On the statistical characteristics of log-compressed Rayleigh signals: theoretical formulation and experimental results

Abstract

This paper addresses the effects of nonlinear log compression on the amplitude of the backscattered signals. The changes in the statistical characteristics of signals were examined and correlated with the compression parameters. We develop an analytical formulation for the probability density function (PDF) of log-compressed amplitude signals. To obtain the theoretical PDF, a Rayleigh distributed signal is subjected to a scaled log-compression of the form n <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> ln(x)+n <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> . Such transformation is common in medical ultrasound image formation as it allows independent control over the dynamic range and gain of the displayed image. The resulting Fisher-Tippett PDF and its statistical parameters are derived and compared with the empirically measured statistics of ultrasound images of scattering phantoms. The comparisons reveal a great degree of similarity between theoretical PDF and the histogram of the image, even though the goodness-of-fit tests indicate statistical mismatch from theoretical model due to factors such as non-ideal log compression transfer, noise, envelope smoothing, etc