Conformal Geometry for Viewpoint Change Representation

Abstract

We propose in this paper a new model for image representation based on the conformal geometry and its powerfulness to encode perspective distortions through the choice of basis of the Minkowski space ${\mathbb{R}^{1,1}}$. This approach allows us to describe an image as a scalar valued function defined on a horosphere corresponding to an embedding of the Euclidean plane into ${\mathbb{R}^{3,1}}$ encoding the latitude angle and the rotation parameter of the camera. This is obtained through a generalization of the conformal model such that it includes representations of perspective planes. In this setting, we describe every viewpoint change as a mapping between two horospheres of the space ${\mathbb{R}^{3,1}}$, each one of these encoding a perspective plane.