Reflection matrix approach for quantitative imaging of scattering media

Abstract

We present a physically intuitive matrix approach for wave imaging and characterization in scattering media. The experimental proof-of-concept is performed with ultrasonic waves, but this approach can be applied to any field of wave physics for which multi-element technology is available. The concept is that focused beamforming enables the synthesis, in transmit and receive, of an array of virtual transducers which map the entire medium to be imaged. The inter-element responses of this virtual array form a focused reflection matrix from which spatial maps of various characteristics of the propagating wave can be retrieved. Here we demonstrate: (i) a local focusing criterion that enables the imaging quality to be evaluated everywhere inside the medium, including in random speckle; (ii) a tomographic measurement of wave velocity, which allows for aberration corrections in the original image; (iii) an highly resolved spatial mapping of the prevalence of multiple scattering, which constitutes a new and unique contrast for ultrasonic imaging. More generally, this matrix approach opens an original and powerful route for quantitative imaging in wave physics.

Distortion matrix approach for ultrasound imaging of random scattering media

“…This information can be obtained from R uθ by applying appropriate time delays to perform focusing in post-processing, both in emission and reception [41]. This focusing can also be easily performed in the frequency domain, where matrix products allow the mathematical projection of R between different mathematical bases [40]. The bases implicated in this work are sketched in Fig.…”

“…This procedure is described in more detail in Ref. [40]. It involves considering only pairs of virtual transducers, r in = (x in , z) and r out = (x out , z), which are located at the same depth z (Fig.…”

“…In a recent work, the broadband focused reflection matrix R xx (z) was shown to be a valuable observable to locally assess the quality of focusing and to quantify the multiple scattering level in the ultrasonic data [40]. In this paper, R xx (z) is used as a basic unit from which: (i ) a confocal image of the sample reflectivity can be built;…”

“…Figure 3B shows R xx (z) at depth z = 25 mm. The brightness of its diagonal coefficients is characteristic of singly-scattered echoes [30,31,40]. In fact, the diagonal of R xx (z) corresponds to the line at depth z of a confocal or compounded [41] ultrasound image I (r) :…”

“…In addition, while R typically contains responses between inputs and outputs in the same basis, e.g. responses between individual ultrasonic transducer elements [38,39] or between focal points inside the medium [40], D is concerned with the 'dual basis' responses between a set of incident plane waves [41] and a set of focal points inside the medium [35]. It has been shown that, for a large specular reflector, a time rever-sal analysis of D yields a one-to-one association between each isoplanatic patch in the focal plane and the corresponding wavefront distortion in the far-field [37].…”