Optimizing Wavelets for the Analysis of fMRI data

Ruttimann et al. have proposed to use the wavelet transform for the detection and localization of activation patterns in functional magnetic resonance imaging (fMRI). Their main idea was to apply a statistical test in the wavelet domain to detect the coefficients that are significantly different from zero. Here, we improve the original method in the case of non-stationary Gaussian noise by replacing the original z-test by a t-test that takes into account the variability of each wavelet coefficient separately. The application of a threshold that is proportional to the residual noise level, after the reconstruction by an inverse wavelet transform, further improves the localization of the activation pattern in the spatial domain. A key issue is to find out which wavelet and which type of decomposition is best suited for the detection of a given activation pattern. In particular, we want to investigate the applicability of alternative wavelet bases that are not necessarily orthogonal. For this purpose, we consider the various brands of fractional spline wavelets (orthonormal, B-spline, and dual) which are indexed by a continuously-varying order parameter α. We perform an extensive series of tests using simulated data and compare the various transforms based on their false detection rate (type I + type II errors). In each case, we observe that there is a strongly optimal value of α and a best number of scales that minimizes the error. We also find that splines generally outperform Daubechies wavelets and that they are quite competitive with SPM (the standard analysis method used in the field), although it uses much simpler statistics. An interesting practical finding is that performance is strongly correlated with the number of coefficients detected in the wavelet domain, at least in the orthonormal and B-spline cases. This suggest that it is possible to optimize the structural wavelet parameters simply by maximizing the number of wavelet counts, without any prior knowledge of the activation pattern. Some examples of analysis of real data are also presented.