2.3.1 Art images.

To ensure a meaningful and consistent artistic message, we chose a planned solo exhibition of Polish visual artist Lidia Kot [11], dedicated to the colour black and presented at the Wozownia Art Gallery, Toruń, Poland. The artistic images used in the laboratory condition were digital copies of the original works on display in the gallery. According to the artist, these works emerged from an “unrestricted creative process”, resulting in a collection of 12 colour art prints prepared specifically for the exhibition. Each artwork shared identical dimensions, cm (W × H). All were artistic fine prints printed using an EPSON STYLUS PRO 11880 printer with an Epson Micro Piezo TFP printhead and Epson UltraChrome K3 ink with Vivid Magenta technology (print resolution: 2880 × 1440 dpi; minimum droplet size: 3.5 pl) on Canson Rag Photographique 210 gsm paper.

Examples of the artworks are shown in Fig 1a, and digital copies of all original artworks used in the study are available in S3 Fig.

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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’.

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2.3.2 Pseudo-art images.

The second exhibition was structured to minimise any potential human involvement that might convey information or intentionality. As the information embedded in art and its reception might be implicit [2], it is difficult, if not impossible, to identify which visual properties communicate meaning or reflect the artist’s intent. Consequently, we refrained from manipulating original artworks and instead employed artificially generated images created through processes specifically designed to minimise the presence of consistent or coherent information. To achieve this, we utilised randomly selected categories from a randomly perturbed BigGAN network [12], generating abstract visual forms derived from initially photorealistic image categories.

In order to ensure that the generated images had an abstract non-figurative character, the weights of all layers in Block 2, as well as those of a layer in the “self-attention” block of the BigGAN network (see Fig 1 in [13]), were substituted by random numbers. All the remaining weights remained unchanged. In this way, we constructed 500 different neural networks by drawing 500 random vectors of neural network weights, which gave us a sufficient variety of image distortion types or styles. Subsequently, a subset of 480 ImageNet categories was specifically chosen to exclude any that featured animals or human faces to ensure a strictly non-figurative character for the pseudo-artistic images. Then, for each of the 500 network modifications, images were generated using 9 categories randomly selected from this ‘non-figurative’ subset. In this way, we obtained 4500 neural network-generated images of size . The images generated by the perturbed neural networks were then enlarged to a resolution by first reducing colour noise in Photoshop (Adobe Systems Inc., Adobe Photoshop 2022) and subsequently upscaling using an SRGAN [14] super-resolution neural network.

Examples of the generated images are shown in Fig 1b, digital copies of all artificially generated images used in the study are available in S4 Fig.

For the purpose of this study, the original artworks were downscaled to a resolution (W × H) using Photoshop. All pseudo-artistic images were rescaled to match the resolution of the artistic images. Next, in order to mitigate differences in audience response arising merely from large disparities in brightness or colour, while minimizing human input, each of the 4500 artificially generated images was compared with each of the artist’s 12 artworks using Matlab’s (The MathWorks, Inc.) imabsdiff function. The imabsdiff function compares two images of the same size by computing the absolute value of the difference between the pixel intensities of the two images (resulting in a matrix of absolute pixel value differences). Next, this matrix was averaged to obtain a single value for each pair of compared images (by comparing each of 12 images with each of 4500 generated images, resulting in 54000 comparisons). The averaged differences of all comparisons were then placed in ascending order, and the 12 images with the smallest difference were selected as the set of pseudo-artistic images used in the present study.

Selected pseudo-artistic images were then printed to the same size as the artistic ones in the same print shop using exactly the same technology and paper. The titles for the pseudo-artistic exhibition were generated using and publicly available painting titles from exhibitions held at the Wozownia Art Gallery. All texts were created using GPT-3 software (beta.openai.com) and initially generated in English before being translated into Polish via Google Translate. From the pool of generated titles, 12 were randomly selected using the Matlab (MathWorks Inc.) randperm function, while 1 pseudo-curatorial text was generated using GPT-3 fine-tuned using original curatorial texts. The pseudo-curatorial text was proofread by an artist. Finally, titles were randomly assigned to pseudo-artistic images using the Matlab (MathWorks Inc.) randi function. The display order of pseudo-artistic images in the gallery was also drawn using the Matlab (MathWorks Inc.) randi function. Each set of images had its own dedicated exhibition open to the public. Both exhibitions ran for the same duration and were held at the same venue. The exhibition of the pseudo-artistic images opened directly after the exhibition of the artist Lidia Kot [11] had ended. In the laboratory, the raw images that were displayed were represented in the JPEG format, with 8 bits for each RGB channel.

