Speckle coherence of piecewise-stationary stochastic targets

Abstract

The van Cittert-Zernike (VCZ) theorem describes the propagation of spatial covariance from an incoherent source distribution, such as backscatter from stochastic targets in pulse-echo imaging. These stochastic targets are typically assumed statistically stationary and spatially incoherent with uniform scattering strength. In this work, the VCZ theorem is applied to a piecewise-stationary scattering model. Under this framework, the spatial covariance of the received echo data is demonstrated as the linear superposition of covariances from distinct spatial regions. This theory is analytically derived from fundamental physical principles, and validated through simulation studies demonstrating superposition and scaling. Simulations show that linearity is preserved over various depths and transmit apodizations, and in the presence of noise. These results provide a general framework to decompose spatial covariance into contributions from distinct regions of interest, which may be applied to advanced imaging methods. While the simulation tools used for validation are specific to ultrasound, this analysis is generally applicable to other coherent imaging applications involving stochastic targets. This covariance decomposition provides the physical basis for a recently described imaging method, Multi-covariate Imaging of Sub-resolution Targets.

I. INTRODUCTION

Coherence in imaging describes the similarity of two points in a wavefield across space and/or time. Many imaging modalities employ coherent processing techniques, which involve the measurement of both the amplitude and phase of the received signal. These methods typically seek to constructively align signal phase on transmission and/or reception to spatially localize energy and improve the signal-to-noise ratio (SNR) of the measurement. Examples of coherent imaging modalities include radar, sonar, ultrasound, stellar interferometry, and monochromatic laser imaging.

Across modalities, many imaging targets contain a combination of deterministic and stochastic components. Deterministic targets are characterized by nonrandom features or boundaries directly resolvable by the imaging system. Stochastic or diffuse targets, however, are described by random, unresolvable microstructure which is rough on the scale of the wavelength. Common examples include biological tissues in medical ultrasound, ground glass diffusers in laser optics, and coarse terrain or vegetative cover on the Earth's surface in radar.

Unlike deterministic targets, stochastic targets are best characterized by average statistical properties. These are typically modeled as statistically stationary and spatially incoherent with uniform scattering strength,^1–4 though more complex models have also sought to incorporate a small but finite coherence length,^4,5 or higher-order structure, such as periodicity, polarization effects, or non-Gaussian scatterer distributions.^6–8 Under coherent radiation, random interference of waves scattered from diffuse targets leads to a granular speckle pattern in the observed image.^2–4,9,10

Given the prevalence of diffuse targets across applications, the study of speckle has garnered much attention in the imaging literature. Early observations of the speckle phenomenon were documented in the 19th-century physics literature,^11,12 however, modern statistical analysis began in the 1960s with the advent of laser optics.^13–17 The first-order statistics of speckle have been well-characterized, drawing many parallels with the classical analysis of thermal light.^{2,4,5,9,18,19} Second-order statistics involving coherence—often quantified in terms of covariance, correlation, or mutual intensity—are described by the van Cittert-Zernike (VCZ) theorem, which provides a mathematical framework expressing the propagation of spatial coherence from an incoherent source distribution.^4,20–22

The VCZ theorem was first applied to the study of speckle by Goodman, who demonstrated an equivalence between the propagation of mutual intensity from an incoherent optical source and from laser light scattered by rough surfaces.⁵ This insight was further extended to pulse-echo ultrasound by Mallart and Fink,²³ who described backscatter from diffuse targets as an incoherent acoustic source, relating the incident energy to the covariance of the backscattered pressure field. Coherence yields myriad insights into in the design and analysis of ultrasound imaging systems.^23–31 Many imaging methods employ insights from spatial coherence in the beamforming process, and have shown improvements in image quality over conventional methods.^32–39

Across modalities, the most common model of diffuse targets assumes stationary statistics with uniform scattering strength. While this model is attractive for analytical calculations (including the derivation of the classical VCZ theorem), it fails to account for local variations in scattering strength common across imaging applications. Under this model, the second-order statistics predicted by the VCZ theorem are a function of the entire source distribution. While nonhomogeneity of the scattering function and changes in the incident energy profile will alter the covariance function,^23,40 the current model does not provide a framework to spatially decompose the covariance into contributions from particular regions of interest.

In this paper, we describe the target distribution as piecewise-stationary, in which scattering strength may vary as a function of absolute position. We apply the VCZ theorem to piecewise-stationary targets to demonstrate that the spatial covariance of the received echo data is the linear superposition of covariances from distinct spatial regions. We derive the mathematical basis of this decomposition from first principles, and validate the results through simulation studies demonstrating superposition and scaling over various depths and transmit apodizations, and in the presence of noise. We discuss the implications of this decomposition for imaging applications, and frame this description in the context of existing covariance models in the literature. To our knowledge, the decomposition of covariance into contributions from distinct spatial regions has not been presented elsewhere in the literature.

While the validation studies are specific to ultrasound, the physical insights presented in this work are generally applicable to coherent imaging applications involving stochastic targets. These insights provide the basis for a recently described estimation-based ultrasound image formation method, Multi-covariate Imaging of Sub-resolution Targets (MIST), which images the statistical properties of stochastic targets by estimating on-axis contributions to the ultrasonic echo data.³⁹

