Algorithms for the decomposition of gray-scale morphological operations

The choice and detailed design of structuring elements plays a pivotal role in the morphologic processing of images. A broad class of morphological operations can be expressed as an equivalent supremum of erosions by a minimal set of basis filters. Diverse morphological operations can then be expressed in a single, comparable framework. The set of basis filters are data-like structures, each filter representing one type of local change possible under that operation. The data-level description of the basis set is a natural starting point for the design of morphological filters. This paper promotes the use of the basis decomposition of gray-scale morphological operations to design and apply morphological filters. A constructive proof is given for the basis decomposition of general gray-scale morphological operations, as are practical algorithms to find all of the basis set members for these operations