A random sets framework for error analysis in estimating geometric transformations — a first order analysis

We consider the problem of estimating the geometric deformation of an object, with respect to some reference observation on it. Existing solutions, set in the standard coordinate system imposed by the measurement system, lead to high-dimensional, non-convex optimization problems. In we proposed a novel framework that employs a set of non-linear functionals to replace this originally high dimensional problem by an equivalent problem that is linear in the unknown transformation parameters. The non-linearity of the employed functionals implies that using standard methods for analyzing the estimation errors is complicated, and is tractable only under a high SNR assumption. In this paper we present an entirely different approach for deriving the statistics of the estimator. The basic principle of this novel approach is based on the understanding that since our goal is to estimate the geometric transformation, the appropriate noise model for the problem is a model that explicitly relates the presence of noise and the measures of the geometric entities in the observed image. This approach naturally leads to very efficient estimation procedures and alleviates the need for restrictive assumptions made in previous work.