A linear and direct method for projective reconstruction

The projective recovery of 3D point structure from multiple images has been one of the classical problems in computer vision. Existing methods for projective reconstruction usually require a priori estimation of a consistent set of projective depths which in turn require the estimation of the projection matrices or the fundamental matrices in advance. Those methods are usually nonlinear, time-consuming, and sometimes inaccurate. This paper presents a direct and linear method for projective reconstruction. First, a 3D point structure is characterized by representing other points as linear combinations of some reference points. Next, cross ratios of projective depths are derived linearly. Then coefficients of the representations scaled by ratios of the projective depths are derived linearly. Projective invariants of these points are ratios of these values.