Ultrasound Matrix Imaging—Part II: The Distortion Matrix for Aberration Correction Over Multiple Isoplanatic Patches

Abstract

This is the second article in a series of two which report on a matrix approach for ultrasound imaging in heterogeneous media. This article describes the quantification and correction of aberration, i.e. the distortion of an image caused by spatial variations in the medium speed-of-sound. Adaptive focusing can compensate for aberration, but is only effective over a restricted area called the isoplanatic patch. Here, we use an experimentally-recorded matrix of reflected acoustic signals to synthesize a set of virtual transducers. We then examine wave propagation between these virtual transducers and an arbitrary correction plane. Such wave-fronts consist of two components: (i) An ideal geometric wave-front linked to diffraction and the input focusing point, and; (ii) Phase distortions induced by the speed-of-sound variations. These distortions are stored in a so-called distortion matrix, the singular value decomposition of which gives access to an optimized focusing law at any point. We show that, by decoupling the aberrations undergone by the outgoing and incoming waves and applying an iterative strategy, compensation for even high-order and spatially-distributed aberrations can be achieved. After a numerical validation of the process, ultrasound matrix imaging (UMI) is applied to the in-vivo imaging of a gallbladder. A map of isoplanatic modes is retrieved and is shown to be strongly correlated with the arrangement of tissues constituting the medium. The corresponding focusing laws yield an ultrasound image with drastically improved contrast and transverse resolution. UMI thus provides a flexible and powerful route towards computational ultrasound.

Imaging the Crustal and Upper Mantle Structure of the North Anatolian Fault: A Transmission Matrix Framework for Local Adaptive Focusing

“…Here, this property does not hold because the NAFZ subsurface exhibits a continuous but fluctuating reflectivity (see Text S3 in Supporting Information S1). In this scattering regime, local distortion matrices should be considered over restricted areas in which the isoplanatic hypothesis is ideally fulfilled (Lambert, Cobus, et al., 2022; Najar et al., 2023).…”

“…By removing the geometrical component of the CLASS wave‐field, spatial correlations are highlighted between distorted wave‐fields induced by neighbor virtual sources $${\mathbf{r}}_{\text{in}}$$ (Badon et al., 2020). Such correlations are a manifestation of a spatial invariance of residual aberrations over areas generally referred to as isoplanatic patches (Lambert, Cobus, et al., 2022).…”

“…Here, we apply it in a new scattering regime since the NAFZ subsurface exhibits, in the frequency range under study, a continuous reflectivity distribution made of specular reflectors and randomly distributed heterogeneities. In this regime, a local time reversal analysis of the distortion matrix can be performed in order to retrieve the transmission matrix between the Earth's surface and any point of the subsurface (Lambert, Cobus, et al., 2022). This transmission matrix is a key tool since its phase conjugate provides the optimized focusing laws that need to be applied to the reflection matrix in order to retrieve a diffraction‐limited image of the subsurface.…”

Phase Aberration Correction: A Convolutional Neural Network Approach

“…As opposed to methods that model the phase aberration effect by a fixed near-field phase screen in front of the transducer, the locally adaptive phase aberration correction technique [14] assumed a spatially varying near-field phase screen and employed multistatic synthetic aperture data to perform the correction at each point adaptively. Lambert et al suggested compensating for the spatially-distributed aberrations by decoupling aberrations undergone by the outgoing and incoming waves utilizing the distortion matrix built from the focused reflection matrix, which contains the responses between virtual transducers synthesized from the transmitted and received focal spots [15], [16], [17].…”

Windowed Radon Transform and Tensor Rank-1 Decomposition for Adaptive Beamforming in Ultrafast Ultrasound

“…To do so, we rely on the concept of change of basis from the canonical sensor basis to a Rx plane waves (PWs) basissometimes denoted as Radon domain-proposed recently [25], [26]. Following [20], [24], the main rationale behind the proposed method is to express the data into a local Rx PW basis to account for the locality of the phase aberrations induced by SoS variations.…”