Parametric estimation of multi-dimensional affine transformations: An exact linear solution

We consider the general framework of planar object recognition based on a set of known templates. While the set of templates is known, the tremendous set of possible affine transformations that may relate the template and the observed signature, makes any detection and recognition problem ill-defined unless this variability is taken into account. Given an observation on one of the known objects, subject to an unknown affine transformation of it, our goal is to estimate the deformation that transforms some pre-chosen representation of this object (template) into the current observation. The direct approach for estimating the transformation is to apply each of the deformations in the affine group to the template in search for the deformed template that matches the observation. We propose a method that employs a set of non-linear operators to replace this high dimensional problem by an equivalent linear problem, expressed in terms of the unknown affine transformation parameters. This solution is further extended to include the case where the deformation relating the observed signature of the object and the template is composed both of the geometric deformation due to the affine transformation of the co-ordinate system and a constant illumination change. The proposed solution is unique and exact and is applicable to any affine transformation regardless of the magnitude of the deformation.