# Enhanced Hybrid Algorithms for Inverse Problem Solutions in Computed Tomography
## 1 Introduction  
In many cases, there is a need to understand the internal structure of certain objects. Such objects may have various physical properties, whose values influence the methods that can be used to explore their structure. However, in most cases, non-invasive methods are required, meaning the examined object must retain the same structure before and after the examination. One type of examination that provides insights into an object's interior is computed tomography (CT). The concepts behind CT emerged long before the construction of the first tomography scanner and even before the invention of the first computer [1]. CT algorithms are computationally complex, and without computers, such examinations would not be possible.  
Theorems ensuring the acquisition of high-quality images of an object's interior require achieving sufficient quality in the scans [2]. This quality is achieved by performing multiple scans from various angles. Some objects have properties that allow for high-quality scans,influenced by the object's size and accessibility. Examples include human bodies (where medical tomography is highly developed) [3], travelers' luggage [4], as well as objects in archaeology [5], art conservation [6], electronics [7], forensic science [8], agriculture [9], and many other fields [10, 11]. However, such favorable conditions are not always available. Often, the size or inaccessibility of an object does not allow for projections that meet the requirements and ensure adequate quality. Examples of such cases include structural elements of buildings (e.g., walls, bridge pillars, large castings like ship screws) or coal seams. In each of these cases, structural flaws or undesirable and potentially hazardous formations within coal seams, such as compressed gas pockets formed during the natural development of the seam, pose significant risks including financial losses and threats to human health and safety. The inability to obtain high-quality projections is referred to as the problem of an incomplete dataset.  
The previous studies conducted by the authors showed that despite the occurrence of an incomplete dataset, it was possible to reconstruct the internal structure of such objects (e.g., a coal seam) [12, 13, 14, 15]. The main drawback of these methods was the reconstruction time--algorithms based on incomplete datasets tended to become slow to converge, and achieving high-quality reconstruction required significant computational effort, which in turn demanded a lot of time and energy. In this paper, we combine two types of algorithms--a heuristic multi-agent algorithm and a deterministic space search algorithm. Their combination, a hybrid algorithm, eliminates the drawbacks of these algorithms and highlights their advantages.  
In Section 2, the considered issues are described, and a fitness function measuring the distances between projection vectors is defined. Section 3 is devoted to describing the applied optimization algorithms, including the metaheuristic multi-agent algorithms, Artificial Bee Colony (ABC) and Jellyfish Search (JS), as well as the deterministic algorithms, Nelder-Mead (N-M) and Hook-Jeeves (HJ). In Section 4, the results of the conducted numerical tests are presented. The results obtained using the hybrid algorithm are also compared with those from a standard metaheuristic algorithm. Finally, Section 6 provides a brief summary of the study conducted.