Wavelets versus resels in the context of fMRI: establishing the link with SPM

Statistical Parametric Mapping (SPM) is a widely deployed tool for detecting and analyzing brain activity from fMRI data. One of SPM's main features is smoothing the data by a Gaussian filter to increase the SNR. The subsequent statistical inference is based on the continuous Gaussian random field theory. Since the remaining spatial resolution has deteriorated due to smoothing, SPM introduces the concept of "resels" (resolution elements) or spatial information-containing cells. The number of resels turns out to be inversely proportional to the size of the Gaussian smoother. Detection the activation signal in fMRI data can also be done by a wavelet approach: after computing the spatial wavelet transform, a straightforward coefficient-wise statistical test is applied to detect activated wavelet coefficients. In this paper, we establish the link between SPM and the wavelet approach based on two observations. First, the (iterated) lowpass analysis filter of the discrete wavelet transform can be chosen to closely resemble SPM's Gaussian filter. Second, the subsampling scheme provides us with a natural way to define the number of resels; i.e., the number of coefficients in the lowpass subband of the wavelet decomposition. Using this connection, we can obtain the degree of the splines of the wavelet transform that makes it equivalent to SPM's method. We show results for two particularly attractive biorthogonal wavelet transforms for this task; i.e., 3D fractional-spline wavelets and 2D+Z fractional quincunx wavelets. The activation patterns are comparable to SPM's.