Direct Least Square Fitting of Paracatadioptric Line Images

Paracatadioptric sensors combine a parabolic shaped mirror and a camera inducing an orthographic projection. Such a configuration provides a wide field of view while keeping a single effective viewpoint. In general the paracatadioptric image of a line is a conic curve. The estimation of line images is an important subject for applications such as reconstruction and visual control of motion. However the estimation of the conic curves where lines are mapped is hard to accomplish. In general only a small arc of the conic is visible in the image and conventional conic fitting techniques are unable to correctly estimate the curve. This paper shows that line images can be accurately estimated by constraining the search space. A conic curve is the paracatadioptric image of a line if, and only if, the image of the circular points lie on the curve and two certain points are conjugate with respect to the conic. Considering the space of all conic curves, the line images lie in a linear subspace which depends on the system calibration. The paracatadioptric projection of a line can estimated by fitting a conic in the subspace to the data points. The proposed approach is computationally efficient since the fitting problem can be solved by an eigensystem