Group Actions, Homeomorphisms, and Matching: A General Framework

Abstract

This paper constructs metrics on the space of images {\cal I} defined as orbits under group actions {\cal G}. The groups studied include the finite dimensional matrix groups and their products, as well as the infinite dimensional diffeomorphisms examined in Trouve (1999, Quaterly of Applied Math.) and Dupuis et al. (1998). Quaterly of Applied Math.). Left-invariant metrics are defined on the product {\cal G} \times {\cal I} thus allowing the generation of transformations of the background geometry as well as the image values. Examples of the application of such metrics are presented for rigid object matching with and without signature variation, curves and volume matching, and structural generation in which image values are changed supporting notions such as tissue creation in carrying one image to another.