2.3.3 Image preprocessing for topological analysis.

Our topological analyses target structural properties such as maximal persistence, cycle density, and perimeter length. Analysing each colour channel separately would often yield redundant spatial information—cycles typically appear in the same location across channels or not at all. Consequently, a greyscale representation effectively captures these topological features while reducing computational complexity and facilitating result interpretation. This approach aligns with prior studies in computational image analysis, including [15], which examined artworks using entropy and complexity measures in greyscale.

For completeness, we demonstrate this in separate RGB channel analyses for 4 images (two from each group and with filtrations in both directions) in S16, S17, S18, S19, S20, S21, S22, and S23 Figs. For a detailed explanation, consult the Limitations part of the Discussion Section.

To maintain methodological consistency and comparability with established literature, we adopted greyscale conversion for all images. All images were converted to greyscale, where every pixel in greyscale was computed as a weighted average of three colours according to the formula [16,17]: Y = 0.299R + 0.587G + 0.114B, where R, G, and B are the pixel intensities in red, green, and blue. This conversion follows the CCIR 601 format originally issued in 1987 by the International Radio Consultative Committee (now the ITU Radiocommunication Sector). All images were displayed on a 21-inch computer screen with a 1920 x 1080 dpi resolution.

2.4.1 Participants.

Participants were either second-year art students or persons of similar age and socio-economic status. The recruitment strategy was designed to control participants’ level of expertise, selecting individuals who were interested in art, familiar with gallery environments, and motivated to engage in artistic exploration. Importantly, the study targeted “young experts”—individuals with an intrinsic curiosity and openness to art — rather than professionals or trained experts, whose responses might be shaped by formal knowledge and contextual frameworks. A total of 58 participants took part in the study (16 males, 27.59%; 42 females, 72.41%), with a mean age of 22.29 years (SD = 2.25 years; age range: 19–27 years). Participants provided informed consent to take part in the study, which was approved by the Research Ethics Committee at the Nicolaus Copernicus University in Toruń (decision # 16/2021/FT).

Participants were randomly allocated to two groups of equal size, with matching procedures ensuring equivalent distributions of sex and age across the groups. Each participant in the first group was required to visit Lidia Kot’s [11] exhibition and then immediately proceed to the laboratory. Participants in the second group followed the same procedure but were instructed to visit the second exhibition, consisting of pseudo-art images instead. The participants were required to revisit the exhibition and the laboratory a second time, not less than one week after their initial visit.

2.4.2 In the gallery.

Each participant attended an exhibition featuring either artistic or pseudo-artistic images at the renowned Wozownia Art Gallery in Toruń. The gallery hall in which the artistic exhibition was held was darkened and devoid of natural daylight, maintaining a low level of ambient illumination. The primary light sources were spotlights directed toward individual artworks, consistent with the artist’s original design. Identical lighting conditions were replicated for the exhibition of pseudo-artistic images.

Both exhibitions were presented in the same gallery space and were announced and open to the public. Participants were equipped with a wearable eye-tracker built into glasses and were instructed to first follow a recommended viewing order before engaging freely with the exhibition.

To measure aesthetic experience, participants completed a six-item questionnaire inspired by the Aesthetic Experience Questionnaire (AEQ, [18]) at the end of their visit. Each question was scored using an unlabelled slider (with no ticks). The slider moved over a 5-hidden-point scale, which recorded the participants’ responses.

2.4.3 In the laboratory.

To ensure comparable illumination between the gallery and the laboratory settings, the laboratory sessions were conducted in a dark room with dim ambient lighting, where the only significant light sources were the computer screens displaying digital reproductions of the artistic and pseudo-artistic images.

