The Semidiscrete Filtered Backprojection Algorithm Is Optimal for Tomographic Inversion

The filtered backprojection algorithm is probably the most often used reconstruction algorithm in two-dimensional computerized tomography. For a semidiscrete version in the parallel scanning geometry we prove optimal L2-convergence rates for density distributions in Sobolev spaces. Additionally we show L2-convergence without rates when the density distribution is only in L2. The key to success is a new representation of the filtered backprojection which enables us to apply techniques from approximation theory. Our analysis provides further a modification of the Shepp--Logan reconstruction filter with an improved convergence behavior. Numerical experiments in the fully discrete setting reproduce the theoretical predictions.

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