Theoretical Foundations of the Echo Envelope Statistical Modeling: A Tutorial

The purpose of this methods and concepts tutorial is to present the homodyned K-distribution (HKD) statistical modeling of the echo envelope of received radio frequency (RF) signals in the context of medical quantitative ultrasound (QUS) imaging, with the aim of explaining its physical, mathematical, and statistical foundations. Several notions and equations are recalled from previous works on HKD modeling and estimation methods. Proofs of claims are presented in Appendices that can be found in Supplementary Materials. Some descriptions have been completed or refined without modifying the main conclusions on HKD or mixtures of HKDs. Mixtures of HKDs are recalled, as well as other models proposed in previous works, such as the generalized KD (GKD), HKD with additive Gaussian noise (HKDN), and the generalized HKD (GHKD), the latter resorting to the generalized central limit theorem (CLT) in the case where the scattering cross section has infinite variance. This article also presents three innovations on the topic: 1) a revised derivation of the HKD model based on Stein's condition to obtain an explicit rate of convergence of the CLT in the case of weakly dependent terms, corresponding to ultrasound (US) scatterers; 2) HKD imaging under frequency-domain filtering of RF signals, yielding information on second-order statistics of the echo envelope; and 3) quantitative results on the Kolmogorov distance between the HKD and other distributions (Nakagami and Rice distributions, GKD, HKDN, and GHKD) together with the domains insuring validity (i.e., statistical equivalence with a confidence level of 0.05).