Fig 1.
Samples of artistic images [11] (a) and pseudo-artistic images (b).
The titles, from left to right, are: (a) ‘Black holes of memory’ (orig. ‘Czarne dziury pamięci’); ‘Lungs of Blackness’ (orig. ‘Płuca czerni’); ‘Ear of Blackness’ (orig. ‘Ucho czerni’); ‘Guts of Blackness’ (orig. ‘Jelita czerni’). (b) ‘Vibrations of time’ (orig. ‘Wibracje czasu’); ‘The Inside’ (orig. ‘Wnętrze’); ‘Cold fire’ (orig. ‘Zimny ognień’); orig. ‘Everything is a thing and nothing is everything’.
Fig 2.
Exploring topology and persistence through filtration.
Visualisation of a pixel-based structure with varying brightness levels, demonstrating the use of grey intensity (on a scale from 0 to 1) as a filtration parameter. At a grey intensity of 0.1, the structure is divided into two distinct parts, outlined by a red contour) with two ‘holes’ of grey intensities 0.3 and 0.5. This results in Betti numbers and
. As the filter transparency (greyness) increases, changes in the visibility of pixels at grey level 0.3, the two distinct shapes are still present, although one of the holes in the bigger part (with intensity r = 0.3) disappears, changing the Betti numbers to
and
. At the intensity r = 0.4, the two shapes merged into one part with one hole (
and
). If we further increase filter intensity to r = 0.5 all pixels will become visible yielding Betti numbers
and
.
Fig 3.
Visualisations of the filtration process.
(a) ‘barcodes’ – each structure corresponds to a line segment parallel to the axis of the filtration parameter r, which begins when the structure appears (i.e., at point rb, birth) and ends when it disappears (at point rd, death), marked by two vertical lines for clarity; (b) persistence diagram – persistence is also representable via a two-dimensional scatter plot diagram with coordinates (rb, rd) representing births and deaths of the shape’s parts and holes on two orthogonal filtration axes marked by the horizontal blue lines orthogonal to respective axes. Naturally, points on this diagram occupy only the area above the main diagonal, (c) and (d) persistence landscape – connecting each of the birth/death points to the diagonal with vertical and horizontal lines, as shown for the uppermost point, green point on b, we get a system of ‘pyramids’ or isosceles right triangles. After rotating the persistence diagram by , it becomes a persistence landscape drawn separately for dim 0 – separate connected components (Fig 2c) and dim 1 – holes (Fig 2d). Persistence landscapes, in general, consist of several layers, where the n-th layer is the n-th highest value at a given point r of filtration. In (c), the top (0-th) layer is constructed from the ‘pyramid’ built on the largest barcode that lasts throughout the filtration (this pyramid is greater than all others at any value of r), the 1st and 2nd layers are constructed from the pyramids built on 2 shorter barcodes – whose corresponding ‘pyramids’ intersect, resulting in 3 layers (a full definition of layers is provided in S3 Appendix).
Fig 4.
Demonstration of the duality of black-to-white (BW) (first row) and white-to-black filtrations (WB) (second row).
The image from which the cubical complexes are derived is to the left of the rows in the middle. Components (and corresponding connections) that comprise a cubical complex at a particular step of the filtration, f (or ), are coloured in black, whereas the pink components are not included in the filtration. Pink components also belong to the dual cubical complex coming from the reverse filtration at step
, where M + 1 is the total number of filtration steps in [0,M].
Table 1.
Number of partitions for filtration steps for the toy example.
Fig 5.
Visualisation of the pipeline for obtaining feature distributions and gaze maps.
After the image is converted to greyscale, the spatial distribution of cycles is obtained from the BW and WB filtrations, resulting in feature maps for each type of topological feature (for example, cycle density, persistence or cycle perimeter). Firstly, for each type of topological feature (for example, cycle density, persistence or cycle perimeter) and each image, the Empirical Cumulative Distribution Function (ECDFs), , is obtained, which is intrinsic to the image. Secondly for a subject s, gaze-heatmaps Gs and its complement heatmap
are used to obtain
(looking) and
(not looking), by weighting the image feature maps with the corresponding gaze-heatmap.
Fig 6.
Visualisation of cycles for BW filtration (middle column) and WB filtration (right column) for an artistic [11] (a) and pseudo-artistic image (b).
Cycles presented in the middle and right-most columns are coloured according to their persistence, as indicated by each legend.
