Physics-Based Learning of the Wave Speed Landscape in Complex Media

TL;DR

The paper tackles the difficulty of recovering large-scale wave-velocity variations from reflection ultrasound data, where conventional methods mainly capture local reflectivity. It introduces Differential Matrix Imaging (DMI), a learning-based framework that models wave propagation as a differentiable multi-layer forward process built from cascaded diffraction phase screens using a split-step Fourier forward model, enabling gradient-based optimization of the speed-of-sound map from reflection data. The focusing quality metric $F(c)$ measures how well numerical focusing aligns energy along the diagonal of the focused reflection matrix, and this metric guides the gradient ascent to update the velocity model $c(\mathbf{r})$ via backpropagation. Validated on tissue-mimicking phantoms, ex vivo tissues, and in vivo breast data, DMI yields quantitative speed-of-sound maps, corrects aberrations, and enhances tissue discrimination, with broad potential to generalize to other wave modalities and imaging contexts such as diffraction tomography and seismic velocity tomography.

Abstract

Wave velocity is a key parameter for imaging complex media, but in vivo measurements are typically limited to reflection geometries, where only backscattered waves from short-scale heterogeneities are accessible. As a result, conventional reflection imaging fails to recover large-scale variations of the wave velocity landscape. Here we show that matrix imaging overcomes this limitation by exploiting the quality of wave focusing as an intrinsic guide star. We model wave propagation as a trainable multi-layer network that leverages optimization and deep learning tools to infer the wave velocity distribution. We validate this approach through ultrasound experiments on tissue-mimicking phantoms and human breast tissues, demonstrating its potential for tumour detection and characterization. Our method is broadly applicable to any kind of waves and media for which a reflection matrix can be measured.

Figures (8)

• Figure 1: Differential matrix imaging.(A) an ultrasound acquisition is performed with an ultrasound probe. (B) Raw data is gathered in a 3D tensor, indexed by emission (Tx), reception (Rx) and time ($t$). (C) A numerical focusing algorithm focuses the signals at point $\color{mblue}\boldsymbol{r}_{in}$ in emission and $\color{morange}\boldsymbol{r}_{out}$ in reception according to a velocity model (H). (D) The repetition of this operation for different depths and lateral positions yields focused reflection matrices. From these matrices, we obtain either (E) a reflection image or (F) a focusing quality computed by averaging the energy on the diagonal, divided by the total energy of reflection matrices. (G) The focusing quality gradient with respect to the velocity model is then computed to update the current velocity model, in a gradient ascent scheme. (I) An initial guess of"' the velocity model (often chosen as uniform) is required to start the optimization process.
• Figure 2: Multi-layer network as a forward model. The propagation of incident and reflected waves is modelled as a succession of diffraction phase screens, $\exp \left [ j 2\pi f \delta n(x,z_i) \delta z / c_0\right ]$, and free-space operators $\mathbf{T}(\delta z,f;c_0/\bar{n}(z_i))$ in homogeneous layers of thickness $\delta z$ and refractive index $\bar{n}(z_i)$. This forward model takes the form of a multi-layer network.
• Figure 3: Ultrasound aberration correction and speed of sound imaging in a 2D in vitro experiment. Data obtained from ali2023sound. (A) Experiment performed in ali2023sound, in which an ultrasound probe is positioned on top of a homemade tissue-mimicking phantom with speed of sound inclusions. (B) Aberrated reflection image obtained with a uniform speed of sound hypothesis of $c=1480$ m.s$^{-1}$. Two zoomed sections are displayed on the right. The dynamic range of the B&W scale is of 50 dB. (C) Examples of focused reflection matrices obtained at initialization and (D) at convergence, at depths of 20 mm, 30 mm and 40 mm, normalized by their maximum. (E) Corrected reflection image obtained at convergence with the speed of sound map displayed in panel F (see Movie S1). (F) Obtained speed of sound map (overlaid on the corrected reflection image), color coded as a function of the speed of sound expressed in m.s$^{-1}$.
• Figure 4: Speed of sound imaging of ex vivo tissues in a 3D configuration. Experiment performed in bureau2023three by imaging a tissue mimicking phantom through muscle and fat with a 1024 transducers matrix probe. Data available at bureau2023three_dataset. (A) Drawing of the experiment performed in bureau2023three. (B) Corrected reflection image obtained with the 3D speed of sound map displayed in D. (C) Examples of focal spots estimated with the focused reflection matrices, at initialization (red) and after correction (green), for depths $z=$30 mm and 40 mm. (D) Obtained 3D speed of sound map, color coded as a function of the speed of sound (in m.s$^{-1}$). (E) Horizontal slices of the 3D speed of sound map, at 5 mm, 10 mm, 15 mm, 30 mm and 45 mm. The speed of sound is displayed in m.s$^{-1}$.
• Figure 5: Sound speed imaging applied to breast ultrasound. (A) Schematic drawing of the acquisition, in which an ultrasound probe is positioned on top of breast tissues. (B, D, F, H, J) Example of reflection images obtained after correction (see Movies S3-S7 for comparison between initial and corrected images). The dynamic range of the B&W scale is of 55 dB. (C, E, G, I, K) Estimated sound speed maps overlaid on the associated reflection images. Speed of sound maps are color-coded from 1420 m.s$^{-1}$ (blue) to 1600 m.s$^{-1}$ (red). White arrows indicate on each image invasive carcinoma (malignant) while orange arrows correspond to benign masses.