Modeling ultrasound imaging as a linear, shift-variant system

We solve the equation that governs acoustic wave propagation in an inhomogeneous medium to show that the radio-frequency (RF) ultrasound signal can be expressed as the result of filtering the scatterer field with a point-spread function. We extend the analysis to make the link between the RF ultrasound signal and the representation of ultrasound scatterers as vectors with small magnitude and random phase in the complex plane. Others have previously performed parts of this analysis. The contribution of the present paper is to provide a single, coherent treatment emphasizing the assumptions that have to be made and the physical consequences of the models derived. This leads to insights into the interaction of monopole and dipole scattering, useful techniques for simulating and analyzing speckle statistics in the complex plane and a new expression for the normalized covariance of the analytic RF ultrasound signal in terms of the complex envelope of the point-spread function.