## 1 Introduction  
Electrical impedance tomography (EIT) is a modality that allows estimating the conductivity distribution inside the human body. Compared to other established imaging techniques, such as magnetic resonance imaging or computed tomography, it has some distinct advantages: firstly, an EIT system can be constructed at a low cost; secondly, it does not depend on potentially harmful X-rays; thirdly, it can be applied long-term at the bedside of the patient [1]. Medical applications of EIT may include the monitoring of the lung, including observing regional lung ventilation and perfusion [2], estimation of the size and volume of the bladder [3], monitoring of lung recruitment and lung collapse [4], tracking of 3D brain activity [5], and usage in breast cancer imaging [6]. The EIT can also be used outside the medical context, for example, for monitoring semiconductor manufacturing [7] or monitoring damage in concrete [8]. The EIT belongs to the class of inverse problems. The solution to such problems requires unique algorithms. These methodologies can be grouped into three major groups. The first set of methods are the variational regularization algorithms. These methods minimize a cost function that is made up of two parts. The first part minimizes the given measurements to a conductivity distribution, while the second part is a regularization term. Two examples of variational regularization algorithms are Thikonov regularization [9] and the total variational regularization [10]. The variational methods are mostly used in the field, as they deliver good results in rather quick time. Commercial devices such as the Draeger PulmoVista(r) 500 use this kind of algorithm. The second group is the statistical inversion methods for EIT. The measurements and conductivities are modeled as random variables. Here, an _a posteriori_ distribution can be estimated, which, in turn, can give estimates of the conductivity distribution [11]. The advantage of these types of algorithm is that not only is a reconstruction generated, but so is a whole statistical distribution of the reconstruction. Some publications also generate better results with such methods [12]. However, a major drawback is the computational cost. The third type of reconstruction algorithms are the direct inversion methods. They analyze the nonlinear inverse problem and try to develop an algorithm based on this analysis. One such algorithm is the D-Bar algorithm [13]. The D-Bar method is a direct, noniterative algorithm used to compute the conductivity distribution inside a domain of interest. The insights and tools for this algorithm date back to [PERSON] [14], who analyzed the inverse conductivity problem. Subsequently, [PERSON] showed that the problem is uniquely solvable in the two-dimensional (2D) domain under certain restrictions [15]. The first implementation of this algorithm was shown by [PERSON] et al. and, thus, opened the door for real-world usage of the algorithm [16]. Nowadays, the D-Bar algorithm is known to have many benefits, such as robustness towards electrode displacement, measurement errors, and errors in the injection current [17]. Significant disadvantages of EIT are its relatively low spatial resolution and sensitivity to noise. Regularization strategies are used to dampen the noise dependence. The regularization is a trade-off between a further loss of spatial resolution and resistance to noise. Due to this fact, there are several strategies to incorporate information from other modalities. This is often carried out by biasing the solution process with the _a priori_ information. [PERSON] et al. modified the sensitivity matrix by using X-ray data and a collection of impedance measurements to improve the reconstruction results [18]. [PERSON] and [PERSON] proposed nudging the solution of the inverse problem towards the desired solution by careful selection of the prior [19]. This approach could also craft a custom regularization matrix [18]. [PERSON] proposed a level set method based on ultrasound as _a priori_ data [20].