Fuzzy system representation of digitized patterns and an edge tracking thinning algorithm

In this paper, we describe how to build a fuzzy system z=S(x, y) to represent digitized patterns consisting of black pixels with value 1 and white pixels with value 0. The numerical representation of digitized images is used as the fuzzy rule table needed to combine input variables x and y. The cubic B-splines are the input fuzzy sets and triangular functions the output fuzzy sets. Center area method is used as the defuzzification method. The resulting function z=S(x, y) is a continuous function whose values at grid points are approximately the same as the original. A pair of fuzzy systems are derived based on this representation to evaluate the x and y components of the curl vector ∇×r where r=(x, y, S(x, y)). These fuzzy systems are used to compute the curl vector ∇×r in an edge tracking algorithm, and they are found to provide a very simple and efficient means for thinning digitized patterns. A few examples are included to verify our algorithm