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Properties of the Zhang-Watari transform

The relationship between the 2D multiple-valued (complex-valued) Haar transform and the 2D real valued Zhang-Watari transform of patterns is studied and a method is disclosed to compute the Haar-(more properly, Watari)-spectrum of a pattern by using only real arithmetic. It is shown that to extend the straight forward 1D results to the 2D case, a special permutation operation has to be introduced. This result is closely related to that known for 2D Chrestenson and Zhang-Hartley transforms, except that a different choice of pattern partition and permutation is required.