Fuzzy inferring mathematical morphology and optical implementation

Fuzzification is introduced into gray-scale mathematical mor- phology by using two-input one-output fuzzy rule-based inference sys- tems. The fuzzy inferring dilation or erosion is defined from the approxi- mate reasoning of the two consequences of a dilation or an erosion and an extended rank-order operation. The fuzzy inference systems with numbers of rules and fuzzy membership functions are further reduced to a simple fuzzy system formulated by only an exponential two-input one- output function. Such a one-function fuzzy inference system is able to approach complex fuzzy inference systems by using two specified pa- rameters within it—a proportion to characterize the fuzzy degree and an exponent to depict the nonlinearity in the inferring. The proposed fuzzy inferring morphological operators tend to keep the object details compa- rable to the structuring element and to smooth the conventional morpho- logical operations. Based on digital area coding of a gray-scale image, incoherently optical correlation for neighboring connection, and optical thresholding for rank-order operations, a fuzzy inference system can be realized optically in parallel. © 1996 Society of Photo-Optical Instrumentation Engi- neers.