Numerical and theoretical explorations in helical and fan-beam tomography

Katsevich's inversion formula for helical tomography is explored in the limit of vanishing pitch, yielding a general reconstruction formula for fan-beam tomography. The relationship of this formula to other formulas in the literature is explored and a rigorous proof of a rebinning formula relating parallel-beam and fan-beam tomography is given. For the case of curved detector coordinates several numerical implementations of this formula and a related fan-beam formula are proposed, numerically implemented, and compared with the standard fan-beam algorithm. This gives insight into some numerical questions also encountered in the three-dimensional case, including a theoretical explanation of the usefulness of a shift in the convolution kernel for removal of ringing artifacts. A new discretization scheme for the derivatives is suggested and shown to be promising in both two and three dimensions. Numerical experiments with simulated as well as real data are presented.

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