Closed form solution for radial distortion estimation from a single vanishing point

Radial lens distortion found in real, most notably off-the-shelf medium to wide angle optics can be quite severe. A-priori calibration remedies the problem, but requires access to the camera. Alternative approaches make use of correspondences in multiple images of a moving camera [3], relying on sufficiently overlapping views. Our paper fits in the category of techniques, where straight scene lines are used to determine distortion parameters from a single image. These plumbline methods [2] critically depend on the existence of long edges and get thrown of easily if a significant portion originates from non-linear structures. Here, the use of vanishing points provides an advantage: When a large number of image edges groups w.r.t. to a common vanishing point, it is very likely that they stem from parallel scene lines. Recent methods for vanishing point detection either ignore radial distortion completely, or assume only weak distortion, which is accounted for after vanishing point estimation [1, 4]. If strong distortion is present the effects are detrimental to the extraction of vanishing points and such algorithms are bound to fail. Furthermore, when dealing with radial distortion effects in refinement stages, the error is assessed in the undistorted image. This creates significant bias towards estimates shrinking the undistorted image [5] Contributions: To overcome this limitation, we suggest a simple RANSAC procedure for vanishing point detection by clustering of distorted lines. The proposed technique builds upon: a) Our newly devised closed-form solution for simultaneously estimating a vanishing point and radial distortion from three distorted image lines. b) A consistency measure which avoid bias by quantifying the error of lines w.r.t. a vanishing point in the distorted image. Specifically, our work bases on the division model [3], mapping a distorted image point x = (x,y) to the undistorted point x by