Department of Medicine/Cardiology Division, Campus Box 8215, Washington University School of Medicine, 660 South Euclid Avenue, St. Louis, Missouri 63110-1093, USA.
J Acoust Soc Am. 2013 Jan;133(1):283-300. doi: 10.1121/1.4770245.
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.
本研究基于对两个随机变量的联合熵概念的扩展，扩展到连续函数，如背向散射超声。对于两个连续的随机变量 X 和 Y，联合概率密度 p(x,y)通常是 x 和 y 的连续函数，其取值在实数平面的二维区域内。然而，在 X=f(t)和 Y=g(t)都是连续可微函数的情况下，X 和 Y 仅集中在 x,y 平面上的一条曲线上，γ(t)=(f(t),g(t))。这种集中只能使用数学上的“奇异”对象（如 Schwartz 分布）来表示。其在成像中的使用需要进行粗粒化操作，本研究对此进行了描述。随后，使用遍历定理去除粗粒化参数。所得到的联合熵表达式应用于多个数据集，展示了该概念在材料特征化和检测靶向液体纳米颗粒超声对比剂方面的应用。在所有情况下，这些技术的灵敏度都与之前使用信号能量或其他熵量的研究相匹配或超过，有时甚至超过两倍。
