| The research for this dissertation is concerned with the investigation of an algebraic structure, known as image algebra, which is used for expressing algorithms in image processing. The major result of this research is the establishment of a rigorous and coherent mathematical foundation of the subalgebra of the image algebra involving non-linear image transformations. In particular, a classification in the image algebra of a set of non-linear image transformations called lattice transforms is presented, using minimax matrix algebra as a tool. Several applications to image processing problems are discussed. Specifically, in addition to describing several non-linear transform decomposition techniques, the subalgebra is used as a model and a tool for the development of methods to compute lattice transforms locally.;The basic operands and operations of the image algebra and minimax algebra are defined, as well as the relationships between the two algebras. Properties of the minimax algebra including the lattice eigenvalue problem are mapped to the image algebra. Mathematical morphology is shown to be embedded in the image algebra as a special subclass of lattice transforms. Networks of processors are modeled as graphs, and images are represented as functions defined on the nodes of the graph. It is shown that every lattice image-to-image transform can be weakly factored into a product of lattice transformations each of which are implementable on the network if and only if the graph corresponding to the network is strongly connected. Necessary and sufficient conditions are given to decompose a rectangular template into two strip templates. A division algorithm is given which is a generalization of a boolean skeletonizing technique. The transportation problem from linear programming is expressed in the image algebra. A method to produce an image complexity measures is discussed. Most results are given both in image algebra and matrix algebra notation. |
