  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.