II. THEORY

A. Pulse-echo response of ultrasonic imaging systems

In scalar diffraction theory, the Huygens-Fresnel principle states that a source function can be represented as an integral over a continuum of spherical radiators. For a monochromatic, two-dimensional complex aperture function, $A({\mathit{X}}_a,f)$, located at depth z = 0, the one-way sensitivity pattern, $P({\mathit{X}}_s,z,f)$, is described by the Rayleigh-Sommerfeld diffraction integral⁴¹

$$P({\mathit{X}}_s,z,f)=\frac{1}{j\lambda }\underset{-\infty }{\overset{\infty }{\iint }}A({\mathit{X}}_a,f)\frac{e^{\mathit{jk}{r}_{\mathit{as}}}}{r_{\mathit{as}}}\text{cos}\hspace{0.17em}(\theta )d{\mathit{X}}_a,$$ (1)

where the coordinates of the source and aperture planes are given by ${\mathit{X}}_s={[x_s,y_s]}^T$ and ${\mathit{X}}_a={[x_a,y_a]}^T$, respectively. The frequency is given by f, the wavelength by λ, and the wavenumber by $k=2\pi /\lambda$. The term $\text{cos}\hspace{0.17em}(\theta )$ is the angle between the outward normal vector $\overline{n}$ and ${\overline{r}}_{\mathit{as}}$, where ${\overline{r}}_{\mathit{as}}$ is the vector from the aperture plane $({\mathit{X}}_a,0)$ to the source plane $({\mathit{X}}_s,z)$ with length r_as.

Assuming r_as is much greater than the wavelength, and using the Fresnel approximation for the distance r_as, Eq. (1) can be simplified to the generalized Fresnel diffraction integral⁴¹

$$\begin{array}{cc}P({\mathit{X}}_s,z,f)\hfill & =\frac{e^{\mathit{jkz}}{e}^{\mathit{jk}({\mathit{X}}_s^{\prime }{\mathit{X}}_s)/2z}}{j\lambda z}\hfill \\ \hfill & \times \underset{-\infty }{\overset{\infty }{\iint }}\left\{A({\mathit{X}}_a,f)e^{\mathit{jk}{\mathit{X}}_a^{\prime }{\mathit{X}}_a/2z}\right\}{e}^{-2\pi j{\mathit{X}}_s^{\prime }{\mathit{X}}_a/\lambda z}d{\mathit{X}}_a,\hfill \end{array}$$ (2)

where a prime represents the transpose operation. Equation (2) is recognized as the scaled Fourier transform of the product of the complex aperture function and a quadratic phase term, evaluated at spatial frequencies $(f_x,f_y)=(x_s/\lambda z,\hspace{0.17em}{y}_s/\lambda z)$. The Fresnel diffraction integral reduces the spherical wavefronts in Eq. (1) to a quadratic model.

In the case of a focused transmitting aperture, Eq. (2) can be expanded as

$$\begin{array}{cc}P({\mathit{X}}_s,z,f)=\hfill & \frac{e^{\mathit{jkz}}{e}^{\mathit{jk}({\mathit{X}}_s^{\prime }{\mathit{X}}_s)/2z}}{j\lambda z}\hfill \\ \hfill & \times \underset{-\infty }{\overset{\infty }{\iint }}\left\{A({\mathit{X}}_a,f)e^{\left(\mathit{jk}{\mathit{X}}_a^{\prime }{\mathit{X}}_a/2\right)\left(z^{-1}-z_f^{-1}\right)}\right\}\hfill \\ \hfill & \times e^{-2\pi j{\mathit{X}}_s^{\prime }{\mathit{X}}_a/\lambda z}d{\mathit{X}}_a,\hfill \end{array}$$ (3)

in which the quadratic phase term becomes a function of both the source plane depth z and the focal depth z_f. When the source plane is selected as z = z_f, the phase term is reduced to unity. This result is characterized by the Fraunhofer diffraction integral⁴¹

$$\begin{array}{cc}P({\mathit{X}}_s,z,f)=\hfill & \frac{e^{\mathit{jkz}}{e}^{\mathit{jk}({\mathit{X}}_s^{\prime }{\mathit{X}}_s)/2z}}{j\lambda z}\hfill \\ \hfill & \times \underset{-\infty }{\overset{\infty }{\iint }}A({\mathit{X}}_a,f)e^{-2\pi j{\mathit{X}}_s^{\prime }{\mathit{X}}_a/\lambda z}d{\mathit{X}}_a,\hfill \end{array}$$ (4)

which is simply the scaled, two-dimensional Fourier transform of the aperture function, valid at the focal plane of a focused aperture, or in the far field of an unfocused aperture. Equation (4) describes a simple Fourier relationship between (1) the transmitting aperture function and the pressure distribution at the focus or, equivalently, (2) the receiving aperture function and its one-way sensitivity pattern.

B. Spatial covariance in medical ultrasound

1. Theory

The van Cittert-Zernike theorem describes the spatial covariance of a propagating wavefield from an incoherent source. This is characterized by a Fourier transform relationship between an incoherent source distribution and the measured spatial covariance between points ${\mathit{X}}_1$ and ${\mathit{X}}_2$ in an observation plane, evaluated at spatial frequency $\Delta \mathit{X}/\lambda z$, where $\Delta \mathit{X}=({\mathit{X}}_1-{\mathit{X}}_2)$, λ is the wavelength and z is the distance between the source and observation planes.⁴

In pulse-echo ultrasound, backscatter from diffuse targets under coherent insonification can be considered a secondary, incoherent source, with intensity distribution equal to the squared magnitude of the product of the transmit beam amplitude (P) and the target function (χ).²³ The VCZ theorem predicts a Fourier relationship between the measured spatial covariance (R) and the source intensity distribution, given by

$$\begin{array}{c}R(\Delta {\mathit{X}}_a,z,f)=\underset{-\infty }{\overset{+\infty }{\iint }}|P({\mathit{X}}_s,z,f)\chi ({\mathit{X}}_s,z,f){|}^2\\ \times e^{-2\pi j({\mathit{X}}_s^{\prime }\Delta {\mathit{X}}_a)/\lambda z}d{\mathit{X}}_s,\end{array}$$ (5)

where the scattering function is characterized by unresolvable, randomly distributed microstructure, with backscatter amplitude described by complex, circular Gaussian statistics.³ Assuming χ to be spatially incoherent and statistically stationary, the autocorrelation of the scattering function at a fixed depth z is given by²³

