Optimal rectification of an affine structure from hypothesized orthogonality relations

Considers the problem of upgrading a 3D structure known only up to an unknown affine transformation to a scaled Euclidean structure by imposing only known or hypothesized orthogonality relations between structure elements. This seemingly simple problem is confounded by the facts that (i) the structure elements assumed to be orthogonal are corrupted by measurement noise, (ii) that the angle measure is not linear in the errors, and (iii) that the assumption of orthogonality may in some cases be unfounded. Assuming normal errors on the structure, we make the following contributions. We show that for all but the shortest direction vectors the distribution of their inner product is indeed well described by a normal distribution, and we give an unbiased estimator of its mean and its variance. Using this distribution we give a likelihood ratio test of the hypothesis of K>5 mutually consistent orthogonality relations, and a maximum likelihood estimator of the absolute conic based on the same relations, thus providing optimal rectification of the structure. We also provide simple means of computing these in a short succession of linear steps, and assess the validity of that approach