Fig 7.
The laboratory comparison between art and pseudo-art of average aesthetic experience questionnaire scores.
Responses were collected using a 5-point slider scale (0%, 25%, 50%, 75%, 100%). The boxplots show medians for a given group (horizontal line) together with the middle 50% of the data denoted by the interquartile range box (the distance between the first and third quartiles) and ranges for the bottom 25% and the top 25% of the data values, excluding outliers (whiskers). Stars above horizontal lines connecting boxplots denote significance levels: *** for p < 0.001; ** for p < 0.01; * for p < 0.05.
Fig 8.
Comparison between art and pseudo-art of average visual intake duration during first (a) and second (b) visit in a gallery.
The boxplots show medians for a given group (horizontal line) together with the middle 50% of the data denoted by interquartile range box (the distance between the first and third quartiles) and ranges for the bottom 25% and the top 25% of the data values, excluding outliers (whiskers). A * above a horizontal line denotes p < 0.05, ns denotes p > 0.05.
Fig 9.
Comparison between art and pseudo-art of average saccade duration during first (a) and second (b) visit in a gallery.
The boxplots show medians for a given group (horizontal line) together with the middle 50% of the data denoted by interquartile range box (the distance between the first and third quartiles) and ranges for the bottom 25% and the top 25% of the data values, excluding outliers (whiskers). A ** above the horizontal line over 1st visit boxplot denotes p < 0.01.
Fig 10.
Comparison between art and pseudo-art of average time spent at image during the first (a) and second (b) visit to the gallery.
The boxplots show medians for a given group (horizontal line) together with the middle 50% of the data denoted by interquartile range box (the distance between the first and third quartiles) and ranges for the bottom 25% and the top 25% of the data values, excluding outliers (whiskers). ** above horizontal line over 1st visit boxplot denotes p < 0.01; *** above horizontal line over 1st visit boxplot denotes p < 0.001.
Fig 11.
Comparison between art and pseudo-art of average fixation duration during first (a) and second (b) visit in a laboratory.
The boxplots show medians for a given group (horizontal line) together with the middle 50% of the data denoted by the interquartile range box (the distance between the first and third quartiles) and ranges for the bottom 25% and the top 25% of the data values, excluding outliers (whiskers). A * above the horizontal line over the 1st visit boxplot denotes p < 0.05, whereas ** denotes p < 0.01.
Fig 12.
Examples of topological properties of art [11] (a) and pseudo-art images (b).
Left column: greyscale-converted original image; middle section: topological results for filtration from black to white; right section: topological results filtration from white to black; within each section – left column: topological characteristic in dimension 0 and right column: topological characteristics in dimension 1. The topological characteristics, the same for both dimensions, are: persistence landscapes(top) and barcodes (bottom). The horizontal axis for both characteristics is pixel intensity filtration steps (in the range ). Vertical axes are: for persistence landscapes – arbitrary units; for barcodes – index of a barcode in the birth sorted list of all barcodes for an image. Every landscape (for every dimension) is constructed from a different number of layers – the layers are coloured according to the legend above each landscape. The barcodes for the pseudo-artistic images are significantly shorter than for the artistic images, which shows a smaller persistence of the former. As can be read from the landscape diagrams, the pseudo-artistic images contained significantly fewer holes and cycles indicting the far richer structure of the artistic image.
Fig 13.
Average group persistence landscapes for BW filtration.
The average persistence landscape was computed for each group: art (a,c) and pseudo-art images (b,d), with results for dimensions 0 shown in the top row and dimension 1 in the bottom row. Every individual landscape was constructed from cycles of persistence greater than 5 pixel intensity values [54]. The average landscapes for artistic and pseudo-artistic images were significantly different for each dimension under the non-parametric Wilcoxon test with permutation, both p < 0.001. A marked difference in structural richness between the two groups of images is again evident, as persistence landscapes in both dimensions contain significantly more layers for artistic images.
Fig 14.
Average Betti curves with 1 standard deviation above and below shaded, together with individual Betti curves for each image for the BW filtration.
The average Betti curve was computed for each group: artistic and pseudo-artistic images, with results for dimension 0 (a) and dimension 1 (b). S8 Fig shows Betti curves for the WB filtration. The numbers of holes and cycles reflected here by the Betti curves is significantly different for artistic and pseudo-artistic images.
Fig 15.
