Fitting conics to paracatadioptric projections of lines

The paracatadioptric camera is one of the most popular panoramic systems currently available in the market. It provides a wide field of view by combining a parabolic shaped mirror with a camera inducing an orthographic projection. Previous work proved that the paracatadioptric projection of a line is a conic curve, and that the sensor can be fully calibrated from the image of three or more lines. However, the estimation of the conic curves where the lines are projected is hard to accomplish because of the partial occlusion. In general only a small arc of the conic is visible in the image, and conventional conic fitting techniques are unable to accurately estimate the curve. The present work provides methods to overcome this problem. We show that in uncalibrated paracatadioptric views a set of conic curves is a set of line projections if and only if certain properties are verified. These properties are used to constrain the search space and correctly estimate the curves. The conic fitting is solved naturally by an eigensystem whenever the camera is skewless and the aspect ratio is known. For the general situation the line projections are estimated using non-linear optimization. The set of paracatadioptric lines is used in a geometric construction to determine the camera parameters and calibrate the system. We also propose an algorithm to estimate the conic locus corresponding to a line projection in a calibrated paracatadioptric image. It is proved that the set of all line projections is a hyperplane in the space of conic curves. Since the position of the hyperplane depends only on the sensor parameters, the accuracy of the estimation can be improved by constraining the search to conics lying in this subspace. We show that the fitting problem can be solved by an eigensystem, which leads to a robust and computationally efficient method for paracatadioptric line estimation.