$$⟨\chi ({\mathit{X}}_1,f){\chi }^{*}({\mathit{X}}_2,f)⟩={\chi }_0(f)\delta ({\mathit{X}}_1-{\mathit{X}}_2),$$ (6)

where $\delta (\mathit{X})$ is a two-dimensional delta function. Under this assumption, the spatial covariance of the received echoes is proportional to the Fourier transform of the two-dimensional transmit beam intensity

$$R(\Delta {\mathit{X}}_a,z,f)\propto \mathcal{F}\{P({\mathit{X}}_s,z,f)P^{*}({\mathit{X}}_s,z,f)\},$$ (7)

which is the well-known result of the VCZ theorem. At the focal plane of a focused transmitting aperture, this reduces to the scaled autocorrelation of the transmitting aperture function. For a conventional linear array with rectangular apodization, the predicted spatial covariance at the transmit focus is described by a triangle function, which linearly decreases in amplitude as element separation (or lag) increases to the full extent of the transmitting aperture.

2. Measurement

In practice, spatial covariance is estimated from discretely sampled array data. The estimated covariance of two zero-mean signals is defined as

$${\widehat{R}}_{\mathit{ij}}\left[t\right]=x_i\left[t\right]x_j^{*}\left[t\right],$$ (8)

where $x_i[t]$ and $x_j[t]$ are the complex signals at time sample t, received at elements i and j, respectively, after quadrature demodulation and focusing time delays have been applied. The complex conjugate operation is given by an asterisk.

For an M-channel array, this generalizes to the outer product of the array data vector

$$\widehat{\mathit{R}}\left[t\right]=\mathit{x}\left[t\right]{\mathit{x}}^H\left[t\right],$$ (9)

where $\widehat{\mathit{R}}[t]$ is the M × M covariance matrix (with entries ${\widehat{R}}_{\mathit{ij}}[t]$) at time sample t, and $\mathit{x}[t]$ is the M × 1 vector of complex array data at time sample t. The conjugate transpose operation is given by the superscript H.

To remove the effects of signal amplitude on the coherence estimate, the spatial covariance can also be normalized to calculate the spatial correlation. The spatial correlation of two signals x_i[t] and x_j[t] is represented by

$${\widehat{\rho }}_{\mathit{ij}}[t]=\frac{{\widehat{R}}_{\mathit{ij}}\left[t\right]}{\sqrt{{\widehat{R}}_{\mathit{ii}}\left[t\right]{\widehat{R}}_{\mathit{jj}}\left[t\right]}},$$ (10)

in which ${\widehat{\rho }}_{\mathit{ij}}[t]$ is the correlation coefficient of the complex signals received at elements i and j at time sample t. The M × M correlation matrix with entries ${\widehat{\rho }}_{\mathit{ij}}[t]$ is given by $\widehat{\mathit{\rho }}[t]$.

While the covariance and correlation measurements in Eqs. (8) and (10) represent M × M matrices, measurements across all (i, j) pairs with common element separation (or lag) are often averaged to generate a one-dimensional (1-D) function, given by

$${\widehat{R}}_{1-\mathrm{D}}[t;m]=\frac{1}{M-m}\sum _{\begin{array}{c}i=1\\ j=i+m\end{array}}^M{\widehat{R}}_{\mathit{ij}}\left[t\right]$$ (11)

and

$${\widehat{\rho }}_{1-\mathrm{D}}[t;m]=\frac{1}{M-m}\sum _{\begin{array}{c}i=1\\ j=i+m\end{array}}^M\frac{{\widehat{R}}_{\mathit{ij}}\left[t\right]}{\sqrt{{\widehat{R}}_{\mathit{ii}}\left[t\right]{\widehat{R}}_{\mathit{jj}}\left[t\right]}},$$ (12)

where ${\widehat{R}}_{1-\mathrm{D}}$ and ${\widehat{\rho }}_{1-\mathrm{D}}$ denote the 1-D covariance and correlation, respectively, given as a function of time sample t and lag m.

While covariance is sensitive to both the amplitude and phase of the received echoes, correlation removes the effects of the former, and is thus only sensitive to the phase.

C. Spatial covariance of piecewise-stationary stochastic targets

1. Theory

Medical ultrasound systems typically operate in pulse-echo mode, such that the total system response is calculated as the product of the one-way responses on transmit and receive. For a complex target distribution $\chi ({\mathit{X}}_s,z,f)$, the received echo signal s(z, f) can be calculated by⁴²

$$s(z,f)=\underset{\mathit{S}}{\iint }{P}_T({\mathit{X}}_s,z,f)P_R({\mathit{X}}_s,z,f)\chi ({\mathit{X}}_s,z,f)d{\mathit{X}}_s,$$ (13)

where $P_T({\mathit{X}}_s,z,f)$ is defined as transmit pressure distribution, and $P_R({\mathit{X}}_s,z,f)$ is defined as one-way receive system response, calculated using Eq. (2). The imaging frequency is given by f and depth by z. The integration surface S is defined as the plane orthogonal to the transducer axis at depth z. Equation (13) provides a means to calculate the received echo signal given arbitrary aperture and scattering functions.

The following analysis considers the monochromatic case at an arbitrary depth z, and is not limited to the focal region of the transducer. For simplicity, the frequency (f) and depth (z) have been removed from the notation, such that $s=s(z,f)$, $P({\mathit{X}}_s)=P({\mathit{X}}_s,z,f)$, and $\chi ({\mathit{X}}_s)=\chi ({\mathit{X}}_s,z,f)$.

From Eq. (13), the echo signal received at a single element i can be calculated as

$$s_i=\underset{\mathit{S}}{\iint }{P}_T({\mathit{X}}_s)P_{R_i}({\mathit{X}}_s)\chi ({\mathit{X}}_s)d{\mathit{X}}_s,$$ (14)

where s_i is the complex pulse echo signal received at element i, and $P_{R_i}$ is the one-way receive system response at element i.

