A Stochastic View of Varying Styles in Art Paintings

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Article A Stochastic View of Varying Styles in Art Paintings

G.-Fivos Sargentis * , Panayiotis Dimitriadis , Theano Iliopoulou and Demetris Koutsoyiannis

Laboratory of Hydrology and Water Resources Development, School of Civil Engineering, National Technical University of Athens, Heroon Polytechneiou 9, 157 80 Zographou, Greece; [email protected] (P.D.); [email protected] (T.I.); [email protected] (D.K.) * Correspondence: ﬁ[email protected]

Abstract: A physical process is characterized as complex when it is difficult to analyze and explain in a simple way, and even more difficult to predict. The complexity within an art painting is expected to be high, possibly comparable to that of nature. Herein, we apply a 2D stochastic methodology to images of both portrait photography and artistic portraits, the latter belonging to different genres of art, with the aim to better understand their variability in quantitative terms. To quantify the dependence structure and variability, we estimate the Hurst parameter, which is a common dependence metric for hydrometeorological processes. We also seek connections between the identified stochastic patterns and the desideratum that each art movement aimed to express. Results show remarkable stochastic similarities between portrait paintings, linked to philosophical, cultural and theological characteristics of each period.

Keywords: stochastic analysis; portrait; art painting; art theory

1. Introduction Citation: Sargentis, G.-F.; Dimitriadis, P.; Iliopoulou, T.; The aesthetic evaluation of art paintings typically involves the physical procedure Koutsoyiannis, D. A Stochastic View of visual examination of an image [1–9]. This approach is considered highly subjective of Varying Styles in Art Paintings. and celebrated as such, with its outcomes being the subject of philosophy of art, and in Heritage 2021, 4, 333–348. https:// particular of the philosophical branch known as aesthetics [10]. In an attempt to conceive doi.org/10.3390/heritage4010021 an alternative and objective procedure for the aesthetic analysis, artificial intelligence and mathematical tools are the subject of a plethora of related publications [11–44]. Academic Editor: Nicola Masini The mathematical field of Stochastics has been introduced as an alternative to deterministic approaches and is used to model the so-called random, i.e., complex, unexplained Received: 14 January 2021 or unpredictable, fluctuations observed in (but not limited to) non-linear geophysical Accepted: 8 February 2021 processes [45–47]. Stochastics helps develop a unified perception for natural phenomena Published: 11 February 2021 and expel dichotomies like random vs. deterministic. Under the viewpoint of stochastics, there is no such thing as a virus of randomness that infects some phenomena to make them Publisher’s Note: MDPI stays neutral random, leaving other phenomena unaffected. Rather, it seems that both randomness and with regard to jurisdictional claims in predictability coexist and are intrinsic to natural systems, which can be deterministic and published maps and institutional affil- random at the same time, depending on the prediction horizon and the time scale [48]. On iations. this basis, the uncertainty in art, in line with other natural processes, could be considered both aleatory and epistemic, since, in principle, in this case as well, we do not know perfectly well the underlying causal mechanisms. By applying the methods of stochastics, we can quantify this uncertainty treating the unpredictable fluctuations of art works as Copyright: © 2021 by the authors. realizations of complex stochastic processes. Licensee MDPI, Basel, Switzerland. A physical process is characterized as complex when it is difficult to analyze or explain This article is an open access article in a completely predictive way. Likewise, constructions by artists (e.g., paintings, music, distributed under the terms and literature, etc.) are also expected to be of high complexity since they are produced by conditions of the Creative Commons numerous human (e.g., logic, intuition, emotions, etc.) and non-human (e.g., quality of Attribution (CC BY) license (https:// paints, paper, tools, etc.) processes interacting with each other in a complex manner. Art creativecommons.org/licenses/by/ paintings are thus likely to resemble a stochastic process of increased uncertainty and 4.0/).

Heritage 2021, 4, 333–348. https://doi.org/10.3390/heritage4010021 https://www.mdpi.com/journal/heritage Heritage 2021, 4 334

variability. Such a process deviates from a purely White-Noise behavior, i.e., unstructured randomness, as its aesthetic attributes are likely to appear in clusters by a highly non- predictive manner. Even though beauty is difﬁcult or even impossible to quantify [49–51] stochastic analysis may offer new insights into aesthetical issues and a basis for their objective evaluation. The stochastic analysis of the examined artworks is performed using the 2D-climacogram (2D-C), a stochastic tool evaluating the degree of variability vs. spatial scale. As case studies, we use multiple images of art paintings depicting human portraits, and seek whether there exist stochastic similarities in the ways the human face has been depicted by various art movements.

