Modern Methods of Image Processing and Pattern Recognition: Part 1. Algebraic Models of Color and Hyperspectral Images

by Valeriy Labunets

We present a theoretical framework for multidimensional image processing using hypercomplex and Clifford algebras. The main goal of the work is to show that commutative hypercomplex algebras and Clifford algebras can be used to solve problems of hyperspectral, multi-color and color image processing and pattern recognition in a natural and effective manner. Primates and animals with different evolutionary histories have color visual systems of different dimensionality. For example, the human brain uses three channel (RGB) images, reptile and tortoise brains use five channel multicolor images, and shrimps use ten channel multicolor images. Our hypotheses are First, brains of primates operate with hypercomplex numbers during image processing and recognition. Secondly, brains use different algebras on two levels, retina and visual cortex. Multicolor images appear on the retina as functions with values in a multiplet K–cycle algebra where K is the number of image spectral channels. But multicolor images in an animal’s visual cortex are functions with values in Clifford algebra. Thirdly, visual systems of animals with different evolutionary history use different hypercomplex algebras for color and multicolor image processing. In the algebraic approach, each pixel is considered not as a multi–dimensional vector, but as a hypercomplex number. For this reason, we assume that the human retina and human visual cortex use 3D hypercomplex numbers and 8D Clifford numbers, respectively, to process color images. While we have no biological evidence in the form of experiments to verify that the brain actually uses any of the algebraic properties arising from the structures of vector spaces or Clifford algebras, we do know that animals are able to recognize objects in an invariant manner and to process multicolor images effectively. We give algebraic models for two general levels of visual systems, retina and visual cortex, using different hypercomplex and Clifford algebras. It is our aim to show that these fit more naturally to the task of recognition of multicolor patterns than does the use of color vector spaces. One can argue that nature has, through evolution, learned to utilize properties of hypercomplex numbers. Thus, the visual cortex might have the ability to operate as a Clifford algebra computing device.

132 pages, Paperback

Published June 3, 2012