The Euler characteristic (EC) of the excursion set of a random field is a tool that has been used over the last decade to analyse positron emission tomography (PET) images and functional magnetic resonance images (fMRI), as well as galaxy density maps and the cosmic microwave background. We present some extensions of the work of Adler (1981) on the expectation of the EC, and use it to detect functional and structural changes in the human brain. The EC is used to test for local increases in cerebral blood flow due to an external stimulus, which allows researchers to find the regions of the brain that are `activated' by the stimulus. Recent work has extended these ideas to detecting changes in brain shape via structure masking, surface extraction, and 3-D deformation fields.

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