The distribution of violation of Alexander duality demonstrated as the normalised differences in the area under the Betti curve for (a) black to white filtration and
and (b) white to black filtration
and
, images used in exhibition – art [11] and pseudo-art.
The vertical axis is Alexander duality violation measure, that is, the difference between the area of the Betti curve one filtration in dimension 0 and the area of the Betti curve reverse filtration in dimension 1, then normalised by their mean area, as in Equation 1. There is a significant difference between the two groups in the second comparison (Mann-Whitney U test, (a) and (b) – statistic value: 103 and 140; two-sided: p = 0.078 and p < 0.001; point estimate: 0.1536 and 0.1435; rank sums: (181, 119) and (218, 82); number of observations in each group: 12 and 12). The statistics for the distributions are presented in S3 Table.
Table 2.
Summary of Hedges’ g effect sizes and Kruskal–Wallis tests.
Fig 16.
The distribution of violation of Alexander duality demonstrated as the normalised differences in the area under the Betti curves for: for (a) and
and (b)
and
, all art images and all pseudo-art.
The vertical axis shows the Alexander duality violation measure ADV, as defined in Equation 1. The data presented are art images extended by a group of paintings of popular abstract painters (we include a list of paintings in S6 Appendix) and all 4500 generated pseudo-art images. There is a significant difference between the two groups in both comparisons (Mann-Whitney U test, (a) and (b) – statistic values: 356695 and 338429; two-sided: p < 0.001 and p < 0.001; point estimate (median of all possible pairwise differences): 0.2764 and 0.2282; rank sums: and
; number of observations in each group: 844500 and 844500). The statistics for the distributions are presented in S3 Table.
Fig 17.
Cycle density comparison for all images (a) and the two closest cross-group samples of cycles visualisation (b,c).
Every curve in (a) represents the ECDF of the cycle density distribution for images for both exhibitions; the density used for ECDF is the sum of densities from BW and WB filtrations; artistic (green) and pseudo-artistic (orange). The vertical axis is the proportion of all windows from a grid mesh. All curves were statistically different under the Kolmogorov-Smirnov test. Images for which cycles were visualised are those marked with ‘*’ in the figure with ECDF plots. The cycle density reflects the texture of an image, i.e., the spatial distribution of changes in contrast.
Table 3.
Results of Mann-Whitney U test.
Fig 18.
Comparison of how topological features were explored in both exhibitions.
MSE between image features distribution ECDF and the features weighted by each participant’s gaze duration data. MSEs between two ECDFs can only lie between 0 and 1. Results for: (a) Persistence; (b) Cycle density; (c) Cycle perimeter. (Exhibitions are indicated on the axis; only laboratory data was used for this comparison). The line-up of plots within each subfigure is as in Fig 11. The Mann-Whitney U test, comparing MSE distribution for the artistic images to MSE distribution for the pseudo-artistic images, yielded significance for a) and c), *** p < 0.001.
Fig 19.
Analysis of topologically derived features that participants were attracted to.
Each point on the plots is derived from the ECDFs obtained from a single participant viewing a single image. For every image and every participant who viewed the image, a weighted feature map distribution was obtained from: (1) the regions where the participant looked (weighted by gaze duration) and (2) the regions where the participant did not look (uniform weighting). (a) results for persistence feature maps. (b) results for cycle density feature maps. (c) results for cycle perimeter derived maps (exhibitions are indicated in the legends). In each plot in the first two columns, we show MSE on the vertical axis and ME on the horizontal axis, both computed from the difference for each participant, i.e., it compares the ECDF of regions that a participant looked at with the regions that the same participant did not look at. Note that the MSEs between two ECDFs can only lie between 0 and 1, whereas the ME can lie between −1 and 1. The third column shows the distribution of MSEs on which significance tests were conducted. Statistics performed with Mann-Whitney U test, comparing MSE distribution of ECDF between the artistic and pseudo-artistic images, *** p < 0.001.
Table 4.
Results of Mann-Whitney U test for MSE of ECDF for looking and not looking.
Table 5.
Summary of image property preferences from Fig 19.
Fig 20.
Linear mixed-effects regression for Cycles density and AEQ–S questionnaire categories with Elements being the reference item for all images (art and pseudo-art) and all 58 participants.
Fig 21.
Linear mixed-effects regression for Maximum persistence and AEQ–S questionnaire categories with Elements being the reference item for all images (art and pseudo-art) and all 58 participants.