# CTprintNet: An Accurate and Stable Deep Unfolding Approach for Few-View CT Reconstruction
## 2 The Numerical Model  
In this section, we briefly introduce a description of the 2D X-ray CT acquisition process, including some mathematical notations. The main focus of CT studies is to develop safer protocols to reduce the radiation dose for the patient since limiting the negative effects produced by the radiation allows the use of the CT technique in a wider class of medical examinations. This goal can be pursued by either decreasing the X-ray tube current at each scan (low-dose CT) or reducing the number of X-ray projections (few-view CT). In the first case, the measured data present high levels of noise; however, in the second case, the incompleteness of the projection data leads to create images with pronounced artifacts. In particular, in this paper, we focus on the few-view CT case, where image reconstruction is particularly difficult. Over the years, different tomographic devices have been designed to fit different medical needs, and this has led to the creation of various protocols and geometries. In particular, in this study, we focus on fan-beam geometry, schematized in Figure 1, since it is among the most widespread geometries nowadays. It is characterized by a source that emits fan-beam X-rays whose intensity is recorded, after passing through a body, by a decoder with \(N_{p}\) elements. The aim of CT imaging is to reconstruct the attenuation coefficient function \(\mu(x,y)\) of the object when passing through a body from the \(N_{\theta}\) projections acquired at equally spaced \(\theta_{k}\) angles, with \(k=1,\ldots,N_{\theta}\), and performed in the angular range \([-\Theta,\Theta]\) (in our case, \(\Theta=\frac{\pi}{2}\)). Each individual projection is modelled by the Radon transform, which is the integral along the line \(L\) describing the path of an X-ray beam at a fixed angle. According to Beer-Lambert's law, given \(l_{0}\) as the intensity emitted by the X-ray source and \(l\) as the intensity measured by the detector, we have  
\[-\ln\left(\frac{l}{l_{0}}\right)=\int_{L}\mu(x,y)\ \ d(x,y). \tag{2}\]  
In the real (discrete) case, the function \(\mu(x,y)\), which describes the object, is discretized in \(N=N_{x}\times N_{y}\) pixels, with values that can be lexicographically re-ordered in a vector \(x\in\mathbb{R}^{N}\). Figure 1: Diagram of a fan-beam projection. During the scanner process, the source \(S\) rotates along a circular trajectory around the discretized object, denoted by \(\{x_{0},\ldots,x_{16}\}\), and the decoder measures the intensities \(\{I_{1},\ldots,I_{7}\}\) of X-rays. Analysing a single projection acquired at a fixed angle \(\theta_{k}\) from \(N_{p}\) cells of the decoder, Equation (2) can be discretized into a sum over all the pixels as  
\[y_{i}^{\theta_{k}}=\sum_{j=1}^{N}H_{i,j}^{\theta_{k}}x_{j}\hskip 28.452756 pti=1, \ldots,N_{p} \tag{3}\]  
where \(y_{i}^{\theta_{k}}=-\ln\left(\frac{I_{i}}{I_{0}}\right)\) and \(I_{i}\) is the intensity measured by the \(i\)-th cell. Repeating for all the angles \(\theta_{k}\), with \(k=1,\ldots,N_{\theta}\), and using a compact notation, we obtain the linear system  
\[y=Hx, \tag{4}\]  
where \(x\in\mathbb{R}^{N}\) denotes the CT image to be reconstructed, \(y\in\mathbb{R}^{N_{d}}\) is the synogram obtained with the projection measurements (with \(N_{d}=N_{\theta}\times N_{p}\)), and \(H\in\mathbb{R}^{N_{d}\times N}\) is a sparse matrix representing the discrete line integrals in (2). Hence, tomographic image reconstruction is mathematically modeled as an ill-conditioned inverse problem, whose solutions can be dominated by noise. In particular, in the few-view CT case, the linear system is also under-determined due to the lack of projections, and, therefore, it might have infinite solutions. The model-based approach is introduced to address these numerical controversies, with the aim to model the CT imaging process as a minimization problem of a suitable cost function.