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

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.

💡 Research Summary

This paper introduces a novel framework for quantitatively mapping the wave‑speed (sound‑speed) distribution inside complex media using only reflection‑matrix measurements. Conventional reflection imaging, which relies on back‑scattered echoes from short‑scale heterogeneities, provides only qualitative reflectivity maps and fails to capture large‑scale variations of the propagation speed. The authors overcome this limitation by exploiting matrix imaging, a technique that synthesises independent transmission and reception focal spots to create virtual transducers throughout the medium. By focusing the recorded signals numerically with an assumed speed model, a focused reflection matrix is obtained; its off‑diagonal energy spread encodes wave‑front distortions caused by speed mismatches.

A “focusing quality” metric F is defined as the ratio of the average diagonal (focused) energy to the total back‑scattered energy. When the assumed speed model matches the true medium, focusing is optimal and F reaches its maximum. Crucially, the authors treat F as a data‑driven loss function and perform gradient ascent on the speed distribution c(x,z) to maximise it.

Wave propagation is modelled as a differentiable multi‑layer forward model built from alternating free‑space propagation operators and diffraction phase‑screen layers (the split‑step Fourier method). Each phase‑screen represents lateral refractive‑index fluctuations δn(x,z) = n(x,z) – ⟨n⟩x, while the free‑space layers use the laterally averaged index ⟨n⟩z. This architecture maps naturally onto modern automatic‑differentiation frameworks (PyTorch, JAX), allowing the entire focusing pipeline to be expressed as a neural‑network‑like graph whose parameters are the phase‑screen values. The gradient ∇cF is computed via the chain rule, and the speed map is updated iteratively. To mitigate the ill‑posed, highly non‑linear nature of the problem, the authors incorporate total‑variation regularisation, spatial filtering, variable splitting, and Lagrangian relaxation.

The methodology is validated through three experimental settings.

1. 2‑D tissue‑mimicking phantom – The phantom contains shallow (≈10 mm) and deep (≈40 mm) sound‑speed inclusions as well as anechoic regions. Using a uniform speed assumption (c = 1480 m/s) yields blurred focal spots and distorted B‑mode images. After optimisation, the focused reflection matrices collapse onto their diagonals, indicating diffraction‑limited focusing, and the recovered speed map reproduces the inclusions with a contrast of ~100 m/s, outperforming previous approaches.

2. 3‑D matrix‑probe experiment – A 1024‑element matrix array images a phantom placed behind a layer of pork fat and muscle. The raw reflection image cannot separate the two tissue types. Optimisation yields a 3‑D speed map that clearly distinguishes fat (≈1520 m/s) from muscle (≈1560 m/s). Corrected reflection images show restored point‑spread functions and improved contrast, although residual multiple‑scattering artefacts remain, highlighting the current forward model’s limitation to long‑range speed fluctuations.

3. In‑vivo breast ultrasound – Retrospective analysis of anonymised clinical scans demonstrates that malignant and benign lesions exhibit distinct sound‑speed signatures (≈20–30 m/s difference). After speed‑map correction, B‑mode images display sharper lesion boundaries and reduced speckle artefacts, suggesting that quantitative sound‑speed imaging could complement conventional B‑mode and shear‑wave elastography for breast lesion characterization.

The authors acknowledge that the present split‑step model does not fully capture complex scattering paths induced by sub‑wavelength heterogeneities; incorporating higher‑order Born series or more sophisticated multiple‑scattering models would improve fidelity but at the cost of increased computational burden and potential convergence issues.

Overall, the work demonstrates that by treating wave physics as a trainable deep network and using the intrinsic focusing quality as a self‑supervised loss, one can retrieve large‑scale speed maps from reflection data alone. This physics‑based learning approach is modality‑agnostic and can be extended to optics, seismology, and any wave‑based imaging where a reflection matrix is measurable, opening new avenues for quantitative, non‑invasive diagnostics.