Morphological Sampling Theorem and its Extension to Grey-value Images

TL;DR

This paper extends the morphological sampling theorem from binary images to grey-value images, incorporating non-flat structuring elements and providing a theoretical foundation for sampling in grey-scale morphology.

Contribution

It generalizes the binary morphological sampling theorem to grey-value images using the umbra notion and non-flat structuring elements, advancing the theoretical understanding.

Findings

Extended the theorem to grey-value images

Incorporated non-flat structuring elements into the theory

Provided illustrative examples of the extended theory

Abstract

Sampling is a basic operation in image processing. In classic literature, a morphological sampling theorem has been established, which shows how sampling interacts by morphological operations with image reconstruction. Many aspects of morphological sampling have been investigated for binary images, but only some of them have been explored for grey-value imagery. With this paper, we make a step towards completion of this open matter. By relying on the umbra notion, we show how to transfer classic theorems in binary morphology about the interaction of sampling with the fundamental morphological operations dilation, erosion, opening and closing, to the grey-value setting. In doing this we also extend the theory relating the morphological operations and corresponding reconstructions to use of non-flat structuring elements. We illustrate the theoretical developments at hand of examples.