Influence of numerical conditioning on the accuracy of relative orientation

We study the influence of numerical conditioning on the accuracy of two closed-form solutions to the overconstrained relative orientation problem. We consider the well known eight-point algorithm and the recent five-point algorithm, and evaluate changes in their performance due to Hartley's normalization and Muehlich's equilibration. The need for numerical conditioning is introduced by explaining the known occurence of the bias of the eight-point algorithm towards the forward motion. Then it is shown how conditioning can be used to improve the results of the recent five-point algorithm. This is not straightforward since the conditioning disturbs the calibration of the input data. The conditioning therefore needs to be reverted before enforcing the internal cubic constraints of the essential matrix. The obtained improvements are less dramatic than in the case of the eight-point algorithm, for which we offer a plausible explanation. The theoretical claims are backed up with extensive experimentation on noisy artificial datasets, under a variety of geometric and imaging parameters.