Participants were instructed to observe these images whilst their eye movements and EEG activity were recorded. The images were presented in the following scheme: first, the title of each image appeared for two seconds, followed by the corresponding image itself for seven seconds. The choice of a seven-second viewing duration was informed by the free viewing study by [19], which showed that after an initial three-second period, characterised by increasing average fixation duration and decreasing saccade duration, both types of eye movements tend to stabilise. Each image presentation was followed by a 2-second mask of white noise and a 2-second grey mask, before the next title and image appeared.

After completing this eye-tracking and EEG session, participants scored each image on a separate computer station located in the same room using the same aesthetic experience questionnaire used in the gallery condition (included in S2 Appendix).

2.4.4 Eye tracking recording and data analysis.

To capture differences in perception of the two sets of images in the laboratory, we used two basic parameters of eye movements: average fixation durations and average saccade amplitudes. Average fixation duration is a measure of attention. Short fixations indicate implicit processing, whereas longer fixations (typically above 70 ms of duration) mark the initiation of perceptual information reception and processing by the brain [20,21]. The amplitude of saccades is a measure of visual space exploration. The data were preprocessed using a positional algorithm for the automated correction of vertical drift in eye-tracking data [22].

For laboratory eye-tracking measurements, the Eyetracker SMI RED 500 was used. This device operates at a sampling frequency of 500 Hz and employs a 5-point calibration procedure for enhanced precision and accuracy. The lighting in the room was kept constant for all participants, and the luminance of the object was the same for everyone.

In the gallery, participants wore eye-trackers built into glasses. To measure attention, we used the average visual intake duration, equivalent to the average fixation duration in the laboratory setting, and the average saccade duration, equivalent to the average saccade amplitude. Both measures were computed for the areas of images extracted from the movie recorded during the visit and for the time spent by participants individually at each image. In the gallery, we used SMI Eye Tracking Glasses 2 Wireless (SMI ETG 2w), dark pupil eye tracking with a binocular sampling rate of 60 Hz, parallax compensation, and accuracy of 0.5°(declared by the manufacturer). After the glasses had been fitted to the study participant and secured, a 1-point calibration procedure was performed. During the study, scene view was recorded with a scene camera (video resolution: 1280x960p and 24FPS), and audio was recorded with an integrated microphone.

All data were curated for outlier treatments using the winsorizing method introduced by Tukey and McLaughlin [23]. That is, any value of a variable above or below 2 standard deviations on each side of the variable’s mean was replaced with the value at plus or minus 2 standard deviations itself and tested for normality using the Shapiro test. Since both laboratory and gallery eye-tracking data failed to conform to the normal distribution (Shapiro test p < 0.001), the comparison was performed using a non-parametric Kruskal-Wallis ANOVA and the post-hoc Dunn test with False Discovery Rate (FDR) correction for multiple comparisons. For other non-parametric comparisons, the Mann-Whitney test was used. The significance of differences in statistical analyses is indicated by the p-value resulting from the applied tests. Differences are assumed to be significant when the p-value is less than 0.05.

2.4.5 EEG recording and data analysis.

Although the primary focus of the present study is the relationship between topological image properties and human perceptual experience, we also aimed to examine whether exposure to artistic and pseudo-artistic images would be reflected in differences in brain activity. To address this question, we analysed brain connectivity and graph theoretic measures derived from EEG signals. Brain connectivity provides an estimate of how different regions of the brain interact during cognitive processing and how these interactions may vary in response to specific experimental conditions. We hypothesized that the two sets of images would elicit different levels of cognitive challenge and engagement, leading to variations in the organization of brain connectivity patterns. To infer EEG-based connectivity, we assessed the statistical dependencies between signals recorded from different brain regions, thereby estimating patterns of interregional communication. Specifically, we computed the weighted Phase Lag Index (wPLI) [24,25] for all pairs of electrodes. Further methodological details of the EEG analysis are presented in S1 Appendix and the results are shown in S1 and S2 Figs.

