A New Multistage Approach to Motion and Structure Estimation: From Essential Parameters to Euclidean Motion Via Fundamental Matrix

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 deened up to a scale factor, which is a linear estimation problem; (ii) reening the motion estimation based on some statistically optimal criteria, which is a nonlinear estimation problem on a ve-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 diicult to obtain a precise initial estimate from the rst stage. This is because we perform a projection of a set of quantities which are estimated in a space of 8 dimensions, much higher than that of the real space which is ve-dimensional. We propose in this paper a novel approach by introducing an intermediate stage which consists in estimating a 33 matrix deened up to a scale factor by imposing the zero-determinant 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 nally to 5. The proposed approach has been tested with synthetic and real data, and a considerable improvement has been observed for the delicate situations mentioned above. 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. Un nouvel algorithme pour l'estimation du mouvement et de la structure: Des parammtres essentiels au mouvement euclidien travers la matrice fondamentale RRsumm : La technique classique pour estimer le mouvement et la structure partir de deux projections perspectives se compose de deux tapes : (i) utiliser l'algorithme des 8-points pour estimer de maniire linnaire les 9 parammtres essentiels dddnis un facteur d''chelle prrs ; (ii) raaner l'estimation du mouvement base d'un crittre statistiquement optimal. Le probllme (ii) est un probllme d'estimation non linnaire sur un espace de 5 dimensions. Malheureusement, les rrsultats obtenus avec cette technique ne sont souvent pas satisfaii sants, surtout quand le mouvement est petit ou quand les points observs …