## 2 Theoretical Background
### Projections, Rays, and Sinograms  
The results of tomography mainly consist of projections. These are sets of measurements produced by integrating along straight lines through an object. This integral is defined as the integral of a certain parameter of the object along a line (Equation (2)) like velocity and attenuation. These straight lines are called rays and are determined using Equation (1):  
\[x\cdot\cos(\theta)+y\cdot\sin(\theta)=t \tag{1}\]  
The definition of the integral line could be written as  
\[P_{\theta}(t)=\int_{(\theta,t)_{line}}f(x,y)dS \tag{2}\]  
where \(f(x,y)\) is a two-dimensional function that models the object under study. Expressing the previous integral as a double integral, we obtain Equation (3), where the function \(\delta\) represents the Dirac delta function.  
\[P_{\theta}(t)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}f(x,y)\cdot\delta( x\cdot\cos(\theta)+y\cdot sin(\theta)-t)dxdy \tag{3}\]  
where function \(P_{\theta}(t)\) is known as the Radon transformation [9] and describes the relationship between the object to be reconstructed and its projections. In this sense, a projection is formed by a combination of line integrals, as seen in Figure 1. The projection is a set of parallel rays for a constant \(\theta\). To obtain the rest of the projections, the source and detector are moved around the object.  
Transmission tomography was used in this work. The selection of this type of tomography restricted the implemented measuring system, in which two transducers face each other, one acting as the transmitter and the other as the receiver. Although the classical projection definition defines parallel rays, it could be redefined as the path connecting the transmitter to the receiver. Likewise, there are different types of projections depending on the direction and distribution of these rays. Although it is simpler to implement the parallel ray structure (see Figure 1a), a fan ray structure was selected due to the circular shape of the specimens. The transmitter and receiver positions were angularly equidistant (Figure 1b). Each projection was achieved by placing the transmitter in one position and moving the receiver by a certain angular increment. This was repeated for all possible positions of the transmitter, around the object under study, but it produced a low density of rays in the central part and a very high density in the outer part. For an object (Figure 2a), if the projections are arranged according to the different directions in which the object is illuminated, the sinogram will be a two-dimensional representation (Figure 2b). Therefore, for a constant \(\theta\), if we went through all the possible positions, we would obtain a representation of the projection for that angle, as seen in Figure 2c, where we can observe the corresponding cuts in the directions of 120 and 40 degrees, respectively.  
Figure 1: (**a**) Representation of a projection for a \(\theta\) angle of an object, mathematically represented as \(f(x,y)\), and the rays for this projection. (**b**) Transmission tomography fan ray structure.  
Figure 2: (**a**) Circular object. (**b**) Sinogram calculated with Radon transform, with 200 rays and 180 directions. (**c**) Representation of projections for two directions (dashed pink lines): \(\theta=120^{\circ}\) and \(\theta=40^{\circ}\).