Algebraically extended representations of multi-dimensional signals

In this article we introduce the Clifford Fourier transform (CFT) which is based on the ta-dimensional Fourier transform but offers a representation of an n-dimensional signal with values in a 2(n)-dimensional Clifford algebra. For n = 2 we get the special case of the quaternionic Fourier transform (QFT). The QFT provides information about the symmetry of a real signal in a more explicit way then the Fourier transform does. The QFT is related to the Fourier transform and to the Hartley transform. Based on the QFT we introduce two-dimensional phase concept. The shift theorem is examined in terms of the QFT and experimental results concerning the estimation of the two-dimensional phase-difference of two images are presented.