## 2 Background
### 3D ECT Image Reconstruction  
3D Electrical Capacitance Tomography reconstruction models generally exploit the relationship between the spatial distribution of the electric field within a capacitance sensor and the spatial distribution of the electric permittivity within the sensor volume, to visualize the contents of a pipe. This concept is illustrated in Figure 3:  
The spatial distribution of an ECT tomography electric field can also be defined from a mathematical and physical perspective. When spatial electric charge is negligible, the electric field behavior in a 3D ECT sensor is governed by the Laplace equation (see Equation (1)):  
\[\
abla[\epsilon(x,y,z)\
abla\phi(x,y,z)]=0\ \ x,y,z\in\ \Omega \tag{1}\]  
In this context, \(\epsilon(x,y,z)\) signifies the electric permittivity distribution in the volume \(\Omega\), while \(\phi(x,y,z)\) represents the electric potential distribution in space. The coordinates \(x,y,\) and \(z\) precisely locate a specific simulated point within the discrete space of the 3D ECT sensor. Knowing the electric potential \(\phi\) at a given point \((x,y,z)\), the induced charge \(Q_{E_{g}}\) on a grounded electrode \(g\) can be determined, using [PERSON]'s law (see Equation (2)):  
\[Q_{E_{g}}=\int_{E_{g}}\epsilon\frac{\delta\phi(x,y,z)}{\delta n}\,\mathrm{d}^{ 3}x, \tag{2}\]  
where n is the unit normal vector to the electrode surface. The initial boundary conditions \(\phi_{0}\) must be specified for all electrodes. At any given time, a single electrode, (\(E_{e}\)), is designated as the excited electrode and a positive electric potential, (i.e., \(\phi_{0}=\phi_{e}\)), is applied to it. Concurrently, the remaining electrodes are grounded (i.e., \(\phi_{0}=0\)). The capacitance \(C_{cg}\) between the electrode pair \(E_{e}\) and \(E_{g}\) can be defined as follows:  
\[C_{cg}=\frac{\iiint_{\Omega}\epsilon(x,y,z)\
abla\phi(x,y,x)\mathrm{d}\Omega}{ \phi_{e}-\phi_{g}} \tag{3}\]  
Figure 3: The general idea of 3D Capacitance Tomography [3]. Figure 2: 3D ECT system and its components [3]. where \(\phi_{e}-\phi_{\bar{g}}\) denotes the electrical voltage difference between the excited electrode \(E_{e}\) and the grounded electrode \(E_{g}\). As 3D ECT tomography is a relative, contrast-based tomography method, the capacitance data for a given object, \(\epsilon_{obj}\), must be normalized to obtain \(C_{m}\) by using pre-measurements for two electric permittivity distributions, \(\epsilon_{high}\) and \(\epsilon_{low}\), that significantly differ in dielectric constant. To calculate the normalized capacitance data \(C_{m}\), the following formula can be used:  
\[C_{m}=\frac{C(\epsilon_{obj})-C(\epsilon_{low})}{C(\epsilon_{high})-C( \epsilon_{low})}, \tag{4}\]  
In the simplest scenario, image reconstruction for a 3D ECT system can be viewed as an inverse problem that necessitates the estimation of the approximate spatial distribution of electric permittivity, \(G_{3D}\), inside an ECT sensor based on normalized capacitance measurements, \(C_{m}\), according to formula [5],  
\[G_{3D}=F(C_{m}) \tag{5}\]  
where \(F\) can be defined as a direct transformation function or an iterative optimization function. For 3D ECT systems, various representations of Jacobian matrices, which serve as sensitivity models, can be utilized [9]. The internal structure of the Jacobian matrix consists of \(M\) weight vectors that operate as sensitivity maps. Each map determines the sensitivity of a given point \(j(x,y,z)\) (from a set of \(N\) points--typically 3D grid nodes or elements) in the measurement space to a non-linear change in capacitance resulting from a perturbation in permittivity at that point. The Jacobian matrix is used to simplify and linearize the inherently non-linear image reconstruction process.