Metric reconstruction of planes utilizing off-the-plane features

Metric reconstruction of a projected plane is equivalent to estimating metric invariants of the plane in projective space. Working with an unknown planar scene taken with an uncalibrated camera, however, estimating the metric invariants may not be possible only with features on the plane, because the human visual system is not good at detecting sufficient information. Although the conventional algorithms use only features on the plane to recover the plane metric, we show that features not on the plane can be utilized. Study about the range of the dual conics and its self-polar triangle verifies that the metric invariant, the conic dual to the circular points of a plane is constrained with the orthogonal vanishing points under the assumption of the same camera. Using the constraint, we can get the orthogonal vanishing points from the given metric invariants, or inversely, the metric invariants from the orthogonal vanishing points. We also show that an image warping based on parallelism and orthogonality gives a physically meaningful parameterization of the metric invariant. With this new parameterization and the self-polar constraint, it is possible to recover the metric of a plane from information that the human visual system can easily detect without explicit camera calibration.