Classification of Lattice Transformations in Image Processing

Abstract

This paper introduces a complete matrix classification of a family of image processing transforms called lattice transformations. Lattice transformations are nonlinear and include mathematical morphology transforms as a subclass. A matrix algebraic structure called minimax algebra provides a rigorous mathematical environment for lattice transforms as used in image processing. This is the first application of minimax algebra to image processing. Minimax algebra was originally developed for solving machine scheduling and other problems in operations research. The relationship between minimax algebra, mathematical morphology, and image algebra, a high-level image processing language, is presented in this paper, and a theoretical foundation in which to analyze image lattice transformations in context of matrices is established. Several examples are presented to exemplify the power of these relationships.