A simplified analysis for predicting the relative performance of tomographic filters

The error in the tomographic inversion process has been studied in the Sobolev space framework by Natterer. Recently, some direct theorems were developed by Munshi et al. that consider certain local Sobolev norms, incorporating certain radially averaged derivatives. The usual assumption in the tomographic inversion process is that of having band-limited projection data. For cross sections having band-limited projection data, using an appropriate convolving function (e.g., band limiting), the inherent error El could be made zero. However, cross sections generally have a finite support; hence, projection data are necessarily not band limited. Nevertheless, due to a finite Fourier frequency cut-off incorporated in the implementation of the convolution backprojection (CBP) algorithm, the approximation (reconstruction) obtained is band limited. The error E1 arises due to this band-limiting aspect. The objective of the study is to consider E1 employing a simplified approach similar to the previous method and predict relative performance of the filter functions used in the area. The factors considered in the study are the smoothness properties of the cross section and the second derivative (Fourier space) of the filter (window) function