Self-Calibration Under the Cayley Framework

The Cayley framework here is meant to tackle the vision problems under the infinite Cayley transformation (ICT), its main advantage lies in its numerical stability. In this work, the stratified self-calibration under the Cayley framework is investigated. It is well known that the main difficulty of the stratified self-calibration in multiple view geometry is to upgrade a projective reconstruction to an affine one, in other words, to estimate the unknown 3-vector of the plane at infinity, called the normal vector. To our knowledge, without any prior knowledge about the scene or the camera motion, the only available constraint on a moving camera with constant intrinsic parameters is the well-known Modulus Constraint in the literature. Do other kinds of constraints exist? If yes, what they are? How could they be used? In this work, such questions will be systematically investigated under the Cayley framework. Our key contributions include: 1. The original projective expression of the ICT is simplified and a new projective expression is derived to make the upgrade easier from a projective reconstruction to a metric reconstruction. 2. The constraints on the normal vector are systematically investigated. For two views, two constraints on the normal vector are derived; one of them is the well-known modulus constraint, while the other is a new inequality constraint. There are only these two constraints for two views. For three views, besides the constraints for two views, two groups of new constraints are derived and each of them contains three constraints. In other words, there are 12 constraints in total for three views. 3. Based on our projective expression and these constraints, a stratified Cayley algorithm and a total Cayley algorithm are proposed for the metric reconstruction from images. It is experimentally shown that they both improve significantly the numerical stability of the classical algorithms. Compared with the global optimal algorithm under the infinite homography framework, the Cayley algorithms have comparable calibration accuracy, but substantially reduce the computational load.