The main goal of the paper is to show that commutative hypercomplex algebras and Clifford algebras can be used to solve problems of multicolor image processing and pattern recognition in a natural and effective manner.
We present a new theoretical framework for multichannel image processing using commutative hypercomplex algebras. Hypercomplex algebras generalize the algebras of complex numbers. The main goal of the work is to show that hypercomplex algebras can be used to solve problems of multichannel (color, multicolor, and hyperspectral) image processing in a natural and effective manner. In this work, we suppose that the animal brain operates with hypercomplex numbers when processing multichannel retinal images. In our approach, each multichannel pixel is considered not as an K–D vector, but as an K–D hypercomplex number, where K is the number of different optical channels. The aim of this part is to present algebraic models of subjective perceptual color, multicolor and multichannel spaces.
In this work we study the harmonic analysis of functions on the 71-D Heisenberg groups H over the Galois field GF(p) for generat,ing Gabor a.toms. Analogous to the Fourier transform, the expmsion of functions on t,he basis of irreducible coinplex matrix representat,ions of the Heisenberg group defines the generalized Fourier transform on this group, or, simply, the Fourier-Heisenberg transform. The f a t algorithms for the n-D Fourier tra.nsforms on t,he Heisenberg and affine groups are developed in this pa,per. A general method of comput,ing the Gabor distribution and Wavelet transform based on the fast Fourier-HeisenbergWeyl transform is also present,ed.