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In this article, we provide an introduction to the theory and techniques of X-ray tomography, and we focus on the case in which we only have X-rays over a limited range of angles. After explaining the mathematical problem, we delve into a study of the Radon transform and its relation to the Fourier transform. Along with microlocal analysis, this theory allows us to make precise the relationship ... read morebetween the smoothness of a function and the behavior of its localized Fourier transform, and it also predicts which singularites of a function will be reconstructed using standard reconstruction methods, even in the limited angle case. We illustrate these ideas with reconstructions of simple shapes, and we explore the effect of modifying the numerical techniques involved in implementation. Our ultimate goal is to understand both the numerical and mathematical obstacles of the problem and develop a method that best accounts for both aspects.read less
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