Radial Multi-focal Tensors Applications to Omnidirectional Camera Calibration

The 1D radial camera maps all points on a plane, containing the principal axis, onto the radial line which is the intersection of that plane and the image plane. It is a sufficiently general model to express both central and non-central cameras, since the only assumption it makes is of known center of distortion. In this paper, we study the multi-focal tensors arising out of 1D radial cameras. There exist no two-view constraints (like the fundamental matrix) for 1D radial cameras. However, the 3-view and 4-view cases are interesting. For the 4-view case we have the radial quadrifocal tensor, which has 15 d.o.f and 2 internal constraints. For the 3-view case, we have the radial trifocal tensor, which has 7 d.o.f and no internal constraints. Under the assumption of a purely rotating central camera, this can be used to do a non-parametric estimation of the radial distortion of a 1D camera. Even in the case of a non-rotating camera it can be used to do parametric estimation, assuming a planar scene. Finally we examine the mixed trifocal tensor, which models the case of two 1D radial cameras and one standard pin-hole camera. Of the above radial multifocal tensors, only the radial trifocal tensor is useful practically, since it doesn’t require any knowledge of the scene and is extremely robust. We demonstrate results based on real-images for this. For the quadrifocal tensor, too, we present a way to do a metric reconstruction of the scene and to undistort the image (given a sufficiently dense set of pointcorrespondences). We also show results on synthetic images. However, it must be noted that currently the quadrifocal and mixed trifocal tensors are useful only from a theoretical stand-point. SriRam Thirthala Dept of Computer Science UNC-Chapel Hill E-mail: [email protected] Present address: Google Engineering, Bangalore Marc Pollefeys Dept of Computer Science UNC-Chapel Hill, Institute of Computational Sciences ETH-Zurich E-mail: [email protected]