Thresholding statistical maps with appropriate correction of multiple testing remains a critical and challenging problem in brain mapping. Since the false discovery rate (FDR) criterion was introduced to the neuroimaging community a decade ago, various improvements have been proposed. However, a highly desirable feature, transformation invariance, has not been adequately addressed, especially for voxel-based FDR. Thresholding applied after spatial transformation is not necessarily equivalent to transformation applied after thresholding in the original space. We find this problem closely related to another important issue: spatial correlation of signals. A Gaussian random vector-valued image after normalization is a random map from a Euclidean space to a high-dimension unit-sphere. Instead of defining the FDR measure in the image's Euclidean space, we define it in the signals' hyper-spherical space whose measure not only reflects the intrinsic "volume" of signals' randomness but also keeps invariant under images' spatial transformation. Experiments with synthetic and real images demonstrate that our method achieves transformation invariance and significantly minimizes the bias introduced by the choice of template images.
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