Self-Portrait of the Focusing Process in Speckle: III. Tailoring Complex Spatio-Temporal Focusing Laws To Overcome Reverberations in Reflection Imaging

This is the third article in a series of three dealing with the exploitation of speckle for imaging purposes. In complex media, a fundamental limit is the multiple scattering phenomenon that completely blurs the imaging process in depth. Matrix imaging can provide a relevant framework for solving this problem. As it proved to be an adequate tool for probing reverberations in speckle [E. Giraudat et al., Part I], we will show how it can be used to tailor complex spatio-temporal focusing laws to monitor the interference between the multiply-reflected paths and the ballistic component of the wave-field. To do so, we extend the distortion matrix concept to the frequency domain. An iterative phase reversal process operated from the space-time Fourier space is then used to compensate for reverberations and optimize both the axial and transverse resolution of the confocal image. Here, we first present an experimental proof-of-concept consisting in imaging a tissue-mimicking phantom through a reverberating plate before outlining the potential and the limits of this strategy for transcranial ultrasound and beyond.

Reflection Matrix Imaging for Wave Velocity Tomography

Besides controlling wave trajectory inside complex media, wave velocity constitutes a relevant bio-marker for medical imaging. In a transmission configuration, wave-front distortions can be unscrambled to provide a map of the wave velocity landscape $c(\mathbf{r})$. However, most in-vivo applications correspond to a reflection configuration for which only back-scattered echoes generated by short-scale fluctuations of $c(\mathbf{r})$ can be harvested. Under a single scattering assumption, this speckle wave-field cannot provide the long-scale variations of $c(\mathbf{r})$. In this paper, we go beyond the first Born approximation and show how a map of $c(\mathbf{r})$ can be retrieved in epi-detection. To that aim, a reflection matrix approach of wave imaging is adopted. While standard reflection imaging methods generally rely on confocal focusing operations, matrix imaging consists in decoupling the location of the incident and received focal spots. Following this principle, a self-portrait of the focusing process can be obtained around each point of the medium. The Gouy phase shift exhibited by each focal spot is leveraged to finely monitor the wave velocity distribution $c(\mathbf{r})$ inside the medium. Experiment in a tissue-mimicking phantom and numerical simulations are first presented to validate our method. Speed-of-sound tomography is then applied to ultrasound data collected on the liver of a difficult-to-image patient. Beyond paving the way towards quantitative ultrasound, our approach can also be extremely rewarding for standard imaging. Indeed, each echo can be assigned to the actual position of a scatterer. It allows an absolute measurement of distance, an observable often used for diagnosis but generally extremely sensitive to wave velocity fluctuations.

Unveiling the deep plumbing system of a volcano by a reflection matrix analysis of seismic noise

In geophysics, volcanoes are particularly difficult to image because of the multi-scale heterogeneities of fluids and rocks that compose them and their complex non-linear dynamics. By exploiting seismic noise recorded by a sparse array of geophones, we are able to reveal the magmatic and hydrothermal plumbing system of La Soufri\`ere volcano in Guadeloupe. Spatio-temporal cross-correlation of seismic noise actually provides the impulse responses between virtual geophones located inside the volcano. The resulting reflection matrix can be exploited to numerically perform an auto-focus of seismic waves on any reflector of the underground. An unprecedented view on the volcano's inner structure is obtained at a half-wavelength resolution. This innovative observable provides fundamental information for the conceptual modeling and high-resolution monitoring of volcanoes.