Recovering the vanishing self-polar triangle from a single view of a planar pattern

We are concerned with the recovery of the metric structure of the 3D space from the metric structure of a plane, given as the world-plane to image homography. Previous works have pointed out that the metric structure of the 3D space can be defined by a set of three mutually conjugated (vanishing) points with respect to the imaged absolute conic (the vanishing self-polar triangle) and three additional factors, defined up to a scale factor. Given that the homography H only allows the image to inherit two of these points and two of these factors, we show how the (three) unknown quantities can be recovered from H, by investigating some camera constraints which are discussed. We propose two solutions in the cases of (i) of known principal point coordinates (ii) of zero-skew and known pixel aspect ratio. We demonstrate that, under these assumptions, the third vanishing point, associated with the direction orthogonal to the reference plane, lies on a line that we call the central line. Adding an additional constraint, zero-skew for (i) and known camera height for (ii), a direct solution is found. Results on real images have proved to be quite accurate and encouraging for the use of our approach.