Basic Morphological Operations, Band-Limited Images and Sampling

Morphological operations are simple mathematical constructs, which have led to effective solution for many problems in image processing and computer vision. These solutions employ discrete operators and are applied to digitized images. The mathematics behind the morphological operators also exists in the continuous domain, the domain where the images came from. We observed that the discrete operators cannot reproduce the results obtained by the continuous operators. The reason for this is that neither the operator (the structuring element) nor the result of the operation are band-limited, and thus cannot be represented by equidistant samples without loss of information. The differences between continuous-domain and discrete-domain morphology are best shown by the dependency of the discrete morphology on sub-pixel translations and rotations of the images before digitization. This article describes an algorithm that applies continuous-domain morphology to properly sampled images. We implemented the dilation for one-dimensional images (signals), and with it constructed the erosion, the closing and the opening. We provide a discussion on a possible extension to higher-dimensional images.