Structure of Mathematical Morphology

For contact surface measurement, estimating the profile of actual surface from the locus of the center point of the stylus is an inverse problem. The measurement process which transforms the profile of actual surface to the locus of the center point is a dilation operation with a disk with error. An erosion operation can be taken as the inverse mapping of the dilation to estimate the actual profile. By using category theory, erosion and dilation can be formulated as two functors between two categories of sets of vectors. Each category has sets of vectors as objects and inclusion functions between sets as
morphisms. The functor of erosion is left adjoint to the functor of dilation