Invariants for 2D and 3D Pattern Recognition Problems - New Results for a Classical Problem

Hans Burkhardt
Albert-Ludwigs-Universitat Freiburg, Germany

In many pattern recognition problems images have to be classified independent of their current position and orientation, which is just a nuisance parameter. Instead of comparing a measured pattern in all possible locations against the prototypes it is much more attractive to extract position-invariant and intrinsic features and to classify the objects in the feature space. Mathematically speaking, patterns form an equivalence class with respect to a geometric coordinate transform describing motion. Invariant transforms are able to map such equivalence classes into one point of an appropriate feature space.

The talk will describe new results for this classical problem and outlines general principles for the extraction of invariant features from images (Haar integrals, Lie-Theory, Normalization techniques). The nonlinear transforms are able to map the object space of image representation into a canonical frame with invariants and geometrical parameters. Beside the mathematical definition the talk will concentrate on characterizing the properties of the nonlinear mappings with respect to completeness and possible ambiguities, disturbance behavior and computational complexity. We especially investigated Haar integrals for the extraction of invariants based on monomial and relational kernel functions.

Examples and applications will be given for problems in 2D and 3D, namely applications in content-based image and object retrieval and classification tasks in 2D and 3D (classification and retrieval of biological objects and structures).