A computational model for colour categorization

This project is based on the research paper "Modeling the emergence of universality in colour naming patterns," authored by Andrea Baronchelli, Tao Gong, Andrea Puglisi, and Vittorio Loreto; Proceedings of the National Academy of Sciences 2010 vol. 107 no. 6 pg. 2403

Abstract

In our model, the artificial human agents learn to categorize colours by playing elementary language games. When all the agents complete 104 games, we collect the data and analyse the final colour categories developed by the entire population consisting of N (generally equal to 50) agents. Finally, we try to correlate our results with those obtained from the World Colour Survey (WCS).

The WCS has observed that there exist some universals in human colour categorization across different cultures. There are certain features in colour naming patterns that remain constant across various human populations. Through our model, we can conclude similar facts such as interaction among independent groups of agents yields a finite number of colour categories.

The Just Noticeable Difference (JND) function

Our perception of colours is nonuniform. The precision with which we perceive a colour depends on its wavelength. The JND is defined as a function of the wavelength of various colours in the visible spectrum. In order to distinguish two stimuli from the same scene, they must be separated by atleast the distance given by the JND function. In the following figure, the JND for x=0.5, dmin(0.5) ~ 0.0143 which means that if the normalized wavelength of a colour in the visible spectrum is 0.5, the other colour stimuli in the scene should have a wavelength difference of atleast 0.0143 from our current colour's wavelength of 0.5.
JND
Human JND function (Image source: Fig. 3 in [1])

The wavelengths in visible spectrum are mapped to the interval [0, 1). In our model, we perform two kinds of simulations:

  1. On human worlds endowed with the human JND function.
  2. On neutral worlds with a uniform JND of 0.0143 = average of the human JND over [0,1).

Why perform two simulations?

The WCS data analysis has confirmed that actual patterns obtained through the survey are more clustered (i.e., less dispersed) than their randomized counterparts [1].
clustering
Dhuman < Dneutral (Image source: Fig. 2 in [1])

In the second simulation, we assumed that JND is uniform. This is equivalent to assuming that the agents are not responsive to changes in JND as the wavelength differs. Even though this language game produces similar number of categories at the end of the simulation, we argue that the relation between its language categories is not "as close as" that between categories obtained from the first simulation. This is proved by using the dispersion values computed using the human JND function instead of a uniform JND. It has been found that Dhuman < Dneutral, which implies that the biological constraint (JND) imposes greater clustering and hence there exists universality in colour categorization. The ratio Dneutral/Dhuman ~ 1.14, which is quite close to the one obtained from the original dataset and randomized datasets in the WCS. Thus, our simulations are in excellent confirmation with the results of WCS.

Pseudocode for the category game model

Each agent has its partition of the interval [0, 1) into contiguous, non-overlapping categories. Each category is associated with an inventory of words. In the beginning, every agent has only single category with no related words. At each individual time step, two agents (speaker and hearer) are chosen randomly. The scene contains M stimuli, where M is at least 2. The current stimuli being communicated is called the topic h.

The speaker assigns every stimulus i to one of its categories. i must be assigned to a unique segment [L, r) such that L <= i < r. A discriminated stimulus k is the one which is the only stimulus from the given scene grouped under a particular category C. But, if a topic h is not already discriminated, it means that there are other stimuli (set 'O') from the scene which are mapped to the same category 'C' as h. In this case, the speaker discriminates his topic h by creating new categories from C. Assume that a = max {i: i < h and i belongs to O} and b = min {i: i > h and i belongs to O}. Now, C is split into new categories by boundaries at (a+h)/2 and (h+b)/2. The newly formed categories are given a new word in addition to the already existing inventory of words from C.

Now, after discriminating every stimulus from the scene, the speaker transmits a word from the discriminating category C of the topic. The chosen word depends on the following:

  1. If the previous communication from category C was successful, the same word that brought the success is used again;
  2. Else the new word added to C when this topic was added to it, is transmitted.
The hearer listens to the word and identifies those categories which contain the word transmitted and are also associated to at least one stimulus from the scene [1].

Now, four mutually exclusive cases arise:

  1. The hearer's set is empty: This is an error check, and the speaker communicates the topic again.
  2. The hearer's set contains one category which corresponds to a single stimulus from the scene: The hearer has no option but to point at it.
  3. Only one category which corresponds to multiple stimuli from the scene: Randomly, hearer points at one stimulus.
  4. Many categories: Once again, hearer randomly points at a single stimulus.
The interaction is said to be successful if both the hearer and the speaker end up pointing at the same stimulus. The interaction fails in all other cases.

Updating: Similar to the speaker, the hearer also discriminates every topic h from the scene. In case of failure, the hearer adds the word transmitted by the speaker to his own discriminating category.

In case of success both agents refine their respective discriminating categories by storing only one word: the transmitted word that yielded the success.

Reference

  1. Research paper by Andrea Baronchelli, Tao Gong, Andrea Puglisi, and Vittorio Loreto; Proceedings of the National Academy of Sciences 2010 vol. 107 no. 6 pg. 2403.