A New Multistage Approach to Motion and Structure Estimation: From Essential Parameters to Euclidean Motion Via Fundamental Matrix
Résumé
The classical approach to motion and structure estimation problem from two perspective projections consists of two stages: (i) using the 8-point algorithm to estimate the 9 essential parameters defined up to a scale factor, which is a linear estimation problem; (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 using this approach are often not satisfactory, especially when the motion is small or when the observed points are close to a degenerate surface (e.g. plane). 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 estimated in a space of 8 dimensions, 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 \emph{zero-determinant constraint} (the matrix has seven independent parameters, and is known as the fundamental matrix). The idea is to \emph{gradually} project parameters estimated in a high dimensional space onto a \emph{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 considerable improvement has been observed for the delicate situations mentioned above. Our conjecture from this work is that the estimation of the uncalibrated epipolar geometry should be used as an intermediate step in order to obtain reliable 3D Euclidean motion and structure estimation from multiple calibrated images.