Spatial color image processing using Clifford algebras: application to color active contour

Abstract

In the literature, the color information of the pixels of an image has been represented by different structures. Recently, algebraic entities such as quaternions or Clifford algebras have been used to perform image processing for example. This paper presents the embedding of color information into the vectorial parts of a multivector. This multivector is an element of the geometric or Clifford algebra constructed from a three-dimensional vector space. This formalism presents the advantage of algebraically separating colors which are handled entities from the geometric operations done to them. We propose to introduce several contributions for color image processing by using this Clifford algebra. First, as colors are represented by 1-vectors, we point out that a color pixel given in the RGB color space can be expressed algebraically by its hue saturation and value using the geometry. Then, we illustrate how this formalism can be used to define color alterations with algebraic operations. We generalize linear filtering algorithms already defined with quaternions and define a new color edge detector. Finally, the application of the new color gradient is illustrated by a new color formulation of snakes. Thus, we propose in this paper the definition and exploitation of a formalism in which we geometrically handle colors with algebraic entities and expressions.