## 3 Electrical Tomography Principles and Applications
### Fundamentals of Electrical Tomography
#### 3.2.3 Image Reconstruction  
For all electrical tomography techniques, the performed measures are sensitive inside a certain volume; this sensitivity is non-constant along the section of interest contained within the volume, and for any given position inside this section, it is also dependent on the spatial resolution of the physical parameter that is being imaged along the entire section of interest. This non-linear behaviour makes obtaining images for the processes under evaluation quite difficult. All electrical tomographic procedures are governed by analogous partial differential equations, and consequently the reconstruction algorithms used share many similarities. Using the data measured, capacitances and potential voltages differences in ECT and ERT/EIT, respectively, in order to construct an image of the domain under evaluation is called inverse problem solving; conversely, inferring the values that will be measured based on a known image of the domain is called forward problem resolution (it is a question of mapping a set of hypothetical parameters onto a set of experimental measurements). For ECT, forward problem-solving means calculating the capacitance values in the domain for a given dielectric permittivity inside the same domain, and this problem has a unique solution. With respect to inverse problem solving, aside from being a non-linear problem, it is also ill-posed and ill-conditioned; mathematically speaking, its solution is based on a matrix inversion approach (in ECT, this corresponds to the determination of the material distribution in the domain by means of capacitance data values). Inverse problem solving is also referred to as a reconstruction problem. Mathematically, the objective is to calculate the reciprocal of the operator F, which describes the original distribution of the dielectric characteristics in the domain, \(\varepsilon=\varepsilon(r)\), using the measured capacitance data values \(C^{*}\):  
\[\varepsilon(r)=F^{-1}[C^{*}] \tag{1}\]  
Due to its non-linearity, there is no analytical answer to this reconstruction problem. The number of variables that are unknown is higher than the collected measurements, and consequently an infinite number of solutions exists matching the measured capacitances. The best solution will be the constructed permittivity distribution that best fits the measured capacitance values. The forward and reconstruction problem solving using EIT follow analogous approaches. In the forward or direct problem, an approximation of the electrical potentials in the boundary of the domain is calculated using the initially estimated conductivity/resistivity distribution. On the other hand, in the reconstruction problem, the conductivity/resistivity distribution in the domain is estimated using the values of the electrical potentials measured along the domain boundary, using an appropriate mathematical algorithm. In the direct problem, and using Maxwell laws [58], a model of the electric field distribution in the domain is constructed. In any case, before addressing the reconstruction problem, it is necessary to calculate the solution for the forward one, which will be performed using a calibration tool. Typically, Finite Element Methods (FEM) are used to solve the forward problem, i.e., to calculate the voltages/capacitances at the boundary of the domains for a known permittivity/conductivity distribution inside the same domain. A sensitivity matrix is obtained from the measured capacitances/electrical potential differences values. The direct problem solution obtained is then fed into the inverse problem algorithm, together with the measured capacitances/electrical potential differences values, which calculates the permittivity/conductivity distribution. The interpretation of the observed permittivity/conductivity distribution is carried out using the sensitivity matrix and the measured data along the boundary.