We present a unified statistical theory for assessing the significance of apparent signal observed in noisy difference images. The results are usable in a wide range of applications, including fMRI, but are discussed with particular reference to PET images which represent changes in cerebral blood flow elicited by a specific cognitive or sensorimotor task. Our main result is an estimate of the P-value for local maxima of Gaussian, t, chi(2) and F fields over search regions of any shape or size in any number of dimensions. This unifies the P-values for large search areas in 2-D (Friston et al. : J Cereb Blood Flow Metab 11:690-699) large search regions in 3-D (Worsley et al. : J Cereb Blood Flow Metab 12:900-918) and the usual uncorrected P-value at a single pixel or voxel.