Parametric Estimation of Affine Transformations: An Exact Linear Solution

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. We propose 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 proposed solution includes the case where the deformation relating the observed signature of the object and the reference template is composed both of the geometric deformation due to the affine transformation of the coordinate system and a constant amplitude gain. The proposed solution is unique and exact and is applicable to any affine transformation regardless of its magnitude.