The radial trifocal tensor: a tool for calibrating the radial distortion of wide-angle cameras

We present a technique to linearly estimate the radial distortion of a wide-angle lens given three views of a real-world plane. The approach can also be used with pure rotation as in this case all points appear as lying on a plane. The three views can even be recorded using three different cameras as long as the deviation from the pin-hole model for each camera is distortion along radial lines. We introduce the 1D radial camera which projects scene points onto radial lines and the radial trifocal tensor which encodes the multi-view relations between radial lines. Given at least seven triplets of corresponding points the radial trifocal tensor can be computed linearly. This allows recovery of the radial cameras and the projective reconstruction of the plane up to a two fold ambiguity. This 2D reconstruction is unaffected by radial distortion and can be used in different ways to compute the radial distortion parameters. We propose to use the division model as in this case we obtain a linear algorithm that computes the radial distortion coefficients and the 3 remaining degrees of freedom of the homography relating the reconstructed 2D plane to the undistorted image. Each feature point that has at least one corresponding point yields one linear constraint on those unknowns. Our method is validated on real-world images. We successfully calibrate several wide-angle cameras.