Conformal Geometry for Viewpoint Change Representation

Ghina El Mir (1) , Christophe Saint-Jean (1) , Michel Berthier (1)

1 MIA - Mathématiques, Image et Applications - EA 3165

We propose in this paper a new model for image representation based on the conformal geometry and its powerfulness to encode perspective distortions through bases of the Minkowski space ${\RR}^{1,1}$. 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 ${\RR}^{3,1}$ 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 ${\RR}^{3,1}$, each one of these encoding a perspective plane.

Mots clés: Computer Vision; Pinhole Camera; Conformal Geometry; Image Representation; Perspective Distortions; Geometric Algebra; Perspective Plane

Domaines: Vision par ordinateur et reconnaissance de formes [cs.CV]; Géométrie métrique [math.MG]; Mathématiques générales [math.GM]

Ghina El Mir, Christophe Saint-Jean, Michel Berthier. Conformal Geometry for Viewpoint Change Representation. Advances in Applied Clifford Algebras, 2014, 24 (2), pp.443-463. ⟨hal-00831744⟩