Projective-space colour filters using quaternion algebra

Instead of mapping colour image pixels into Euclidean vectors as is conventionally done in colour image processing, we present the idea of using projective space mapping based on homogeneous coordinates. This approach offers a much richer set of geometric operations in the colour space compared to the Euclidean geometry operations that exist in classical colour spaces. The projective geometry of points (pixel values) is described and compared to the classical Euclidean view of pixel values as vectors. The use of homogeneous coordinates is introduced and the geometric operations that are possible are outlined. Then it is shown how colour image pixel values may be transformed into and out of homogeneous coordinates, based on a representation in both cases using quaternions. We then show some examples of colour image operations that offer potential for new types of vector filter and we discuss the possibilities.