# EPTlib: An Open-Source Extensible Collection of Electric Properties Tomography Techniques
## 3 EPT Techniques
### Convection-Reaction-EPT  
Convection-reaction-EPT overcomes the homogeneity assumption of Helmholtz-EPT by elaborating Equation (4) with no assumptions on the distribution of the EPs. The linearly combination of the \(x\) and \(y\) components of the vector Equation (4) according to the Tx sensitivity (5) leads to the following equation [19],  
\[\left(\frac{\partial\gamma}{\partial y}-\mathrm{i}\frac{\partial\gamma}{ \partial x}\right)\frac{1}{2}\left(\frac{\partial B_{1,y}}{\partial x}-\frac{ \partial B_{1,x}}{\partial y}\right)-\frac{\partial\gamma}{\partial z}\frac{ \partial B_{1}^{+}}{\partial z}=\gamma\
abla^{2}B_{1}^{+}+\omega^{2}\mu_{0}B_{ 1}^{+}\,, \tag{14}\]  
Figure 3: Savitzky–Golay filter windows of different shapes assuming a size equal to [2,2,1]. where the variable \(\gamma=\tilde{\epsilon}^{-1}\) has been introduced and the transverse derivatives of the longitudinal component of \(B_{1}\) (i.e., \(\partial B_{1,z}/\partial x\) and \(\partial B_{1,z}/\partial y\)) have been assumed to be negligible. This assumption holds, for instance, around the mid-plane of a birdcage coil, in which case also the longitudinal derivative of \(B_{1,z}\) would be negligible, and so  
\[\
abla\cdot B_{1}\cong\frac{\partial B_{1,x}}{\partial x}+\frac{\partial B_{1,y}}{\partial y}\cong 0\quad\Rightarrow\quad\frac{\partial B_{1}^{+}}{ \partial x}-\mathrm{i}\frac{\partial B_{1}^{+}}{\partial y}\cong\mathrm{i} \frac{1}{2}\left(\frac{\partial B_{1,y}}{\partial x}-\frac{\partial B_{1,x}}{ \partial y}\right). \tag{15}\]  
By substituting (15) in (14), the following equation, in which only the measurable component \(B_{1}^{+}\) of the magnetic flux density appears, is obtained [19],  
\[\left(\frac{\partial\gamma}{\partial x}+\mathrm{i}\frac{\partial\gamma}{ \partial y}\right)\left(-\frac{\partial B_{1}^{+}}{\partial x}+\mathrm{i} \frac{\partial B_{1}^{+}}{\partial y}\right)-\frac{\partial\gamma}{\partial z }\frac{\partial B_{1}^{+}}{\partial z}=\gamma\
abla^{2}B_{1}^{+}+\omega^{2} \mu_{0}B_{1}^{+}\,. \tag{16}\]  
The conservative form of the latter equation is [41]  
\[\boxed{\
abla\cdot\left(\gamma\mathbf{\beta}^{+}\right)=-\omega^{2}\mu_{0}B_{1}^{ +}}\,, \tag{17}\]  
being \(\mathbf{\beta}^{+}=\
abla B_{1}^{+}-\mathrm{i}\
abla\times\left(B_{1}^{+}\mathbf{z}\right)\) the convective term. The latter equation is treated as a complex-valued partial differential equation (PDE) with respect to \(\gamma\) and complemented with Dirichlet boundary conditions, namely the value of \(\gamma\) at the boundary of the computational domain is established a priori. Since, in case of a birdcage coil, Equation (17) holds only around the mid-plane of the coil, where \(\
abla B_{1,z}\cong\mathbf{0}\), the determination of the boundary conditions on the top and bottom slices of the region of interest could be tricky. This issue could be avoided by solving the PDE in a single slice with the two-dimensional approximation obtained assuming the longitudinal invariance of the EPs (i.e., \(\partial\gamma/\partial z\cong 0\)),  
\[\boxed{\
abla_{xy}\cdot\left(\gamma\mathbf{\beta}_{xy}^{+}\right)+\gamma\frac{ \partial^{2}B_{1}^{+}}{\partial z^{2}}=-\omega^{2}\mu_{0}B_{1}^{+}}\,. \tag{18}\]  
Similarly to Helmholtz-EPT, also convection-reaction-EPT admits a phase-based approximation [23]. It can be deduced by observing that if the Tx sensitivity \(|B_{1}^{+}|\) is almost homogeneous, then Equation (17) is approximated by  
\[\
abla\cdot\left(\gamma\mathbf{\beta}_{\varphi}^{+}\right)+\mathrm{i}\gamma|\
abla \varphi^{+}|^{2}=\mathrm{i}\omega^{2}\mu_{0}\,, \tag{19}\]  
being \(\mathbf{\beta}_{\varphi}^{+}=\
abla\varphi^{+}-\mathrm{i}\
abla\times\left( \varphi^{+}\tilde{z}\right)\). A similar equation can be deduced for the Rx sensitivity under the assumption of a Rx magnetic flux density whose longitudinal component has negligible derivatives and of an almost homogeneous Rx sensitivity \(|B_{1}^{-}|\).