Motion and structure from two perspective views: from essential parameters to Euclidean motion through the fundamental matrix

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The standard approach to the motion and structure estimation problem consists of two stages: (1) using the eight-point algorithm to estimate the nine essential parameters defined up to a scale factor and (2) 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 one performs a projection of a set of quantities that are estimated in a space of eight dimensions (by neglecting the constraints on the essential parameters), a much higher dimension than that of the real space, which is five dimensional. A novel approach is proposed by the introduction of 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 project parameters estimated in a high-dimensional space gradually onto a slightly lower-dimensional space, namely, from eight dimensions to seven and finally to five. The proposed approach has been tested with synthetic and real data, and a considerable improvement has been observed. The conjecture is that the imposition of the constraints arising from projective geometry should be used as an intermediate step to obtain reliable three-dimensional Euclidean motion and structure estimation from multiple calibrated images. The software is available on the Internet.