Reconstructing a 3D Line from a Single Catadioptric Image

TitleReconstructing a 3D Line from a Single Catadioptric Image
Publication TypeConference Paper
Year of Publication2006
AuthorsLanman, D., M. Wachs, G. Taubin, and F. Cukierman
Keywords3D line reconstruction axial noncentral optical systems enumerative geometry feature extraction image reconstruction Plucker coordinates single catadioptric image singular value decomposition spherical catadioptric camera SVD

This paper demonstrates that, for axial non-central optical systems, the equation of a 3D line can be estimated using only four points extracted from a single image of the line. This result, which is a direct consequence of the lack of vantage point, follows from a classic result in enumerative geometry: there are exactly two lines in 3-space which intersect four given lines in general position. We present a simple algorithm to reconstruct the equation of a 3D line from four image points. This algorithm is based on computing the Singular Value Decomposition (SVD) of the matrix of Plucker coordinates of the four corresponding rays. We evaluate the conditions for which the reconstruction fails, such as when the four rays are nearly coplanar. Preliminary experimental results using a spherical catadioptric camera are presented. We conclude by discussing the limitations imposed by poor calibration and numerical errors on the proposed reconstruction algorithm.