2. Quantifying the Variability in Art Painting through the 2D Climacogram A stochastic computational tool called 2D-C [52–58] is used to analyze images of art paintings quantifying the variability of the brightness vs. scale in a specific image and in a group of images as well. Here, we refer to spatial (not time) scale, defined as the ratio of the area of k × k adjacent cells (i.e., scale k) that are averaged to form the (scaled) spatial field, over the spatial resolution of the original field (i.e., at scale 1). The term climacogram [59,60] comes from the Greek word κλιµαξ (pronounced: climax meaning scale). It is defined as the (plot of) variance of the averaged process (assuming stationary) versus averaging scale k, and is denoted as γ(k). The climacogram is useful for detecting the long-term change (or else dependence, persistence, clustering) of a process, which emerges particularly in complex systems as opposed to white noise (absence of dependence) of even Markov (i.e., short-term persistence) behavior [61]. In order to obtain a quantitative characterization of the artwork, its image is digitized in 2D and each pixel is assigned a grayscale color intensity (white = 1, black = 0). Assuming that our sample is an area n∆ × n∆, where n is the number of intervals (e.g., pixels) along each spatial direction and ∆ is the discretization unit determined by the image resolution, (e.g., pixel length), the empirical classical estimator of the climacogram for a 2D process can be expressed as: n/κ n/κ 1 ( ) 2 γˆ(κ) = x κ − x (1) 2 2 − ∑ ∑ i,j n /κ 1 i=1 j=1

(κ) where the ‘ˆ’ over γ denotes estimation, κ is the the dimensionless spatial scale, xi,j = 1 κj κi x κ2 ∑ψ=κ(j−1)+1 ∑ξ=κ(i−1)+1 ξ,ψ represents a local average of the space-averaged process at (n) scale κ, at grid cell (i, j), and, x ≡ x1,1 is the global average of the process of interest. Note that the maximum available scale for this estimator is n/2. The difference between the value in each element and the field mean is raised to the power of 2, since we are mostly interested in the magnitude of the difference rather than its sign. Thus, the climacogram expresses in each scale the diversity in the greyscale intensity among the different elements. In this manner, we may quantify the uncertainty of the brightness intensities at each scale by measuring their spatial variability. An important property of stochastic processes is the Hurst–Kolmogorov (HK) dynamics (usually known in hydrometeorological processes as Long-Term Persistence, or LTP), which can be summarized by the Hurst parameter as estimated in large scales. This parameter is estimated by minimizing the fitting error of the logarithmic slope of the tail (i.e., the last 50 scales in the presented applications) of the observed (from data through Equation (1)) and the modelled (Equation (2)) climacogram. The isotropic HK process with an arbitrary marginal distribution (e.g., for the Gaussian one, this results in the well- known fractional-Gaussian-noise, as described by Mandelbrot and van Ness [62]), i.e., the power-law decay of variance as a function of scale, is defined for a 1D or 2D process as:

λ ( ) = γ k ( − ) (2) (k/∆)2d 1 H Heritage 2021, 4 FOR PEER REVIEW 3

𝜆 𝛾(𝑘) = (2) (𝑘/𝛥)() where 𝜆 is the variance at scale k = κΔ, d is the dimension of the process/field (i.e., for a 1D process d = 1, for a 2D field d = 2, etc.), and H is the Hurst parameter (0 < H < 1). For 0 < H < 0.5 the HK process exhibits an anti-persistent behavior, H = 0.5 corresponds to the white noise process, and for 0.5 < H < 1 the process exhibits LTP (clustering). In the case Heritage 2021, 4 of clustering behavior, due to the non-uniform heterogeneity 335of the brightness of the painting, the high variability in brightness persists even in large scales. This clustering

