Four-view Geometry with Unknown Radial Distortion

We present novel solutions to previously unsolved prob-lems of relative pose estimation from images whose calibration parameters, namely focal lengths and radial distortion, are unknown. Our approach enables metric reconstruction without modeling these parameters. The minimal case for reconstruction requires 13 points in 4 views for both the calibrated and uncalibrated cameras. We describe and implement the first solution to these minimal problems. In the calibrated case, this may be modeled as a polynomial sys-tem of equations with 3584 solutions. Despite the apparent intractability, the problem decomposes spectacularly. Each solution falls into a Euclidean symmetry class of size 16, and we can estimate 224 class representatives by solving a sequence of three subproblems with 28, 2, and 4 solutions. We highlight the relationship between internal constraints on the radial quadrifocal tensor and the relations among the principal minors of a
$4\times 4$
matrix. We also address the case of 4 upright cameras, where 7 points are minimal. Finally, we evaluate our approach on simulated and real data and benchmark against previous calibration-free solutions, and show that our method provides an efficient startup for an SfM pipeline with radial cameras.