Dual distributions of multilinear geometric entities

In this paper, we propose how the parameter distributions of multilinear geometric entities can be dualised. The dualisation concern, for example, the parameter distributions of conics, multiple view tensors, homographies, or as simple entities as points, lines, and planes. The dual distributions are related to Triggs´ joint feature distributions but our approach is different in certain fundamental aspects. Our starting point is in the assumption that the maximum likelihood estimate, or the corresponding robust estimate, and the covariance matrix of the parameters of the geometric entity are available. We then use the asymptotic normality property of the MLE which allows us to transform the parameter uncertainty distribution in a dual form. The dualisation of the parameter distribution allows us, for instance, to look at the uncertainty distributions in feature distributions, which are essentially tied to the distribution of training data, and helps us to derive conditional distributions for point or line transfer and characterise confidence intervals of the estimates. Applications of the proposed approach are thus uncertainty analysis, statistical prediction, probabilistic transfer, etc.