Parametric estimation of multi-dimensional affine transformations: an exact linear solution [image recognition applications]

We consider the general framework of planar object recognition based on a set of known templates. 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 to 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-D 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 coordinate 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.