Motion and Structure From Two Perspective Views: From Essential Parameters to Euclidean Motion Via Fundamental Matrix

The standard approach consists of two stages: (i) using the 8-point algorithm to estimate the 9 essential parameters defined up to a scale factor; (ii) refining the motion estimation based on some statistically optimal criteria, which is a nonlinear estimation problem on a five-dimensional space. Unfortunately, the results obtained are often not satisfactory. The problem is that the second stage is very sensitive to the initial guess, and that it is very difficult to obtain a precise initial estimate from the first stage. This is because we perform a projection of a set of quantities which are estimated in a space of 8 dimensions (by neglecting the constraints on the essential parameters), much higher than that of the real space which is five-dimensional. We propose in this paper a novel approach by introducing an intermediate stage which consists in estimating a 3 × 3 matrix defined up to a scale factor by imposing the rank-2 constraint (the matrix has seven independent parameters, and is known as the fundamental matrix). The idea is to gradually project parameters estimated in a high dimensional space onto a slightly lower space, namely from 8 dimensions to 7 and finally to 5. The proposed approach has been tested with synthetic and real data, and a considerable improvement has been observed. Our conjecture from this work is that the imposition of the constraints arising from projective geometry should be used as an intermediate step in order to obtain reliable 3D Euclidean motion and structure estimation from multiple calibrated images. The software is available from the Internet.