The Geometry of Basis Sets for Morphologic Closing

Maragos [1] has recently provided an elegant frame- work for the decomposition of many morphologic operations into orthogonal components or basis sets. Using this framework, a method is described to find the minimal basis set for the important operation of closing in 2-D. The closing basis sets are special because their elements are members of an ordered, global set of closing shapes or primitives. The selection or design of appropriate individual or multiple structuring elements for image filtering can be better understood, and sometimes implemented more easily, through consideration of the orthogonal closing de-composition. Partial closing of images using ordered fractions of a closing basis set(s) may give a finer texture or roughness measure than that obtained from the conventional use of scaled sets of shapes such as the disc. The connection between elements of the basis set for closing and the complete, minimal representation of arbitrary logic functions is analyzed from a geometric viewpoint.