Ultrasound Matrix Imaging. I. The focused reflection matrix and the F-factor

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.