effectwhere λ mayis the substantially variance at scale kincrease = κ∆, d is thethe dimension diversity of thebetween process/field the (i.e.,brightness for a in each pixel of the image,1D process a dphenomenon= 1, for a 2D field alsod = 2, observed etc.), and H isin the hydr Hurstometeorological parameter (0 < H < 1processes). For (such as temperature,0 < H < precipitation, 0.5 the HK process wind exhibits etc.), an anti-persistent natural landscapes behavior, H = and 0.5 corresponds music [63]. to the white noise process, and for 0.5 < H < 1 the process exhibits LTP (clustering). In the case of clusteringThe behavior, algorithm due to that the non-uniform generates heterogeneity the climacogram of the brightness in 2D of thewas painting, developed in MATLAB for rectangularthe high variability images in brightness [64]. In persists particular, even in large for scales.the current This clustering analysis, effect maythe images are cropped to 400substantially × 400 pixels, increase 14.11 the diversity cm × 14.11 between cm, the in brightness 72 dpi in(dots each per pixel inch). of the image,In particular, for the current a phenomenon also observed in hydrometeorological processes (such as temperature, analysis,precipitation, the wind images etc.), natural are landscapescropped andto music400 × [ 63400]. pixels, 14.11 cm × 14.11 cm, in 72 dpi (dots per Theinch). algorithm The thatvalues generates assigned the climacogram to each in pixel 2D was are developed the grayscale in MATLAB color for intensity (white = 1, rectangular images [64]. In particular, for the current analysis, the images are cropped to black400 × 400 = 0) pixels, of the 14.11 image. cm × 14.11 In cm,Figure in 72 dpi1, we (dots present per inch). three In particular, images for the for current benchmark image analysis: (a)analysis, white the noise; images are(b) cropped image to with 400 × clustering;400 pixels, 14.11 and cm ×(c)14.11 an cm,art inpainting. 72 dpi (dots In Figure 2, we describe theper inch).steps The in values our analysis, assigned to and each pixelshow are how the grayscale grouped color pixels intensity at (white scales = 1,k = 2, 4, 8, 16, 20, 25, 40, black = 0) of the image. In Figure1, we present three images for benchmark image analysis: 50,(a) white 80, 100 noise; and (b) image 200, withare clustering;used to andcalculate (c) an art the painting. clim Inacogram Figure2, we in describe Figure 3. In Figure 3, we also depictthe steps the in our standardized analysis, and show climacogram, how grouped pixels which at scales is definedk = 2, 4, 8,as 16, the 20, 25,ratio 40, 𝛾(𝑘)/𝛾(1), and is also a50, function 80, 100 and of 200, scale are used k but to calculate without the having climacogram any in effect Figure3 .from In Figure the3, wemarginal also variance of the pro- depict the standardized climacogram, which is defined as the ratio γ(k)/γ(1), and is also a cess.function of scale k but without having any effect from the marginal variance of the process.

(a) (b) (c)

Figure 1. 1.Benchmark Benchmark of image of image analysis; analysis; (a) White noise, (a) White average noise, brightness averag 0.5; (be) Imagebrightness with 0.5; (b) Image with clustering, average average brightness brightness 0.54; (c) Photo 0.54; of Audrey(c) Photo Hepburn, of Audrey average brightnessHepburn, 0.57. average brightness 0.57. The presence of clustering is reﬂected in the climacogram, which shows a marked differenceThe compared presence to independent of clustering pixels asis inreflected white noise in processes the climacogram, (Figure3). Speciﬁ- which shows a marked cally, the variance of the clustered images is notably higher than that of white noise at all differencescales, indicating compared a greater degree to independent of variability and pixels uncertainty as in of white the process. noise Likewise, processes (Figure 3). Specif- ically,comparing the the variance clustered imageof the and clustered the art painting, images the latteris notably has the mosthigher pronounced than that of white noise at all scales,clustering indicating behavior and a a highergreater degree degree of variability. of variability and uncertainty of the process. Likewise, comparing the clustered image and the art painting, the latter has the most pronounced clustering behavior and a higher degree of variability.

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Figure 2. Example of stochastic analysis of 2D picture; steps. Grouped pixels at different scales used to calculate the cli- Figure 2. Example of stochastic analysis of 2D picture; steps. Grouped pixels at different scales used to calculate the Figuremacogram; 2. Example (a) ofWhite stochastic noise; ( banalysis) Image with of 2D clustering; picture; ( csteps.) Photo. Grouped pixels at different scales used to calculate the cli- climacogram; (a) White noise; (b) Image with clustering; (c) Photo. macogram; (a) White noise; (b) Image with clustering; (c) Photo.

(a) (b) Figure 3. (a) Climacograms of the benchmark images; (b) Standardized climacograms of the benchmark images.

( a) (b)

Figure 3. ((a)) Climacograms Climacograms of the benchmark images; ( b) Standardized climacograms of thethe benchmarkbenchmark images.images.

