6.3 Methodology
6.3.2. Fractal stimulus & Measuring Complexity
The Stimulus: Professor Richard Taylor and Colleagues from the University of Oregon, USA, developed the stimuli sets used within this thesis (See Figure 6.5 for example). The patterns were generated using a mid-point displacement technique, which allows generation of fractal images, by manipulating the core image and controlling for 9 levels of fractal dimension. This control ensures a complete range of fractal images are developed from very low, to very high and set point between to accurately represent to full range of fractal patterns. This control over a full level of fractal dimension (from a core image) was an essential requirement of the stimulus as the lack of full range has been suggested to account for some of the variance found within the optimal preference in early fractal aesthetics studies.
When exploring the aesthetic response to the fractal pattern with the linear mixed- effect models 3 different models (and grouping of the images) will be tested. They are outlined below.
The Mid-Range Hypothesis: Richard Taylor and colleagues in a series of studies (Taylor et al., 2009;2011) found evidence to suggest three groups within the fractal continuum that appear to have significantly different aesthetic responses. Taylor et al outlined the peak preference lying within the ‘mid’-range of fractal dimension, which they defined, as 1.3-1.5 and distinguished 2 other groups, ‘low’ 1.1 & 1.2 and ‘high’ 1.7-1.9 which were preferred less over the images falling within the mid-range.
The Equalised 3 Level Model: An alternative method of distinguishing the grouping could be an equalised model which still demonstrates 3 categories of fractal dimension; within an equalised model these are low (1.1-1.3), mid (1.4-1.6) and high (1.7-1.9). As the first study within this thesis attempts to explore if Taylor and colleagues ‘mid-range’ hypothesis is an accurate classification of aesthetic responses to fractal patterns, and alternative but similar classification was developed to allow strength comparisons to be carried out.
Binomial Complexity grouping: The final distinction between levels of fractal dimension during analysis will be classifying the images as more or less complex than it’s paired comparison image. Within this grouping, a higher fractal dimension is representative of a image of higher complexity, therefore analysis will be done exploring if choices are made between the higher or lower images in terms of visual complexity. This link between fractal dimensions has been discussed in previous chapter (chapter 3) however this thesis aims to quantify this relationship statistically. To explore if this categorisation is representative of visual complexity (as well as fractal dimension) the thesis stimulus will be measured using computational complexity compression measures which has been demonstrated to provide reliable measures of human judgments of complexity (Forsythe et. al., 2008) in Chapter 7.
Measuring visual complexity:
To assess the classification between fractal dimension and visual complexity for the stimulus used within this thesis comparisons were made between fractal dimension and computational measures of complexity (GIF).
Computational measures of complexity have been used to quantify the visual complexity of many different stimulus including art, abstract patterns or realistic photographs. Measuring the visual complexity of a stimulus differs significantly from measures of fractal dimension that explore the roughness and underlying order or self-similarity of an image. Visual complexity, with compression measures, takes into account the whole of the image (rather than only fractal complexity) and compression breaks the whole image down to composite parts depending on the amount of information within the image. The resulting compression information becomes a string of symbols representing parts of the image including elements, shapes, contrasts and colours. The generalized method means that complexity compression measures can be widely applied to different stimulus sets; the same cannot be said for fractal measurement techniques as although the methods will provide a score for fractal dimension (for example using the box-counting method) for any image analysed, these cannot always be considered reliable when measuring non-fractal images.
GIF Compression: The GIF compression ratio provides a measure of the size of an image after compression and this is divided by the original size of the image (in .BMP format). This method was chosen over other computational compression measures such as JPEG as it works best with mono-chrome and geometric shapes over photographs or art scan which are better suited to JPEG compression techniques. The analysis between fractal dimension and the computational measure of visual complexity will be discussed in the following chapter (see Chapter 7).
Stimulus Summary:
Fractals offer a way of selecting one of the many facets of visual complexity and by using pure fractal stimulus we can truly explore the role this plays on visual
judgments. The stimulus are a ‘pure’ form of fractal dimension and allow real and controlled predictions to be made about aesthetic responses to fractal patterns, more so than previous studies have been able to do with other stimulus. Although the application could be seen as less ecologically valid than the use of other stimulus include art or photographs, the use of pure fractal images allow assertions to be made on the basis that judgment are based on fractal dimension and visual complexity alone rather than any other variables that have been found to powerfully influence aesthetic preference such as familiarity, meaningfulness and colour.
Images selection (2A-FC Design):
The images were developed as detailed above. In total there was 9 sets of 9 images developed by Richard Taylor and colleagues. To run a full forced-choice design this would involved comparing all 81 images to each other, resulting in 6,480 possible pairs. This amount of forced-choice pairs is obviously not a feasible number of pairs to ask participants to rate.
In a bid to make the design more usable when faced with a large stimulus set, Prof Chris McManus from UCL developed a method of reducing the number of pairings required when using a forced-choice design in aesthetics research (C McManus, 2009). McManus’s method samples an entire range of stimulus but also provides detailed information on closely similar rectangles. Established analysis of 2A-FC involves summing across each column when using complete paired comparisons. McManus champions the use of a regression model approach in which dummy variables allow comparison of the preference.
McManus’ study justified a modified method of 2A-FC and this thesis developed a further modification to the traditional design. The method adopted was justified because Taylor et al (2011) found no significant differences between aesthetic response patterns to the different sets of fractal images set (developed in the same way to those used in the current thesis) demonstrating it can securely be assumed that the fractal dimension, rather than any other individual structural differences that contributed to aesthetic findings.
