Digital morphology in the 3-D space

A logical framework for digital 3-D treatment is given steming from the point of view of the set theory. Two different structures can equip the digital space, namely that of a module and that of a grid. The notions of size distributions, filterings, erosions, openings, thinnings, convexity, derive from the first structure, those of rotations, homotopy and connectivity from the second one. To digitalize, it is not enough merely to replace a Euclidean figure by a mosaic of pixels. We also have to replace Euclidean translations, rotations, convex hulls, connected particles, etc. by analogous versions on a grid of points. A number of umbrella notions result. The first is the notion of a module, classical in linear algebra, which governs increasing transformations, projections and convexity analyses. Then there is cur notion of a grid, which is the key to all the rotation and connectivity problems. Finally, one can also equip the grid with a metric, but we shall not do it here.