The book "The Mathematics of Computerized Tomography" by F. Natterer provides a comprehensive overview of the mathematical theory and algorithms of computerized tomography (CT). CT is a technique used to reconstruct a function from its line or plane integrals, with applications in various fields including diagnostic radiology. The book aims to be accessible to both research mathematicians and practitioners, covering the state-of-the-art in the mathematical theory of CT as it has evolved since the early 1970s. Key topics include: - **Introduction to CT**: Basic examples and applications. - **Radon Transform and Related Transforms**: Definitions, inversion formulas, uniqueness, ranges, Sobolev space estimates, and the attenuated Radon transform. - **Sampling and Resolution**: The sampling theorem, resolution, and two-dimensional sampling schemes. - **Ill-Posedness and Accuracy**: Ill-posed problems, error estimates, and the singular value decomposition of the Radon transform. - **Reconstruction Algorithms**: Filtered backprojection, Fourier reconstruction, Kaczmarz's method, and algebraic reconstruction techniques. - **Incomplete Data**: General remarks, limited angle problem, exterior problem, interior problem, restricted source problem, and reconstruction of homogeneous objects. - **Mathematical Tools**: Fourier analysis, integration over spheres, special functions, Sobolev spaces, and the discrete Fourier transform. The book is self-contained, with necessary mathematical background reviewed in an appendix. It is based on courses taught at the Universities of Saarbrücken and Münster, and includes contributions from several scholars who provided valuable feedback and support.The book "The Mathematics of Computerized Tomography" by F. Natterer provides a comprehensive overview of the mathematical theory and algorithms of computerized tomography (CT). CT is a technique used to reconstruct a function from its line or plane integrals, with applications in various fields including diagnostic radiology. The book aims to be accessible to both research mathematicians and practitioners, covering the state-of-the-art in the mathematical theory of CT as it has evolved since the early 1970s. Key topics include: - **Introduction to CT**: Basic examples and applications. - **Radon Transform and Related Transforms**: Definitions, inversion formulas, uniqueness, ranges, Sobolev space estimates, and the attenuated Radon transform. - **Sampling and Resolution**: The sampling theorem, resolution, and two-dimensional sampling schemes. - **Ill-Posedness and Accuracy**: Ill-posed problems, error estimates, and the singular value decomposition of the Radon transform. - **Reconstruction Algorithms**: Filtered backprojection, Fourier reconstruction, Kaczmarz's method, and algebraic reconstruction techniques. - **Incomplete Data**: General remarks, limited angle problem, exterior problem, interior problem, restricted source problem, and reconstruction of homogeneous objects. - **Mathematical Tools**: Fourier analysis, integration over spheres, special functions, Sobolev spaces, and the discrete Fourier transform. The book is self-contained, with necessary mathematical background reviewed in an appendix. It is based on courses taught at the Universities of Saarbrücken and Münster, and includes contributions from several scholars who provided valuable feedback and support.