Calibrating focal length for paracatadioptric camera from one circle image

Camera calibration from circles has great advantages, but for paracatadioptric camera, the estimation of intrinsic parameters using circle images is still an open and challenging problem. Previous work proved that the paracatadioptric projection of a circle is a quartic curve. But due to the partial occlusion, only part of the quartic curve is visible on the image plane. Consequently, circle image cannot be directly estimated using image points extracted from the visible part and camera parameters cannot be calibrated. To solve this problem, In this paper, we study the properties of paracatadioptric circle image and application in calibrating the focal length for the case that aspect ratio is 1 and skew is 0. Firstly, we derive the necessary and sufficient conditions that must be satisfied by paracatadioptric circle image. Next, based on these conditions, a new object function is presented to correctly estimate the circle image. Then, we show that the focal length can be computed from the estimated paracatadioptric circle image and the principal point that is estimated from the projected contour of parabolic mirror. Experimental results on both simulated and real image data have demonstrated the effectiveness of our method.