Towards a minimal solution for the relative pose between axial cameras

An axial camera is a particular case of a non-central camera where every back-projection ray intersects a line in 3D (the axis). The axial camera can be used to model vision systems and imaging situations of practical interest. Examples include any catadioptric system that combines a revolution mirror with a central camera for which the viewpoint is aligned with the mirror axis (e.g. a pinhole looking at a spherical mirror) [8]; the situation of a perspective camera looking through multiple flat refractive mediums [1]; or a multi-camera rig composed by two or more pinhole cameras with collinear optical centers [3]. This paper addresses the problem of estimating the translation t and the rotation R between two axial cameras using point correspondences. Pless showed that this problem can be linearly solved from a minimum of 17 point correspondences using a DLT like approach [4]. Later in [3, 7] it was observed that for the case of axial cameras this linear estimation could be accomplished from 16 point correspondences. The relative pose problem has 6 unknowns meaning that in theory 6 point correspondences provide enough information for determining the relative rotation and translation of the axial camera. Stewenius et al. proposed in [6] a minimal solution for the relative pose between generalized cameras. However, their algorithm is complex, provides a large number of possible solutions (up to 64), and, as reported in [3], it degenerates for most axial camera configurations. This article does not provide a minimal solution for the relative pose between axial cameras, but shows how the motion can be computed using as few as 10 point correspondences. Our 10-point method is an advance with respect to the previous 16-point [3, 7]. Given that all back-projection rays of an axial camera intersect its axis, they belong to a linear line congruent of dimension 4 [5]. This means that all rays can be represented by 5 dimensional coordinate vectors λi that are a linear combination of 5 base lines aligned with the axes x, y, z, ŷ, ẑ in Fig. 1(a). Given a set of intersecting ray correspondences (λi,λ ′ i ), we can establish linear relations with the form