  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 image quality and the wave velocity to be evaluated everywhere inside the
medium, including in random speckle, and (ii) an highly resolved spatial
mapping of the prevalence of multiple scattering, which constitutes a new and
unique contrast for ultrasonic imaging. The approach is demonstrated for a
controllable phantom system, and for in vivo imaging of the human abdomen. More
generally, this matrix approach opens an original and powerful route for
quantitative imaging in wave physics.
