Vision Theory in Spaces of Constant Curvature

In this paper, vision theory for Euclidean, spherical and hyperbolic spaces is studied in a uniform framework using diierential geometry in spaces of constant curvature. It is shown that the epipolar geometry for Euclidean space can be naturally generalized to the spaces of constant curvature. In particular, it is shown that, in the general case, the bilinear epipolar constraint is exactly the same as in the Euclidean case; also, there are only bilinear, trilinear and quadrilinear constraints associated with multiple images of a point. Diierential (continuous) case is also studied. For the structure from motion problem, 3D structure can only be determined up to a universal scale, the same as the Euclidean case. Approaches are proposed to reconstruct 3D structure with respect to a normalized curvature. Spaces of constant curvature are Riemannian manifolds with constant sectional curvature. In diierential geometry, they are also referred to as space forms. A Riemannian manifold of constant curvature is said to be spherical, hyperbolic or at (or locally Euclidean) according as the sectional curvature is positive, negative or zero. Geometry about spaces of constant curvature is also called absolute geometry 1], due to one of the co-founders non-Euclidean geometry: Janos Bolyai. Not until Einstein's general relativity theory, non-Euclidean geometry, or Riemannian geometry in general, is just a pure mathematical creation rather than geometry of physical spaces. In general relativity theory, the physical space is typically described as a (3 dimensional) Riemannian manifold (with possibly non-zero curvature). In such a space, light travels the geodesics of the manifold