Please summarize these statistical properties:

1
Description of Statistical Image Properties
1.1
PHOG Complexity, Self-similarity, and Anisotropy
The three SIPs were calculated using the Pyramid Histogram of Oriented Gradients (PHOG) method,
which was initially developed for object recognition and image classification in computer vision
(Bosch et al., 2007). After converting an image into its representation of luminance and color
gradients (called gradient images), histograms of oriented gradients (called HOG descriptors; Dalal
and Triggs, 2005) were calculated for pyramidal subdivisions of the gradient image (for a detailed
description of the method, see the Appendix in Braun et al., 2013).
Image Complexity was expressed as mean gradient strength for all subdivisions of the
gradient image. It overlaps with other SIPs that are known to reflect image Complexity, including
edge density, the overall signal strength in Gabor-filtered images and the fractal dimension (for
reviews, see Machado et al., 2015; Van Geert and Wagemans, 2020). The degree of Complexity
contributes to image preference of a large variety of visual patterns. Different groups of beholders
prefer different degrees of Complexity (Bies et al., 2016; Güclütürk et al., 2016; Spehar et al., 2016).
Self-similarity of an image is calculated by determining how similar the HOG descriptors for
the subsections are to those of other subsections at different levels of the pyramid or to the ground
level histogram (Bosch et al., 2007; Amirshahi et al., 2012). Higher Self-similarity values indicate
that the subsections of an image contain gradients whose orientation histograms are more similar to
other subsections of the image. In the present study, we compared histograms of all levels up to level
3 of the pyramid (8 ✕ 8 grid, resulting in 64 subdivisions at pyramid level 3) with the histogram of
the ground image (32 orientation bins). An intermediate to high degree of Self-similarity is
associated with traditional paintings and other visually pleasing stimuli (Braun et al., 2013).
The HOG approach can also be used to determine how different the strength of the gradients
is across orientations in an image ("anisotropy"). If the strength of all oriented gradients is uniform
across orientations, i.e. all orientations are of about equal strength, HOG anisotropy decreases with a
lower limit of zero. If particular orientations predominate in an image, e.g. horizontal and vertical
orientations in an architectural photograph, anisotropy is high. PHOG anisotropy correlates highly
inversely with 2nd-order entropy of edge orientations (Spearman coefficient r = -0.72; see next
paragraph) and was therefore excluded from the final set of 8 variables (Table 1).
1.2 Entropy of Edge Orientations
Summary statistics for the prevalence of luminance edges and their orientations in an image can also
be obtained by filtering an image with a set of oriented Gabor filters, which can be viewed as models
of simple cell responses in the primary visual cortex. In histograms that cover all orientations at equal
intervals, we plotted the strength of the response as a function of filter orientation (Geisler et al.,
2001). As a measure of histogram uniformity, Shannon entropy was calculated. Entropy is maximal
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if response strength is the same across all orientations. It decreases as particular orientations become
more prominent in the histograms. We computed first-order and second-order entropy of edge
orientations. First-order entropy refers to global histograms for Gabor responses (oriented edges) that
are summarized for the entire image. Second-order entropy reflects pairwise statistics of edge
orientations across an image. To obtain this measure, we compared the orientation of all edges to the
orientation of all other edges by pairwise comparison in each image (Geisler et al., 2001; Redies et
al., 2017). Entropy is maximal if all orientation differences are equally likely to occur, i.e. the
orientations of the edge pairs are independent of each other across an image. Entropy is lower in
images where particular combinations of orientations abound, for example, edges of parallel or
orthogonal orientations. Pairwise statistics of edge orientations have been studied in the context of
contour grouping and natural scene statistics (Geisler et al., 2001; Sigman et al., 2001).
In a previous study, we described that 2nd-order edge orientation is relatively high in large
subsets of traditional artworks of different cultural provenance (Redies et al., 2017) and in other types
of visually pleasing images (Grebenkina et al., 2018). Because 1st-order entropy is generally high in
images with high 2nd-order entropy, the two measures correlate strongly. Indeed, the Spearman
coefficient r for the 150 style-transferred images was 0.84. Therefore, we excluded 1st-order Entropy
from the set of SIPs used the final analysis (Table 1).
1.3
Variances of Feature Responses in Convolutional Neural Networks (CNNs)
CNNs can predict the aesthetic quality of images in rating tasks (Lu et al., 2015; Rafegas and
Vanrell, 2016; for a review, see Brachmann and Redies, 2017). A draw-back of CNNs is the lack of
interpretability of individual feature responses, especially at higher levels. To classify visual
artworks, we proposed to study variances of filter responses at lower levels of the AlexNet CNN
model, as described in detail by Brachmann and Redies (2017). This method captures the distribution
of luminance and color edges (Rafegas and Vanrell, 2016) as well as spatial frequency filters across
images and allows to classify traditional artworks with high accuracy. At the same time, the feature
responses can be interpreted in terms of low-level neural responses up to cortical area V1.
