Recursive operations in image Algebra and their applications to image processing
Dong Li · University of Florida Digital Collections (University of Florida) · 1991
The result of this research is a continuation of the development of image algebra, an algebraic structure for digital image processing. A study of the use of image algebra as a mathematical model and tool for the development of linear and nonlinear recursive image processing algorithms is presented. Specifically, the notion of a recursive template and of recursive template operations are introduced, which allow image algebra to express a set of linear and nonlinear recursive transformations. Algebraic properties of these recursive template operations are given, which provide a mathematical basis for recursive template composition and decomposition. The basic operands and operations of image algebra are defined as well as the new recursive template and recursive template operations. Relationships between the extended image algebra and other algebraic structures are described. In the linear case, the extended image algebra may be mapped to rings; in the nonlinear case, it may be mapped to a subalgebra of the minimax algebra. Recursive filters used in signal and image processing are shown to be a special subclass of linear recursive transforms in the extended image algebra. Necessary and sufficient conditions for the decomposition of both recursive and nonrecursive separable templates are given. Algorithms are provided to decompose a set of symmetric convex morphological templates. Template decomposition techniques based on the factorization of max-polynomials are also described. Finally, some applications of recursive template operations in image processing are presented.