Euclidean Structure from Confocal Conics: Theory and Application to Camera Calibration

Abstract

Plane-based calibration is now a very popular procedure because of its flexibility. One key step consists in detecting a set of coplanar features, from which the Euclidean structure of the corresponding 3D plane has to be computed. We suggest to use confocal conics as calibration targets, as they offer undeniable advantages over other ones (e.g., points or lines) in terms of detection and estimation, especially in the presence of partial occlusion. We introduce important projective and Euclidean properties of the linear family of conics (i.e., the confocal conic range), spanned by two confocal conics. In particular, we rely on the fact that the circular point-envelope - a rank-2 conic that encodes the 2D Euclidean structure - is a degenerate member of any confocal conic range. This allows us to give closed-form solutions in three cases: one conic with known foci, two confocal conics with known product of ratios of semi axes, and two unknown confocal conics. The performances of the proposed algorithms (consisting of a few lines of Matlab-like code) show up high accuracies for both intrinsic and extrinsic camera parameters. In addition to experiments with synthetic data, a video sequence is processed, showing off the interest of using confocal conics as calibration targets, for augmented reality purposes.