We present an algorithm for plane-based self-calibration of cameras with radially symmetric distortions given a set of sparse feature matches in at least two views. The projection function of such cameras can be seen as a projection with a pinhole camera, followed by a non-parametric displacement of the image points in the direction of the distortion center. The displacement is a function of the points distance to the center. Thus, the generated distortion is radially symmetric. Regular cameras, fish-eyes as well as the most popular central catadioptric devices can be described by such a model. Our approach recovers a distortion function consistent with all the views, or estimates one for each view if they are taken by different cameras. We consider a least squares algebraic solution for computing the homography between two views that is valid for rectified (undistorted) point correspondences. We observe that the terms of the function are bilinear in the unknowns of the homography and the distortion coefficient associated to each point. Our contribution is to approximate this non-convex problem by a convex one. To do so, we replace the bilinear terms by a set of new variables and obtain a linear least squares problem. We show that like the distortion coefficients, these variables are subject to monotonicity constraints. Thus, the approximate problem is a convex quadratic program. We show that solving it is sufficient for accurately estimating the distortion parameters. We validate our approach on simulated data as well as on fish-eye and catadioptric cameras. We also compare our solution to three state-of-the-art algorithms and show similar performance.
Tardif J.-P., Sturm P. Roy S., Plane-based Self-calibration of Radial Distortion, 11th IEEE International Conference on Computer Vision, Rio de Janeiro, Brazil, October 14-20, 2007