**Abstract**

If the noise component of image data is non-isotropic, that is, if it has non-constant smoothness or effective point spread function, then theoretical results for the P-value of local maxima and the size of supra-threshold clusters of a statistical parametric map (SPM) based on random field theory are not valid. This assumption is reasonable for PET or smoothed fMRI data, but not if this data is projected onto an unfolded, inflated or flattened 2D cortical surface. Anatomical data such as structure masks, surface displacements and deformation vectors are also highly non-isotropic. The solution proposed in this paper is to suppose that the image can be warped or flattened (in a statistical sense) into a space where the data is isotropic. The subsequent corrected P-values do not depend on finding this warping -- it is only sufficient to know that such a warping exists.