Histograms for fuzzy color spaces

Abstract

In this paper we introduce two kinds of fuzzy histograms on the basis of fuzzy colors in a fuzzy color space and the notion of gradual number by Dubois and Prade. Fuzzy color spaces are a collection of fuzzy sets providing a suitable, conceptual quantization with soft boundaries of crisp color spaces. Gradual numbers assign numbers to values of a relevance scale, typically [0,1]. Contrary to convex fuzzy subsets of numbers (called fuzzy numbers, but corresponding to fuzzy intervals as an assignment of intervals to values of [0,1]), they provide a more precise representation of the cardinality of a fuzzy set. Histograms based on gradual numbers are particularly well-suited for serving as input to another process. On the contrary, they are not the best choice when showing the information to a human user. For this second case, linguistic labels represented by fuzzy numbers are a better alternative, so we define linguistic histograms as an assignment of linguistic labels to each fuzzy color. We provide a way to calculate linguistic histograms based on the compatibility between gradual numbers and linguistic labels. We illustrate our proposals with some examples. © 2011 Springer-Verlag Berlin Heidelberg.