2.6.1 Persistent homology method.

Central to the analysis of topological features in datasets, regardless of their representation (for example, as an image), are the concepts of a filtration and persistence. To illustrate the construction of a 1-parameter filtration for our application, consider an image composed of pixels with varying shades of grey. A filter that permits only shades above a certain intensity to pass through might inadvertently exclude certain geometric structures formed by darker pixels. By varying the filter’s threshold, some structures may either emerge or vanish. In our example, this visual filtering results in a mathematical filtration – whereby we obtain a sequence of cubical complexes (see Subsection 4.6.2 below) ordered by inclusion, that is, a complex obtained from thresholding at a lower level of the filtration is included in the complex obtained at any higher level of the filtration. The span of filtration parameter values over which a particular topological structure exists is termed its persistence. The lowest parameter value for which a structure exists is called its birth, and the highest, its death. Persistence, therefore, is a structure’s lifetime. Structures characterised by extended persistence periods are deemed the most salient, typically offering distinctive insights into the object under examination. We have just described informally a tool used in computational topology [34], persistent homology [10].

When this method is applied to two-dimensional objects such as digital images, each pixel can form vertices of elementary (e.g., in 2D, squares), and two pivotal topological characteristics come to the forefront (as further described in Section 2.6.2). The first is the number of connected components of the parts of an image that have exceeded the filtration threshold, i.e., discrete, disconnected segments within a given structure which are regions of the same or darker colour, hereinafter referred to as the ‘dimension 0’ or ‘dim 0’ cycles (formally, cycle representatives of homology classes). The second is the number of ‘holes’ present in a designated area, for example, the count of regions of a certain colour entirely encircled by areas of a different colour, hereinafter referred to as the ‘dimension 1’ or ‘dim 1’ cycles. In algebraic topology, these integers correspond to the Betti numbers (formally, ranks of homology groups), denoted and respectively, as illustrated in Fig 2.

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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 .

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Through persistent homology, it is possible to capture the emergence (birth) and disappearance (death) of components and voids relative to filtration values. The evolution of those topological properties is commonly presented in the form of barcodes [34]. The barcodes are often transformed into persistence diagrams [35] and persistence landscapes [36]. These techniques are elucidated and depicted in Fig 3, offering graphical insights that effectively highlight variations in topological properties among the objects under scrutiny.

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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 r_b, birth) and ends when it disappears (at point r_d, death), marked by two vertical lines for clarity; (b) persistence diagram – persistence is also representable via a two-dimensional scatter plot diagram with coordinates (r_b, r_d) 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).

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As well as providing valuable graphical representations for qualitative assessment, persistence landscapes (sequences of piecewise linear functions) also enable us to use standard tools from statistics [36]. Without a transformation to persistence landscapes, a multiset of barcodes is inadequate for defining even simple notions such as mean and median. In addition to comparing landscapes using L1 distance (see S10 Fig), we also used simple derived measures such as the total area beneath the ‘pyramids’ in the persistence landscape (this can also be considered the L1-norm of the persistence landscape). Even though this quick quantification entails some information loss concerning each image’s topological structure, it provided an adequate first glimpse into distinguishing between the two sets of images in our application.

However, our goal was not just to investigate differences in art pieces, but also to explore whether human perception was guided by the topological properties of the images uncovered via persistent homology. For this purpose, we derived measures that could bind these topological structures with their spatial distribution, namely, feature maps. We then investigated their relationship with the gaze-derived heatmaps, as explained in the following sections.

2.6.2 Cubical complexes.

An important aspect of the topological investigation is the decision on how to decompose the object of study, i.e., the image, using topological structures. Here, we shall compute persistent homology from a filtration of cubical complexes [31,32] with the V-construction [37].

To obtain a cubical complex, we exploit the fact that the digital representation of an image is a matrix, where the pixels are laid out in a grid, where we take the neighbourhood of a pixel to be limited to the 4 compass directions (some of which are not possible, for example, the pixels at the periphery of the image). Using this grid structure, a cubical complex is formed as a topological space constructed from a union of vertices and edges, where a vertex is assigned to a pixel, and edges are formed from adjacent pixels which satisfy a condition, for example being above (or below) a certain value in the greyness scale as is demonstrated in Figs 2 and 4. Each cubical complex thus created (one for each value of the filtration) is dependent on the relationship between neighbouring pixels. Hence, a cubical complex could be, for example, a set of disjoint pixels located in the image matrix, pixels covering some extent of an image, pixels encircling an area containing pixels that are not yet included in the filtration (thus forming a hole), or any combination of the above.

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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].

