An efficient method for correcting the edge artifact due to smoothing

Spatial smoothing is a common pre-processing step in the analysis of functional brain imaging data. It can increase sensitivity to signals of specific shapes and sizes (Rosenfeld and Kak [1982]: Digital Picture Processing, vol. 2. Orlando, Fla.: Academic; Worsley et al. [1996]: Hum Brain Mapping 4:74-90). Also, some amount of spatial smoothness is required if methods from the theory of Gaussian random fields are to be used (Holmes [1994]: Statistical Issues in Functional Brain Mapping. PhD thesis, University of Glasgow). Smoothing is most often implemented as a convolution of the imaging data with a smoothing kernel, and convolution is most efficiently performed using the Convolution Theorem and the Fast Fourier Transform (Cooley and Tukey [1965]: Math Comput 19:297-301; Priestly [1981]: Spectral Analysis and Time Series. San Diego: Academic; Press et al. [1992]: Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. Cambridge: Cambridge University Press). An undesirable side effect of smoothing is an artifact along the edges of the brain, where brain voxels become smoothed with non-brain voxels. This results in a dark rim which might be mistaken for hypoactivity. In this short methodological paper, we present a method for correcting functional brain images for the edge artifact due to smoothing, while retaining the use of the Convolution Theorem and the Fast Fourier Transform for efficient calculation of convolutions.