We consider the case where the total system output at element i is composed of both the complex pulse echo signal (s_i) and additive noise (n_i), such that $x_i=s_i+n_i$. The covariance of the received signals at elements i and j is then given by

$$R_{\mathit{ij}}=⟨x_i{x}_j^{*}⟩=⟨(s_i+n_i){(s_j+n_j)}^{*}⟩$$ (15)

where $⟨·⟩$ is the expected value operator. Assuming the signal and noise are uncorrelated, Eq. (15) can be expanded as

$$R_{\mathit{ij}}=⟨s_i{s}_j^{*}⟩+⟨n_i{n}_j^{*}⟩.$$ (16)

We then define the signal and noise covariance terms as

$$A_{\mathit{ij}}=⟨s_i{s}_j^{*}⟩$$ (17)

and

$$N_{\mathit{ij}}=⟨n_i{n}_j^{*}⟩,$$ (18)

respectively, such that the expected covariance is given by

$$R_{\mathit{ij}}=A_{\mathit{ij}}+N_{\mathit{ij}}.$$ (19)

We first consider the covariance of the pulse echo signal. Substituting Eq. (14) into Eq. (17), this is given by

$$A_{\mathit{ij}}=⟨\underset{\mathit{S}}{\iint }{P}_T({\mathit{X}}_1)P_{R_i}({\mathit{X}}_1)\chi ({\mathit{X}}_1)d{\mathit{X}}_1\times \underset{\mathit{S}}{\iint }{P}_T^{*}({\mathit{X}}_2)P_{R_j}^{*}({\mathit{X}}_2){\chi }^{*}({\mathit{X}}_2)d{\mathit{X}}_2⟩.$$ (20)

Rearranging the order of integration and expected value yields

$$\begin{array}{c}{A}_{\mathit{ij}}=\underset{\mathit{S}}{\iint }\underset{\mathit{S}}{\iint }{P}_T({\mathit{X}}_1)P_{R_i}({\mathit{X}}_1)P_T^{*}({\mathit{X}}_2)P_{R_j}^{*}({\mathit{X}}_2)\\ \times ⟨\chi ({\mathit{X}}_1){\chi }^{*}({\mathit{X}}_2)⟩d{\mathit{X}}_{1}d{\mathit{X}}_2.\end{array}$$ (21)

The autocorrelation of an incoherent scattering function with stationary statistics is given by Eq. (6). Here, we relax the assumption of stationarity to a piecewise-stationary model, and allow the scattering strength to vary as a function of absolute position, such that the autocorrelation of the target function is given by

$$⟨\chi ({\mathit{X}}_1){\chi }^{*}({\mathit{X}}_2)⟩={\chi }_0({\mathit{X}}_1){\chi }_0({\mathit{X}}_2)\delta ({\mathit{X}}_1-{\mathit{X}}_2),$$ (22)

where ${\chi }_0(\mathit{X})$ is the spatially varying scattering strength of the target region. Substituting Eq. (22) into Eq. (21), the covariance is given by

$$\begin{array}{c}{A}_{\mathit{ij}}=\underset{\mathit{S}}{\iint }\underset{\mathit{S}}{\iint }\hspace{1em}{P}_T({\mathit{X}}_1)P_{R_i}({\mathit{X}}_1)P_T^{*}({\mathit{X}}_2)P_{R_j}^{*}({\mathit{X}}_2)\hfill \\ \hspace{1em}\times {\chi }_0({\mathit{X}}_1){\chi }_0({\mathit{X}}_2)\delta ({\mathit{X}}_1-{\mathit{X}}_2)d{\mathit{X}}_{1}d{\mathit{X}}_2.\hfill \end{array}$$ (23)

Using the sifting property of the δ-function, one of the ranges of integration can be eliminated to yield

$$A_{\mathit{ij}}=\underset{\mathit{S}}{\iint }{P}_T({\mathit{X}}_s)P_{R_i}({\mathit{X}}_s)P_T^{*}({\mathit{X}}_s)P_{R_j}^{*}({\mathit{X}}_s){\chi }_0^2({\mathit{X}}_s)d{\mathit{X}}_s.$$ (24)

The integration surface S can then be broken up into K separate spatial regions ${\mathit{S}}_k$. Under the piecewise-stationary definition of the target function, we assume the scattering strength ${\chi }_0^2({\mathit{X}}_s)$ to be constant across each region ${\mathit{S}}_k$. Equation (24) can then be represented by a summation over K spatial regions

$$A_{\mathit{ij}}=\sum _{k=1}^K\hspace{0.17em}\hspace{0.17em}\underset{{\mathit{S}}_k}{\iint }{P}_T({\mathit{X}}_s)P_{R_i}({\mathit{X}}_s)P_T^{*}({\mathit{X}}_s)P_{R_j}^{*}({\mathit{X}}_s){\chi }_{0,k}^{2}d{\mathit{X}}_s,$$ (25)

where the constant value of ${\chi }_0^2({\mathit{X}}_s)$ in region ${\mathit{S}}_k$ is given by ${\chi }_{0,k}^2$, which represents the variance in region ${\mathit{S}}_k$. The ${\chi }_{0,k}^2$ term can be pulled out of the integral

$$A_{\mathit{ij}}=\sum _{k=1}^K\hspace{0.17em}\hspace{0.17em}{\chi }_{0,k}^2\underset{{\mathit{S}}_k}{\iint }{P}_T({\mathit{X}}_s)P_{R_i}({\mathit{X}}_s)P_T^{*}({\mathit{X}}_s)P_{R_j}^{*}({\mathit{X}}_s)d{\mathit{X}}_s.$$ (26)

