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The semidiscrete filtered backprojection algorithm is optimal for tomographic inversion. (English) Zbl 1052.65118
The filtered backprojection algorithm (FBA) is probably the most often used reconstructing algorithm in two-dimensional computerized tomography. The reformulation of the FBA leading to optimal $$L^2$$-convergence rates in a semidiscrete setting where the discrete backprojection operator is replaced by the continuous one. It is observed that the reconstruction filter, the interpolation process and the Sobolav regularity of the density distribution $$f$$ influence the convergence rate.
As a by product of this analysis, they discover a new reconstruction filter with an improved convergence behavior compared to the widely use Shepp-Logan filter. This modified Shepp-Logan filter yields optimal convergence for Sobolev orders up to 5/2, whereas the convergence order of the original Shepp-Logan filter saturates at 2. Numerical experiments in the fully discrete setting reproduce the theoretical predictions.

##### MSC:
 65R10 Numerical methods for integral transforms 44A12 Radon transform 92C55 Biomedical imaging and signal processing
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