# Review on Electrical Impedance Tomography: Artificial Intelligence Methods and its Applications
## 4 Solving the Inverse Problem  
[PERSON] defined that for vector spaces \(X\) and \(Y\) such that \(T\): \(X\to Y\). For a given general problem \(T(x)=y\) where \(x\) is unknown. This problem is defined as the well-posed problem only if it satisfied the following conditions [6]. * The inverse mapping is continuous. * The solution does exist. * The solution is unique. By contrast, if the problem does not satisfy any of the conditions as mentioned above the problem is an ill-posed or ill-conditioned problem. For solving \(T(x)=y\) for the unknown \(x\) it is called as the inverse problem. It is generally used when \(T\) is continuous and exists for evaluating \(T\) such that the forward problem is a classical or least well-known problem. The term inverse problem is only used for an ill-posed problem according to the Hadamard conditions. EIT is an ill-conditioned problem since there is not only a unique solution (image reconstruction) of this inverse problem for the given potential border distribution. [PERSON] et al. have concluded that in EIT inverse problem the image generated is not always consistent or well defined [7]. As a result, different methods are adopted to solve the EIT inverse problem as an optimization problem. The objective is to reduce the relative error between the calculated border potential of the solution candidate and measure border potential of an object. Many researchers have worked on different artificial intelligence methods and evolutionary computational techniques to solve this inverse problem. In the beginning, many linear algorithms for image reconstruction have been proposed including the sensitivity matrix technique, equipotential back projection, and regularization approach. On the other hand, the linear hypothesis is effective only if the perturbation from the linearization point is slightly small in which there are no large areas where conductance varies not too much from the background. Many reconstruction approaches have been used for EIT, the standard method is the use of regularized Newton-type techniques. The application of this method to solve the inverse problem is quite clear since the EIT problem is non-linear so the first step is to linearize it since the linear problem is an ill-posed problem, and therefore, regularize it; the linear approximation cannot reconstruct complex geometries or large contrasts so the process must be applied iteratively. The most common non-linear Newton-type method is the Newton-Raphson method. This technique works by searching the conductivity distribution that reduces the difference between applied potentials and the calculated potential from the forward problem by finite element method. The drawback of this algorithm is that it is computationally complex, requires numerous orders of magnitude and computational time is high as compared to the linear methods. These non-linear methods give good results if the hypotheses about the problem are simplified. A Neural Network (NN) is also used to solve the linear problem as compared to linear algebraic methods. This is because the problem is ill-posed. The advantage of using the NN is that the inverse of the matrix is taken directly from the finite element method and Signal to Nosie Ratio (SNR) is achieved without any initial hypotheses and calculations. However, the training of the NN is highly affected due to the ill-conditioning of the problem. During the training, the NN responds to training patterns near the boundary of the domain and at the center, it responds to the training patterns only when the performance on the boundary is approximately optimal. Moreover, the under-determination indicates that spatial resolution is poorer to the size of the training patterns [8]. As a result, the NN can never replicate the training patterns precisely, and the error will never tend to zero, but will only decrease to a fraction of the initial error.