Closed-Form Solutions for Affine Reconstruction under Missing Data

The known factorization algorithm for the maximum likelihood affine reconstruction requires that all the feature points used must be visible in all views. We derive here a closed-form-expression for the 3D coordinates of the feature points and translation vectors given the inhomogeneous affine projection matrices, but no single feature point is required to be visible in all views. The expression represents a closed-form maximum likelihood affine triangulation under missing data and unknown translations and it implies an expectation maximization algorithm for the maximum likelihood affine reconstruction where all the measured data may be used. The solution also has applications in bundle adjustment and identifying degenerate motion.