Estimation of multi-dimensional homeomorphisms for object recognition in noisy environments

We consider the general problem of object recognition based on a set of known templates, where the available observations are noisy. While the set of templates is known, the tremendous set of possible transformations and deformations between the template and the observed signature, makes any detection and recognition problem ill-defined unless this variability is taken into account. We propose a method that reduces the high dimensional problem of evaluating the orbit created by applying the set of all possible transformations in the group to a template, into a problem of analyzing a function in a low dimensional Euclidian space. In this setting, the problem of estimating the parametric model of the deformation is transformed using a set on non-linear operators into a set of equations which is solved by a linear least squares solution in the low dimensional Euclidian space. For the case where the signal to noise ratio is high, and the non-linear operators are polynomial compositions, a maximum-likelihood estimator is derived, as well.