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6. H. Schulz-Mirbach Anwendung von Invarianzprinzipien zur Merkmalgewinnung in der Mustererkennung. Dissertation, Technische Universitiit Hamburg-Harburg, September 1994. Zur Veroffentlichung akzeptiert als VDI Fortschrittbericht, Reihe 10, VDI-Verlag, 1995. 7. H. Schulz-Mirbach Invariant features for gray scale images. Tagungsband Mustererkennung 1995 (17. DAGM Symposium), Bielefeld 1995. 8. I. Weiss Geometric Invariants and Object Recognition. International Journal of Computer Vision, 10:3, 201-231, 1993.

This can be reformulated by saying that the features should be optimal with respect to a given (application specific) criterion. The question is how to choose the function I so that the group average A[/] is optimal. The basic idea is to consider parameterized families of functions. Averaging such parameterized functions according to equation (4) over the transformation group 26 m[O] m[l] A[f](m) m[2] m[3] Fig.!. Calculating invariant features by evaluating a local function fl and summing the results of the local computations by traversing a binary tree.

5 How to choose the parameters With hard limiter functions for all functions tj and Uj the mapping of the network can be described as F := 10 0 ... 0 lid N with lid N : JRN -+ f)· '. {O, {O, l}N/2 1}N/(2 1dN - i ) -+ 10: {O, 1}2 -+ {O,l}. (8) {O, 1}N/(21dN-i-l) for j = ld N - 1, ... , 1 (9) (10) Sigmoid transfer functions are smoother and therefore show better robustness with respect to distortions in the pattern. Nevertheless, according to our experience the final network output is similar to that for hard limiter functions.