Abstract
Epipolar geometry of a stereopair can be expressed either in 3D, as the relative orientation (i.e. translation and rotation) of two bundles of optical rays in case of calibrated cameras or, in case of unclalibrated cameras, in 2D as the position of the epipoles on the image planes and a projective transformation that maps points in one image to corresponding epipolar lines on the other. The typical coplanarity equation describes the first case; the Fundamental matrix describes the second. It has also been proven in the Computer Vision literature that 2D epipolar geometry imposes two independent constraints on the parameters of camera interior orientation. In this contribution these constraints are expressed directly in 3D Euclidean space by imposing the equality of the dihedral angle of epipolar planes defined by the optical axes of the two cameras or by suitably chosen corresponding epipolar lines. By means of these constraints, new closed form algorithms are proposed for the estimation of a variable or common camera constant value given the fundamental matrix and the principal point position of a stereopair.
Highlights
In the general case ≥3 images are required in order to fully calibrate a camera only from image point correspondences
If the principal point position is known it is possible to estimate the camera constant from 2 images even when it is not common for their two cameras. This has been the subject of several contributions in the field of Computer Vision, where closed form solutions have been proposed for the estimation of a variable or common camera constant value from the fundamental matrix assuming known principal point
Pan et al (1995) derived a 3rd degree equation in the values of c2. They presented a linear solution in c2 for the cases of identical and different camera constants (Newsam et al, 1996). They found two critical geometries which do not allow the computation of varying c values from the fundamental matrix: when the optical axes are coplanar with the base or when one optical axis is perpendicular to the plane defined by the other axis and the base
Summary
In the general case ≥3 images are required in order to fully calibrate a camera only from image point correspondences. Ronda & Valdés (2007) have examined the Kruppa equations in the case of a stereopair and, based on a projective geometry theorem of the French mathematician Poncelet, propose a parameterization of all possible solutions for camera calibration In this contribution the constraints that the fundamental matrix imposes on the interior orientation parameters are derived in 3D Euclidean space. A second independent constraint is derived in a similar way from the equality of the dihedral angle of the epipolar planes that correspond to two suitably chosen epipolar lines By means of these constraints four new closed form algorithms are developed for the computation of a common and variable camera constant from the fundamental matrix assuming known principal point
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