We then define the covariance of echoes from region ${\mathit{S}}_k$ received at elements i and j as

$$A_{\mathit{ij}}^{(k)}=\underset{{\mathit{S}}_k}{\iint }{P}_T({\mathit{X}}_s)P_{R_i}({\mathit{X}}_s)P_T^{*}({\mathit{X}}_s)P_{R_j}^{*}({\mathit{X}}_s)d{\mathit{X}}_s,$$ (27)

such that Eq. (26) can then be represented as

$$A_{\mathit{ij}}=\sum _{k=1}^K{\chi }_{0,k}^2{A}_{\mathit{ij}}^{(k)}.$$ (28)

Equation (28) implies that the covariance of the complex signals received at channels i and j can be expressed as a weighted superposition of constituent covariance components $A_{\mathit{ij}}^{(k)}$ from K spatially independent regions ${\mathit{S}}_k$.

Defining the full pulse echo data vector as $\mathit{s}={[s_1,s_2,\dots ,s_M]}^T$, and ${\mathit{s}}_k$ as the subset of echoes from region ${\mathit{S}}_k$, it can be shown that Eq. (28) generalizes to the full covariance matrix $\mathit{A}=⟨\mathit{s}{\mathit{s}}^H⟩$

$$\mathit{A}=\sum _{k=1}^K{\chi }_{0,k}^2{\mathit{A}}_k,$$ (29)

where ${\mathit{A}}_k=⟨{\mathit{s}}_k{\mathit{s}}_k^H⟩$ is the M × M covariance matrix of echoes from spatial region ${\mathit{S}}_k$. The full pulse echo data covariance A is the superposition of K echo covariance matrices ${\mathit{A}}_k$, weighted by variances ${\chi }_{0,k}^2$. This result, and the relationship to the scattering target models of Eqs. (6) and (22) are shown in Fig. 1.

FIG. 1.

(Color online) Models of diffuse scattering targets. (a) Spatially incoherent, stationary scattering target with constant scattering variance. The covariance A incorporates echoes from the entire target region, consistent with the scattering target model presented by Mallart and Fink (Ref. 23). (b) Piecewise-stationary scattering target. The covariance A represents a weighted superposition of covariances from each locally homogeneous component of the target region.

We now consider the covariance of the noise term. Assuming L uncorrelated sources of additive noise, Eq. (18) can be similarly represented by a superposition of covariances,

$$\mathit{N}=\sum _{l=1}^L{\beta }_l^2{\mathit{N}}_l,$$ (30)

where ${\mathit{N}}_l=⟨{\mathit{n}}_l{\mathit{n}}_l^H⟩$ is the $M\times M$ covariance matrix of additive noise source ${\mathit{n}}_l$, and N is the superposition of L noise covariance matrices ${\mathit{N}}_l$, weighted by variances ${\beta }_l^2$. Example additive noise sources may include thermal noise and superimposed wavefronts due to multipath or off-axis scattering.

Based on the spatial and noise decompositions of Eqs. (29) and (30), we consider the $M\times 1$ vector of received aperture data x as a weighted superposition of pulse echo signal vectors ${\mathit{s}}_k$ and additive noise vectors ${\mathit{n}}_l$, given by

$$\mathit{x}=\sum _{k=1}^K{\chi }_{0,k}{\mathit{s}}_k+\sum _{l=1}^L{\beta }_l{\mathit{n}}_l.$$ (31)

The M × M covariance matrix is given by

$$\mathit{R}=⟨\mathit{x}{\mathit{x}}^H⟩.$$ (32)

Substituting (31) into (32), the covariance is given by

$$\mathit{R}=⟨\left(\sum _{k=1}^K{\chi }_{0,k}{\mathit{s}}_k+\sum _{l=1}^L{\beta }_l{\mathit{n}}_l\right)\times {\left(\sum _{k=1}^K{\chi }_{0,k}{\mathit{s}}_k+\sum _{l=1}^L{\beta }_l{\mathit{n}}_l\right)}^H⟩.$$ (33)

Assuming the pulse echo signal ${\mathit{s}}_k$ and noise ${\mathit{n}}_l$ are uncorrelated, the cross terms vanish, yielding the generalized form of Eq. (19) [via Eqs. (29) and (30)]

$$\mathit{R}=⟨\sum _{k=1}^K{\chi }_{0,k}^2{\mathit{s}}_k{\mathit{s}}_k^H⟩+⟨\sum _{l=1}^L{\beta }_l^2{\mathit{n}}_l{\mathit{n}}_l^H⟩$$ (34a)

$$=\sum _{k=1}^K{\chi }_{0,k}^2{\mathit{A}}_k+\sum _{l=1}^L{\beta }_l^2{\mathit{N}}_l$$ (34b)

$$=\mathit{A}+\mathit{N}.$$ (34c)

By combining terms, Eq. (34) can be represented as the superposition of P spatial and additive noise covariances. This result is given by

$$\mathit{R}=\sum _{p=1}^P{\alpha }_p^2{\mathit{R}}_p,$$ (35)

where ${\mathit{R}}_p$ is the set of all ${\mathit{A}}_k$ and ${\mathit{N}}_l,\hspace{0.17em}{\alpha }_p^2$ is the set of all ${\chi }_{0,k}^2$ and ${\beta }_l^2$, and $P=K+L$. Equation (35) implies the measured covariance matrix R can be represented as the superposition of P constitutive covariances ${\mathit{R}}_p$ of echo data from K spatial regions ${\mathit{S}}_k$ and L additive noise components.

2. Example application: Medical ultrasound

In medical ultrasound, the total spatial region S is given by the transmit intensity distribution. Restricting the analysis to the one-dimensional case, a simple theoretical example is presented in Fig. 2. The transmit intensity distribution for a 3 MHz, 1.92 cm, 64-element linear array with rectangular apodization is shown at the top. Assuming a uniform, diffuse target, this is given by a ${\text{sinc}}^2$ function at the focus.

