# EPTlib: An Open-Source Extensible Collection of Electric Properties Tomography Techniques
## 3 EPT Techniques
### Gradient-Ept  
Gradient-EPT is developed starting from the same equation of convection-reaction-EPT (17), obtained without the assumption of homogeneity of the EPs used by Helmholtz-EPT. However, the computation is significantly different between the two techniques. In particular, gradient-EPT assumes that a parallel transmission (pTx) system with multiple Tx channels is used. In order to distinguish the quantities associated to each Tx channel, they are denoted in the following by the subscript \(i\) with reference to the \(i\)-th Tx channel. Being based on Equation (17), gradient-EPT works under the hypothesis that the gradient of the longitudinal component of \(B_{1,i}\) is negligible, for any channel \(i\). This is verified around the mid-plane of longitudinally symmetric sources, which are usually adopted for volume coils also in pTx systems. By noting that  
\[\frac{1}{\gamma}\
abla\gamma=-\frac{1}{\tilde{\epsilon}}\
abla\tilde{\epsilon }=-\
abla\log(\tilde{\epsilon})\,, \tag{24}\]  
Equation (16) can be conveniently arranged in the equivalent form [20]  
\[\
abla B_{1,i}^{+}\cdot(g_{+},-{\rm i}g_{+},g_{z})-\omega^{2}\mu_{0}\tilde{ \epsilon}B_{1,i}^{+}=\
abla^{2}B_{1,i}^{+}\,, \tag{25}\]  
where \(g=\
abla\log(\tilde{\epsilon})\) and \(g_{+}=g_{x}+{\rm i}g_{y}\) have been introduced. In a pTx system, the positively rotating component \(B_{1,i}^{+}\) cannot be estimated easily, since the TRx phase does not approximates, in general, the Tx phase. However, assuming that the same Rx coil, or combination of Rx coils, has been used to acquire the signal of each Tx channel, the measured TRx phases \(\varphi_{i}^{\pm}=\varphi_{i}^{+}+\varphi^{-}\) can be combined to estimate the relative Tx phases \(\varphi_{i}^{+,x}\) with respect to a reference Tx phase \(\varphi_{0}^{+}\),  
\[\varphi_{i}^{+,x}=\varphi_{i}^{\pm}-\varphi_{0}^{\pm}=\varphi_{i}^{+}-\varphi _{0}^{+}\,. \tag{26}\]  
Thus, the positively rotating component of the \(i\)-th channel can be written as  
\[B_{1,i}^{+}=B_{1,i}^{+,x}{\rm e}^{{\rm i}\varphi_{0}^{+}}\,, \tag{27}\]  
being \(B_{1,i}^{+,x}=|B_{1,i}^{+}|^{{\rm i}\varphi_{i}^{+,x}}\) the measurable part and \(\varphi_{0}^{+}\) the unknown reference phase. Substituting Equation (27) in (25) and cancelling out the common factor \({\rm e}^{{\rm i}\varphi_{0}^{+}}\), the following equation is obtained [20],  
\[\boxed{-{\rm i}2\
abla B_{1,i}^{+,x}\cdot\
abla\varphi_{0}^{+}+\
abla B_{1,i}^ {+,x}\cdot(g_{+},-{\rm i}g_{+},g_{z})+B_{1,i}^{+,x}\theta^{+}=\
abla^{2}B_{1, i}^{+,x}}\,, \tag{28}\]where \(\theta^{+}=-\omega^{2}\mu_{0}\bar{\varepsilon}+|\
abla\rho_{0}^{+}|^{2}-\mathrm{i} \
abla^{2}\rho_{0}^{+}+\mathrm{i}\
abla\varphi_{0}^{+}\cdot(g_{+},-\mathrm{i}g _{+},g_{z})\) combines the EPs with the other unknown terms. A system of linear equations in the algebraically independent unknowns \(\
abla\varphi_{0}^{+}\), \(g_{+}\), \(g_{z}\) and \(\theta^{+}\) is obtained in each voxel by writing Equation (28) for each Tx channel \(i\) and solved in the least-squares sense. In order to reduce the presence of numerical artefacts, a linear system is solved using each Tx channel as the reference one; the results are then averaged weighted by the Tx sensitivity magnitude of the corresponding reference channel [20]. An estimate of the EPs can be deduced by combining the unknown terms. The solution of (28) is the local step of gradient-EPT, which can be followed by a noise-filtering global step, consisting in the solution in the computational domain \(\Omega\) of the minimization problem [20]  
\[\boxed{\min_{\bar{\varepsilon}}\left(\left\|g_{+}(\bar{\varepsilon})-g_{+ \beta}\right\|_{L^{2}(\Omega)}^{2}+\left\|g_{z}(\bar{\varepsilon})-g_{z,0} \right\|_{L^{2}(\Omega)}^{2}\right)}\,, \tag{29}\]  
where \(g_{+,0}\) and \(g_{z,0}\) denotes the solutions of the local step in each voxel, whilst \(g_{+}(\bar{\varepsilon})\) and \(g_{z}(\bar{\varepsilon})\) are the derivatives of the guess distribution \(\bar{\varepsilon}\).