By P.W. Becker

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Invariant attributes can be generated by moments (Hu 1962), autocorrelations (Mc Laughlin and Raviv 1968), integral geo-diffraction-pattern sampling (Lendaris and Stanley 1970), integral geometry (Tenery 1963) and by application of Lie group theory (Hoffmann 1965). Examples oftransformation invariant attributes may be found in papers indicated by descriptor 0 12 the in Minsky 1963. Often the patterns can be expanded using a set of orthonormal functions; in such cases the coefficients to the p most important terms of the expansion may be used as attrib utes.

In Figure 4 will give perfect separation if and only if the means for categorization can establish a separation surface consisting of two vertical and two horizontal lines. In the remaining part ofthis subsection it will be assumed that the de signer has selected an IP and a means for categorization which are suited for the problem at hand. Selection of a Set of p Attributes 51 Fig. 4. A Bivariate Disbibution. The members of CI8811 A are located in the three 3 x 2 rectangles marked A ; the tri-modal, bi8 variate probability density is constant and equal A B C 6 to 1/18.

Izer, a categorizer would have been obtained that was optimum in the sense chosen by the designer. In practice the approach described above - the construction of an "optimal recognition function" (Marill and Green 1960) - seldomly works out for three reasons. l). (2) There is usually not enough time and funding to process the many patterns even if they were available. (3) Unless the separation surfaces are of fairly simple shape or the dimensionality p, of the space is very low, the number of constants needed to describe the surfaces easily becomes so large that the information storage in the categorizer presents problems; in the extreme case where the micro-regions do not fuse at all it becomes necessary to store the class membership for each of the NM micro-regions.