FIG. 2.

(Color online) Theoretical intensity distribution, correlation and covariance curves predicted at the focus for a narrowband array with rectangular transmit apodization. The superposition of the mainlobe and sidelobe covariances is equivalent to the covariance of the full PSF.

If the source distribution is then segmented into two components—chosen here as the mainlobe and sidelobe regions, respectively, delimited by the first zero crossing of the ${\text{sinc}}^2$ function—the predicted covariance and correlation from each distinct region can be determined numerically. These results are plotted at the bottom of Fig. 2, in which the covariances have been normalized to the maximum value of the covariance of the full point spread function (PSF) curve. The mainlobe and sidelobe components show distinct curves, which superimpose in the covariance domain to recover the triangle function from the full ${\text{sinc}}^2$ intensity distribution.

III. METHODS

To validate the linearity of spatial covariance presented in Eq. (35), Field II^43,44 was used to simulate radiofrequency (RF) channel data from a 64-element, 3 MHz linear array with a 50 mm transmit focus. The simulation parameters are presented in Table I. A uniform scattering phantom (20 scatterers per resolution cell) measuring 60 mm in depth, 60 mm in azimuth, and 15 mm in elevation was generated, and decomposed into three distinct imaging targets based on the nominal mainlobe width at the transmit focus with rectangle apodization, given by $\lambda z/D=1.3\hspace{0.17em}\text{mm}$, where λ is the wavelength, z is the focal depth, and D is the azimuthal aperture extent. The three segmented scattering targets were defined as follows:

• (1)The portion of the target in which all scatterers outside the mainlobe width were removed (center region).
• (2)The portion of the target in which all scatterers inside the mainlobe width were removed (outer region).
• (3)The full scattering target (i.e., the superposition of targets 1 and 2).

TABLE I.

Field II simulation parameters.

Parameter	Value
f0	3 MHz
Bandwidth	70%
fs	1 GHz
Element count	64
Element pitch	300 μm
Array footprint	1.92 × 1 cm
Transmit focus	5 cm
Transmit F/#	2.6
Transmit apodization	Rectangle, Hamming,
	inverse Hamming
Receive apodization	Rectangle

For each of the three scattering targets, a single beam with rectangular transmit apodization was transmitted on-axis (x = 0), and the received, unfocused RF echo data were collected. The sampling frequency was set to 1 GHz to minimize the potential for numerical artifacts in the simulation process. The RF echo data were downsampled to 100 MHz for processing, and focused using conventional dynamic receive processing. In-phase and quadrature (I/Q) data were generated using the Hilbert transform, and the spatial covariance $(\mathit{R})$ and correlation $(\mathit{\rho })$ matrices were calculated at each pixel along the beam. The superposition of measured covariance from the center and outer regions was compared to the measured covariance of the full scattering target. For all simulations, covariances and correlations were averaged over a 5 mm axial window. A total of 100 speckle realizations were simulated for each configuration to obtain a statistically reliable result. The covariance and correlation matrices were collapsed into 1-D functions by Eqs. (11) and (12) for display purposes. All covariance plots were normalized by the maximum of the total scattering target covariance.

Simulations were performed to evaluate the linearity of spatial covariance for the following criteria.

A. Superposition

Correlation and covariance curves of the three scattering targets were calculated at the transmit focal depth (50 mm) for each of the three segmented scattering targets. The cross-correlation and cross-covariance were also calculated between the center and outer regions.

B. Scaling

The simulations of the segmented scattering targets were repeated, but an additional scaling term was added to scale the scattering amplitude of the center region relative to the outer region. Simulations were performed using native contrasts of [–12, −6, 0, +6, +12] dB between the center and outer regions. Measurements of spatial covariance and correlation were performed for each native contrast.

C. Depth dependence

Correlation and covariance curves were calculated at regions shallow (30 mm) and deep (70 mm) to the transmit focal depth for each of the three unscaled, segmented scattering targets.

D. Transmit apodization

Simulations were repeated using Hamming and inverse Hamming transmit apodization windows, in which the latter was defined as $[1-\text{Hamming}+\text{min}(\text{Hamming})]$. Correlation and covariance curves were calculated at the transmit focus for each of the three unscaled, segmented scattering targets.

E. System noise

Analysis of the segmented scattering targets was repeated with an additional component representing additive electronic or incoherent noise. Echo data was collected from the center, outer, and full scattering targets as described above. The random noise component was calculated by bandpass-filtering white noise to the transducer bandwidth. The noise was scaled in amplitude such that the root-mean-square noise power was equal to that of echoes received from the full scattering target at each channel (0 dB channel SNR). A unique noise realization was generated for each speckle realization. Covariance and correlation curves were calculated at the transmit focal depth for the echoes from the (1) center and (2) outer region targets, (3) the random noise, and (4) the superposition of the full scattering target echoes with the random noise.

IV. RESULTS

A. Superposition

Images of the segmented scattering targets, and measured covariance and correlation curves at the transmit focus are shown in Fig. 3. For the full target region, the correlation and covariance curves show the triangle function predicted by the VCZ theorem for a rectangular aperture. Similar to the theoretical results for a narrowband transducer presented in Fig. 2, the center, mainlobe region is highly correlated, while the outer, sidelobe region decorrelates much more quickly with increasing lag. The cross-covariance of the center and outer region echoes is essentially zero across all lags. The sum of the mainlobe and sidelobe covariance functions is the triangle function, equivalent to the covariance measured from the full scattering target.

FIG. 3.

