Waish-Hadamard Power Spectra Invariant to Certain Transform Groups

Changes in Walsh-Hadamard power spectrum of an input pattern are investigated under several transformations: 1) shifting the elements of the input pattern cyclically, 2) enlarging and reducing the input pattern, 3) rotating the input pattern by multiples of 90°, and so on. Then the Walsh-Hadamard power spectrum is developed to be unchangeable by all the transformations through an introduced composing process. It may be considered as one of geometrical features. Every interesting geometrical property is generally invariant under some transformation groups. The composing process is available for obtaining functions having such group-invariant properties. The main idea is to make a linear combination of group-equivalent functions. First a G1-invariant power spectrum and next a permutation group on the G1-invariant power spectrum caused by G2 operating on the input pattern is found, thus arriving at a power spectrum being constant under both G1 and G2 through the composing process, where G1 and G2 are some transformation groups. Continuing this process, a power spectrum can be derived from the Walsh-Hadamard power spectrum that is invariant under a more general group of transformations.