We provide conditions for exact reconstruction of a bandlimited function from irregular polar samples of its Radon transform. First, we prove that the Radon transform is a continuous L2-operator for certain classes of bandlimited signals. We then show that the Beurling-Malliavin condition for the radial sampling density ensures existence and uniqueness of a solution. Moreover, Jaffard's density condition is sufficient for stable reconstruction.
Bandlimitedness conditions also appear implicitly in reconstruction techniques based on discretizations of the inverse Radon transform. Filtered backprojection tacitly assumes that the Radon transform is bandlimited and periodic in the radial coordinate (for computation of a so-called absolute derivative operator) and that quadrature rules for the angular integral (for the backprojection) are exact---for example by assuming that the angular component of the Radon transform has a finite Fourier series representation @cite_14 .
@cite_14
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The Mathematics of Computerized Tomography covers the relevant mathematical theory of the Radon transform and related transforms and also studies more practical questions such as stability, sampling, resolution, and accuracy. Quite a bit of attention is given to the derivation, analysis, and practical examination of reconstruction algorithm, for both standard problems and problems with incomplete data.