A Differential Geometric Approach to Multiple View Geometry in Spaces of Constant Curvature

Abstract

Based upon an axiomatic formulation of vision system in a general Riemannian manifold, this paper provides a unified framework for the study of multiple view geometry in three dimensional spaces of constant curvature, including Euclidean space, spherical space, and hyperbolic space. It is shown that multiple view geometry for Euclidean space can be interpreted as a limit case when (sectional) curvature of a non-Euclidean space approaches to zero. In particular, we show that epipolar constraint in the general case is exactly the same as that known for the Euclidean space but should be interpreted more generally when being applied to triangulation in non-Euclidean spaces. A special triangulation method is hence introduced using trigonometry laws from Absolute Geometry. Based on a common rank condition, we give a complete study of constraints among multiple images as well as relationships among all these constraints. This idealized geometric framework may potentially extend extant multiple view geometry to the study of astronomical imaging where the effect of space curvature is no longer negligible, e.g., the so-called “gravitational lensing” phenomenon, which is currently active study in astronomical physics and cosmology.