Decomposition and construction of neighbourhood operations using linear algebra

Abstract

In this paper, we introduce a method to express a local linear operated in the neighbourhood of each point in the discrete space as a matrix transform. To derive matrix expressions, we develop a decomposition and construction method of the neighbourhood operations using algebraic properties of the noncommutative matrix ring. This expression of the transforms in image analysis clarifies analytical properties, such as the norm of the transforms. We show that the symmetry kernels for the neighbourhood operations have the symmetry matrix expressions. © 2008 Springer-Verlag Berlin Heidelberg.