Relation-based variations of the discrete radon transform

The finite Radon transform was introduced by Bolker around 1976. Since then, many variations of the discrete Radon transform have been proposed. In this paper, we first propose a variation of the discrete Radon transform which is based on a binary relation. Then, we generalize this variation to weighted Radon transformation based on a weighted relation. Under such generalization, we show that discrete convolution is a special case of weighted Radon transformation. To further generalize Radon transformation to be defined on lattice-valued functions, we propose two nonlinear variations of Radon transformation. These two nonlinear variations have very close relations with morphological operations. Finally, we generalize Matheronʹs representation theorem to represent translation-invariant operations on functions from an abelian group to a complete lattice.