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3.3. StochasticStochastic EvaluationEvaluation in in Arts Arts Art’sArt’s functionsfunctions areare vital vital for for our our society society on on multiple multiple levels. levels. AA workwork ofof art art is is not not only only (and(and perhapsperhaps notnot necessarily) something something pleasing pleasing to to the the eye, eye, but but also also a medium a medium that that por- portraystrays our our emotions. emotions. Although Although each each artist’s artist’s work work contains contains a a unique message, thethe fieldfield of of artart creates creates a a cultural cultural process; process; aesthetics: aesthetics: a a communication communication channel channel between between the the artist artist and and thethe observer. observer. CartesianCartesian rationalism, rationalism, which which was was derived derived from from the Frenchthe French philosopher philosopher Rene Descartes,Rene Des- doescartes, not does regard not aesthetic regard aesthetic quality as quality an inherent as anquality inherent of aquality physical of object,a physical but insteadobject, but as theinstead distinction as the ofdistinction mind and of nature, mind and paving nature the, waypaving for the humans way for to appreciatehumans to the appreciate role of theirthe role own of subjective their own feelings subjective in determining feelings in theirdetermining aesthetic their preferences. aesthetic Other preferences. philosophers, Other suchphilosophers, as Leibniz, such believed as Leibniz, that therebelieved is a that norm there behind is a norm every behind aesthetic every feeling, aesthetic which feeling, we simplywhich dowe notsimply know do yet not how know to measureyet how to [65 measure]. [65]. TheThe most most common common stochastic stochastic attribute attribute in naturalin natural processes processes is the is long-termthe long-term persistence persis- behaviortence behavior or HK-behavior, or HK-behavior, which which is identified is identified in global-scale in global-scale analyses analyses including including billions bil- oflions records, of records, and in and over-centennial in over-centennial timeseries time ofseries the mostof the important most important hydrometeorological hydrometeoro- processeslogical processes (i.e., temperature, (i.e., temperature, humidity, humidity, wind, solar wind, radiation, solar radiation, river discharge, river discharge, atmospheric at- pressure,mospheric and pressure, precipitation), and precipitation), [58,59], and [58,59], expressed and throughexpressed the through climacogram the climacogram for large scales as shown in Equation (2). Remarkably, the shape of the dependence structure, as for large scales as shown in Equation (2). Remarkably, the shape of the dependence struc- visualized through the climacogram, exhibits similarities among natural processes, having ture, as visualized through the climacogram, exhibits similarities among natural pro- a Markov-type behavior at small scales and a long-term persistence behavior (i.e., a power- cesses, having a Markov-type behavior at small scales and a long-term persistence2d behav-(1−H) lawior function(i.e., a power-law of scale) at function large scales, of scale) described at large by scales, the expression describedγ (byk) the= λ expression(1 + k/q) 𝛾(𝑘) =, where𝜆(1q is 𝑘/𝑞) now( a scale-parameter), where q is now indicative a scale-parameter of the transition indicative point of between the transition the short-term point be- (roughness)tween the short-term and long-term (roughness) (persistence) and long-term behaviours (persistence) in the climacogram behaviours (Figure in the4). clima- The lattercogram behavior (Figure can 4). The be quantified latter behavior by the can Hurst be quantified parameter, by whichthe Hurst is defined parameter, as the which semi is H s s log-logdefined slope as the plus semi one, log-log i.e., slope= plus/2+1, one, where i.e., H =is s/2+1, the slopewhere in s is logarithmic the slope in axes logarithmic of the climacogram at large scales (as shown in Equation (2)). axes of the climacogram at large scales (as shown in Equation (2)).

FigureFigure 4. 4.Illustration Illustration of of the the Hurst-Kolmogorov Hurst-Kolmogorov dynamics dynamics model model visualized visualized through through the the climacogram, climacogram, γ, vs. scale, k, and expanding from micro (GHK behavior) to macro (HK behavior) scales. γ, vs. scale, k, and expanding from micro (GHK behavior) to macro (HK behavior) scales.

