Hypercomplex operators and vector correlation

Auto-, cross- and phase-correlation are standard techniques in image and signal processing. Recently published hypercomplex forms have been applied to vector images using quaternion Fourier transforms. The various permutations of ordering and conjugation of the basic definition are compounded by realization of a closed set of hypercomplex operators leading to multiple derivations and the possibility of unintentional mirror responses. This paper identifies the necessity of carefully choosing the form of the correlation function, presents for the first time the set of equivalent forms for hypercomplex correlation and demonstrates how these can be validated using hypercomplex operator algebra.