## 3 Electrical Tomography Principles and Applications
### Fundamentals of Electrical Tomography
#### 3.2.3 Image Reconstruction
The methodologies used for solving the inverse problem, and consequently to obtain the 2D cross-sections of the domains, can be classified into three categories: linear; non-linear iterative, and heuristic multivariate methods. * Linear methods (single-step and iterative methods): In this approach, images are generated by simply multiplying the measurements by a pre-calculated matrix, and consequently they are quite fast in terms of computational effort. Among the linear methods, Linear back-projection (LBP) is the most widely used (the used matrix is the transpose of an estimated solution of the forward problem [82]). Nevertheless, they are usually non-iterative methods. Images obtained by means of LBP are low-resolution, heavily smoothed ones, because the transpose of the forward problem solution is a poor estimate of the solution to the inverse problem. To circumvent some of these limitations, an iterative back projection algorithm was proposed by [PERSON] et al. [83]. Linear methods have been improved by including approaches based on ridge-regression and eigenanalysis [84]. * Non-linear methods: in this approach, numerical forward solvers that predict the measurements, together with sensitivity maps that make it possible to calculate the measurement residuals, are used to estimate the image. Then, in a second stage, that image is updated via a non-linear iterative scheme, such as the modified Newton Raphson (NR) [85] or adaptive mesh regenerating techniques [86]. However, because NR can introduce artificial errors into the solution, regularisation procedures, such as the Marquard and the Tikhonov procedures, are applied to obtain a better approximation at each iteration [73]. In cases where direct inverse solution algorithms are used, there is a search for the minimised residual vector, making it possible to obtain images with improved accuracy [73; 74]. In cases where higher measurement flexibility is desired, non-linear iterative methods are to be recommended; however, the considerable computational effort makes image reconstruction difficult and too slow. Real-time reconstruction of images using non-linear iterative methods is their major drawback, although this may change through a combination of efficient algorithms [87] ([PERSON] et al. [88] decreased the time needed to reconstruct images using an adaptive mesh grouping method based on a fuzzy genetic algorithm). The increase in computing performance will also help to decrease the time required for image reconstruction. In FEM methods (Finite Element Methods), which are mesh-based reconstruction algorithms, in order to obtain higher-resolution images, mesh optimisation is also of major importance [74]. * Heuristic methods: these methods can be linear or non-linear, and they model empirically the relationship between calibration sets of images and measured ones. Methods making use of multiple linear regressions [89] belong in the linear sub-class. Self-organised maps and artificial neural networks [90] are examples of methods belonging to the non-linear sub-class. Heuristic models also allow the mathematical relations between the measurements and other variables of interest to be obtained [91]. In many cases, image reconstruction in process applications is just an intermediate stage towards the calculus of other variables.