Error estimates and convergence rates for filtered back projection

Abstract:

We consider the approximation of target functions from fractional Sobolev spaces by the method of filtered back projection (FBP), which gives an inversion of the Radon transform. The objective of this paper is to analyse the intrinsic FBP approximation error which is incurred by the use of a low-pass filter with finite bandwidth. To this end, we prove $\mathrm {L}^2$-error estimates on Sobolev spaces of fractional order. The obtained error bounds are affine-linear with respect to the distance between the filter’s window function and the constant function $1$ in the $\mathrm {L}^\infty$-norm. With assuming more regularity of the window function, we refine the error estimates to prove convergence for the FBP approximation in the $\mathrm {L}^2$-norm as the filter’s bandwidth goes to infinity. Further, we determine asymptotic convergence rates in terms of the bandwidth of the low-pass filter and the smoothness of the target function. Finally, we develop convergence rates for noisy data, where we first prove estimates for the data error, which we then combine with our estimates for the approximation error.