Conditional solutions for the affine reconstruction of N-views

The known factorisation algorithm for the maximum likelihood affine reconstruction requires that all the feature points used must be visible in all views. We derive here a conditional solution for the 3D coordinates of the feature points and translation vectors given the inhomogeneous affine projection matrices, but no single feature point needs to be visible in all views. The solution represents closed-form maximum likelihood, generalised affine triangulation under missing data and unknown translations, and it implies two iterative algorithms for the maximum likelihood affine reconstruction where all the measured data may be used. For the case where at least four non-coplanar points are visible in all views, a non-iterative solution for affine reconstruction with missing data is also proposed. As reported, the closed-form-expression additionally has applications in affine bundle adjustment and identifying degenerate configurations.