Dimensional measurements and operators in mathematical morphology
Pierre Soille, Jean C. Serra, Jean-François Rivest · Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE · 1992
In mathematical morphology, grey tone images are often considered as 3D Euclidean sets through their umbra or subgraph. This model allows one to extend measurements for sets to grey tone images. It has been shown that any valid measurement on Euclidean sets should satisfy some basic constraints such as invariance to displacements and to magnifications. However, when applied to subgraphs, these measurements may be meaningless as the image plane is not homogeneous with the grey tone axis. An additional constraint is introduced called dimensionality. This property holds for the inhomogeneity of image dimensions. A measurement on a grey tone image will be dimensional if the same measurement applied to this image after a magnification of its image plane and an affine transform of its grey tone axis can be related to the initial measure. The authors first recall valid measurements on sets and their properties. Then it is shown how to generalize to grey tone images and the dimensionality constraint is introduced. Set measurements are then reviewed to determine those satisfying the dimensionality criterion and consideration is given to the measure of the fractal dimension in the light of this new criterion. Eventually, dimensionality must also be considered when processing images. This is illustrated by a segmentation experiment.