AlthoughAlthough artart isis aa mixmix ofof determinism determinism (e.g.,(e.g., certaincertain rulesrules havehave to to be be followed) followed) and and stochasticitystochasticity (e.g.,(e.g., creativity creativity and and inspiration), inspiration), we we treat treat artistic artistic works works as as a a natural natural process process andand testtest whetherwhether art paintings paintings share share a asimilar similar structure structure of ofdependence dependence as do as dovarious various nat- naturalural processes. processes. In art paintings portraying human faces, the face is a focus point which condenses the feeling of the painting. In order to evaluate how nature depicts in 2D we first examine

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Heritage 2021, 4 FOR PEER REVIEW 6 the climacogram of photographs of human faces [66–68], shown in Figure 5. Then, we analyzeIn portraits art paintings from portraying various artistic human faces,genres the dating face is from a focus medieval point which to recent condenses times (data from:the feeling [69–77]), of the i.e., painting. of Byzantine In order art, to evaluateRenaissance how natureand Baroque, depicts in and 2D we20th ﬁrst century examine modern thethe climacogram of of photographs of human faces [[66–68],66–68], shown inin FigureFigure5 5.. Then,Then, wewe art. We also analyze the self-portraits of Rembrandt Harmenszoon van Rijn [78] and Pablo analyze portraits from various artistic genres dating from medieval to recent times (data Picassofrom:from: [69–77]), [ 69[79]–77 made]), i.e., in of different Byzantine periods art, RenaissanceRena ofissance their andlives. Baroque, Each 2D and photo 20th centuryand painting modern is digit- izedart. Webased We also also on analyze analyzea grayscale the the self-portraits self-portraitscolor intensity of Rembrandt of Rembrandtand the Harmenszoon climacogram Harmenszoon va isn Rijnestimated van [78] Rijn and [78 based Pablo] and on the geometricPicassoPablo Picasso [79] scale made [79 rather] in made different than in different the periods average periods of their one of lives. their(Figures Each lives. 5–16).2D Each photo 2D The photoand Hurst painting and parameter painting is digit- is is esti- matedizeddigitized based for basedeach on a image ongrayscale a grayscale set colorand colorshown intensity intensity in and Table andthe 1. climacogram the climacogram is estimated is estimated based based on the on geometricthe geometric scale scale rather rather than than the average the average one one(Figures (Figures 5–16).5– 16The). TheHurst Hurst parameter parameter is esti- is matedestimated for each for each image image set and set and shown shown in Table in Table 1. 1.

FigureFigure 5. 5. Images ImagesImages of of of portraitportrait portrait photography. photography. PhotosPhot Photosos courtesy courtesy of Kostasof Kostas Mountrichas. Mountrichas.

(a) (b)

Figure 6. (a) Climacograms(a) and (b) standardized climacograms of the images in (Figureb) 5 (Hurst parameters ranging from 0.76 for the lower left image to 0.91 for the upper right, averaging to 0.87). FigureFigure 6. (a 6.) Climacograms(a) Climacograms and and (b (b) )standardized standardized climacogramsclimacograms of of the the images images in Figurein Figure5 (Hurst 5 (Hurst parameters parameters ranging ranging from from 0.76 0.76for the for thelower lower left left image image to to 0.91 0.91 fo forr the the upper right,right, averaging averaging to to 0.87). 0.87).

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Figure 7. Images of portrait paintingspaintings fromfrom thethe RenaissanceRenaissance andand BaroqueBaroque periods.periods. FigureFigure 7. Images7. Images of ofportrait portrait pain paintingstings from from the the Renaissance Renaissance and and Baroque Baroque periods. periods.

(a) (b) (a)( a) (b)( b) Figure 8. (a) Climacograms and (b) standardized climacograms of the paintings in Figure 7 (Hurst parameters ranging FigureFigureFigure 8.8. ((8.aa) )( ClimacogramsaClimacograms) Climacograms and and and (b ()b standardized)( bstandardized) standardized climacograms climacograms climacograms of theof ofthe paintings the paintings paintings in Figure in inFigure 7Figure (Hurst 7 (Hurst7 parameters(Hurst parameters parameters ranging ranging fromranging from 0.90 for the lower left image to 0.94 for the lower right, averaging to 0.92). 0.90fromfrom for 0.90 the0.90 for lower for the the lower left lower image left left image to image 0.94 to for to0.94 the0.94 for lower for the the lower right, lower right, averaging right, averaging averaging to 0.92). to to0.92). 0.92).

Figure 9. Self-portraits of Rembrandt in chronological order FigureFigure 9. 9. Self-portraits Self-portraits of ofRembrandt RembrandtFigure 9. inSelf-portraits inchronological chronological of order Rembrandt order in chronological order.

