On weighting and choosing constraints for optimally reconstructing the geometry of image triplets

Abstract

Optimally reconstructing the geometry of image triplets from point correspondences requires a proper weighting or selection of the usedconstrain ts between observed coordinates andunkno wn parameters. By analysing the ML-estimation process the paper solves a set of yet unsolved problems: (1) The minimal set of four linearily independent trilinearities (Shashua 1995, Hartley 1995) actually imposes only three constraints onto the geometry of the image triplet. The seeming contradiction between the number of used constraints, three vs. four, can be explainednaturally using the normal equations. (2) Direct application of such an estimation suggests a pseudoinverse of a 4 × 4-matix having rank 3 which contains the covariance matrix of the homologeous image points to be the optimal weight matrix. (3) Insteadof using this singluar weight matrix one could select three linearily dependent constraints. This is discussed for the two classical cases of forward andlateral motion, andclarifies the algebraic analyis of dependencies between trilinear constraints by Faugeras 1995. Results of an image sequence with 800 images andan Euclidean parametrization of the trifocal tensor demonstrate the feasibility of the approach.