Parametric Estimation of Multi-Dimensional Homeomorphic Transformations: Solving a Group-Theory Problem as a Linear Problem

We consider the general framework of planar object registration and recognition based on a set of known templates. Whereas the set of templates is known, the tremendous set of possible 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 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 homeomorphism 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 with an equivalent linear problem, expressed in terms of the unknown parameters of the transformation model. The proposed solution is unique and is applicable to any homeomorphic transformation regardless of its magnitude. In the special case where the transformation is affine the solution is shown to be exact. The effectiveness of the proposed solution will be demonstrated using various examples.