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

TL;DR

This work introduces the time-focused reflection matrix (TFRM) as a matrix-imaging framework to visualize the spatio-temporal evolution of wave packets inside complex media using speckle. By measuring the ultrasound reflection matrix $\mathbf{R}_{u\theta}(t)$ and forming a time-dependent focused matrix $\mathbf{R}_{rr}(\tau)$, the method decouples input and output focusing to reveal axial aberrations, reverberations, and dispersion that degrade focusing. It demonstrates two complementary pathways to extract coherent wave-packets in speckle: (i) a SVD-based synthesis of a coherent guide star from multiple incoherent virtual sources, and (ii) an iterative phase reversal (IPR) approach that yields a broadband, high-resolution transmittance estimate across space, time, and frequency. The results show how ballistic timing, depth-shifts, and reverberation-induced echoes manifest in the self-portrait, enabling potential compensation strategies and even speed-of-sound tomography, with broad applicability to other wave fields beyond ultrasound.

Abstract

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.

Figures (11)

• Figure 1: Experimental configurations and procedure.a, Experimental set up: An ultrasonic linear probe is placed in contact of a tissue-mimicking phantom composed of a random distribution of sub-resolved diffusers schematized by the gray background texture, two hyper-echoic cylinders, and a set of bright nylon rods. A set of plane waves ($\bm{\theta}_{\textrm{in}}$) is emitted by the probe and the time-dependent back-scattered wave-field $R(\mathbf{u}_{\textrm{out}},\bm{\theta}_{\textrm{in}},t)$ recorded by each transducer ($\mathbf{u}_{\textrm{out}}$) is stored in the so-called reflection matrix $\mathbf{R}_{\mathbf{u}\bm{\theta}}(t)=[R(\mathbf{u}_{\textrm{out}},\bm{\theta}_{\textrm{in}},t)]$ sketch in panel $\mathbf{b}$. $\mathbf{c}$, $\mathbf{d}$ Ultrasound image deduced from a beamforming process applied to $\mathbf{R}_{\mathbf{u}\bm{\theta}}(t)$ using $c_0=1540$ and 1440 m.s$^{-1}$, respectively. e, Same experimental configuration as in panel a except that a reverberating layer is placed between the probe and the tissue-mimicking phantom. f, Corresponding ultrasound image ($c_0=1540$ m.s$^{-1}$).
• Figure 2: Principle of ultrasound imaging.a, Case of a medium of homogeneous speed-of-sound. Left: The excitation of each transducer by a time-delayed pulse generates a cylindrical wave-front focusing at a given point $\mathbf{r}_{\textrm{in}} = (x_{\textrm{in}}, z_{\textrm{in}})$. Right: In reception mode, the echoes back-scattered by reflactors in the vicinity of the same point are time-shifted using the same delay law as that applied in transmit. This allows the signals induced by a single scattering event at $\mathbf{r}_{\textrm{out}} =\mathbf{r}_{\textrm{in}}$ to be constructively summed. The pixel of the ultrasound image corresponds to the resulting signal measured at the expected ballistic time $t = 2z_{\textrm{in}}/c_0$. The set of points generating the backscattered echoes at time $t$ defines the isochronous volume. When the wave velocity model and the medium speed-of-sound coincide, the focusing plane of the incident wave-front belongs to the isochronous volume. b, For a medium with an heterogeneous speed-of-sound distribution, this coincidence is no longer checked: The focusing plane and the isochronous volume are axially shifted in reverse directions. The delay-and-sum beamforming process that assumes an homogeneous wave velocity $c_0$ is hampered by a defocus. c, For a reverberating medium, the situation is even worse since each multiply-reflected wave-front is associated with a different isochronous volume and a distinct focusing plane. Reverberation phenomena undergone by the wave at input and output gives rise to a series of echoes for each scatterer on the beamformed image.
• Figure 3: Time-focused reflection matrix.a The matrix $\mathbf{R_{rr}}(\tau)$ is constructed in post-processing by decoupling the position of the emission focusing points $\mathbf{r}_{\textrm{in}} = (x_{\textrm{in}},z_{\textrm{in}})$ and reception focal points $\mathbf{r}_{\textrm{out}} = (x_{\textrm{out}},z_{\textrm{out}})$, thus forming a set of virtual sources and receivers inside the medium. The temporal responses between each virtual source $\mathbf{r}_{\textrm{in}}$ and each virtual receiver $\mathbf{r}_{\textrm{out}}$ are obtained by introducing an additional variable time $\tau$, corresponding to a shift from the expected ballistic time $t = \left(z_{\textrm{in}}+ z_{\textrm{out}} \right)/c_0$. This time delay $\tau$ allows the wave to propagate virtually in the medium from $\mathbf{r}_{\textrm{in}}$ to $\mathbf{r}_{\textrm{out}}$. b Time focused reflection matrix $\mathbf{R_{rr}}(\tau)$ displayed at different time lapses $\tau$ in the ultrasound phantom (Fig. \ref{['fig1']}a). c Same matrix $\mathbf{R_{\Delta r}}(\tau)$ expressed in the de-scanned basis. d Example of a reflected wave-fields reshaped in de-scanned coordinates $\left(\Delta x, \Delta z \right)$ extracted from one column of the de-scanned matrix $\mathbf{R}_{\bm{\Delta} \mathbf{r}}(\tau)$ designated by a black arrow in panel c and associated with a given point $\mathbf{r}_{\textrm{in}}$ corresponding to a bright point-like scatterer. e Same as in panel d but in speckle.
• Figure 4: Coherent wave-packet associated with a bright scatterer inside an homogeneous phantom.a,b Ultrasound image of the phantom beamformed at the phantom speed-of-sound ($c_0=1540$ m.s$^{-1}$) and at a wrong speed-of-sound ($c_0=1440$ m.s$^{-1}$), respectively: The position of the virtual source $\mathbf{r}_{\textrm{in}}$ corresponds to a bright scatterer indicated by an orange cross and the set of virtual receivers $\mathbf{r}_{\textrm{out}}$ surrounding it is delimited by a blue rectangle. c-g, Normalized amplitude of the wave-field $R(\Delta \mathbf{r},\mathbf{r}_{\textrm{in}},\tau)$ produced by the virtual source $\mathbf{r}_{\textrm{in}}$ at the phantom speed-of-sound (a). h-l Same as in (c-g) but for a wrong speed-of-sound (b). In panels (c-l), each focused wave-field is displayed in de-scanned coordinates for various lapse times $\tau$. The propagation of the coherent wave-packet shown in panels (c-g) and (h-l) are also displayed in Supplementary Movies 1 and 2.
• Figure 5: Coherent wave-packet associated with a bright scatterer in presence of reverberations.a, Ultrasound image of the phantom through the Plexiglas layer (Fig. \ref{['fig1']}e): The position of the virtual source $\mathbf{r}_{\textrm{in}}$ corresponds to a bright scatterer indicated by an orange circle and the set of virtual receivers $\mathbf{r}_{\textrm{out}}$ surrounding it is delimited by a blue rectangle. b-f, Normalized amplitude of the wave-field produced by the virtual source $\mathbf{r}_{\textrm{in}}$ in de-scanned coordinates displayed for various lapse times $\tau$. The propagation of the coherent wave-packet is also displayed in Supplementary Movie 2.