Self-Portrait of the Focusing Process in Speckle: II. Gouy Phase Shift for Defocus Correction and Pixel Depth Reassignment

TL;DR

The paper tackles axial aberrations in ultrasound caused by spatial variations in speed-of-sound within heterogeneous media. It introduces ultrasound matrix imaging (UMI) and analyzes the self-portrait of the focusing process by examining the de-scanned focused-reflection matrix, isolating incoherent and coherent reflection point spread functions (RPSFs). By exploiting the Gouy phase shift observed in the coherent RPSF, it proposes a precise, local optimization of the speed-of-sound at each image patch, enabling depth reassignment of scatterers and axial aberration compensation. The approach is validated on a tissue-mimicking phantom, supported by numerical simulations, and demonstrated on in-vivo liver data, showing improved contrast, resolution, and the potential for quantitative depth measurements. The work sets the stage for integrating depth-averaged speed-of-sound mappings with refraction-aware beamforming and complementary imaging modalities to enhance diagnostic utility of ultrasound and other echo-based systems.

Abstract

This is the second article in a series of three dealing with the exploitation of speckle for aberration correction and reverberation compensation in reflection imaging. When probing heterogeneous media with waves, we have to cope with multi-scale fluctuations of the wave velocity. On the one hand, short-scale heterogeneities induce back-scattered echoes whose random interference generate a speckle pattern on the beamformed image. On the other hand, large-scale fluctuations of the wave-velocity can distort the focused wave-fronts, resulting in aberrations on the same image. In this paper, we show how the self-portrait of the wave evolves as a function of the speed-of-sound model. Strikingly, a Gouy phase shift is observed when the speed-of-sound model is optimal. This particularly sensitive feature enables: (i) an optimization of the speed-of-sound model for each pixel of the image; (ii) a local and fine compensation of defocus across the field-of-view, thereby compensating for most aberrations in the image. Experiment in a tissue-mimicking phantom and numerical simulations are first presented to validate our method. It is then applied to in-vivo liver data of a difficult-to-image patient. The speed-of-sound optimization allows an axial compensation of aberrations and a depth-reassignment of each singly-scattered echo to the actual position of the associated scatterer. As distance measurement is often critical for diagnosis, such a wave speed optimization can be crucial for ultrasound but also for any other imaging methods based on the principle of echo-location.

Figures (8)

• Figure 1: Impact of an incorrect speed-of-sound model in ultrasound imaging. (A) The acquisition of the reflection matrix consists in insonifying the medium with a set of plane waves emitted by the ultrasonic probe. The recorded wave-fronts are stored in the reflection matrix. The contribution of one scatterer located at depth $z_t$ is highlighted in green. For sake of simplicity, the wave velocity $c$ is considered as homogeneous. (B) The numerical focusing process can be seen as a fictive time reversal experiment in a medium of wave velocity $c_0$. If $c_0=c$, the time-reversed wave-front back-focuses exactly at the initial scatterer location. If $c_0\neq c$, a mismatch exists between the focusing plane ($z_f=cz_t/c_0$), the isochronous plane ($z_t=c t/2$) and the imaging plane ($z_0=c_0 t/2$). (C) Experimental configuration: A linear array of transducers is placed on top of an ultrasound phantom. (D) Scheme of the phantom with nylon rods (white), random distribution of unresolved scatterers (gray) and a more echogene cylinder displaying a stronger concentration of scatterers (light gray). (E, F) Corresponding ultrasound image for $c = c_0 = 1540$ m.s$^{-1}$ and $c\neq c_0=1800$ m.s$^{-1}$, respectively.
• Figure 2: The de-scan focused basis. (A) Schematic view of the input and output focal spots. (B) Reflection matrix $\mathbf{R}=[R(x_\textrm{in},x_\textrm{out},t,c_0)]$ expressed in the conventional focused basis. (C) Reflection matrix $\mathbf{R}_{\mathcal{D}}=[R(x_{\textrm{in}},t,\Delta x,c_0)]$ expressed in the de-scanned basis, with $\Delta x=x_\textrm{out}-x_\textrm{in}$. The sub-panels B$_2$ and C$_2$ are examples of reflection matrices sketched in sub-panels B$_1$ and C$_1$, respectively. They correspond to the tissue mimicking phantom experiment (Table 1) for time $t=32\mu$s and speed-of-sound $c_0=1540$ m.s$^{-1}$.
• Figure 3: Self-portrait of the focusing process. ($A$) Matrix imaging consists in splitting the input and the output focusing points during the beamforming process. The focused reflection matrix allows the monitoring of the focusing process with respect to the wave velocity model $c_0$. ($B$) Such a matrix can be expressed in a de-scanned basis in order to provide the dependence of the RPSF shown here in amplitude for three speckle spots of the ultrasound image. ($C$) The three speckle grains considered in panel B and the area $\mathcal{P}$ considered for the local averaging of the RPSF in panels D-H are superimposed to the ultrasound image of the phantom. ($D$) Incoherent RPSF. ($E$) Amplitude of the coherent RPSF. ($F$) Imaginary part of the coherent RPSF. ($G$) Phase of the coherent RPSF as a function of $c_0$ at $\Delta x=0$. ($H$) Magnitude of the incoherent RPSF (black line), of the coherent RPSF (blue line) and its real part (dashed blue) as a function of $c_0$ at $\Delta x=0$. Spatial averaging is here performed with a window of size $(p_x,p_t)=(10$ mm, $1.3$$\mu$s) centered around $(x,t)=(0$ mm, $43$$\mu$s).
• Figure 4: Numerical validation of defocus compensation. (${A}, {B}$) Simulated speed-of-sound distributions $c(\mathbf{r})$. (C, D) Optimized wave velocity $\hat{c}(\mathbf{r})$. (${E}, {F}$) Estimation of the local speed-of-sound map ${c}(\mathbf{r})$ with each pixel reassigned to its estimated position. (${G}, {H}$) Original ultrasound image. (${I},{J}$) Corrected image with each pixel reassigned to its estimated position.
• Figure 5: Speed of sound optimization in the liver experiment. (A, B) Original and optimized ultrasound images, respectively. Both images are normalized by the global maximum between the two images and are displayed along the same depth axis, which is estimated with a constant speed of sound ($c_0=1540$ m/s). (B,C) Zoom on specific areas of the field-of-view containing either muscle fibers or veins before and after optimization, respectively. Subscripts "1" and "2" refer to two different areas of the field of view. (E,F) Focused reflection matrix corresponding to $t=90.9$$\mu$s ($\rho_0=c_0t/2 \sim70$mm) before and after optimization respectively. (G) Incoherent RPSF before (red curve) and after correction (green curve).