can be used for applications such as arrival time estimation and coherence-based imaging. The presented theory provides a framework for analyzing the correlations of arbitrary pulse-echo configurations for applications in which scatterer motion is expected.

APPENDIX: CORRELATION OF BULK MOTION

Substituting Eq. (10) into Eq. (9) and integrating over the delta function such that ${\mathit{x}}_s={\mathit{x}}_{s_1}={\mathit{x}}_{s_2}-{\mathit{x}}_d$ and $z_d=z_2-z_1$, we have

$$\begin{array}{c}{\Gamma }_{12}\approx \frac{|S_0{|}^2}{{(2\pi z)}^4}{e}^{-j4\pi z_d/\lambda }\int \iint \iint T_1({\mathit{x}}_{t_1})T_2^{*}({\mathit{x}}_{t_2})\\ \hspace{1em}\times R_1({\mathit{x}}_{r_1})R_2^{*}({\mathit{x}}_{r_2})\hspace{0.17em}\text{exp}\hspace{0.17em}\left[\frac{j2\pi }{\lambda z}{\mathit{x}}_s^T({\mathit{x}}_s-{\mathit{x}}_{t_1}-{\mathit{x}}_{r_1})\right]\\ \hspace{1em}\times \text{exp}\hspace{0.17em}\left[-\frac{j2\pi }{\lambda z}{({\mathit{x}}_s+{\mathit{x}}_d)}^T({\mathit{x}}_s+{\mathit{x}}_d-{\mathit{x}}_{t_2}-{\mathit{x}}_{r_2})\right]\\ \hspace{1em}\times \text{exp}\hspace{0.17em}\left[\frac{j2\pi }{\lambda z}\frac{z_d}{z+z_d}{({\mathit{x}}_s+{\mathit{x}}_d)}^T({\mathit{x}}_s+{\mathit{x}}_d-{\mathit{x}}_{r_2}-{\mathit{x}}_{t_2})\right]\\ \hspace{1em}\times d{\mathit{x}}_{s}d{\mathit{x}}_{t_1}d{\mathit{x}}_{t_2}d{\mathit{x}}_{r_1}d{\mathit{x}}_{r_2},\end{array}$$ (A1)

where we approximate $z\approx z_1\approx z_2$ for amplitude terms.

By assumption 3, let us assume $z_d/(z+z_d)\approx 0$, allowing higher order phase terms to be ignored. Collecting terms containing ${\mathit{x}}_s$,

$$\begin{array}{c}{\Gamma }_{12}\approx \frac{|S_0{|}^2}{{(2\pi z)}^4}{e}^{-j4\pi z_d/\lambda }\iint \iint T_1({\mathit{x}}_{t_1})T_2^{*}({\mathit{x}}_{t_2})\\ \hspace{1em}\times R_1({\mathit{x}}_{r_1})R_2^{*}({\mathit{x}}_{r_2})\hspace{0.17em}\text{exp}\hspace{0.17em}\left[\frac{j2\pi }{\lambda z}{\mathit{x}}_d^T({\mathit{x}}_{t_2}+{\mathit{x}}_{r_2}-{\mathit{x}}_d)\right]\\ \hspace{1em}\times \int \hspace{0.17em}\text{exp}\hspace{0.17em}\left[\frac{j2\pi }{\lambda z}{\mathit{x}}_s^T({\mathit{x}}_{t_2}+{\mathit{x}}_{r_2}-{\mathit{x}}_{t_1}-{\mathit{x}}_{r_1}-2{\mathit{x}}_d)\right]d{\mathit{x}}_s\\ \hspace{1em}\times d{\mathit{x}}_{t_1}d{\mathit{x}}_{t_2}d{\mathit{x}}_{r_1}d{\mathit{x}}_{r_2}.\end{array}$$ (A2)

