  This is the first article in a series of two dealing with a matrix approach
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 confocal
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 fitting
process enables the quantification of the single scattering, multiple
scattering and noise components in the image. From the single scattering
contribution, a focusing criterion is defined, and its evolution used to
quantify the amount of aberration throughout the ultrasound image. In contrast
to the state-of-the-art coherence factor, this new indicator is robust to
multiple scattering and electronic noise, thereby providing a contrasted map of
the focusing quality at a much better transverse resolution. After a validation
of the proof-of-concept based on time-domain simulations, UMI is applied to the
in-vivo study of a human calf. Beyond this specific example, UMI opens a new
route for speed-of-sound and scattering quantification in ultrasound imaging.
