The Statistics of Normalized Data in Electrical Impedance Tomography

This study focuses on the noise statistics of the normalized data, in time-difference imaging. The commonly practiced normalization procedure, represents the normalized data as the ratio of two random variables. We provide closed form expressions for computing estimates of the first, second order moments, and the covariance matrix of normalized data. The expressions are derived using Taylor series expansion method for the function of random variables (propagation of uncertainty, delta method), and accurate provided that the signal to noise ratio of the reference measurement is larger than 16 dB. The analysis presented here, provides an explanation as well as insight, into the potential cause and presence of spikes in normalized data. Furthermore, we demonstrate that the statistics of normalized data, is significantly different to that of the raw measurements. We validate the closed form expressions using a Monte-Carlo method. We also conclude, that a minimum SNR of 13 dB is required for the reference measurement, in order to guarantee the existence of moments, and to ensure the accuracy of reconstructed images.