Parametric estimation of multi-dimensional affine transformations in the presence of noise: a linear solution

We consider the general framework of planar object registration and 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 a noisy 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. We propose a method that employs a set of nonlinear operators to replace the original high dimensional and non-linear problem by an equivalent linear least-squares problem, expressed in terms of the unknown affine transformation parameters. The proposed solution is unique and is applicable to any affine transformation regardless of the magnitude of the deformation