Introduction to the Mathematics of Medical Imaging [Epstein]

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Introduction to the Mathematics of Medical Imaging: Second Edition, published by SIAM, develops the mathematical foundations behind modern medical imaging, using X-ray computed tomography (CT) as its main example. The book begins by modeling how X-rays interact with matter and then connects the resulting measurements to linear algebra, integral equations, the Radon transform, Fourier analysis, convolution, sampling theory, and noise. A central theme is the reconstruction of a three-dimensional object from two-dimensional measurements: the Radon transform represents the measured line integrals, while the central slice theorem connects these measurements to the Fourier transform.

The later chapters move from mathematical theory toward practical reconstruction algorithms. In particular, the book derives the filtered back-projection formula, an exact inversion of the Radon transform that underlies many CT reconstruction methods, and then examines the complications introduced by discrete sampling, filtering, measurement errors, and noise. 

The second edition also adds material on magnetic resonance imaging (MRI), the relationship between continuous and discrete Fourier transforms, gridding, and MRI noise analysis. Overall, it is a bridge between mathematical analysis and computational medical imaging, showing how abstract tools such as Fourier transforms and integral transforms become concrete algorithms for producing diagnostic images. 

BOOK