Recently, we have introduced an integrated framework that combines wavelet-based processing with statistical testing in the spatial domain. In this paper, we propose two important enhancements of the framework. First, we revisit the underlying paradigm; i.e., that the effect of the wavelet processing can be considered as an adaptive denoising step to "improve" the parameter map, followed by a statistical detection procedure that takes into account the non-linear processing of the data. With an appropriate modification of the framework, we show that it is possible to reduce the spatial bias of the method with respect to the best linear estimate, providing conservative results that are closer to the original data. Second, we propose an extension of our earlier technique that compensates for the lack of shift-invariance of the wavelet transform. We demonstrate experimentally that both enhancements have a positive effect on performance. In particular, we present a reproducibility study for multisession data that compares WSPM against SPM with different amounts of smoothing. The full approach is available as a toolbox, named WSPM, for the SPM2 software; it takes advantage of multiple options and features of SPM such as the general linear model. © 2007 Elsevier Inc. All rights reserved.Keywords: Wavelets; Wavelet thresholding; Statistical testing; Bias reduction; Shift-invariant transform; Reproducibility study
IntroductionStatistical parametric mapping (SPM) Frackowiak et al., 1997) is probably the most popular parametric hypothesis-driven method for the analysis of fMRI data. To control the multiple testing problem, SPM considers the data as a lattice representation of a continuous Gaussian Random Field (GRF). To conform with this hypothesis, the data are pre-smoothed with a Gaussian filter with fixed size (Worsley et al., 1996;Poline et al., 1997). The user has the option to adjust the smoothing strength for optimal detection (compromise between SNR enhancement and spatial definition).The discrete wavelet transform (DWT) has also been applied to the analysis of fMRI data for parametric hypothesis testing. Three important properties justify the use of wavelets in this application. First, they typically encode activation patterns with a small set of wavelet coefficients; this is referred to as the DWT's sparsifying property. Second, an orthogonal DWT leaves the noise evenly distributed in the wavelet domain; it therefore increases the SNR. Third, the DWT acts (approximately) as a decorrelator. Therefore, a conservative Bonferroni correction for multiple hypothesis testing is closer to optimal in the wavelet domain than in the spatial domain. Basically, the standard wavelet approach to parametric hypothesis testing consists in statistical testing the wavelet domain representation of the parameter map (Ruttimann et al., 1998;Turkheimer et al., 2000). The remaining difficulty is how to fully exploit the reconstruction of the parameter map after thresholding in the wavelet domain. Several approaches have been proposed, s...