Self-Portrait of the Focusing Process in Speckle: I. Spatio-Temporal Imaging of Wave Packets in Complex Media

This is the first article in a series of three dealing with the exploitation of speckle for imaging purposes. Speckle is the complex interference wave-field produced by a random distribution of un-resolved scatterers. In this paper, we show how these scatterers can be used as virtual microphones to monitor the spatio-temporal propagation of a wave-packet inside the medium. To do so, the concept of matrix imaging is particularly useful. It consists in decoupling the location of the transmitted and received focal spots in a standard beamforming process. By scanning the wave-field with the output focal spot that then acts as a virtual transducer, one can image the spatio-temporal evolution of the wave-packet inside the medium. This unique observable will allow us to highlight the imperfections of the focusing process, in particular the defocus and reverberations induced by a strong aberrating layer. As a proof-of-concept, we will consider ultrasound experiments on tissue-mimicking phantoms. In the next two papers, we will show how this observable can be leveraged to compensate for these phenomena that hamper wave focusing and imaging in all fields of wave physics. Our method is indeed broadly applicable to different types of waves beyond ultrasound for which multi-element technology allows a reflection matrix to be measured.

💡 Research Summary

This paper introduces a novel matrix‑imaging framework that turns the ubiquitous speckle field—normally regarded as a source of image degradation—into a diagnostic tool for visualizing the internal propagation of a wave packet in a complex medium. The authors focus on ultrasound experiments but emphasize that the method applies to any wave system where a multi‑element array can record a reflection matrix.

The core idea is to acquire a full reflection matrix R(u_out, θ_in, t) by transmitting a set of plane waves at many incidence angles (θ_in) and recording the back‑scattered signals on each transducer element (u_out). Conventional delay‑and‑sum beamforming of this matrix yields a “confocal” image, which assumes that the speed‑of‑sound model c₀ used in the beamforming matches the true medium speed. When this assumption fails, or when a strongly reflecting layer (reverberation) is present, the confocal image suffers axial shifts, loss of resolution, and ghost artifacts.

To overcome these limitations, the authors decouple the transmit and receive focusing points. By digitally focusing the transmitted field at an arbitrary point r_in and, independently, digitally focusing the received field at another point r_out, they construct a five‑dimensional “time‑focused reflection matrix” R(r_out, r_in, τ), where τ is a time offset relative to the expected ballistic travel time between the two virtual points. This matrix can be interpreted as the impulse response between two virtual transducers placed inside the medium.

A further change of variables, Δr = r_out − r_in, yields a “de‑scanned” matrix R(Δr, r_in, τ). In this representation each column corresponds to a fixed virtual source r_in, and the rows map the spatial response of all surrounding virtual receivers as a function of the relative displacement Δr and the additional delay τ. Visualizing these columns in Cartesian coordinates (Δx, Δz) provides a spatio‑temporal movie of how the wave packet emanates from the source, propagates through the heterogeneous medium, and is recorded by the receivers.

Two experimental configurations are examined. In the first, a tissue‑mimicking phantom with a known speed of sound (~1542 m/s) is imaged without any additional layer. The resulting R(r_out, r_in, τ) shows a single dominant peak at τ = 0, confirming that the virtual source and receiver are correctly aligned with the iso‑chronous volume and that the wave packet propagates without distortion. In the second configuration, a Plexiglas plate is inserted between the probe and the phantom, creating a strong reverberating layer and a mismatch between the assumed speed c₀ (1540 m/s) and the actual local speed. The time‑focused matrix now exhibits multiple peaks at different τ values, each corresponding to a distinct echo path (single, double, etc.). Moreover, the axial position of the focal plane drifts relative to the iso‑chronous volume, producing an observable τ‑shift that quantifies the defocus caused by the speed mismatch.

To extract a coherent wave field from the noisy speckle background, the authors apply an iterative phase‑reversal algorithm across the full bandwidth (2–10 MHz). This algorithm reverses the phase of the measured matrix at each τ, effectively “time‑reversing” the wave packet and reinforcing the coherent component while suppressing random speckle fluctuations. The result is a clean reconstruction of the underlying wave packet, revealing both its temporal dispersion (due to multiple scattering) and spatial defocus (due to speed errors).

Key insights from the study include:

1. Speckle as Virtual Microphones – The random scatterers that generate speckle act as an array of passive receivers, enabling the measurement of the medium’s impulse response without any active internal sources.
2. Decoupled Input/Output Focusing – By separating transmit and receive focal points, the method isolates the contributions of the incident and reflected wave fronts, allowing direct observation of axial aberrations and reverberation artifacts that are invisible in conventional confocal imaging.
3. Time‑Focused Matrix as a Diagnostic Tool – The additional τ dimension provides a direct handle on the temporal evolution of the wave packet, making it possible to quantify the magnitude of defocus (τ‑shift) and the number of reverberant echoes (multiple τ peaks).
4. De‑scanned Representation for Efficient Storage – Transforming to Δr coordinates reduces the sampling density while preserving the essential physics, facilitating practical storage and processing of the high‑dimensional data.
5. Broad Applicability – Although demonstrated with ultrasound, the framework requires only a multi‑element array capable of measuring a reflection matrix, making it suitable for optics, radar, seismic imaging, and other wave‑based modalities.

The authors conclude that this “self‑portrait” of the focusing process opens a new avenue for wave‑physics imaging. In the subsequent papers of the series they will exploit the extracted information to (i) design adaptive compensation schemes that correct axial aberrations and reverberations in the speckle regime, and (ii) develop quantitative speed‑of‑sound mapping from the reflection matrix itself. Ultimately, the approach promises higher‑resolution, artifact‑free imaging in strongly scattering or aberrating media across a wide range of scientific and medical applications.