A novel histogram definition for fuzzy color spaces

Abstract

Abstract-In this paper we introduce fuzzy naturals-based histograms on fuzzy color spaces. In our approach, histograms are fuzzy probability distributions on a fuzzy color space, where the fuzzy probability is calculated as the quotient between a fuzzy natural number and the number of pixels in the image, the former being a fuzzy (non-scalar) cardinality of a fuzzy set. This approach to histograms avoids the well-known disadvantages of the ordinary sigma-count as an estimation of the probability. We illustrate the potential application of the proposal by applying it to the problem of dominant color selection.

Histograms for Fuzzy Color Spaces

“…Several alternative proposals that comply with this idea are available [23,2]. In [1] we introduced histograms on fuzzy color spaces in which the cardinality associated to each fuzzy color was a fuzzy subset of the naturals calculated by using method ED, proposed in [2]. This fuzzy set is not convex in general.…”

“…Gradual numbers assign numbers to values of a relevance scale, typically [0,1]. The cardinality of a fuzzy set can be represented by a gradual number in which the cardinality of the α-cut of the fuzzy set is assigned to α.…”

“…In [1] we employed non-convex fuzzy subsets of the non-negative integers for defining fuzzy histograms. This approach represents a compromise between the advantages and disadvantages of using fuzzy numbers and gradual numbers.…”

“…In [1], we introduced an approach to fuzzy histograms based on the concept of fuzzy natural number introduced in [3]. This approach represented a kind of compromise between precision and understandability.…”

“…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.…”