Detection and characterization of targets in complex media using fingerprint matrices

TL;DR

This work presents the fingerprint operator, $\mathbf{\Gamma}=\mathbf{R}\times\mathbf{R}_0^{\dagger}$, to detect, localize, and characterize targets within strongly scattering media by exploiting correlations in surviving wavefields. A calibrated free-space reference $\mathbf{R}_0(\mathbf{q})$ encodes target state parameters and, when combined with a measured reflection matrix $\mathbf{R}$, yields a likelihood map $\gamma(\mathbf{q})$ that sharply localizes targets even under heavy multiple scattering. The authors demonstrate three ultrasound scenarios (granular suspensions with elastic spheres, lesion markers in tissue-mimicking foam, and in vivo muscle-fiber mapping) and provide a theoretical framework showing a potential contrast gain $G\sim N_SN_T$ relative to conventional confocal imaging, along with CRB-based localization precision that scales with $\mathcal{C}_{\gamma}$. They also establish a robust detection criterion with a formal false-alarm probability and connect the likelihood mapping to standard confocal beamforming via a Hadamard product in the plane-wave basis. The method is broadly applicable to other wave systems where reflection matrices can be measured, offering a computationally efficient, calibration-light route to quantitative imaging and material characterization in complex media.

Abstract

When waves propagate through a complex medium, they undergo several scattering events. This phenomenon is detrimental to imaging, as it causes full blurring of the image. Here we describe a method for detecting, localizing and characterizing any scattering target embedded in a complex medium. We introduce a fingerprint operator that contains the specific signature of the target with respect to its environment. When applied to the recorded reflection matrix, it provides a likelihood index of the target state. This state can be the position of the target for localization purposes, its shape for characterization or any other parameter that influences its response. We demonstrate the versatility of our method by performing proof-of-concept ultrasound experiments on elastic spheres buried inside a strongly scattering granular suspension and on lesion markers, which are commonly used to monitor breast tumours, embedded in a foam mimicking soft tissue. Furthermore, we show how the fingerprint operator can be leveraged to characterize the complex medium itself by mapping the fibre architecture within muscle tissue. Our method is broadly applicable to different types of waves beyond ultrasound for which multi-element technology allows a reflection matrix to be measured.

Figures (21)

• Figure 1: Multiple scattering fog.a, Experimental configuration: A 2D array of 1024 ultrasound transducers acting both as transmitters and receivers is used to detect two metal spheres that are embedded in strongly scattering glass beads below water. b, The reflection matrix is measured by insonifying the medium with a set of incident plane waves ($\bm{\theta}_{\textrm{in}}$). c, Each back-scattered wave field $R(\mathbf{u}_{\textrm{out}},\bm{\theta}_{\textrm{in}},t)$ is recorded on each transducer $\mathbf{u}_{\textrm{out}}$ of the same probe. d, Volumetric visualization of the probed medium by projecting the intensity maximum of the confocal image obtained by a delay-and-sum beamforming applied to the recorded reflection matrix (a dynamic view of this is shown in Supplementary Movie 1). With this conventional imaging technique, the specular echo of the interface between water and the glass bead suspension largely predominates and the targets are completely invisible on the image.
• Figure 1: Experimental parameters.
• Figure 1: Elastic target signatures encoded in the reference reflection matrix. a, The reference reflection matrix $\mathbf{R}_0$ is measured on the target sphere placed in water. The confocal beamforming process applied to $\mathbf{R}_0$ selects not only the ballistic echo of the sphere but also its reverberations resulting from multiple reflections of bulk elastic waves (depicted by red arrows) at its inner surface. b, Matrix imaging decouples the input and output focal spots Lambert2020a, $\mathbf{r}_{\textrm{in}}$ and $\mathbf{r}_{\textrm{out}}$, to highlight the contribution of circumferential waves (depicted by a black arrow) generated by the incoming wave at a specific angle of incidence with respect to sphere surface Prada1998. c, Cross-section of the focused reflection matrix, $\mathbf{R}_{0,xx}(y,z)$, in the plane $y=0$ and at depth $z=21$ mm showing the diagonal contribution of the ballistic echo. d, Same matrix but at depth $z=28.5$ mm showing the off-diagonal contribution of circumferential waves. e, $(x,z)$-section of the confocal image in the plane $y=0$ showing the spatio-temporal dispersion of the target echo. f, Likelihood index map $\textcolor{black}{\gamma}(\mathbf{r})$ (Eq. \ref{['eq P']}) built from the fingerprint operator indicating that we can accurately locate the target inside the reference environment (the sphere surface is highlighted by a red dashed line in panels e and f). Since in this case $\mathbf{R}\equiv\mathbf{R}_0\textcolor{black}{(\mathbf{r}_0)}$, this result serves as a consistency check for the formalism.
• Figure 2: Detecting and localizing target metal spheres hidden in the multiple scattering fog. Different cross-sections of the $\gamma$-map for each sphere (diameter $d_1 = 10$ mm in red, $d_2 = 8$ mm in green) are superimposed to the corresponding confocal image (B&W scale in dB). a$(x,z)-$cross-section at $y=y_1=y_2$ (scale bar: 10 mm). b$(y,z)-$cross-section at $x=x_1$. c$(y,z)-$cross-section at $x=x_2$. The confocal images correspond to cross-sections of Fig. \ref{['config']}d, the dashed circles correspond to the known radius of each sphere. The value of the contrast $\mathcal{C}_\gamma$ at the target's position with respect to the average value of $\gamma$ outside each target is also indicated. While the targets are completely invisible in the confocal images, they appear very well localized through the calculation of their associated likelihood index $\gamma$.
• Figure 3: Localizing a lesion marker in ultrasound speckle.a, Experimental configuration: The $32\times 32$ probe is used to image a lesion marker embedded into foam soaked in water. The position $\mathbf{r}_m=(x_m,y_m,z_m)$ of the lesion marker center is $(3.5,4.5,32)$ mm. b, Photography of the lesion marker (credit: Arthur Le Ber). c, Longitudinal cross-section of the confocal ultrasound image in the plane $y=y_m$(scale bar: 5 mm). d, Corresponding likelihood map of the lesion marker superimposed onto the confocal image in transparency.