# Enhanced Hybrid Algorithms for Inverse Problem Solutions in Computed Tomography
## 2 The Issue of Computed Tomography with an Incomplete Dataset
### Methodology of the Solution  
The main objective of the research conducted in this paper is to reduce the time required to achieve satisfactory reconstruction quality. This acceleration occurs on multiple levels. The first fact on which we have based our research is the precision of the reconstruction itself.  
Another approach to acceleration involves simulating the location of potential inclusions. This time, we represent such objects as circles. This approach offers dual benefits. The first advantage is the reduced number of parameters required to uniquely describe a circle, which is one less than the number needed for the previously considered rectangles.  
A rectangle is defined by the following five characteristics: the position of the bottom-left vertex (two numbers), the top-right vertex (two numbers), and the type of material (e.g., coal, compressed gas, rock) from which it is made. The material type is represented by a single number corresponding to its ability to absorb the energy of the penetrating radiation beam used for projections. A circle is defined by the following four numbers: the position of the center (two numbers), the radius length (one number), and the type of material it is made of. Reducing the number of parameters by one may seem like a small improvement, but when searching for four such circles, the problem's dimensionality decreases from 20 to 16, which is already a significant change. The second benefit of using circles instead of rectangles is the speed of obtaining simulated projections. The complexity of finding the length of the segment that is the common part of a line and a rectangle is much greater than the complexity of finding the length of the segment that is the common part of a line and a circle (where it is enough to calculate the distance from the point to the line, which is equivalent to using a single formula). It should be mentioned that such simulations in heuristic algorithms are performed hundreds, if not thousands, of times, which results in a significant overall time savings. Another acceleration of the reconstruction process is achieved through the use of a hybrid algorithm, which searches the space in a significantly more efficient manner. As a result, the computation time is shorter.  
Based on the previously mentioned property that each type of matter has a unique ability to absorb energy, we aim to reconstruct the internal structure of the studied object based on the projections, which form the vector \(p=[p_{1},p_{2},\ldots,p_{n}]\) of these losses. To do this, we search for a distribution of circles inside the studied object, for which the simulated projection vector \(\hat{p}\) will be as close as possible to the vector \(p\), and the measure of this fit will be the value of the minimized functional, as follows:  
\[\mathcal{F}=\|p-\hat{p}\|=\sqrt{\sum_{i=1}^{n}(p_{i}-\hat{p}_{i})^{2}}. \tag{1}\]  
After setting a certain accuracy \(\varepsilon_{h}\) for the heuristic algorithm, the solution obtained from such an algorithm with this accuracy is passed as the starting solution to a deterministic algorithm, which seeks a solution with the corresponding accuracy \(\varepsilon_{d}<\varepsilon_{h}\).