Group morphology
J.B.T.M. Roerdink. Group morphology. Pattern Recognition 33 (6), pp. 877-895, 2000.
In its original form, mathematical morphology is a theory of binary image transformations which are invariant under the group of Euclidean translations. This paper surveys and extends constructions of morphological operators which are invariant under a more general group G, such as the motion group, the affine group, or the projective group. We will follow a two-step approach: first we construct morphological operators on the space P(G) of subsets of the group G itself; next we use these results to construct morphological operators on the original object space, i.e. the Boolean algebra P(E^n) in the case of binary images, or the lattice Fun(E^n,T) in the case of grey value functions F:E^n -> T, where E equals R or Z, and T is the grey value set. G-invariant dilations, erosions, openings and closings are defined and several representation theorems are presented. Examples and applications are discussed.
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