(Color online) Covariance superposition simulations. (Left) Images of segmented targets containing scatterers in the center region (defined by the $\lambda z/D$ mainlobe), scatterers outside the mainlobe region, and the full field of scatterers. Images are log-compressed and displayed on 60 dB dynamic range. (Right) Plots of the correlation and covariance curves of each region averaged across 100 speckle realizations. The superposition of the center and outer region covariances (“Center + Outer”) is equivalent to the covariance of the full target. The cross covariance of the center and outer regions (“Center/Outer”) is essentially zero.

B. Scaling

In Fig. 4(a), sample A-lines at the focal depth from one speckle realization of each target region are shown as a function of native contrast between the center and outer regions. With increasing contrast, the strength of the echoes from the outer region decreases relative to the center region.

FIG. 4.

(Color online) Simulation results demonstrating the scaling of spatial covariance. The centrally located scatterers were scaled in amplitude such that the native contrast between center and outer ranged from −12 to + 12 dB. (a) A-lines from the center, outer and full regions showing the amplitude scaling. In all cases, the superposition of the center and outer traces recover the full trace. (b) Corresponding covariances for each region, where individual central and outer regions superimpose to recover the full covariance trace across all native contrasts.

The corresponding covariances are shown in Fig. 4(b). At low contrast, the covariance from the full scattering target is nearly identical to that of the outer region. Similar behavior between the full target and center region is observed with high contrast. As the echoes from the center region constitute a larger fraction of the received signal, the covariance from the center region becomes increasingly similar to the covariance from the full target until the two nearly overlap at +12 dB contrast. Across all contrasts, the superposition of covariance from the center and outer regions yields the covariance of the full scattering target.

C. Depth dependence

Measurements of the covariance and correlation were performed at 30 and 70 mm (20 mm away from the 50 mm transmit focus), where the quality of transmit focusing is significantly degraded. These results are shown in Fig. 5. The transmit intensity is shown in (a), with overlaid contours representing the lateral −6, −12, and −20 dB beam widths normalized through depth. The A-lines from a single realization of the segmented speckle targets are shown in (b). The axial windows used to calculate each set of correlations and covariances (centered at 30, 50, and 70 mm) are highlighted in (a) and (b). The ensemble average correlation and covariance curves of the three targets are plotted for each depth in (c) and (d). The superposition of the measured center and outer covariances align with the measured covariance of the full scattering target at all measured depths.

FIG. 5.

(Color online) Simulation results for spatial covariance through depth. (a) Transmit intensity profile of a 64-element array focused at 5 cm (F/2.6) with overlaid contours of the −6, −12, and −20 dB lateral beam widths. (b) A-lines of the segmented speckle targets through depth. Ensemble average (c) correlation and (d) covariance curves over a 5 mm window at (top) 30 mm, (middle) 50 mm, and (bottom) 70 mm depth. Superposition of the center and outer region covariances is shown to be equivalent to the full region covariance at all depths.

D. Transmit apodization

Figure 6 presents the simulation results for the segmented scattering targets with varying transmit apodization schemes. The apodization profiles are shown in Fig. 6(a), with the corresponding covariances in Fig. 6(b). For each apodization case, the superposition of measured covariance from the center region and the outer region yields the measured covariance from the full scattering target.

FIG. 6.

(Color online) Linearity of covariance for rectangle, Hamming, and inverse Hamming transmit apodization. (a) Individual transmit apodization profiles. (b) Covariance profiles from the center, outer, and full scattering targets, as well as the superposition of the center and outer covariances for each transmit apodization, shown to be equivalent to the full covariance for each apodization.

E. System noise

Correlation and covariance curves for the system noise simulations are plotted in Fig. 7. The noise is spatially incoherent, and is represented by a delta function at lag zero. The center and outer region correlation curves are unchanged from Fig. 3. The full scattering target correlation with the added noise shows a triangle function (predicted by the VCZ theorem) plus a delta function at lag zero, consistent with prior results presented by our group.^31,45 The morphology of the center and outer region covariance curves is also unchanged, though the relative amplitudes are reduced due to the normalization. The three individual covariance curves superimpose to match the measured covariance of the full scattering target with added noise.

FIG. 7.

(Color online) Correlation and covariance superposition studies in the presence of random noise (0 dB channel SNR) averaged across 100 speckle realizations. The incoherent noise is represented by a delta function at lag zero. Superposition of the center, outer, and noise covariances is equivalent to the covariance of the full target.

V. DISCUSSION

The results presented in Sec. IV demonstrate the application of the van Cittert-Zernike theorem to a piecewise-stationary diffuse scattering model. These results reflect an important insight: the covariance of echo data from stochastic targets can be described as the superposition of covariances from distinct spatial regions in the imaging field. Superposition was validated though simulation studies in which the spatial regions corresponded to inside and outside the nominal mainlobe beam width.

A. Superposition of covariance

Superposition (Fig. 3) and scaling (Fig. 4) were demonstrated at the transmit focus across a range of relative scattering amplitudes, and in the presence of random noise. Superposition was also demonstrated through depth, both shallow and deep to the transmit focus (Fig. 5), where imperfect transmit focusing causes the predicted spatial covariance to deviate from the triangle function predicted at the focus. This relationship is described by the quadratic phase term in the Fresnel diffraction integral in Eq. (3).

Away from the transmit focus, the covariance of the outer, sidelobe region largely mirrors the behavior of the full scattering target, while the mainlobe covariance remains relatively constant through depth. This is likely due to the limited lateral extent of the mainlobe region, where the transmit focal error is insignificant for such a spatially narrow region. Qualitatively, these results suggest the covariance functions of the mainlobe and sidelobe regions for a rectangular aperture may become more orthogonal away from the transmit focus.

The choice of transmit apodization dictates the lateral beam pattern, and the predicted spatial covariance of the backscattered echoes. Simulation studies performed using rectangle, Hamming, and inverse Hamming apodization profiles demonstrated robustness of the theory to a range of transmit intensity distributions, shown in Fig. 6. As was the case with measurements away from the transmit focus, the covariance of the constituent spatial regions also may become more orthogonal with particular transmit apodization profiles.

