## 2 Background
### 3D ECT Image Reconstruction
The sensitivity map \(J_{j}\) for a given 3D point \(j(x,y,z)\) can be computed using the electrical field response to changes in capacitance \(C\) (and electric potential \(\phi\)), according to the following formula:  
\[J_{j}=-\int_{A_{j}}\frac{E_{e_{1}}}{\phi_{e_{1}}}\frac{E_{e_{2}}}{\phi_{e_{2}} }\,\mathrm{d}j \tag{6}\]  
where \(E_{e_{1}}\) and \(E_{e_{2}}\) are the electrical field intensity values at point \(j\), while the electrodes \(e_{1}\) and \(e_{2}\) store potentials \(\phi_{e_{1}}\) and \(\phi_{e_{3}}\), respectively, and \(A_{j}\) denotes the volume of point \(j\) in the imaging region. Calculating sensitivity across all 3D grid points or elements for all unique measurement electrode combinations yields a sensitivity matrix, \(J\). Directly employing matrix \(J\) in the image reconstruction process is mathematically problematic, due to its non-rectangular nature. Additionally, the number of unknowns, \(N\), corresponding to image elements is typically significantly larger than the number of equations, \(M\) (the number of capacitance measurements). As a result, multiple solutions are anticipated, leading to various reconstructed image quality issues, such as artifacts, blurred regions, limited axial or spatial resolution, heterogeneity, and poor close objects separation capabilities, especially in 3D mode. To address these numerical challenges, numerous research efforts have focused on optimizing the reconstruction process and minimizing the final image errors during image construction [10]. One straightforward solution involves replacing the non-invertible matrix \(J^{-1}\) with its transposed approximation, \(J^{T}\). Equation (5) with Equation (6) can then be reformulated in a discrete matrix form:  
\[G_{3D}=J^{T}C_{m}. \tag{7}\]  
To address the inverse nature of the ECT reconstruction process defined by Equation (7), numerous algorithms have been developed to mitigate the ill-posed and ill-conditioned nature of this problem in Electrical Capacitance Tomography. These algorithms can be broadly classified into three main groups: one-step, fast-but-less-preciseimage reconstruction algorithms, statistical and stochastic methods, and accurate-but-computationally expensive iterative algorithms. The one-step method family includes Linear Back Projection (LBP), Singular Value Decomposition, Tikhonov Regularization, Multiple Linear Regression with Regularization, and other algorithms investigated by various researchers [6; 10; 11]. One-step approaches are known to significantly limit 3D ECT imaging, resulting in low spatial and axial resolution, numerous image artifacts, and poor image quality in specific regions of a 3D ECT scan. However, these limitations can sometimes be acceptable for specific applications, such as emergency state detection or tracking fast-moving objects with high permittivity. Linear direct reconstruction methods offer near-real-time computation using modern PC hardware, such as GPU CUDA parallel computing [12], which is often required in industrial applications. Statistical and stochastic methods, including Bayesian inference, Monte Carlo Markov chain sampling, and the Metropolis-Hastings algorithm, have also been developed to improve the ECT imaging process [13; 14]. Another group of reconstruction algorithms involves iterative optimization methods such as the Gauss-Newton algorithm [15], Landweber iterations, the Simultaneous Iterative Reconstruction Technique (SIRT) [6], and non-linear (FEM)-based algorithms. These methods are especially crucial for the research presented here. They generate a finite element grid where the nodes represent the electric permittivity at the corresponding spatial locations within the pipe. This finite element grid enables visualization of the pipe's contents, using specialized tools. Non-linear inversion approaches, driven by numerical algorithms and a comprehensive numerical sensor model, significantly enhance the efficiency of 3D ECT inversion [9]. Known for their superior quality, these methods excel in both 2D and 3D ECT image reconstruction.