Total error in the discrete convolution backprojection algorithm in computerized tomography

Abstract

The CBP algorithm in computerized tomography (CT) is a discrete realization of a well-known tool from approximation theory; namely, the approximation of a function f in R n by its convolution with a (bandlimited) peaked kernel. The steps in a computer implementation (e.g., in medical CT scanners) of the algorithm evaluate an n-dimensional convolution by (a) interpolation of projection data (line integrals in a two-dimensional case), (b) a one-dimensional discrete convolution, and (c) interpolation of the convolved data, required in (d) a discrete backprojection (integration over a unit sphere). The total error in the algorithm is due to the discretization steps (a)–(d) and (e) the truncation error in the basic convolution approximation. In this work we augment the known error estimates for steps (b) and (b) with those for (a), (c) and (e) to arrive at a total error profile of the algorithm, which may be summarized as follows. In a discrete b-bandlimited CBP reconstruction f d b of f, under appropriate conditions in (a)–(e), the total error f − f d b is essentially of the order of ε(f, b) = sup θϵS n−1∫ [brvbar];σ[brvbar]b [brvbar]σ[brvbar] n−1[brvbar] f ̂ (σθ)[brvbar]dσ, b→∞ .