# Algorithms in Tomography and Related Inverse Problems--A Review
## 1 Introduction
### Tomographic Reconstruction Techniques, Principles, and New Approaches  
In the realm of tomographic reconstruction, the journey begins with a fundamental categorization of algorithms, as detailed by [PERSON] and [PERSON] [1]. This classification neatly segments the methods for reconstructing objects from their projections into four distinct categories. These categories include the summation technique, the Fourier transform utilization, solving integral equations analytically, and employing series expansion methodologies. This initial categorization not only provides a structured approach to the field but also offers a basis for understanding the diversity of techniques at play. As we delve into this domain, according to [PERSON] [2], a pivotal study is provided where four key algorithms are adapted to directly reconstruct three-dimensional objectsfrom projections. This transformative step showcases the practical application of these methods. Among the selected algorithms, the Algebraic Reconstruction Technique, the Simultaneous Iterative Reconstruction Technique, and the Iterative Least Squares Technique are featured. This comprehensive approach exemplifies the potential for substantial streamlining of calculations and introduces the concept of tomographic projections. Diving further into the fine distinction of tomography reconstruction, [PERSON] and [PERSON] [3] distinguish between the 2D and 3D reconstruction problems. To comprehend the foundations of computed tomography image reconstruction, [PERSON], [PERSON], and [PERSON] provide essential insights in their work [4]. These foundational principles are paramount for understanding the core concepts and techniques applied in the field. In the realm of classical tomography, the 2D problem is prevalent, but as we move into 3D, considerations expand to density functions and lines with arbitrary orientations in space. Building upon this knowledge, the research conducted by [PERSON], [PERSON] et al. [5] offers an overview of cutting-edge 3D modeling algorithms developed over the past four decades. This rich resource equips researchers and practitioners with the latest advancements in tomographic reconstruction techniques. This differentiation is crucial, as it influences the choice of reconstruction methods in various applications. Of particular significance is the enhanced temporal resolution for cardiac imaging and the ability to acquire dual-energy information by operating the two tubes at varying voltages. The synergy of technological innovation and practical application is exemplified by [PERSON] and [PERSON] [6]. This reference introduces an automated technique capable of converting a series of tomographic image slices into an isotropic volume dataset. By establishing correspondence between points in consecutive slices and employing linear interpolation, this method offers a substantial enhancement in data accessibility. As reported by [PERSON] [7], the research delves into algorithms tailored for reconstructing attenuation images from transmission scans characterized by a low count of photons per beam. In situations where the average number of photons per beam is modest enough to warrant caution when employing traditional filtered backprojection imaging techniques, these algorithms offer a critical solution. The ongoing evolution underscores the dynamic nature of tomographic image reconstruction. As reported by [PERSON] and [PERSON] [8], their research indicates a critical advancement by integrating nonlocal boundary information into the regularization approach. This incorporation stems from the insights drawn from the reference and demonstrates the growing importance of statistical methods in tomographic image reconstruction. Beyond mere system modeling, these methods offer statistical models and adherence to physical constraints that surpass the traditional filtered backprojection method. Addressing specific challenges, [PERSON] et al.