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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. As a proof-of-concept, ultrasound matrix imaging (UMI) is applied to the in-vivo imaging of a human calf. A map of isoplanatic patches 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 an optimal contrast and a transverse resolution close to the ideal value predicted by diffraction theory. UMI thus provides a flexible and powerful route towards computational ultrasound.

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This is the first article in a series of two dealing with a matrix approach \alex{for} aberration quantification and correction in ultrasound imaging. Advanced synthetic beamforming relies on a double focusing operation at transmission and reception on each point of the medium. Ultrasound matrix imaging (UMI) consists in decoupling the location of these transmitted and received focal spots. The response between those virtual transducers form the so-called focused reflection matrix that actually contains much more information than a raw ultrasound image. In this paper, a time-frequency analysis of this matrix is performed, which highlights the single and multiple scattering contributions as well as the impact of aberrations in the monochromatic and broadband regimes. Interestingly, this analysis enables the measurement of the incoherent input-output point spread function at any pixel of this image. A focusing criterion can then be built, and its evolution used to quantify the amount of aberration throughout the ultrasound image. In contrast to the standard coherence factor used in the literature, this new indicator is robust to multiple scattering and electronic noise, thereby providing a highly contrasted map of the focusing quality. As a proof-of-concept, UMI is applied here to the in-vivo study of a human calf, but it can be extended to any kind of ultrasound diagnosis or non-destructive evaluation.

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