  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.
