Additional Implementation Details for Single Image 3D Without a Single 3D Image

We produce metric rectifications of the scene facing a particular vanishing point using the method of Zaheer, Rashid, and Khan to generate the homographies. The method gives a homography for metric rectification under the assumption of square pixels and known principal point but leaves behind an ambiguity consisting of a similarity transform (i.e., in-plane rotation, isotropic scale, translation). In order to use it, we need to be able to determine three parameters – one for focal length, and two corresponding to rotation angles. Similarity Ambiguity: We handle the remaining similarity in a brute-force manner by multi-scale detection (handling translation and isotropic scaling) and by rotating scenes consistently (i.e., ensuring the vertical vanishing point is above/below the image). Parameters: We determine the free parameters of transformation using the mutual-orthogonality of the vanishing points’ corresponding directions by ensuring: (1) that the surface normals corresponding are orthogonal as possible given the assumed principal point and (2) that rays drawn from the vanishing points not being rectified (i.e., corresponding to orthogonal normals) form right angles in the rectified image. Essentially, this is the technique of Zaheer et al., but using virtual lines as opposed to detected lines.