A Fuzzy System Representation of Functions of Two Variables and its Application to Gray Scale Images
Byung-Soo Moon, Young-Taek Kim, Jang-yeol Kim · Journal of Korean institute of intelligent systems · 2001
An approximate representation of discrete functions 〈수식기호참조〉 in two variables by a fuzzy system is described. We use the cubic B-splines as fuzzy sets for the input fuzzification and spike functions as the output fuzzy sets. The ordinal number of f i , j in the sorted list is taken to be the out put fuzzy set number in the (i,j)th entry of the fuzzy rule table. We show that the fuzzy system is an exact representation of the cubic spline function 〈수식기호참조〉 and that the approximation error S(x,y)-f(x,y) is surprisingly O(h²) when f(x,y) is three times continuously differentiable. We prove that when j(x, y) is a gray scale image, then the fuzzy system is a smoothed representation of the image and the original image can be recovered exactly from its fuzzy system representation when it is a digitized image.