Physical significance of measurements in images

In mathematical morphology, images are represented with sets. On binary images, sets delineate portions in the image plane, creating binary shapes. Measurements can be done on these shapes. It has been demonstrated that, for digital images, there exist very few basic measurements. These measurements, along with image transformations, generate all the possible measurements that can be done on an image. For binary images, all the measurements are based on the area, perimeter and connectivity number (number of connected components minus the number of holes inside them). Gray-tone images can also be modeled as 3-D sets. However the 5-D space is not homogeneous: units along the image plane are not the same as those on the intensity axis. This causes problems because not all the basic measurements are physically valid. The basic measurements on gray-tone images are the volume, surface, norm (or mean curvature) and the connectivity number. In this paper, we present a criterion to assess the physical validity of the basic measurements. This allows us to further limit the number of useful basic measurements. On gray-tone images, these are the volume and the connectivity number. We illustrate our findings with an experiment using physically valid and invalid measurements. These measurements are applied to texture characterization and segmentation. We study the behavior of these measurements under illumination changes. We show that a physically invalid measurement gives erratic answers under such circumstances. This has important consequences on the robustness of image analysis algorithms