A closed-form solution for paraperspective reconstruction

We address the problem of 3D reconstruction from image features tracked along a sequence. The most precise algorithms compute the Maximum Likelihood (ML) estimate and are iterative. They need an approximate 3D reconstruction as starting position. For that purpose, we propose a closed-form expression of paraperspective reconstruction. A matrix that approximately verifies the properties of a paraperspective projection matrix is first built, as in Christy and Horaud [1] or Poelman and Kanade [3]. Our contribution lies in showing how to transform this matrix so that it exactly verifies the properties of paraperspective projection matrices. This is done by a closed form expression, in which the depth of the camera is also retrieved. The camera position is then found directly, instead of being obtained as the solution of a non-linear optimization problem, like in [3]. As in [1, 5, 3], we assume that calibration is known.