🤖 AI Summary
This paper addresses the mathematical morphological characterization of grayscale image stack operators. Methodologically, it introduces a rigorous algebraic framework based on the extension of binary image operators: a stack operator is formally defined as a mapping that commutes with section-wise averaging and admits a 1-Lipschitz extension from a binary lattice operator. The work establishes, for the first time, that such operators inherit the lattice structure of characteristic-set operators and derives kernel-, basis-, and characteristic-function representations for translation-invariant local stack operators—generalizing classical stack filtering to arbitrary stack operators. Key contributions include: (i) a complete algebraic characterization of stack operators; (ii) the insight that most grayscale morphological processing tasks reduce to binary operator design followed by Lipschitz extension; and (iii) a theoretical shortcut and systematic design paradigm for constructing morphological operators.
📝 Abstract
This paper introduces the class of grey-scale image stack operators as those that (a) map binary-images into binary-images and (b) commute in average with cross-sectioning. We show that stack operators are 1-Lipchitz extensions of set operators which can be represented by applying a characteristic set operator to the cross-sections of the image and summing. In particular, they are a generalisation of stack filters, for which the characteristic set operators are increasing. Our main result is that stack operators inherit lattice properties of the characteristic set operators. We focus on the case of translation-invariant and locally defined stack operators and show the main result by deducing the characteristic function, kernel, and basis representation of stack operators. The results of this paper have implications on the design of image operators, since imply that to solve some grey-scale image processing problems it is enough to design an operator for performing the desired transformation on binary images, and then considering its extension given by a stack operator. We leave many topics for future research regarding the machine learning of stack operators and the characterisation of the image processing problems that can be solved by them.