The trinion Fourier transform of color images

Any color may be represented in terms of three components (RGB or HSL) or four components (CMYK). For the four component color representation the use of quaternions, with one real and three imaginary components, is natural. By setting one component to zero, quaternions have been used in RGB or HSL representation of colors and color images. In this paper a new quantity, trinion, with one real and two imaginary components, is introduced and its use in color image representation is examined. The goal is to see if significant efficiencies in representation, analysis and computation involving three component color images accrue with the use of trinions. Two versions of the trinion Fourier transform (TFT) are introduced and it is shown that using TFT is preferable for combined analysis of three component color images rather than separate monochromatic analysis of each component and use of quaternions. Joint space-wavenumber localized trinion S (TS) transform with a two-dimensional Gaussian window function that scales with wavenumbers is also presented. Invertibility, rotation invariance, and computational aspects of the TS transform are discussed.