The inner integral evaluates to a delta function

$$\begin{array}{c}{\Gamma }_{12}\approx \frac{|S_0{|}^2}{{(2\pi z)}^4}{e}^{-j4\pi z_d/\lambda }\iint \iint T_1({\mathit{x}}_{t_1})T_2^{*}({\mathit{x}}_{t_2})\\ \hspace{1em}\times R_1({\mathit{x}}_{r_1})R_2^{*}({\mathit{x}}_{r_2})\hspace{0.17em}\text{exp}\hspace{0.17em}\left[\frac{j2\pi }{\lambda z}{\mathit{x}}_d^T({\mathit{x}}_{t_2}+{\mathit{x}}_{r_2}-{\mathit{x}}_d)\right]\\ \hspace{1em}\times \delta \left({\mathit{x}}_{t_2}+{\mathit{x}}_{r_2}-{\mathit{x}}_{t_1}-{\mathit{x}}_{r_1}-2{\mathit{x}}_d\right)d{\mathit{x}}_{t_1}d{\mathit{x}}_{t_2}d{\mathit{x}}_{r_1}d{\mathit{x}}_{r_2}.\end{array}$$ (A3)

Letting ${\mathit{x}}_t={\mathit{x}}_{t_2}+{\mathit{x}}_{r_2}-{\mathit{x}}_{r_1}-2{\mathit{x}}_d$ and integrating over the delta function, we have

$$\begin{array}{c}{\Gamma }_{12}\approx \frac{|S_0{|}^2}{{(2\pi z)}^4}{e}^{-j4\pi z_d/\lambda }\iint R_1({\mathit{x}}_{r_1})R_2^{*}({\mathit{x}}_{r_2})\\ \hspace{1em}\times \int T_1({\mathit{x}}_t)T_2^{*}({\mathit{x}}_t-{\mathit{x}}_{r_2}+{\mathit{x}}_{r_1}+2{\mathit{x}}_d)\\ \hspace{1em}\times \hspace{0.17em}\text{exp}\hspace{0.17em}\left[\frac{j2\pi }{\lambda z}{\mathit{x}}_d^T({\mathit{x}}_t+{\mathit{x}}_{r_1}+{\mathit{x}}_d)\right]d{\mathit{x}}_{t}d{\mathit{x}}_{r_1}d{\mathit{x}}_{r_2}.\end{array}$$ (A4)

Collecting ${\mathit{x}}_{r_2}$ terms inside the innermost integral and letting ${\mathit{x}}_{r_1}={\mathit{x}}_1$ and ${\mathit{x}}_{r_2}={\mathit{x}}_2$, the expression can be rearranged as

$$\begin{array}{c}{\Gamma }_{12}\approx \frac{|S_0{|}^2}{{(2\pi z)}^4}{e}^{-j4\pi z_d/\lambda }\iint R_1({\mathit{x}}_1)T_1({\mathit{x}}_t)\\ \hspace{1em}\times \int R_2^{*}({\mathit{x}}_2)T_2^{*}({\mathit{x}}_t-{\mathit{x}}_2+{\mathit{x}}_1+2{\mathit{x}}_d)\hspace{0.17em}d{\mathit{x}}_2\\ \hspace{1em}\times \hspace{0.17em}\text{exp}\hspace{0.17em}\left[\frac{j2\pi }{\lambda z}{\mathit{x}}_d^T({\mathit{x}}_t+{\mathit{x}}_1+{\mathit{x}}_d)\right]d{\mathit{x}}_{t}d{\mathit{x}}_1.\end{array}$$ (A5)

Further, letting ${\mathit{x}}_t=\mathit{x}-{\mathit{x}}_1$, we have

$$\begin{array}{c}{\Gamma }_{12}\approx \frac{|S_0{|}^2}{{(2\pi z)}^4}{e}^{-j4\pi z_d/\lambda }{e}^{j2\pi {\mathit{x}}_d^T{\mathit{x}}_d/(\lambda z)}\\ \hspace{1em}\times \int \hspace{0.17em}\text{exp}\hspace{0.17em}\left[\frac{j2\pi }{\lambda z}{\mathit{x}}_d^T\mathit{x}\right]\left(\int R_1({\mathit{x}}_1)T_1(\mathit{x}-{\mathit{x}}_1)\hspace{0.17em}d{\mathit{x}}_1\right)\\ \hspace{1em}\times {\left(\int R_2({\mathit{x}}_2)T_2(\mathit{x}+2{\mathit{x}}_d-{\mathit{x}}_2)\hspace{0.17em}d{\mathit{x}}_2\right)}^{*}d\mathit{x}.\end{array}$$ (A6)

