It has become generally accepted in image processing to apply differential invariants from Euclidean space to the "image surface" (in "image space", that is the picture plane times intensity space) and indeed many well known algorithms are based on this usage. Yet the usage makes strictly no sense since these invariants are with respect to Euclidian isometries, e.g. rotations. But clearly you can't rotate the image surface to see its other side, or the intensity to a spatial direction! Thus the angle measure cannot be periodic in planes other than the picture plane. One needs to set up the proper transformation group to arrive at a set of invariants that makes sense. This yields a new, generic geometrical framework for image processing. I show how most of the well known global image transformations are movements, similarities or conformal transformations in this framework. The differential geometry yields novel definitions for many features such as ruts and ridges.

j.j.koenderink@phys.uu.nl