Fast recursive algorithms for morphological operators based on the basis matrix representation

A real-time implementation method for the most general morphological system, the so-called grayscale function processing (FP) system is presented. The proposed method is an extension of our previous works (1993, 1995) using the matrix representation of the FP system with a basis matrix (BM) and a block basis matrix (BBM) composed of grayscale structuring elements (GSE). In order to further improve the computational efficiency of the basis matrix representation, we propose recursive algorithms based on the observation of the BM and BBM. The efficiency of the proposed algorithms is gained by avoiding redundant steps in computing overlapping local maximum or minimum operations. It is shown that, with the proposed scheme, both opening and closing can be determined in real time by 2N-2 additions and 2N-2 comparisons, and OC and CO by 4N-4 additions and 4N-4 comparisons, when the size of the GSE is equal to N. It is also shown that the proposed recursive opening and closing require only 3N-3 memory elements.