On the representation of stack operators by mathematical morphology
This work provides a theoretical foundation for designing image operators, potentially simplifying grey-scale image processing tasks by reducing them to binary image problems, though it is incremental in extending known concepts.
The paper tackles the representation of grey-scale image stack operators by showing they are 1-Lipchitz extensions of set operators that inherit lattice properties from characteristic set operators, enabling the design of grey-scale image processing solutions by first solving binary image transformations.
This paper introduces the class of grey-scale image stack operators as those that (a) map binary-images into binary-images and (b) commute on average with cross-sectioning. Equivalently, 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 adding. 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.