Dimensionality of morphological operators and cluster analysis
Pierre Soille, Jean-François Rivest · Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE · 1993
The concept of dimensionality has been introduced in image analysis to assess the validity of image measurements. In this paper, we extend the notion of dimensionality to image operators and present formal definitions for a dimensional operator. We make a distinction between dimensional operators for unknown image plane scalings and dimensional operators for unknown intensity axis scalings. A dimensional operator is an operator that commutes with these scalings. Morphological operators are then reviewed to determine whether they are dimensional. Finally, we show that new dimensionality problems arise when the image plane itself has inhomogeneous units. This lead us to define dimensional image operators for image plane anamorphosis (i.e., stretching or shrinking of the image plane in one direction). Multivariate histograms are typical n-dimensional images whose image plane is not homogeneous. It is shown that some clustering techniques applied to these histograms encounter dimensionality problems.