Observe that the integral in Eq. (A6) is essentially the Fourier transform of two convolutions. At this point, we can utilize the convolution theorem to reduce the triple integral into a single integral using the Fourier transform. Letting A_i denote the convolution of T_i and R_i, the Fourier transform $\widetilde{A_i}$ can be expressed as

$$\begin{array}{c}\widetilde{A_i}(\mathit{u})=\int \left(\int T_i({\mathit{x}}_i)R_i(\mathit{x}-{\mathit{x}}_i)d{\mathit{x}}_i\right)e^{-j2\pi {\mathit{u}}^T\mathit{x}}d\mathit{x}\\ =\iint T_i({\mathit{x}}_i)R_i(\mathit{x})e^{-j2\pi {\mathit{u}}^T(\mathit{x}+{\mathit{x}}_i)}d\mathit{x}d{\mathit{x}}_i\\ =\widetilde{T_i}(\mathit{u})\widetilde{R_i}(\mathit{u}).\end{array}$$ (A7)

The integral in Eq. (A6) can be simplified as

$$\begin{array}{c}\int \hspace{0.17em}\text{exp}\hspace{0.17em}\left[\frac{j2\pi }{\lambda z}{\mathit{x}}_d^T\mathit{x}\right]A_1(\mathit{x})A_2^{*}(\mathit{x}+2{\mathit{x}}_d)\hspace{0.17em}d\mathit{x}\\ =\int \hspace{0.17em}\text{exp}\hspace{0.17em}\left[\frac{j2\pi }{\lambda z}{\mathit{x}}_d^T\mathit{x}\right]\left(\int \widetilde{A_1}(\mathit{u})e^{j2\pi {\mathit{u}}^T\mathit{x}}d\mathit{u}\right)\\ \hspace{1em}\times {\left(\int \widetilde{A_2}(\mathit{v})e^{j2\pi {\mathit{v}}^T(\mathit{x}+2{\mathit{x}}_d)}d\mathit{v}\right)}^{*}d\mathit{x}\\ =\iint \widetilde{A_1}(\mathit{u}){\widetilde{A_2}}^{*}(\mathit{v})e^{-j4\pi {\mathit{v}}^T{\mathit{x}}_d}\\ \hspace{1em}\times \int \hspace{0.17em}\text{exp}\hspace{0.17em}\left[j2\pi {\mathit{x}}^T\left(\frac{{\mathit{x}}_d}{\lambda z}+\mathit{u}-\mathit{v}\right)\right]d\mathit{x}\hspace{0.17em}d\mathit{u}d\mathit{v}\\ =\iint \widetilde{A_1}(\mathit{u}){\widetilde{A_2}}^{*}(\mathit{v})e^{-j4\pi {\mathit{v}}^T{\mathit{x}}_d}\hspace{0.17em}\delta \left(\frac{{\mathit{x}}_d}{\lambda z}+\mathit{u}-\mathit{v}\right)d\mathit{u}d\mathit{v}\\ =\int \widetilde{A_1}(\mathit{u}){\widetilde{A_2}}^{*}\left(\mathit{u}+\frac{{\mathit{x}}_d}{\lambda z}\right)e^{-j4\pi {(\mathit{u}+{\mathit{x}}_d/(\lambda z))}^T{\mathit{x}}_d}\hspace{0.17em}d\mathit{u}.\end{array}$$ (A8)

Therefore,

$$\begin{array}{c}{\Gamma }_{12}\approx \frac{|S_0{|}^2}{{(2\pi z)}^4}{e}^{-j4\pi z_d/\lambda }\int e^{-j2\pi {\mathit{x}}_d^T({\mathit{x}}_d/(z\lambda )+2\mathit{u})}\\ \hspace{1em}\times \widetilde{T_1}(\mathit{u})\widetilde{R_1}(\mathit{u}){\left[\widetilde{T_2}\left(\mathit{u}+\frac{{\mathit{x}}_d}{\lambda z}\right)\widetilde{R_2}\left(\mathit{u}+\frac{{\mathit{x}}_d}{\lambda z}\right)\right]}^{*}\hspace{0.17em}d\mathit{u}.\end{array}$$ (A9)