In the framework of mathematical morphology, we study in two particular cases how morphological measurements characterize a set. The first one concerns the geometrical covariogram and we show that in the generic polygonal (non necessarily convex) case, the geometrical covariogram is characteristic up to a translation and reflection about the origin. The second one concerns the surface area of the dilation by compacts. We show that in the random case, the mean value of the measurements allows a characterization up to a random translation. In the deterministic case the theorem holds. Procedures to retrieve the original set from its measurements is given in both cases.
"Two inverse problems in mathematical morphology", Proc. SPIE 1568, Image Algebra and Morphological Image Processing II, (1 July 1991); doi: 10.1117/12.46123; https://doi.org/10.1117/12.46123