Interval-valued fuzzy sets constructed from matrices: Application to edge detection

In this paper we present a method to construct interval-valued fuzzy sets (or interval type 2 fuzzy sets) from a matrix (or image), in such a way that we obtain the length of the interval representing the membership of any element to the new set from the differences between the values assigned to that element and its neighbors in the starting matrix. Using the concepts of interval-valued fuzzy t-norm, interval-valued fuzzy t-conorm and interval-valued fuzzy entropy, we are able to detect big enough jumps (edges) between the values of an element and its neighbors in the starting matrix. We also prove that the unique t-representable interval-valued fuzzy t-norms and the unique s-representable interval-valued fuzzy t-conorms that preserve the length zero of the intervals are the ones generated by means of the t-norm minimum and the t-conorm maximum.