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From this, it naturally follows that the topological structures we can investigate are limited to 0-dimensional (connected components) and 1-dimensional (loops) cycles. We record the locations in a filtration where such structures emerge (births) and disappear (deaths). The list of all such births and deaths results in the multi-set of barcodes to which we can associate a persistence landscape (see Fig 3). More details on obtaining persistence landscapes from cubical complexes derived from image data can be found in S3 Appendix. Finally, we also plot Betti curves , functions defined in (S3 Appendix) for each dimension d, which show how the Betti numbers (the number of barcodes present at any given point of the filtration) vary throughout the filtration.

Apart from the natural and easily definable transition from image to a nested sequence of cubical complexes, an advantage of this cubical representation of the image is that once the topological content of the image is established, it is possible to study not only the properties of the cycles but also their spatial distribution and spatial extent within an image. This allowed us to compare the images from the exhibition in two ways – by following a standard approach of analysing persistence properties using persistence landscapes and, as introduced in our work, by exploring the spatial distribution of the topological structures using feature maps.

2.6.3 Duality of persistent homology of cubical complexes on images.

Lastly, the persistent homology groups of cubical complexes thus defined exhibit a simple but interesting duality. Topological invariants computed for the filtration starting from black and finishing on white (BW) can be related to invariants computed from the reverse filtration starting from white and finishing on black (WB), namely, persistent homology classes of dimension 1 can be (injectively) mapped to dimension 0 classes (connected components) of the reverse filtration. This is similar to results obtained in [37,38] and is an example of Alexander duality [39]. A proof can be easily deduced from looking at Fig 4; a sketch proof is provided in S3 Appendix.

The advantage of this is that by using information from both filtrations (BW and WB) and only dimension 1 cycles, we capture almost all information about cycles in images – this was utilised in the analysis of feature maps. The only discrepancies are due to cycles which comprise or interact with cycles at the boundary of the image.

Fig 4 illustrates the duality of filtrations via a simple example. At each filtration step (different steps of the dual filtrations), the underlying grid is partitioned by the connected components (0-cycles of one cubical complex and its complement with respect to the grid, which we shall denote the ‘dual’ complex). Let f and be the steps in the BW and WB filtrations, respectively, where . For this toy example, the total length of a filtration, M, is 4 (for a general greyscale image, M = 255). Hence, the black components for f = 0 are pink components for and vice versa. Hence, as shown in Table 1, for any given pair , the connected components from both filtrations partition the underlying grid; hence, the number of partitions of the grid is also the sum of the zeroth Betti numbers.

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Table 1. Number of partitions for filtration steps for the toy example.

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For the BW filtration at a grey intensity level, f = 0 (on a scale from 0 to 4), the cubical complex consists of three distinct parts marked with black dots. This results in the Betti numbers being for the three components and , indicating the lack of holes. As the filter transparency increases to 1 (f = 1), one more disconnected component is added, increasing the zeroth Betti number to 4. In the next filtration step f = 2, there are 2 changes: (1) two components are joined (bottom right corner), decreasing by 1, and (2) another component merges with new elements joining the filtration, forming a hole, thus increasing to 1. At intensity f = 3, all 3 shapes merge into a single connected component with one hole ( and ). If we further increase filter intensity to f = 4, all pixels will be added to the cubical complex, yielding Betti numbers and .

Now, for every filtration step f with non-zero , we can find the dual step, , in the reverse filtration, where there exists a dimension 0 component, which we call its dual. For example, as shown in Fig 4 for f = {2, 3} top row, a cycle exists in the top left corner, and in WB filtration for , bottom row, there is a connected component present in that corner. If we study the reverse filtration, the same rule applies – for the two steps, in which is not zero, we can find dual components in steps f = {1,0}. In all cases, the persistence of dual components is equal (unless there is an interaction with a cycle on the periphery). This can also be seen within the barcodes and persistent landscape, as demonstrated with another example where most cycles do not interact with the periphery of the image in S9 Fig.

Altogether, with this duality, we have shown that the topological properties derived from a single filtration in both dimensions are similar to those from both filtrations but in the same dimension, and we exploit this fact in constructing feature maps.

1. Violations of Alexander duality

Finally, we also investigated to what extent Alexander duality was violated by each image, and whether there were differences between the art and the pseudo-art. We hypothesise that artists would be very aware of the existence of the rectangular frame around any image they create, which encloses their composition. Artists will be deliberate about which shapes touch the frame, which will affect the distribution of cycles touching the frame and thereby impacting Alexander duality.

