Fourier transforms are a fundamental tool in signal and image processing, yet, until recently, there was no definition of a Fourier transform applicable to color images in a holistic manner. In this paper, hypercomplex numbers, specifically quaternions, are used to define a Fourier transform applicable to color images. The properties of the transform are developed, and it is shown that the transform may be computed using two standard complex fast Fourier transforms. The resulting spectrum is explained in terms of familiar phase and modulus concepts, and a new concept of hypercomplex axis. A method for visualizing the spectrum using color graphics is also presented. Finally, a convolution operational formula in the spectral domain is discussed.

Hamilton's hypercomplex. or quaternion. extension to the complex numbers provides a means to algebraicly analyze systems whose dynamics can be described by a system of partial differential equations. The Quaternion-Fourier transformation, defined in this work, associates two dimensional linear time-invariant (2D-LTI) systems of partial differential equations with the geometry of a sphere.This transform provides a generalized gain-phase frequency response analysis technique. It shows full utility in the algebraic reduction of 2D-LTI systems described by the double convolution of their Green's functions. The standard two dimensional complex Fourier transfer function has a phase associated with each frequency axis and does not describe clearly how each axis interacts wiih the other. The Quaternion-Fourier transfer function gives an exact measure of this interaction by a single phase angle that may be used as a measure of the relative stability of the system. This extended Fourier transformation provides an exquisite tool for the analysis of 2D-LTI systems.

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An involution or anti-involution is a self-inverse linear mapping. In this paper we study quaternion involutions and antiinvolutions. We review formal axioms for such involutions and anti-involutions. We present two mappings, one a quaternion involution and one an anti-involution, and a geometric interpretation of each as reflections. We present results on the composition of these mappings and show that the quaternion conjugate may be expressed using three mutually perpendicular anti-involutions. Finally, we show that projection of a vector or quaternion can be expressed concisely using three mutually perpendicular antiinvolutions.