Error estimates for filtered back projection

Abstract

Computerized tomography allows us to reconstruct a bivariate function from its Radon samples. The reconstruction is based on the filtered back projection (FBP) formula, which gives an analytical inversion of the Radon transform. The FBP formula, however, is numerically unstable. Therefore, suitable low-pass filters of finite bandwidth are employed to make the reconstruction by FBP less sensitive to noise. The objective of this paper is to analyse the reconstruction error occurring due to the use of a low-pass filter. To this end, we prove L2 error estimates on Sobolev spaces of fractional order. The obtained error estimates are affine-linear with respect to the distance between the filter's window function and the constant function 1 in the L∞-norm. Our theoretical results are supported by numerical simulations, where in particular the predicted affine-linear behaviour of the error is observed.