We investigate this by computing the difference in the areas under the Betti curves (or integrated Betti numbers) for the opposing filtrations and dimensions and normalizing by dividing by the mean area under those two Betti curves (denoted and , where f is the filtration parameter). Hence the area under the Betti curve for dimension 0 cycles in the black to white filtration (BW), is given by the integral, . Since always, we used the following positive measure for violations of Alexander duality:

(1)

and also its counterpart with BW and WB interchanged, ADV_WB. This measure is bounded between 0 (no violation) and a maximum of 2.

2.6.4 Topological feature maps.

In visual art, artistic impressions are created with objects, colours, and textures. Whereas the first two are directly impacted throughout the artistic process, texture is controlled at the earliest stages, when the technique is chosen. Furthermore, texture has a dual nature – for images, the texture is the spatial distribution of colours or changes in contrast on the canvas, whereas, for physical objects, texture refers to the smoothness or roughness of an object’s surface. In painting, texture is a combination of these two aspects, as both are combined (by the artist’s selection of canvas and paint) to create the overall artistic impression.

In this space of objects, colours, and textures, our method is inherently sensitive not only to shapes of various sizes but also to their spatial distribution, which makes it suitable for investigating both objects and textures. The motivation is that, in areas with rich texture, there are many small patches of colour, which translate into a large number of cycles per unit area. On the other hand, smooth areas do not vary as abruptly in terms of colour, so they shall contain fewer distinct cycles per unit area (low cycle density).

To explore textures and differences in the spatial extent of visual structures between the two sets of images and their effect on perception, we introduce topological feature maps. The pipeline visualising this process is shown in Fig 5. Topological feature maps are topographic maps conveying the spatial location of properties of various topological features of an image that were obtained by persistent homology calculations.

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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 G_s and its complement heatmap are used to obtain (looking) and (not looking), by weighting the image feature maps with the corresponding gaze-heatmap.

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Starting with the image depicting the location of all 1-dimensional cycle representatives (Step A3 in Fig 5, samples shown in Fig 6), one can overlay a square grid mesh consisting of non-overlapping windows of fixed size. Then, for each window, a feature is derived (A4 in Fig 5) – it could be cycle density (computed as the total number of cycles in each grid window), maximal 1-dimensional cycle perimeter (the maximal number of pixels belonging to the perimeter of any cycle lying within the grid window), or maximal persistence (given by the largest persistence of all cycles found in the grid window). A feature map can therefore be specified by a matrix M^image where, depending on the choice of topological descriptor, will be the feature map value in the square in the ith row and the jth column of the grid over that image. Full details can be found in the S4 Appendix.

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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.

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These three topologically derived measures, cycle density, maximal persistence, and cycle perimeter, are proxies for texture in the image, local contrast, and shape size, respectively.

Feature maps are derived from the locations of 1-dimensional cycle representatives, and this, at first glance, excludes the information about the topology in dimension 0. To overcome this, we exploit the duality feature of cycles and incorporate the information about the cycles in dimension 0 from BW filtration by studying their dual in dimension 1 in WB filtration. As demonstrated in S9 Fig and also shown in S10 Fig, very little is lost by only using dimension 1 cycles.

Finally, we devised composite feature maps derived from both filtrations (WB and BW, step C1 in Fig 5) and used these composite feature maps in our study. The feature maps were as follows: (1) cycle density, which was obtained from summing the densities per unit grid area from both filtrations; (2) maximum cycle persistence, which took the maximum persistence of all the cycles from both filtrations present per unit grid area; (3) maximum cycle perimeter, which took the maximum perimeter of all the cycles from both filtrations present in any particular square in the grid.

All values from a feature map M^image yield a distribution that was specified using its Empirical Cumulative Distribution Function (, where U is a uniform weight, thus not favouring any regions, C2 in Fig 5). This function is the cumulative sum of the count of windows having a given property value (for example, a cumulative sum of a histogram showing the counts of windows having a given cycle density). Even though some 2D spatial information is lost, we shall show that the ECDF provides a convenient way of comparing and visualising distributions derived from the feature maps of the images combined with the gaze maps of individuals, as will be seen below.

