Self-Portrait of the Focusing Process in Speckle: III. Tailoring Complex Spatio-Temporal Focusing Laws To Overcome Reverberations in Reflection Imaging

TL;DR

This work tackles reverberation in reflection imaging by extending matrix-based focusing to the frequency domain and introducing iterative phase-reversal to extract a dispersive phase law $\phi_H(f)$, yielding a broadband impulse response $H(\tau)$ that temporally recompresses echoes. Building on this, the authors develop a spatio-temporal correction via a frequency-aware distortion matrix $D_{kx}(z,f)$ and a local transmittance $\overline{T}_{\mathcal{A}}(k_x,f)$, enabling phase-conjugated focusing $R'(\Delta x, r_{in}\in\mathcal{A},\tau)$. The approach is demonstrated experimentally in a tissue-mimicking phantom behind a reverberating Plexiglas plate and then applied to a head phantom, achieving up to ~10 dB improvements in contrast and substantial sharpening of focal spots, while revealing limitations due to skull heterogeneity and limited isoplanicity in speckle. The results suggest a promising, post-processing-compatible pathway to mitigate reverberations in transcranial ultrasound and potentially extendable to optical and seismic imaging, where multi-channel measurements permit a reflection-matrix formulation. Overall, the paper presents a rigorous, frequency- and space-time-resolved framework for tailoring focusing laws to complex media and paves the way for noninvasive, high-resolution deep imaging in challenging environments.

Abstract

This is the third article in a series of three dealing with the exploitation of speckle for imaging purposes. In complex media, a fundamental limit is the multiple scattering phenomenon that completely blurs the imaging process in depth. Matrix imaging can provide a relevant framework for solving this problem. As it proved to be an adequate tool for probing reverberations in speckle [E. Giraudat et al., Part I], we will show how it can be used to tailor complex spatio-temporal focusing laws to monitor the interference between the multiply-reflected paths and the ballistic component of the wave-field. To do so, we extend the distortion matrix concept to the frequency domain. An iterative phase reversal process operated from the space-time Fourier space is then used to compensate for reverberations and optimize both the axial and transverse resolution of the confocal image. Here, we first present an experimental proof-of-concept consisting in imaging a tissue-mimicking phantom through a reverberating plate before outlining the potential and the limits of this strategy for transcranial ultrasound and beyond.

Figures (8)