In the present study, we calculated filter response variances at convolutional layer 1 (conv1)
of the CNN where each response map is partitioned into n ✕ n equally sized subregions. Responses
were then recorded for each subregion and every filter by a max-pooling operation over the response
maps. Two types of variances were calculated. First, we calculated the total variance over all 96 filter
entries of the n ✕ n subregions of conv1. This variance is henceforth called Variance Pa(n)
(Brachmann et al., 2017). Low values of this variance indicate that a large number of filters respond
at many image positions (richness of filter responses). High values indicate few responses in a small
number of subregions (sparseness of filter responses). The variances can be calculated for a different
number of n ✕ n subregions, which correspond to different levels of spatial resolution. In the present
study, the number of subregions were 2 ✕ 2, 4 ✕ 4, 8 ✕ 8, 16 ✕ 16, and 30 ✕ 30, resulting in
variances henceforth called Pa(2), Pa(4), Pa(8), Pa(16), and Pa(30). Second, we calculated the median
over the variances of each of the 96 filters, again for different numbers of n ✕ n subregions at conv1.
This variance is henceforth termed Variance Pf(n) and was calculated for the same number of
subregions as Variance Pa(n), resulting in variances henceforth called Pf(2), Pf(4), Pf(8), Pf(16), and
Pf(30). Low values of this variance indicate a high degree of self-similarity of filter responses across
the response map, whereas high values indicate a greater variability of filter responses (Brachmann et
al., 2017).
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Traditional artworks of different cultural provenance show a particularly high richness of
filter responses (i.e. low Variance Pa[22] values) and a low to intermediate degree of variability (i.e.,
low Variance Pf[12] values; Brachmann et al., 2017). Because of the high degree of redundancy of
the variances of each type, we used only one variance each in the final analysis (Pa[2] and Pf[30];
Table 1).
1.4
Color Measures
Color is an important attribute of visual artworks and photographs (Li and Chen, 2009; Yanulevskaya
et al., 2012; Rafegas and Vanrell, 2017). Artists seem to have an intuitive knowledge of chromatic
image composition, which matches viewers' preferences, and implement this knowledge in their
creations (Nascimento et al., 2017). Preferred color gamuts are perceived as natural (Nascimento et
al., 2021) and colorful (Altmann et al., 2021). For abstract artworks, individual differences in color
preferences have been reported (Mallon et al., 2014).
We measured nine chromatic features. For the L*a*b* color space, we included the mean
values for each channel. For the HSV color space, we included both the mean values and the
Shannon entropy for all three channels. In our final model, we retained three of the measures only to
reduce collinearities: the mean value of the b channel (blue-yellow axis) of the L*a*b* color space
(henceforth abbreviated Lab [b]); the mean value of the S-channel (saturation) of the HSV color
space (HSV [S]); and the Shannon entropy of the H channel of the HSV color space (HSV [H]
entropy; Table 1). HSV (H) entropy represents the uniformity of the color spectrum. Highest values
are encountered if all hues are present at equal strength in the image. This measure thus represents a
type of colorfulness.
1.5
Fourier Spectral Properties (Slope and Sigma)
The Fourier spatial frequency spectrum of natural and visually preferred images, such as artworks,
shows statistical regularities (Graham and Field, 2007; Redies et al., 2007). Two characteristics of the
spectrum, here called Fourier slope and Fourier sigma, were included in the initial set of 29 variables.
The Fourier slope is the slope of log-log plots of spectral power versus spatial frequency and
indicates the relative prominence of low versus high spatial frequencies (relation of coarse to fine
structure) in an image. Grayscale images of natural scenes and traditional artworks fall off linearly
with increasing spatial frequency and share a Fourier power slope of about -2 (Graham and Field,
2007; Redies et al., 2007). Fourier slope is correlated with Complexity (Table 1) because more
shallow slopes (i.e., higher slope values) indicate more fine detail in an image. Fourier sigma is
defined as the deviation of the Fourier spectral curve from a straight line in the log-log plots (Redies
et al., 2007). It assumes small values for natural images and large sets of traditional artworks. When
the deviation of the spectral curve from a straight line is large and Fourier sigma is high, images can
be perceived as unpleasant (Fernandez and Wilkins, 2008).
1.6
Symmetry
The role of symmetry as an aesthetic primitive that is universally associated with visual preference
(Bode et al., 2017) has recently been contested (Leder et al., 2019). In the present work, we
calculated up-down mirror symmetry, left-right mirror symmetry as well as a combination of both
symmetries, according to the algorithm described by (Brachmann and Redies, 2016). We use first-
layer (conv1) filter responses of a CNN to measure higher-order symmetry. This measure relies not
only on color, edges and texture depicted in an image, but also reflects the symmetry of simple
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shapes and objects of an image, as does the human visual system. The three symmetry measures for
the style-transferred images correlate moderately to highly with Self-similarity and Variance Pf(30)
(Spearman's coefficients |r| = 0.61 to 0.72), which represents the variability of filter responses (see
above). Because of this redundancy and the contested role of symmetry as a universal mediator of
beauty (Leder et al., 2019), the symmetry measures were not included in the final set of variables
(Table 1).