Robust Non-Parametric Estimation of Speckle Probability Densities and gCNR

In ultrasound imaging, speckle originates from a large amount of sub-resolution scatterers within the medium. In idealized cases, the speckle envelope statistics follow a Rayleigh distribution, but in practical pulse-echo imaging, the distribution depends on both the imaging system and the underlying tissue structure. Estimating envelope statistics is part of quantitative ultrasound workflows and is also important for image quality assessment as it relates to lesion and tissue detectability. A concrete example is the generalized contrast-to-noise ratio (gCNR), which is a functional of two pixel-value probability density functions (PDFs) from different speckle regions. Such speckle PDFs have, by convention, been estimated from data using histograms, but the accuracy of these estimates can be affected by the nontrivial selection and tuning of the binning parameters. However, the statistics literature widely advocates kernel density estimation (KDE) as a better alternative to histogram-based approaches. In this article, we propose applying a KDE-based method to estimate speckle PDFs in medical ultrasound imaging. The method is practically tuning-free and leverages the Box-Cox transformation to achieve best-in-class performance across a wide range of test cases, and is also robust in cases where gCNR estimation may otherwise fail, such as for skewed distributions that may arise with adaptive beamformers. Furthermore, this work highlights theoretical aspects related to the estimation of PDFs and derived quantities, including the gCNR.