• Figure 1: Artifacts caused by reverberations in ultrasound. a A medium is insonified by a plane wave. After passing through a reverberating layer, the plane wave is split into a tail of multiply-reflected waves. b The wave reflected by a strong diffuser in the medium produces a set of echoes, that are multiplied again when they pass back through the reverberating layer. c During the image reconstruction stage using a homogeneous velocity model, the multiple reflected echoes, associated with longer times-of-flight, are back-propagated to overestimated depths, giving rise to ghost images, or replicas, of the strong scatterer. Only a dispersive focusing law could correct for these artifacts. d, Example of a confocal image obtained on an ultrasound head phantom containing a bright target, appearing tripled due to reverberations induced by the skull.
• Figure 2: Extraction of a frequency dispersion law in speckle. a Acquisition of the reflection matrix $\mathbf{R}_{\mathbf{u}\bm{\theta}}(t)$: (i) each incident plane wave is multiply-reflected by the reverberating layer before being scattered by sub-resolved heterogeneities; (ii) the reflected waves also undergo reverberations before being recorded by the ultrasonic probe. b Confocal delay-and-sum beamforming process at each point $\mathbf{r}$ in the medium. The temporal dispersion of the echoes dispersed is captured by investigating the confocal signal $\mathcal{I}_0(\mathbf{r},\tau)$ as a function of time lapse $\tau$ with respect to the expected ballistic time. c An IPR analysis conducted on the correlation matrix between the backscattered echoes at each frequency leads to the synthesis of a coherent virtual source and the access to the associated reverberated signal $H(\tau)$ that accumulates the axial aberrations of the incident and reflected paths. d Time reversal of this signal yields a wave-front that can compensate for the axial aberrations of the reflection matrix, leading to a temporal recompression of the echoes associated with each point $\mathbf{r}$ of the medium.
• Figure 3: Dispersive compensation of reverberations in speckle.a Sketch of an incident focused wave-field in presence of a reverberating layer. The ballistic and multiply-reflected components focus at different depths shifted with respected to their respective isochronous volume. b Confocal image obtained for the homogeneous model $c_0 = 1540$ ms in the phantom - Plexiglas experiment. c Modulus and phase of the frequency dispersion law $\bar{H}(f)$ (Eq. \ref{['transfer']}) extracted by IPR in the speckle on the blue area in panel b. d Normalized real part of the temporal dispersion law $H(\tau)$ (Eq. \ref{['psf']}), highlighting a first peak at $\tau_{0,0}^{*} \sim -12.4$$\mu$s associated with the ballistic echo and secondary peaks associated with multiple reflections. e Corrected confocal image $\mathcal{I}_1(\mathbf{r},\tau=0)$ obtained by applying a dispersive focusing law (Eqs. \ref{['correction']} and \ref{['I1']}). f Sketch of the dispersive focusing law tailored and applied in post-processing to make interfere constructively each multiply-reflected path with the ballistic component of the wave-field.
• Figure 4: Local spatio-temporal correction of aberrations in the phantom-plexiglass experiment.a Confocal image $\mathcal{I}_0$ obtained for the homogeneous model $c_0 = 1540$ m.s$^{-1}$ in the phantom - Plexiglas experiment. b Image $\mathcal{I}_1$ (Eqs. \ref{['correction']}-\ref{['I1']}) obtained after global frequency compensation of reverberations. c Image $\mathcal{I}_2$ (Eqs. \ref{['correct']}-\ref{['I2']}) after spatio-temporal refocusing of ultrasound data. d Transverse cross-section of each image at the depth corresponding to the horizontal line of scatterers in the phantom. e Longitudinal cross-section of each image at the lateral position corresponding to the vertical line of scatterers in the phantom. In panels d and e, the original image $\mathcal{I}_0$ is displayed as a red continuous line while the corrected images $\mathcal{I}_1$ and $\mathcal{I}_2$ correspond to the yellow and green continuous lines, respectively.
• Figure 5: Extraction of a spatio-temporal focusing law in speckle. a Acquisition of the reflection matrix $\mathbf{R}_{\mathbf{u}\bm{\theta}}(t)$: (i) each incident plane wave is multiply-reflected by the reverberating layer before being scattered by sub-resolved heterogeneities; (ii) the reflected waves also undergo reverberations before being recorded by the ultrasonic probe. b A delay-and-sum beamforming process decoupling the input and output focal spots is applied to $\mathbf{R}_{\mathbf{u}\bm{\theta}}(t)$ to compute a time focused reflection matrix $\mathbf{R}_{xx}(z,\tau)$ at each depth $z$. A de-scan operation followed by a 2D Fourier transform yields the frequency-dependent distortion matrix $\mathbf{D}_{kx}(f)$ in the plane wave basis. c A correlation matrix is computed between the distorted wave-fields associated with each virtual source $\mathbf{r}_{\textrm{in}}$ belonging to the selected area $\mathcal{A}$ centered around the point $\mathbf{r}_{\mathcal{A}}$. An IPR analysis conducted at each spatio-temporal frequency leads to the synthesis of a coherent virtual source and the estimation of the reverberated wave-field $\overline{T}(k_x, f)$ in the Fourier space, or equivalently, to ${T}(\Delta x, \tau)$ in real space. d Time reversal of this spatio-temporal wave-field yields a focusing law that can compensate for reverberations, leading to a spatio-temporal recompression of the echoes associated with each point $\mathbf{r}$ of the medium.