Affine-Invariants That Distribute Uniformly and Can Be Tuned to Any Convex Feature Domain Case II: Three-Dimensional Feature Domains

.We derive and discuss a set of parametric equations which, when given a convex 3D 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]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]x[0,1]. The features of all recognizable objects (models) are assumed to be three-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). Obvious enhancements produce rigid-transformation and similarity-transformation invariants with the same good distribution properties, making this approach generally applicable. We have tested these equations and present results for a variety of convex domains using synthetic data; we also demonstrate a potential use of the affine invariants in the real-world problem of docking a ligand (i.e. a drug molecule) to a receptor (i.e. a protein) by using actual crystallographic data for the HIV-protease and Abbott's A-74704.

Published under the title "Well-Behaved, Tunable 3D Affine Invariants"