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(a) (b) (a()a ) (b(b) ) Figure 10. (a) Climacograms and (b) standardized climacograms of the paintings in Figure 9 (Hurst parameters ranging FigureFigure 10. 10. (a ()a Climacograms) Climacograms and and (b ()b standardized) standardized climacograms climacograms of of the the paintingspainting paintings sin in Figure Figure 99 (Hurst 9(Hurst (Hurst parametersparameters parameters rangingranging ranging from 0.87 for the lower right image to 0.95 for the upper left, averaging to 0.92). fromfrom 0.87 0.87 for for the the lower lower right right image image to to 0.950.95 0.95 forfor for thethe the upperupper upper left,left, left, averagingaveraging averaging toto to 0.92).0.92). 0.92).

Figure 11. Images of portrait paintings of 20th century art. FigureFigure 11. 11. ImagesImages Images of of of portrait portrait paintings paintings of of 20th 20th century century art. art.

(a) (b) (a()a ) (b(b) ) Figure 12. (a) Climacograms and (b) standardized climacograms of the paintings in Figure 11 (Hurst parameters ranging FigureFigure 12. 12. (a ()a Climacograms) Climacograms and and (b ()b standardized) standardized climacograms climacograms of of the the painting paintings sin in Figure Figure 11 11 (Hurst (Hurst parameters parameters ranging ranging Figurefrom 12. 0.88(a )for Climacograms the lower left and image (b) standardizedto 0.93 for the climacogramsupper left, averaging of the paintings to 0.90). in Figure 11 (Hurst parameters ranging fromfrom 0.88 0.88 for for the the lower lower left left image image to to 0.93 0.93 for for the the upper upper left, left, averaging averaging to to 0.90). 0.90). from 0.88 for the lower left image to 0.93 for the upper left, averaging to 0.90).

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FigureFigure 13. 13. Self-portraits Self-portraits of of Pablo PabloFigure Picasso Picasso 13. Self-portraits in in chronological chronological of Pablo order order Picasso in chronological order. Figure 13. Self-portraits of Pablo Picasso in chronological order

(a()a ) (b()b ) (a) (b) FigureFigure 14. 14. (a ()a Climacograms) Climacograms and and (b ()b standardized) standardized climacograms climacograms of of the the painting paintings ins in Figure Figure 13 13 (Hurst (Hurst parameters parameters ranging ranging Figure 14. (a) Climacograms and (b) standardized climacogramsclimacograms of the paintingspaintings in Figure 1313 (Hurst(Hurst parametersparameters rangingranging fromfrom 0.73 0.73 lower lower second second from from the the left left image image to to 0.91 0.91 lower lower in in the the middle, middle, averaging averaging to to 0.83). 0.83). fromfrom 0.730.73 lowerlower secondsecond fromfrom thethe leftleft imageimage toto 0.910.91 lowerlower inin thethe middle,middle, averagingaveraging toto 0.83).0.83).

FigureFigure 15. 15. Images Images of of portrait portrait paintings paintings of of Byzantine Byzantine art, art, fresco. fresco. Figure 15. Images of portrait paintings of Byzantine art,art, fresco.fresco.

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(a) (b)

