On invariant sets of a certain class of fast translation-invariant transforms

The paper gives a complete description of the mapping properties of a general class of fast translation-invariant transforms which are used for position independent pattern classification problems. The complete set of transformation invariants are deduced allowing to generate all patterns of the original space which are mapped into one point of the feature space, thus clarifying structural ambiguities. The results are extended to the special case of the R-transform. The power spectrum of the modified Walsh-Hadamard transform is deduced as a particular case of the general class of translation invariant transforms. The results are valid for one- and two-dimensional patterns.