Calibration of a stereo system with small relative angles

Because of their relative ease in solving the correspondence problem, stereo systems without relative rotation are popular. However, in practice, mechanical difficulties will lead to a small, unknown relative rotation between stereo cameras. In this paper we present an algorithm for the calibration of a stereo system with small relative angles in an uncontrolled environment. This algorithm has two advantages: (a) It is more accurate than the existing algorithms in the computer vision and photogrammetry literatures (b) It provides useful insight into the problem of camera calibration and relative orientation. This is done by deriving explicit analytical solutions for the relative pan, tilt, and roll angles in terms of the world pan angle (gaze angle) and the coordinates of the feature points used in their computations. These solutions allow us a better understanding of the problem of calibration in general by providing us with insight as to how errors due to quantization and uncertainty in the location of image centers affect the computation of rotation angles. It is shown that as the distance of feature points from the center of the image decreases, the error due to quantization in the relative pan angle increases quadratically, that of the relative roll angle increases linearly, while that of the tilt angle does not change appreciably. Likewise, it is shown that the errors in the locations of principal points (image centers) do not affect the computation of relative pan and roll angles appreciably, whereas the impact on the relative tilt angle is significant. These findings are likely to be of use even when the relative rotation angles are not small. All of the analytical findings have been supported by extensive simulation.