Figure 16. (a) Climacograms and (b) standardized climacograms of the paintings in Figure 15 (Hurst parameters ranging Figure 16. (a) Climacograms and (b) standardized climacograms(b) of the paintings in Figure 15 (Hurst parameters ranging from 0.72 for the lower left image to 0.86 for the upper right, averaging to 0.79). from 0.72 for the lower left image to 0.86 for the upper right, averaging to 0.79). Figure 16. (a) Climacograms and (b) standardized climacograms of the paintings in Figure 15 (Hurst parameters ranging from 0.72 for the lower left image to 0.86 for theFor upperFor each eachright, painting, painting, averaging we weto calculated0.79). calculated thethe coefficientcoefficient of variation variation of of the the original original image image at at scalescale 1, which,1, which, by by definition, definition, is is the the standard standard deviation deviation of of brightness brightness divided divided by by the the average. aver- For each painting,age. We wethen calculated calculated the the coeffici averageent of of all variation the images of theof each original set. Aimage high atvalue of the coef- We then calculated the average of all the images of each set. A high value of the coefficient scale 1, which, byficient definition, of variation is the shows standard that, deviation in each set of therebrightness is a broad divided range by of the brightness aver- in paintings of variation shows that, in each set there is a broad range of brightness in paintings for the age. We then calculatedfor the same the average average of brightness. all the images The of average each set. co Aefficient high value of variation of the coef- of each set is pre- same average brightness. The average coefficient of variation of each set is presented in ficient of variationsented shows in Table that, in1. each set there is a broad range of brightness in paintings for the sameTable average1. brightness. The average coefficient of variation of each set is pre- Table sented1. Average in Table Hurst 1.parameter H and coefficient of variation of the climacograms g(k) of the exanimated sets. Table 1. Average Hurst parameter H and coefficient of variation of the climacograms g(k) of the exanimated sets. Portrait Photog- Renaissance and Self-Portraits of Portrait Paintings of Self-Portraits of Byzantine Art, Table 1. Average Hurst parameter H and coefficient of variation of the climacograms g(kPortrait) of the exanimated sets. Portraitraphy BaroqueRenaissance Periods and Self-PortraitsRembrandt of 20th Century Art Self-PortraitsPablo Picasso of ByzantineFresco Art, Paintings of 20th HurstPortrait parameter Photog- PhotographyRenaissance 0.87 and BaroqueSelf-Portraits 0.92 Periods of PortraitRembrandt0.92 Paintings of Self-Portraits0.90 of PabloByzantine0.83 Picasso Art, 0.79Fresco Century Art Coefficientraphy of Baroque Periods Rembrandt 20th Century Art Pablo Picasso Fresco Hurst parameter 0.870.726 0.609 0.92 0.647 0.92 0.524 0.90 0.445 0.83 0.379 0.79 Hurst parameterCoefficient variation 0.87 of 0.92 0.92 0.90 0.83 0.79 0.726 0.609 0.647 0.524 0.445 0.379 Coefficient of variation 0.726 0.609 0.647 0.524 0.445 0.379 variation In order to evaluate the different sets, averages of climacograms thereof are plotted (Figure 17a). To quantify the range of fluctuations of the climacograms, for each scale k we In order to evaluate the different sets, averages of climacograms thereof are plotted In order tocalculated evaluate thethe differentcoefficient sets, of avvariatioeragesn ofof climacogramsall standardized thereof climacograms, are plotted g (k). Figure 17b (Figure 17a). To quantify the range of fluctuations of the climacograms, for each scale k we (Figure 17a). Toshows quantify the the graphs range of of g (fluctuk) vs.ations k for allof examinedthe climacograms, cases, indicating for each scalethe range k we of fluctuations calculated the coefficient of variation of all standardized climacograms, g(k). Figure 17b calculated the coefficientfrom the average of variatio climacogramn of all standardized of each set. climacograms, g(k). Figure 17b shows the graphsshows of theg(k) graphs vs. k for of allg( kexamined) vs. k for cases, all examined indicating cases, the range indicating of fluctuations the range of fluctuations from the averagefrom climacogram the average of climacogram each set. of each set.

(a) (b)

Figure 17. (a) Average of standardized climacograms of different sets; (b) Coefficient of variation (a) (b) of the climacograms of the different image sets. Figure 17. (a) AverageFigure 17. of (a standardized) Average of standardized climacograms climacograms of different sets; of different (b) Coefﬁcient sets; (b) of Coefficient variation of variation the climacograms of the of the climacograms of the different image sets. different image sets.

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Results show an interesting insight. Renaissance/Baroque and Rembrandt’s self- portraits which follow the most naturalistic aesthetic approach, have a strong dependence structure similar to that of photos and of various hydrometeorological processes (estimated H ≈ 0.9). A strong dependence structure is also observed in the portraits of 20th century (estimated H ≈ 0.9) and Picasso’s self-portraits (estimated H ≈ 0.85), yet with a bigger range of fluctuations, indicative of the artistically freer and nonrepresentational approach of modern art. The most interesting insight, however, is that the portraits belonging to the Byzantine art period show a weaker dependence structure (estimated H ≈ 0.8), which is closer to white noise. This attribute seems consistent with the theology of Christian’s Orthodox dogma dictating a specific type of representation of saints and religious figures.

