The Gray-Scale Soft Morphological Filters and the Idempotency

Abstract

Gray-scale soft mathematical morphology is the natural extension of binary soft mathematical morphology which has been proved to be less sensitive to additive noise and to small variations. But gray-scale soft morphological operations are difficult to implement in real time. In this paper, a superposition property called threshold decomposition and another property called stacking are introduced and have been found to apply successfully to gray-scale soft signals and structuring elements to be decomposed into their binary sets, respectively operated by only logic gates in new VLSI architecture, and then these binary results are combined to produce the desired output as the time-consuming gray-scale processing. The theorems developed may significantly improve speed as well as give new theoretical insight into the operations. At last, the new class of idempotent soft morphological filters are presented . Theorems and proofs are provided.