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

Ultrasonic characterization of backscatter from human cancellous bone with a renyi entopy metric: Correlation with x-ray bone mineral density

“…The observed inverse relationship between these two metrics is consistent with the definition of the entropy metric and with previous measurements in soft tissue. [8][9][10][11][12] The lack of any apparent trends in the curve-fitting residuals supports the general linear relationship for these two log-compressed metrics and the bone mineral density. For ease of comparison, the R 2 values are represented as a bar chat in Figure 3.…”

“…The general range of the correlation coefficients in comparable to previously reported values. [7,11] It is interesting to note that the Renyi entropy values appear to have a slightly improved correlation with the BMD values for the two higher frequency cases. The apparent equivalence in the correlations for the 2.25 MHz data is also of interest and suggestive for future studies and simulations designed to improve the interpretation of these metrics.…”

“…Additional details of the mathematical and computation properties of the limiting form of the Renyi entropy are described by Hughes, et al [12]. Table I.…”

“…[9] A useful generalization of this entropy metric is the Renyi entropy, If(r) , which is defined for r < 2 and is identical to H f when (r = 1). [10,11] Further increases sensitivity and in time-to-detection, for the accumulating nanoparticles was found for values of r close to 2 (i.e., r = 1.99); however, there was also an increased computational demand [11]. Subsequently, a limiting form of the Renyi entropy has been developed; denoted I f ,∞ , which is amenable to real-time calculation and equivalent to the Renyi entropy I f r ( ) of a segment of radiofrequency data, as r → 2 − .…”

“…Subsequently, a limiting form of the Renyi entropy has been developed; denoted I f ,∞ , which is amenable to real-time calculation and equivalent to the Renyi entropy I f r ( ) of a segment of radiofrequency data, as r → 2 − . [12] This computationally efficient form, I f,∞ , is applied to the cancellous bone data in the current paper.…”