Maximum Likelihood 3D Reconstruction from One or More Images under Geometric Constraints

We address the 3D reconstruction of scenes in which some planarity, collinearity, symmetry and other geometric properties are known a-priori. Our main contribution is a reconstruction method that has advantages of both constraintbased and model-based methods. Like in the former, the reconstructed object needs not be an assemblage of predefined shapes. Like in the latter, the reconstruction is a maximum likelihood estimate and its precision can be estimated. Moreover, we improve on other constraint-based methods by using symmetry and other forms of regularity in the scene, and by working indifferently with one or more images. A second contribution is a method for parameterising a configuration of 3D points subject to geometric constraints. Using this parameterisation, the maximum likelihood reconstruction is obtained by solving an unconstrained optimisation problem. Another contribution lies in validating experimentally the assumption under which the maximum likelihood estimator was defined, namely, that the errors in hand-identified 2D points behave approximately like identically distributed independent Gaussian random variables. With this assumption validated, benchmarking is performed on synthetic data and the precision obtained on real-world data is shown. These experiments show that the maximum likelihood estimator is well-behaved and give insight on the precision obtained in real-world situations.