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.
Preface to the second edition Preface How to use this nook Notational conventions 1. Measurements and modeling 2. Linear models and linear equations 3. A basic model for tomography 4. Introduction to the Fourier transform 5. Convolution 6. The radon transform 7. Introduction to Fourier series 8. Sampling 9. Filters 10. Implementing shift invariant filters 11. Reconstruction in X-ray tomography 12. Imaging artifacts in X-ray tomography 13. Algebraic reconstruction techniques 14. Magnetic resonance imaging 15. Probability and random variables 16. Applications of probability 17. Random processes A. Background material B. Basic analysis Bibliography Index. In image reconstruction there are techniques that use analytical formulae for the Radon transform to recover an image from a continuum of data. In practice, however, one has only discrete data available. Thus one often resorts to sampling and interpolation methods. This article presents an approach to the inversion of the Radon transform that uses a discrete set of samples which need not be completely regular. In this paper, we suggest a new Fourier transform based algorithm forthe reconstruction of functions from their nonstandard sampled Radon transform. The algorithm incorporates recently developed fast Fourier transforms for nonequispaced data. We estimate the-corresponding aliasing error in dependence on the sampling geometry of the Radon transform and confirm our theoretical results by numerical examples. We consider the problem of reconstructing a 2D bandlimited signal from its nonuniform samples taken in polar coordinates. We introduce two nonuniform sampling strategies in polar coordinates and develop algorithms for reconstructing the signal from these samples. The proposed methods collect nonuniform samples along concentric circles or radial lines, where the circles or lines are nonuniformly distributed. We then apply these methods to the problem of reconstruction of tomographic images and show through simulations that they result in a higher quality of reconstruction with respect to the traditionally used algorithms.