Theory of vector filters based on linear quaternion functions

Vector filtering of signals and images has many applications, but there is little theoretical framework underpinning rather ad-hoc approaches to the development of such filters. In this paper we make a significant step towards improving this position by showing that the geometric operations possible on samples or pixels can be expressed in a canonic form. In the formalism of quaternions, this canonic form has at most 4 quaternion coefficients. In the formalism of matrices and groups, the coefficients are 4×4 matrices, or members of the General Linear Group of order 4. We show how to combine series and parallel filters and reduce the result to the canonic form, and we discuss the set of geometric operations possible on vector samples or pixels, thus generalising the notion of sample scaling in classical DSP to a geometric concept of sample modification through linear operations.