Correlation Between Human Aesthetic Judgement and Spatial Complexity Measure

Abstract

The quantitative evaluation of order and complexity conforming with human intuitive perception has been at the core of computational notions of aesthetics. Informational theories of aesthetics have taken advantage of entropy in measuring order and complexity of stimuli in relation to their aesthetic value. However entropy fails to discriminate structurally different patterns in a 2D plane. This paper investigates a computational measure of complexity, which is then compared to a results from a previous experimental study on human aesthetic perception in the visual domain. The model is based on the information gain from specifying the spacial distribution of pixels and their uniformity and non-uniformity in an image. The results of the experiments demonstrate the presence of correlations between a spatial complexity measure and the way in which humans are believed to aesthetically appreciate asymmetry. However the experiments failed to provide a significant correlation between the measure and aesthetic judgements of symmetrical images.

Appendix

In this example the pattern is composed of two different colours $S=\{ white,black\}$ where the set of permutations with repetition is $\{ ww,wb,bb,bw\}$. Considering the mean information gain (Eq. 8) and given the matrix M (Eq. 7), the calculations can be performed as follows:

$\begin{array}{lcl} white-white \\ P{(w,w_{(i,j+1)})} = \frac{5}{6} \\ P{(w|w_{(i,j+1)})} =\frac{4}{5} \\ P(w,w_{(i,j+1)})= \frac{5}{6} \times \frac{4}{5}=\frac{2}{3}\\ {G}{(w,w_{(i,j+1)})} = \frac{2}{3} \log _2P (\frac{4}{5}) \\ {G}{(w,w_{(i,j+1)})} = 0.2146 \; bits \\ white-black \\ P{(w,b_{(i,j+1)})} = \frac{5}{6} \\ P{(w|b_{(j+1)})} =\frac{1}{5} \\ P(w,b_{(i,j+1)})= \frac{5}{6} \times \frac{1}{5}=\frac{1}{6}\\ \overline{G}{(w,b_{(i,j+1)})} = \frac{1}{6} \log _2P{\frac{1}{5}} \\ \overline{G}{(w,b_{(i,j+1)})} = 0.3869 \; bits \\ \end{array}$ $\begin{array}{lcl} black-black \\ P{(b,b_{(i,j+1)})} = \frac{1}{6}\\ P{(b|b_{(i,j+1)})} =\frac{1}{1} \\ P(b,b_{(i,j+1)})= \frac{1}{6} \times \frac{1}{1}=\frac{1}{6}\\ {G}{(b,b_{(i,j+1)})} = \frac{1}{6} \log _2P(1) \\ {G}{(b,b_{(i,j+1)})} = 0 \; bits\\ black-white \\ P{(b,b_{(i,j+1)})} = \frac{1}{6} \\ P{(b|w_{(i,j+1)})} =\frac{0}{1} \\ P(b,w_{(i,j+1)})= \frac{1}{6} \times {0} \\ {G}{(b,w_{(i,j+1)})} = 0 \; bits \\ \\ \end{array}$

$\begin{array}{lcc} \overline{G}= {G}{(w,w_{(i,j+1)})}+{G}{(w,b_{(i,j+1)})} +{G}{(b,b_{(i,j+1)})} +{G}{(b,w_{(i,j+1)})} \\ \overline{G} = 0.6016 \; bits \\ \end{array}$

In $white-white$ case G measures the uniformity and spatial property where $P{(w,w_{(i,j+1)})}$ is the joint probability that a pixel is white and it has a neighbouring pixel at its $(i,j+1)$ position, $P{(w|w_{(i,j+1)})}$ is the conditional probability of a pixel is white given that it has white neighbouring pixel at its $(i,j+1)$ position, $P{(w,w_{(i,j+1)})}$ is the joint probability that a pixel is white and it has neighbouring pixel at its $(i,j+1)$ position, ${G}{(w,w_{(i,j+1)})}$ is information gain in bits from specifying a white pixel where it has a white neighbouring pixel at its $(i,j+1)$ position. The same calculations are performed for the rest of cases; black-black, white-black and black-white.