Real-time calculation of a limiting form of the Renyi entropy applied to detection of subtle changes in scattering architecture

Abstract

Previously a new method for ultrasound signal characterization using entropy H f was reported, and it was demonstrated that in certain settings, further improvements in signal characterization could be obtained by generalizing to Renyi entropy-based signal characterization I f ͑r͒ with values of r near 2 ͑specifically r = 1.99͒ ͓M. S. Hughes et al., J. Acoust. Soc. Am. 125, 3141-3145 ͑2009͔͒. It was speculated that further improvements in sensitivity might be realized at the limit r → 2. At that time, such investigation was not feasible due to excessive computational time required to calculate I f ͑r͒ near this limit. In this paper, an asymptotic expression for the limiting behavior of I f ͑r͒ as r → 2 is derived and used to present results analogous to those obtained with I f ͑1.99͒. Moreover, the limiting form I f,ϱ is computable directly from the experimentally measured waveform f͑t͒ by an algorithm that is suitable for real-time calculation and implementation.

Joint entropy of continuously differentiable ultrasonic waveforms

“…Our response to this fact has been to post-process the resulting H f ;g images using all analysis thresholds and to determine if there exists a rational set of criterion for differentiating the targeted from control groups. The results of the present study, as well as previous studies, 8,15 advance one strategy for doing so. The investigation of other strategies for choice of an optimum reference given specific detection requirements is the subject of ongoing research.…”

“…In the exceptional case, 11 sensitivities were essentially equivalent. (c) The most recent version of entropy published 8,15 is amenable to real-time implementation. a) Author to whom correspondence should be addressed.…”

Estimation of an Entropy-based Functional

“…One strategy would be to first construct an approximating function g (using the observations {(t j , f (t j ))}), and then calculate h(g) and use this as our estimate of h(f ). In Hughes et al [7], this method is applied to real-time imaging calculations in a laboratory setting; instead of using just the values {f (t j )} directly, they first construct an approximating function g and then calculate the desired entropy value using g. In [7], they are estimating a variant of the Rényi entropy of ρ f , not h(f ) as we have defined it, but the same authors have demonstrated the utility of h(f ). Incidentally, the quantity that is eventually computed in [7] is…”

“…A standard functional in the industry is the "energy" in the signal, ∫ 1 0 |f | 2 , or (more often) its logarithm. However, there is a series of papers by Hughes and others ( [1][2][3][4][5][6][7]) demonstrating the utility of using the continuous entropy of the "density function" of f as the functional, instead of the log-energy. Suppose that f is a real-valued function on [0,1].…”

“…The effectiveness of using the h functional has been shown in several different settings in ultrasound signal processing, ranging from defect detection in plexiglass [1] to detection of contrast agents in the tissue of live animals ( [5,7]). In some situations, the h functional produced images of objects which, because the material was engineered by the experimenter, were known to be present but were not detected by the log-energy functional.…”