This study is based on an extension of the concept of joint entropy of two random variables to continuous functions, such as backscattered ultrasound. For two continuous random variables, X and Y, the joint probability density p(x,y) is ordinarily a continuous function of x and y that takes on values in a two dimensional region of the real plane. However, in the case where X=f(t) and Y=g(t) are both continuously differentiable functions, X and Y are concentrated exclusively on a curve, γ(t)=(f(t),g(t)), in the x,y plane. This concentration can only be represented using a mathematically "singular" object such as a (Schwartz) distribution. Its use for imaging requires a coarse-graining operation, which is described in this study. Subsequently, removal of the coarse-graining parameter is accomplished using the ergodic theorem. The resulting expression for joint entropy is applied to several data sets, showing the utility of the concept for both materials characterization and detection of targeted liquid nanoparticle ultrasonic contrast agents. In all cases, the sensitivity of these techniques matches or exceeds, sometimes by a factor of two, that demonstrated in previous studies that employed signal energy or alternate entropic quantities.
Hughes, M S.; McCarthy, John E.; Marsh, J N.; and Wickline, S A., "Joint entropy of continuously differentiable ultrasonic waveforms" (2013). Mathematics Faculty Publications. 11.