Chapter Six - Perspectives on Color Image Processing by Linear Vector Methods Using Projective Geometric Transformations

Abstract

Development of processing methods for color images has been slow, but over the past 10–15 years ideas have been developed employing geometric operations implemented with quaternion algebra in the color space of the image pixels as generalizations of the fundamental scaling and spatial shift operations of linear grey-scale image processing. The goal has been filters that are sensitive in some way to both color and spatial features. In Euclidean color space the set of linear operations is very limited. Recent developments, however, have shown that there is a simple connection between geometric operations expressed using quaternion equations and those expressed as 4×44×4 matrices using homogeneous coordinates. These operations include projective transformations as well as all the classical Euclidean operations. Remarkably, they are all linear in homogeneous coordinates. This realization greatly expands the possibilities of linear vector filtering methods and means that the insight gained by working in quaternion algebra can be applied to the design of algorithms implemented using matrix methods. This paper (1) illustrates the use of simple quaternion equations to express geometric operations and (2) shows how the use of homogeneous coordinates makes it possible to work with quite difficult geometric transformations relatively easily. The possibilities for using projective and other transformations for devising new types of color image filter are shown and the difficulties discussed.