# EPTlib: An Open-Source Extensible Collection of Electric Properties Tomography Techniques
## 4 Examples of Application
### 1.5 T Body-Coil
#### 4.1.2 Convection-Reaction-EPT  
The convection-reaction-EPT technique applied on the complex input data of the simulated 1.5 T body-coil led to extremely ill-conditioned linear systems unless an over-regularization was introduced, making the method unfeasible even in the noiseless case. The results reported in Figure 7 are obtained by the phase-based approximation of convection  
Figure 6: Electric conductivity distributions recovered by complete Helmholtz-EPT. (**a**) Actual distribution at 64 MHz in the considered section. (**b**) Distribution recovered from noiseless input with a cross window of size [1,1,1] for the Savitzky–Golay filter. (**c**) Distribution recovered from input with SNR 200 using a cubic window of size [2,2,2] for the Savitzky–Golay filter. (**d–f**) Distributions recovered from input with SNR 100 using windows of size [3,3,3] and different shapes (cross, ellipsoid and cuboid, respectively) for the Savitzky–Golay filter.  
reaction-EPT with artificial diffusion regularization applied on a three-dimensional portion of data. Precisely, 21 slices (10 above and 10 below the mid-plane) were considered and found to be enough to reduce the influence of the boundary conditions on the top and bottom slices on the reconstruction in the mid-plane. Figure 7 collects the distribution of \(\sigma\) recovered for different SNR in the input data and different values of the artificial diffusion coefficient \(\lambda\). All the results were obtained using the minimal Savitzky-Golay filter, namely the cross of size [1,1,1]. The reference distribution of \(\sigma\) was the one reported in Figure 6a.  
The reconstructions obtained operating on noiseless data show that a negligible loss of details occurred when the coefficient \(\lambda\) was increased from 0.001 up to 0.003. In particular, a small reduction in the value of the cerebrospinal fluid (CSF) conductivity was observed, with no sensible consequences on the effective image resolution. Moreover, the distributions of \(\sigma\) obtained by convection-reaction-EPT with noiseless input (cf. Figure 7) were significantly improved with respect to the distribution estimated by Helmholtz-EPT (cf. Figure 6b), especially because of the absence of large errors at the tissue boundaries.  
When the input data were noisy, the conditioning of the problem at the basis of convection-reaction-EPT worsened, making its resolution more time-consuming and affecting the quality of the result. However, the artificial diffusion term restored the conditioning with a regularization effect that became more relevant when the coefficient \(\lambda\) increased, as can be observed in Figure 7 for the cases with SNR 200 and, more evidently, with SNR 100. It is worth noting that, differently from the increased windows of the Savitzky-Golay filter  
Figure 7: Electric conductivity distributions recovered by the phase-based approximation of convection-reaction-EPT with artificial diffusion regularization. All the distributions are recovered using a cross window of size [1,1,1] for the Savitzky–Golay filter. Different artificial diffusion coefficients are used (\(\lambda=0.001\), 0.002 and 0.003) on input data with different amounts of noise (noiseless, signal-to-noise ratio—\(\text{SNR}=200\) and \(\text{SNR}=100\)). To be compared with the actual distribution at 64 MHz reported in Figure 6a.  
used for Helmholtz-EPT, the artificial diffusion term managed the noise propagation without affecting significantly the image resolution. The adoption of a large window for the Savitzky-Golay filter in conjunction with the artificial diffusion regularization could help when dealing with very noisy input data.