Multi-projective Parameter Estimation for Sets of Homogeneous Matrices

A number of problems in computer vision require the estimation of a set of matrices, each of which is defined only up to an individual scale factor and represents the parameters of a separate model, under the assumption that the models are intrinsically interconnected. One example of such a set is a family of fundamental matrices sharing an infinite homography. Here an approach is presented to estimating a general set of interdependent matrices defined to within separate scales. The input data is assumed to consist of individually estimated matrices for particular models, which when considered collectively may fail to satisfy the constraints representing the inter-model relationships. Two cost functions are proposed for upgrading, via optimisation, the data of this sort to a collection of matrices satisfying the inter-model constraints. One of these functions incorporates error covariances. Each function is invariant to any change of scale for the input estimates. The proposed approach is applied to the particular problem of estimating a set of fundamental matrices of the form of the example set above. Experimental results are given which demonstrate the effectiveness of the approach.