Probing the speckle to estimate the effective speed of sound, a first step towards quantitative ultrasound imaging

In this paper, we present a mathematical model and analysis for a new experimental method [Bureau et al., arXiv:2409.13901, 2024] for effective sound velocity estimation in medical ultrasound imaging. We perform a detailed analysis of the point-spread function of a medical ultrasound imaging system when there is a mismatch between the effective sound speed in the medium and the one used in the back-propagation imaging functional. Based on this analysis, an estimator for the speed of sound error is introduced. Using recent results on stochastic homogenization of the Helmholtz equation, we provide a representation formula for the field scattered by a random multi-scale medium (whose acoustic behavior is similar to a biological tissue) in the time-harmonic regime. We then prove that statistical moments of the imaging function can be accessed from data collected with only one realization of the medium. We show that it is possible to locally extract the point-spread function from an image constituted only of speckles, and build an estimator for the effective sound velocity in the micro-structured medium. Some numerical illustrations are presented at the end of the paper.