B. Implications for imaging

An ideal ultrasonic imaging system would only interrogate the on-axis component of the target. However, diffraction effects and a finite pulse length dictate a geometric spreading of acoustic energy throughout the field. The received echoes incorporate scattering from the entire isochronous volume.⁴⁶ Assuming a stationary, uniform scattering target, the classical VCZ theorem relates the transmit intensity distribution to the covariance of the received echoes.²³

By expanding this model to incorporate piecewise-stationary statistics, the spatial covariance can be further decomposed into contributions from distinct spatial regions of the field. While the classical VCZ theorem describes the second order statistics of backscatter from the entire source plane, the results presented in this work imply a framework to spatially localize backscatter energy based on known statistical properties.

For the purposes of imaging, an intuitive decomposition may be on- and off-axis echo contributions (as presented in this work), but other models may also be valuable, as the piecewise-stationary approach can be used to represent nearly any stochastic target distribution. While these results were presented in the context of ultrasound, this insight may have applications in a number of coherent imaging modalities where diffuse scattering environments are nearly universal.

We have recently presented an estimation-based, ultrasonic imaging method derived from the insights of this work, Multi-covariate Imaging of Sub-resolution Targets, or MIST.³⁹ Using the framework of Eq. (35), the echo data covariance is assumed a linear superposition of covariance models corresponding to (1) on-axis, (2) off-axis, and (3) incoherent noise contributions. MIST seeks to estimate the contribution of each model to the measured echo data covariance, forming directly from the on-axis component. Images demonstrated consistent improvements in contrast-to-noise ratio and speckle SNR, while preserving both native target contrast and resolution. These results were consistent across simulation, phantom, and in vivo datasets.

C. Relation to literature covariance models

The general concept of covariance decomposition is common in the statistical signal processing literature.^47,48 However, this decomposition almost universally involves the superposition of signal and noise components, given by $\mathit{R}={\mathit{R}}_s+{\mathit{R}}_n$ [this assumption is used in Eq. (19) of this work to incorporate additive noise into the model], where noise is typically described by spatially white statistics or discrete interference sources. Many methods in the ultrasound and sonar literature have incorporated similar frameworks representing covariance or correlation as a superposition of signal and noise.^31,45,49,50

Alternative techniques decompose covariance using various forms of principal component analysis. A classical example is the multiple signal classification (or MUSIC) algorithm, which takes advantage of the orthogonality of the signal and noise subspaces to optimize a set of steering vectors for direction-of-arrival estimation.⁵¹ Similar approaches have been described in the ultrasound literature involving eigendecomposition or singular value decomposition.^52–54

A related approach, the minimum variance distortionless response (MVDR) beamformer, seeks to minimize the total beamformer output power subject to a distortionless constraint in the look direction.^55,56 The result is a directional filter that uses adaptive array weights to suppress off-axis contributions to the received array data. While the MVDR approach and the implications of this work share a similar goal of suppressing off-axis contributions, MVDR does not involve a decomposition of covariance matrices into distinct spatial contributions.

To the extent that covariance has been modeled as a superposition of components in the literature, to our knowledge none have presented a decomposition based on distinct spatial regions in the field. Many methods have been developed for direction-of-arrival estimation or noise suppression applications, but these primarily concern discrete targets without an underlying statistical basis. The approach presented in this work, however, is derived from fundamental physical principles to describe the statistical properties of stochastic targets.

D. Limitations and future work

While this work assumes a piecewise-stationary, incoherent scattering function, many imaging environments contain a combination of stochastic and deterministic structures, such as specular or point targets, or reflective boundaries, which violate the assumption of incoherence. These effects are not captured by the stochastic target model, and may result in non-zero cross-covariances between the echoes from each region ${\mathit{S}}_k$. Additionally, sources of wavefront degradation such as aberration or multiple scattering will also result in deviations of the beam profile from the theoretical predictions, which may be challenging to model and estimate for the purposes of imaging.^29,30,42

The covariance models presented in Fig. 2 (and used in MIST³⁹) represent the simple case derived from the Fraunhofer diffraction integral [Eq. (4)], valid only within the depth of field for a focused transmission. However, it is important to note that the results of Eq. (35) can be applied to more complex models of the field, which may depend on depth, frequency, or other parameters. The incorporation of more complex models may be a target of future work.

The estimation of the covariance matrix is an inherently noisy measurement, such that a single-sample estimate is unlikely to reflect the true underlying statistics. Across a large ensemble of measurements, the cross-covariance terms average out to zero, and the covariance estimates approach the theoretical predictions. Accordingly, the plots presented in this work represent an average over a large number of realizations to capture the statistics across that large ensemble. In imaging applications, a small degree of spatial averaging may be beneficial to better estimate the true covariance matrix.

The segmentation of the target region in this work used the nominal mainlobe beam width of the transmitting aperture as a spatial cutoff, with two constituent components. The model, however, is not restricted to two components, to the $\lambda z/D$ cutoff, or to a laterally symmetric model. Future work may explore the limits of the theory with additional spatial components, though recent results with MIST imaging have demonstrated even this simple decomposition consistently out-performs conventional imaging methods.³⁹

VI. CONCLUSION

The van Cittert-Zernike theorem was applied to a piecewise-stationary model of diffuse scattering targets, demonstrating the spatial covariance of echo data as the linear superposition of covariances from distinct spatial regions. The model was shown to be robust to target scattering amplitude, transmit focal depth, transmit apodization, and random noise. These results demonstrate the covariance of backscatter from diffuse targets can be decomposed into contributions from regions of interest, providing a framework to improve performance in a number of imaging applications. While the validation studies were performed in the context of ultrasound, these results are generally applicable to coherent imaging modalities involving stochastic targets.