Dimensional measurements and operators in mathematical morphology

Pierre Soille; Jean C. Serra; Jean-Francois Rivest

doi:10.1117/12.58372

1 April 1992 Dimensional measurements and operators in mathematical morphology

Pierre Soille, Jean C. Serra, Jean-Francois Rivest

Proceedings Volume 1658, Nonlinear Image Processing III; (1992) https://doi.org/10.1117/12.58372
Event: SPIE/IS&T 1992 Symposium on Electronic Imaging: Science and Technology, 1992, San Jose, CA, United States

Abstract

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.

Citation Download Citation

Pierre Soille, Jean C. Serra, and Jean-Francois Rivest "Dimensional measurements and operators in mathematical morphology", Proc. SPIE 1658, Nonlinear Image Processing III, (1 April 1992); https://doi.org/10.1117/12.58372

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