Affine-Invariants that Distribute Uniformly and Can be Tuned to any Convex Feature Domain Case I: Two-Dimensional Feature Domains

We derive and discuss a set of parametric equations which, when given a convex 2D feature domain, K, will generate affine invariants with the property that the invariants' values are uniformly distributed in the region [0,1]x[0,1]. Definition of the shape of the convex domain K allows the computation of the parameters' values and thus the proposed scheme can be tuned to a specific feature domain. Moreover, this mechanism will guarantee a uniform distribution of the invariants over the mapping's range, i.e. the region [0,1]x[0,1]. The features of all recognizable objects (models) are assumed to be two-dimensional points and uniformly distributed over the assumed feature domain K. The scheme leads to improved discrimination power, improved computational-load and storage-load balancing and can also be used to determine and identify biases in the database of recognizable models (over-represented constructs of object points). We have tested these equations and present results for a variety of convex domains using both synthetic data and real databases. Obvious enchancements produce rigid-transformation and similarity-transformation invariants with the same good distribution properties, making this approach generally applicable. An extension to the case of affine invariants for feature points in three-dimensional space, with the invariants now being uniformly distributed in the region [0,1]x[0,1]x[0,1], has also been carried out and a basic outline is discussed briefly.

Published under the title "2D Affine-Invariants That Distribute Uniformly and Can Be Tuned to Any Convex Feature Domain"