Note that all images were the same size, and the same grid mesh size (50 pixels by 50 pixels) was used for each image. Although grid size may seem arbitrary, we verified that the choice makes very little difference to the structure of the ECDFs by testing multiple versions of the grid size (see S30 Fig) and Section 3.8 for a more detailed discussion).

2.6.5 Computation of cycles and software.

Cycle representatives for dimension 1 homology classes used in this study were obtained using the ‘Ripserer.jl’ Julia library [40–43] and they are shown in S12, S13, S14, and S15 Figs for art, pseudo-art, and using both filtrations.

Further details, together with the algorithmic framework for this procedure, are presented in S4 Appendix. While the ECDF curves characterize the underlying nature of the images’ textures, gaze-weighted ECDF curves, introduced in the next section, will provide insight into which features the participants were interested in while interacting with art (or pseudo-art).

2.6.6 Feature maps’ robustness to image distortions.

Our exploration of topologically derived properties is grounded in their inherent independence from arbitrarily defined coordinates and metric attributes of perceived objects [44]. It allows for direct comparisons between different images by examining their intrinsic features. In the context of 2D visual art, persistent homology was used here to analyse images based on either a single greyscale channel or three colour channels. The use of a common scale of 256 points for pixel intensity ensures a fair and unbiased comparison across all analysed images. Topological analysis using pixel-based cubical complexes is relatively fault-tolerant to differences in image resolution. Our analyses show that the output of topologically derived features is only affected after a large reduction in the resolution of analysed images (for details, see S11 Fig).

Furthermore, topological properties exhibit a notable resilience to disturbances, such as variations in visual acuity or noise [45], including changes in illumination [32], as well as image degradation [46]. Moreover, since these features reflect the distribution of local contrast rather than its absolute global value, they are expected to remain stable under global contrast variations. To confirm that the outcomes of our topological analyses are not artifacts of global contrast differences, we conducted persistent homology computations on images with systematically altered contrast levels (consult S5 Appendix for full details). The resulting topological descriptors remained largely invariant across a wide range of contrast manipulations (see S24, S25, S26, S27, S28, and S29 Figs), supporting the contrast insensitivity of the method. The numbers of barcodes are largely preserved, although these can shrink or be stretched.

Importantly, the relation between image sets is preserved – this is shown in S28 and S29 Figs, where we demonstrate the L1 distance computed between all pairs of images, after the transformation was applied to all images independently. In all cases, the distance between pseudo-art images was lower than for art images.

2.6.7 Analysis of the relationship between feature maps and fixation patterns.

Potential differences in the eye movements of the investigated groups could have resulted from the different topological properties of the observed image sets. To verify this hypothesis, we compared information from eye-tracking movements and topologically derived properties of the images.

To this end, fixation sequences lasting over 75 ms were extracted from the eye-tracking data of each participant and used to generate fixation heatmaps (B2 in Fig 5). These heatmaps, created by applying a Gaussian kernel to qualifying fixation positions, were produced for each participant, image, viewing, and session (a total of 4 viewings, 2 per session). More details on heatmap generation can be found in S4 Appendix.

Apart from the feature distributions intrinsic to the image, we studied two weighted- weighted with individual heatmap G_s and weighted with a uniform weight where a participant did not look (C4 and C6 in Fig 5 respectively). The main idea of weighting via gaze maps is as follows – the more time a participant spent looking at a specific window in the mesh grid, the more important that window was and the higher the weight given to it; this also means that the topological properties of that window are more important. For example, if a person only looks at the parts of an image with highly persistent cycles and ignores other parts of the image, then the distribution obtained will be highly skewed and very different from the underlying distribution of persistent cycles in the image that is obtained from the neutral or ‘uniform gaze’. An electronic scanner will have a uniform gaze – all parts of the image are given equal weight; there are no preferred areas.

1. ECDF comparison

Two comparisons were made between datasets to investigate what kinds of features participants’ eyes were drawn to. The first comparison was about the “ground truth” for an image and the areas where subjects look at. This was done by comparing the distributions of the mean squared ECDF errors (MSE), computed as a mean squared difference of and (looking), that is a mean squared difference of the ECDF from feature map of an image and ECDF derived from a feature map of the same image weighted with a heatmap of subject s viewing it. The statistical tests were performed by comparing the distribution of MSE for artistic images with the distribution of MSE for pseudo-artistic images.

