On the phase aberration estimation using common mid-angle correlations

TL;DR

The paper addresses how common-mid-angle correlation phases in ultrasound relate to phase aberrations in speckle-dominated tissue. It develops a rigorous speckle-based theory that extends the point-reflector linear relation to diffuse scattering, providing a closed-form expression for the expected correlation and a model for correlation-phase fluctuations. The analysis reveals that correlation-phase variance scales with coherence loss, increasing approximately linearly with the square of the correlation phases, and validates these predictions with experimental measurements in a tissue-mimicking phantom. This work establishes a theoretical accuracy limit for common-mid-angle correlation phases and sets benchmarks for evaluating advanced aberration-estimation methods in speed-of-sound imaging.

Abstract

Phase aberrations, despite degrading ultrasound images, also encode valuable information about the spatial distribution of the speed of sound in tissue. In pulse-echo ultrasound, we can quantify them by exploiting speckle correlations. Among existing strategies, correlations between steered acquisitions that share a common mid-angle have proven particularly effective for inferring the speed of sound. Their phases can be linearly related to the phase aberrations undergone by both the incident and reflected wavefronts. This relationship has so far been demonstrated only through geometric arguments based on point reflectors. Here, we develop a rigorous theoretical formalism that extends this relationship to the speckle regime, completing the previously established linear model and clarifying its underlying assumptions. More importantly, we build on this formalism to analyze correlation-phase fluctuations arising from aberration-induced speckle decorrelation. The analysis reveals that phase variance is governed by the relative loss of coherence, which increases approximately linearly with the square of the correlation phases. Local correlation-phase estimates therefore become increasingly uncertain as their magnitude grows. Experimental measurements in a uniform tissue-mimicking phantom show excellent agreement with the predicted variance. Beyond providing a theoretical basis for advancing speed-of-sound imaging, this formalism establishes the accuracy limit of common-mid-angle correlation phases, offering a benchmark for evaluating more advanced aberration-estimation techniques.

Figures (2)

• Figure 1: Example of reflectivity images reconstructed from single transmit-receive plane-wave measurements. A coherently compounded image is shown at the top for reference. Data were acquired in a uniform tissue-mimicking phantom (see Sec. \ref{['sec:experiment']}). Each row corresponds to a pair of transmit and receive propagation directions (solid arrows) that share a common mid-angle (dashed lines). The images are spatially invariant along the direction perpendicular to $\hat{\mathbf{k}}_\text{m}$; therefore, when $\hat{\mathbf{k}}_\text{m}$ is unchanged, the images remain well correlated.
• Figure 2: Schematic of phase-aberration estimation using common mid-angle correlations. (a) A discrepancy between the true travel times $t$ and assumed times $t_0$ displaces the location of reconstructed echoes $r$ from their true positions $x$ along the mid-angle direction $\hat{\mathbf{k}}_\text{m}$. The shift $\delta x$ depends on the travel-time error $\Delta t$ (phase aberrations) and the angle difference $\varphi_\text{d}$ between the transmit and receive directions. (b) Different acquisitions with the same mid-angle yield different echo displacements. (c) The correlation phase $\phi(r)$ corresponds to the phase difference of the reconstructed echoes, evaluated at the peak location of their PSF product.