Fuzzy Logic-based methods

The classification of a colour is not a simple task, mainly because it is a subjective process. On this page we present the four main methods for colour segmentation based on fuzzy sets.

The background framework: Fuzzy logic and fuzzy sets

Traditionally, logic has been based on binary statements, with no need for further specification. However, many everyday situations require a greater degree of detail; for example, determining whether a person is tall or short, whether a process is efficient or not, or, specifically, whether a colour is red, orange or yellow. Because of the subjectivity of these processes, fuzzy logic is a good framework for modelling uncertainty.

Given a collection $X$, the fuzzy set $F_X$ associated to $X$ is the collection of pairs $F_X = \left\{\left.(x,\mu_X(x))\,\right|\, x\in X\right\}$, where $\mu_X: X\to [0,1]$ is the membership function. For brevity, since the fuzzy set is uniquely determined by its membership function, we denote the fuzzy set by $\mu_X$. Of all the membership functions in the literature, the most important are:

• Triangular functions. They are defined by a lower limit $a$, an upper limit $b$ and a value $m$ which satisfies $a<m<b$. Its expression is given by:

$$\mu_X(x) = \begin{cases} 0, & \text{if $x\leq a$ or $x\geq b$,} \\ \dfrac{x-a}{m-a}, & \text{if $a<x\leq m$,} \\ \dfrac{b-x}{b-m}, & \text{if $m<x\leq b$.} \\ \end{cases}$$

Graphical representation of the triangular membership function.

• Trapezoidal functions. They are defined by a lower limit $a$and an upper limit $b$, and a lower support limit $m$ and an upper support limit $n$. The three parameters satisfy $a<m<n<b$. Its expression is given by:

$$\mu_X(x) = \begin{cases} 0, & \text{if $x< a$ or $x>d$,} \\ \dfrac{x-a}{m-a}, & \text{if $a\leq x\leq m$,} \\ 1, & \text{if $m\leq x\leq n$,} \\ \dfrac{b-x}{b-n}, & \text{if $m<x\leq b$.} \\ \end{cases}$$

Graphical representation of the trapezoidal membership function.

• Gaussian functions. They are defined by a central value $m$ and a standard deviation $k>0$. When $k$ goes smaller, the narrower the bell is. Its expression is given by:

$$\mu_X (x) = \exp\left\{-\dfrac{(x-m)^2}{2\cdot k^2}\right\}$$

Graphical representation of the Gaussian membership function.

Colours as fuzzy sets

As shown, color grading is a very subjective process. Therefore, the concepts of fuzzy sets will be applied for segmentation.

Due to the separation of the chromatic component, the implemented methods are based on the HSV colour space. The H component concentrates the pure colour, while the S and V components represent, respectively, the saturation and the level of white present in that colour. In this module, each color will represent a fuzzy set, so that using its associated membership function over the H channel the degree of chromaticism can be studied. Currently, there are four methods based on fuzzy sets:

• Amante-Fonseca, using fuzzy trapezoidal membership functions.

• Chamorro-Martínez, using fuzzy trapezoidal membership functions.

• Liu-Wang, using fuzzy trapezoidal membership functions.

• Shamir, using fuzzy triangular membership functions.

The difference between each of the methods is the colour palette they define, and their membership functions. The overall process can be summarised in the following diagram.

The method for determining whether a colour is chromatic or not from its S and V components is defined in Amante-Fonseca.