Image Space

Abstract

“Image processing” is a purely syntactical discipline that transforms images into other images. As such it is distinct from such fields as “image recognition,” “feature detection,” and so forth, which are semantically oriented. If semantics plays a role, the “meaning” is relative to a user’s world model. In the case of purely syntactical operations there is some hope to put the discipline on a foundation of first principles. Image processing in its present state is more of a bag of hat tricks though, and many of its methods are easily shown to be inconsistent. This contribution is an attempt to put image processing on a solid geometrical basis. In order to do so the (currently only vaguely defined though much used) concept of “image space” is formalized. Departing from a few commonly acknowledged properties one arrives at a three-dimensional Cayley-Klein space with a single isotropic dimension that can be understood as a limit of either Euclidean or Minkowskian three-space. In this geometry, planes containing an isotropic line are isomorphic with the dual number plane, whereas any other plane is isomorphic with the conventional complex number plane. The elements of the group of similarities are readily identified with image transformations that “don’t change the image,” thus an “image” can be defined as an invariant under similarities. The differential invariants are a substitute for the “feature operators” of current image processing. Their formal structure is rather simpler than the corresponding Euclidean differential invariants, but the differential geometry is just as rich as that of conventional Euclidean space.