In this paper, we present a new construction method for interval-valued fuzzy relations (interval-valued fuzzy images) from fuzzy relations (fuzzy images) by vicinity. This construction method is based on the concepts of triangular norm ( t -norm) and triangular conorm ( t -conorm). We analyze the effect of using different t -norms and t -conorms. Furthermore, we examine the influence of different sizes of the submatrix around each element of a fuzzy relation on the interval-valued fuzzy relation. Finally, we apply our construction method to image processing, and we compare the results of our approach with those obtained by means of other, i.e., fuzzy and nonfuzzy, techniques.

An important problem in working with fuzzy sets is the correct construction of the membership functions that represent the objects of the system. Different experts construct different membership functions to represent the same object. In this paper, we construct an interval type-2 fuzzy set (IT2FS) with different fuzzy sets such that the length of the (membership) interval represents the uncertainty of the expert with respect to the choice of the membership function. We analyze this problem in the context of image segmentation. We propose a new version of the classical fuzzy thresholding algorithm, in which an expert can select multiple membership functions, to avoid the problem of selecting only one to represent the image. From these membership functions, we construct an IT2FS, and by minimizing its entropy, we find a threshold with which to binarize the image. We present experimental results that show that it is advisable to use this methodology when it is not known which membership function is the most suitable.