This paper presents an axiomatic approach to corner detection. In the first part of the paper we review five currently used corner detection methods (Harris-Stephens, Forstner, Shi-Tomasi, Rohr, and Kenney et al ) for graylevel images. This is followed by a discussion of extending these corner detectors to images with different pixel dimensions such as signals (pixel dimension one) and tomographic medical images (pixel dimension three) as well as different intensity dimensions such as color or LADAR images (intensity dimension three). These extensions are motivated by analyzing a particular example of optical flow in pixel and intensity space with arbitrary dimensions. Placing corner detection in a general setting enables us to state four axioms that any corner detector might reasonably be required to satisfy. Our main result is that only the Shi-Tomasi (and equivalently the Kenney et al. 2-norm detector) satisfy all four of the axioms.