The pairing matrix below (Table 6.2) demonstrates how the images for each pairing were chosen. Each stimulus set included 9 images, which would result in 81 pairs per set. As literature demonstrates the presence of 3 distinct groups of fractal dimension, ‘Low’, ‘Mid’, ‘High’ the images were grouped to match the current findings, comparisons were not made between fractal patterns falling within the same group. This reduced design resulted in 26 potential pairing, and because of the number of sets two pairs allowing a variety of sets to be used. The individual images selection was done using a (quasi) randomly assigned design. For each pair, the stimulus sets were labelled 1-9. Using a random number generator the FD paired comparison was made up from one image (matching the required Fractal Dimension outlined in the matrix) from the first randomly selected set and a second image selected using the same method using a different set. A quasi-random design was chosen to avoid repetition of sets within each category.
This modified design meant that there was an equal chance and probability of participants choosing each fractal dimension point within the scale. Therefore allowing judgments to be made about differences in preference between the 3 fractal groups (low, mid, high) and complexity scales (higher or lower complexity).
The design outlined above was used for studies 3, 4, 5 and 6 of this thesis. The same selection was used in each to allow an overall comparison of the sample in the final stages of the analysis within this thesis. A regression model analysis was developed in addition as advocated by McManus (2009).
Table 6.2 Image Selection Matrix D1.1 D1.2 D1.3 D1.4 D1.5 D1.6 D1.7 D1.8 D1.9 D1.1 D1.2 D1.3 D1.4 D1.5 D1.6 D1.7 D1.8 D1.9
Image Selection (Rating Design):
Image Selection (Version one): Two images from each set were chosen to be included within the sample for Study 2 (Chapter 8). The first repetition of images to be included was chosen using a simple 1-9 numbering system based on the image set order. The 2nd repetition was done using a split number sample in which the number was chosen on the basis that there were at least 4-5 FD scores between each image. Given this the same numerical system was used, however the order began with set 5, ensuring that there was at least 4 FD points between the images chosen from the same sets. Images from each set are all similar structure but vary only in FD therefore to avoid preference affected by familiarity/structure rather than fractal dimension. At 4-5 points apart it is difficult to detect strong similarities between the images (see Table 6.3 for selection and difference information).
Table 6.3- Survey Type 1
Image Set FD 1st image FD 2nd Image Diff FD
A Set01- 1116 1 6 5 B Set02- 1135 2 7 5 C Set03- 1161 3 8 5 D Set04- 3003 4 9 5 E Set05- 3056 5 1 4 F Set06- 3077 6 2 4 G Set07- 3091 7 3 4 H Set08- 1043 8 4 4 I Set09- 1048 9 5 4
Developing versions 2-4: In total 4 versions of the study was developed to ensure that preferences were a function of FD rather than the specific image sets used. Above outlines how version 1 was created, versions 2-4 was developed in a similar way however before image selection the Image Set order was randomised each time using a random number generator which output a unique random number generator order from 1-9 to arrange the stimulus sets. The same process of image selection was used on the second, third and fourth sample, ensuring again that images did not too closely resemble the images within the same set, leaving at least 4 points between them (table 6.4 demonstrates version 2 selections).
Table 6.4- Differences between FD measures
Image Set FD 1st image FD 2nd image Diff
H Set08- 1043 1 6 5 G Set07- 3091 2 7 5 F Set06- 3077 3 8 5 A Set01- 1116 4 9 5 B Set02- 1135 5 1 4 C Set03- 1161 6 2 4 I Set09- 1048 7 3 4 E Set05- 3056 8 4 4 D Set04- 3003 9 5 4
Proposed Analyses:
Within the thesis there is a mix of SPSS and R analysis software used to explore the data. The rationale behind these choices are explored below.
Correlation & ANOVA in SPSS:
Correlation is used in this thesis to explore the relationship between the fractal dimension of the stimuli and the computational measurement of complexity (GIFratio). In addition Repeated Measures Analysis of Variance (ANOVA) was used to explore the mean scores for a selection of stimuli within study 3 in which participants were asked to rate on a scale (of 0-10) about how much they like the fractal images. Repeated measured ANOVA was also used to explore the differences amongst the frequency data based on the 2A-FC methods in studies 4, 5 & 6. The models used above have notable limitations because although variance is accounted their individual differences based on participants and stimulus are considered noise in this model. Serious problems have been identified with the use of ANOVA’s in categorical variables, such as the forced-choice design and other categorical outcome variables (Jaeger, 2008). Despite the use of transformation, there are continued problems with using ANOVA’s on categorical variable outcomes, justifying the use of mixed-effect models when using this type of data and offer advantages over using ANOVA.
Linear Mixed-Effect Modelling using R:
Although commonly used within linguistics, linear mixed-effect models (LMM) are a flexible and powerful tool for understanding and analyses response to the environment. LMM is a type of regression model that takes into account variables that would be considered of attributed as ‘noise’ in fixed-effects approaches. The model uses both fixed-effects such as the independent variables such as Age, Gender and stimulus as well as random-effects that are specific to the data sample, including individual variations in judgment and variances between stimulus used (as only a small selection of all possible stimulus that could be used). The analysis will be a logistic regression model with mixed effect as the dependent variable (fractal dimension image choice) is a binary variable. The model uses 3 different models exploring the classifications of the fractal images outlined above.