Reconstruction of a 3D point moving along a line under a varying focal length

This paper consider the problem of reconstructing the 3D coordinates of a moving point seen from a monocular moving camera with varying focal length, i,e., to reconstruct moving objects and recover the focal length together from line of sight measurements only which extends the camera calibration from static points to dynamic points. Since the point is moving while the camera is moving, then even if the camera motion is known, it is impossible to reconstruct the 3D location of the point under general circumstances. However, we show that if the point is moving along a straight line, and the camera focal is allowed to vary, then the parameters of the line (and hence the 3D position of the point) and the camera focal can be uniquely recovered from at least 6 views. We first form the measurement matrix with the unknown focal length, then the focal length is estimated iteratively so that the rank of the measurement matrix is made to be as close as possible to 5; From the measurement matrix, the line of the moving point can be recovered; at last, the 3D points are reconstructed by intersecting each ray with the line. Thus we propose a novel method in the auto-calibration of camera field of computer vision that is based on the dynamic points. Our theory is validated by the synthetic and real image experiments.