The second comparison was about the areas that subjects looked at and areas where they did not look. This was computed as mean squared difference and mean difference between (looking) and the ECDF derived from the complement gaze, (not looking), thereby comparing features that a viewer, s, looked at with the features in the image that the (same) viewer did not explore. The first measure was used to assess if there are group differences between the art images and the pseudo-art images, while the second was necessary for interpretation of the results.

The comparisons were performed between artistic and pseudo-artistic images. In all comparisons, data was used from all images for all visits and sessions and repeated for three feature maps – maximal cycle persistence, maximal cycle perimeter, and cycle density. More details of this procedure, including a precise definition of these feature maps, can be found in S4 Appendix.

2.7.1 Topological data analyses.

1. Empirical cumulative distribution function analysis

In the study of the spatial distribution of cycle density, the shapes of the ECDF curves generated from the grid of images were compared using the Kolmogorov-Smirnov (KS) statistic [47–49]. The choice of grid size was motivated by the achieved total number of windows (smaller grid size) – the more windows, the better the statistical power of the KS test. However, this was limited by the fact that the window size needed to be large enough to contain a range of cycles.

The Friedman test was used to assess the difference between regions where subjects looked at and did not look at (using combined data from two exhibitions, one test for each of the feature maps), and in the post hoc analysis, the Wilcoxon signed-rank test was used to determine differences within blocks (with one block being data for the art exhibition and another for the exhibition of pseudo-artistic images).

2.7.2 Statistical Image Properties (SIP).

Many statistical measures can be applied to the analysis of images, such as artworks. As a benchmark for the persistent homology method, we have chosen statistics commonly used for artistic image analyses, representing a wide variety of parameters, from spectral to non-linear measures such as complexity or entropy. The final set of SIPs used for comparison with persistent homology analyses included: Fourier slope – the least squares fit of a line to the log-log power spectrum of the images transformed into the frequency domain [50], Self-similarity – self-similarity assessment based on the Pyramid Histogram of Oriented Gradients (PHOG) that is of occurrences of gradient orientation computed on a dense grid of uniformly spaced cells on an image to facilitate edge detection [51], Complexity – the mean norm of the Histogram of Oriented Gradients (HOG) across all orientations [52], Anisotropy – the variance of gradient strength in the HOGs across its bin entries [52] and the edge density [53]. The details of the statistical methods can be found in the cited articles. The analyses were performed using scripts provided by the authors of the relevant methods found in the Supplementary Materials to the cited articles.

The p-value represents the probability of obtaining the results by chance alone. In this study, a value below 0.05 was used to indicate the statistical significance of the results.

3.1.1 Aesthetic experience differences in the gallery.

Aesthetic experience scores were gathered at the conclusion of each participant’s visit to the gallery, using a paper-and-pencil questionnaire. Participants evaluated their overall exhibition experience.

Analysis of the collected data with two-way mixed-design MANOVA: Exhibition showed no significant differences: F(6,46) = 1.748, p = 0.131, .

3.1.2 Aesthetic experience differences in the laboratory.

Aesthetic experience scores were collected in the laboratory for each image individually, independently of the eye-tracking. Subsequently, the scores were averaged across images for each participant. Two-way mixed-design Multivariate Analysis of Variance, MANOVA yielded a significant effect of the exhibition: Exhibition, F(6,49) = 4.14, p = 0.002, . To further investigate the multivariate effect of exhibition type, separate one-way ANOVAs were conducted for each dependent variable. The results revealed significant effects of exhibition type on most measures of aesthetic experience: Elements perception: F(1, 54) = 22.57, p < 0.001, ; Emotional intensity: F(1, 54) = 12.01, p = 0.001, ; Understanding the artist’s intent: F(1, 54) = 23.48, p < 0.001, ; Context clarity: F(1, 54) = 4.02, p = 0.050, ; Understanding flow: F(1, 54) = 17.89, p < 0.001, ; Personal experience of flow: F(1, 54) = 10.18, p = 0.002, See Fig 7.

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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.

https://doi.org/10.1371/journal.pcbi.1014156.g007