Estimating the variability of reconstructed PET data: a technique based on approximating the reconstruction filter by a sum of Gaussian kernels

The estimation of variability in reconstructed Positron Emission Tomography (PET) images is an important goal. Exact and approximate formulae for variances of average activity over regions-of-interest (ROI) have been developed. We introduce a new approach which relies on approximating the convolution filter of the reconstruction by a weighted sum of Gaussian kernels with different full-width-at-half-maxima (FWHM). Results obtained for a one-dimensional model deconvolution problem show that the percent errors in approximated standard deviations for reconstructed ROI values of varying size are essentially zero. The method is also applied to obtain convolution formulae for pixelwise variances of reconstructed PET images. The approach works remarkably well. These results are insensitive to the amount of smoothing used in the reconstruction process. Relative to other approximation techniques, the Gaussian approximation provides substantially improved accuracy with negligible increase in compute time. Thus, this approach looks quite promising.