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Degenerate case

The computation of the two-view geometry requires that the matches originate from a 3D scene and that the motion is more than a pure rotation. If the observed scene is planar, the fundamental matrix is only determined up to three degrees of freedom. The same is true when the camera motion is a pure rotation. In this last case -only having one center of projection- depth can not be observed. In the absence of noise the detection of these degenerate cases would not be too hard. Starting from real -and thus noisy- data, the problem is much harder since the remaining degrees of freedom in the equations are then determined by noise.

A solution to this problem has been proposed by Torr et al. [155]. The methods will try to fit different models to the data and the one explaining the data best will be selected. The approach is based on an extension of Akaike's information criterion [1] proposed by Kanatani [64]. It is outside the scope of this text to describe this method into details. Therefore only the key idea will briefly be sketched here.

Different models are evaluated. In this case the fundamental matrix (corresponding to a 3D scene and more than a pure rotation), a general homography (corresponding to a planar scene) and a rotation-induced homography are computed. Selecting the model with the smallest residual would always yield the most general model. Akaike's principle consist of taking into account the effect of the additional degrees of freedom (which when not needed by the structure of the data end up fitting the noise) on the expected residual. This boils down to adding a penalty to the observed residuals in function of the number of degrees of freedom of the model. This makes a fair comparison between the different models feasible.


next up previous contents
Next: Three and four view Up: Two view geometry computation Previous: Robust algorithm   Contents
Marc Pollefeys 2002-11-22