To calculate each component of current density, derivatives of two perpendicular components of the induced should be used. The derivatives are calculated using derivative templates.10 These derivative templates rely on an averaging operation in a neighborhood to approximate the derivatives, therefore, their application compromises the spatial resolution and itself acts like denoising. If a template is used, the nine values of in the neighborhood of each point are used to calculate the derivative at each point and two derivatives. Therefore, the final distribution is the summation of 18 scaled distributions of the form provided in Eq. (12). Using the fact that the distribution of summation is a convolution of the distributions, we can find the final distribution of the current density (). It is also known that the convolution of Gaussians is Gaussian,19 therefore, if the distribution in Eq. (12) is close to Gaussian, we can conclude the noise distribution in the measured current density is also Gaussian. Figures 4 and 5 show the distribution for various values of , , and . The values chosen for and in these figures are arbitrarily chosen in the range of observed values for real and imaginary images. These figures show the mean is dependent on and , while the variance depends on , , and . We can also conclude that the distribution in Eq. (12) can be approximated by a Gaussian for each point. The Gaussian claim for can be quantified using kurtosis and skewness measures.20 Furthermore, the normality of distribution based on its samples can be classified using the D’Agostino test20 which is based on kurtosis and skewness. The results for Figs. 4 and 5 are shown in Table 1. The significance level used for the D’Agostino test in this table is 0.01. The table shows that for all the cases investigated in Figs. 4 and 5 except for one case, the distributions can be safely approximated by a Gaussian. However in order to find the ranges of , , and for which the Gaussian approximation holds, the kurtosis, skewness, and D'Agostino test results for various ranges of and (between 1 and 100 with step of 5) and are shown in Fig. 6. The significance level used for the D’Agostino test is 0.01. It can be seen that the ratio of magnitude to determines whether the Gaussian assumption holds. As this ratio increases, the distribution gets closer to a Gaussian. In both of our datasets, this ratio is close to 50 for a of approximately 10, which means the distribution at each point can be approximated by a Gaussian. We can verify the distribution of noise in the phantom by performing a baseline analysis.21 In this case, we record real and imaginary images twice when there is no current in each orientation. The current is then calculated by considering repeated images in each orientation as two phase cycles of the CDI method.21 The resulting current density map in and its histogram are shown in Figs. 7(a) and 7(b). This histogram is reflective of the noise distribution because there is not much change in real and imaginary images in the absence of injected current, thus the real and imaginary images should be almost constant. Figures 7(c) and 7(d) and 7(e) and 7(f) show the calculated current density maps and their histograms when current is injected for Dataset 1 and Dataset 2, respectively. It can be seen that the noise distribution is only shifted when the current is injected. The red curves on the histograms show the Gaussian distributions fitted on the samples. For the no current case, the mean of the fitted Gaussian is 0.03 and its standard deviation is 3.62. The Gaussian fitted for Dataset 1 has mean of 13.24 and standard deviation of 3.67, while for Dataset 2 the Gaussian has mean of 29.45 and standard deviation of 3.19. This matches the fact that the convolution of 18 Gaussians (for the case of a derivative template) is Gaussian itself and, therefore, the final has a noise which is Gaussian while its mean and variance depend on the , , and noise in the real and imaginary images. In all three cases in Fig. 7, the background noise has almost the same zero mean Gaussian distribution with a standard deviation of approximately 10 and the noise in the current density images is also zero mean Gaussian with a standard deviation around 3. For the cases where a current was injected, and are not constant because of phase changes induced by the current. However, the magnitude is constant for all three cases in Fig. 7 because the phantom has the same homogeneous structure. Looking back at Eq. (12), we can see that if the background noise is constant and the magnitude is constant, then the distribution only depends on . Functions in Eq. (6) for various s and s satisfying a constant are the same; they are only shifted. This can be shown mathematically as Display Formula
(13)where , , and . This can also be verified by plotting for a constant which is shown in Fig. 8.