4. Discussion Since prehistoric art creations, art was related to, and inspired by natural forms. Aristotle said that “art takes nature as its model”, and “ ... not only imitates nature, but also completes its deficiencies” [80]. Furthermore, Plato connected arts with ideas, the soul and the idealistic expression [81,82]. Around the 20th century, artists embraced an alternative way of expression by focusing more on modern society’s habits and deviating from naturalistic themes and related representation. Even more, artists then found Nature’s complexity obsolete. Paul Klee noted that “Nature is garrulous to the point of confusion, let the artist be truly taciturn.” [83]. One may argue however that the uncertainty within ourselves that drives all forms of art is the same inherent uncertainty of nature that was used subconsciously as inspiration. If this is true, then we should be able to find stochastic similarities between art projects and natural processes imprinted in the estimated variability vs. scale. The results of this study provide grounds to this hypothesis as patterns can be observed in the dependence structure in photographs and portraits of Renaissance/Baroque which represent the most naturalistic approach, having similar moderate structure (photo, average H ≈ 0.87; Renaissance/Baroque, average H ≈ 0.92). A similar Hurst parameter was estimated by Hurst in his pioneering work on the stage of the river Nile [84–86] and, additionally, in global-scale analyses of the long annual records of key hydrometeorological processes such as temperature and wind speed as well as of small-scale processes recorded in the laboratory such as grid-turbulence and turbulent jets [59]. An overall Hurst parameter (H) equal to 0.9 is observed in all hydrometeorological processes (after adjusting for bias; a task that is often neglected in stochastic analysis [87–89]). Patterns can also be observed in the art portraits of 20th century yet with a bigger range of fluctuation as modern artists wanted to express their ‘own’ view of nature away from what they perceived as “nature’s monotony” [90,91]. Interestingly, though, although the aesthetic means employed are indeed markedly different from older naturalistic paintings, the stochastic variability of modern art portraits is still on average close to the one found both in the most naturalistic paintings and in natural processes itself, revealing an overall Heritage 2021, 4 FOR PEER REVIEW 16 Hurst parameter (H) equal to 0.9. Another related validation is the evaluation of self-portraits during the lifetime of Rembrandt’s and Picasso. Rembrandt’s works exhibit stochastic similarities and a strong dependence91. Sargentis, structure G.-F. Aesthetic (average ElementH = 0.92) in Wat whereaser, Hydraulic Picasso’s Works self-portraits and Dams. showMaster’s a widerThesis, National Technical University of range ofAthens, fluctuation Athens, (Figure Greece, 17 1998;b). doi:10.5281/zenodo.3770806. 92. SantSant Gregory Gregory ofof NyssaNyssa (Άγιοςγιoς ΓρηγόριοςΓρηγóριoς ΝύσσηςNυσσης´ ), )Α (diedbout Christian c. 395 AD) perfection describes (Περί how Χριστιανικής Τελειότητος), Tertios, the creationKaterini. of 1980. Byzantine Available art online: is connected https://en.wikipedia.or with the spiritualityg/wiki/Gregory_of_Nyssa of the artist “man (accessed is the on 4 February 2021). painter93. Gerontos of his life, Paisiou creator Agioritou of paintings (Γέροντος is artists’Παϊσίουmotive”« Αγιορείτουτης), For ιδιας´ prayerεκαστ´ (Περίoς ζω Προσευχήςης´ εστι´ ), Ieron Isichastirio Evangelistis ζωγραϕÍwannisoς, τεχν o ιτης´ Theologos δε της [(ζωγραϕικΙερόν Ησυχαστήριοης´ ] δηµιo υργ«Ευαγγελιστήςιας´ ταυτης´ εστΙωάννηςιν´ η πρ οo αΘεολόγοςιρεσις´ »[»92), ] Souroti. 2012. Available online: Byzantinehttps://www.politeianet.gr/ekdotis/ieron-is art, is a religious art serving the purposesuchastirion-euaggelistis-ioannis-theolo of the Christian’s Orthodoxgos-2852 dogma. (accessed on 4 February 2021). 94. Holy Bible, New Testament, Mathew (Κατά Ματθαίον) 22,37; Mark (Κατά Μάρκον) 12,30; Luke (Κατά Λουκά) 10,27, Available An interesting note is that Byzantine art portraits correspond to a weaker dependence online: http://www.apostoliki-diakonia.gr/bible/bible.asp?contents=new_testament/contents.asp&main= (accessed on 4 structure (0.7 < H < 0.85) than that of the other genres, with wide range of fluctuation at February 2021). small95. scalesPhilocalia and ( smallΦιλοκαλία range), ofedt. fluctuation Ioannou Mavrogordatou at large scales, ( closerΙωάννου to Μαυρογορδάτου white noise. This), Venice might 1782, Greek translation. Available be explainedonline: consideringhttps://greekdownloads.wordpress.com/ that these paintings depictφιλοκαλία religious/ figures(accessed being on 2 inJanuary divine 2021). states

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small scales and smallHeritage range 2021 of, 4 FORfluctuation PEER REVIEW at large scales, closer to white noise. This might 16

be explained considering that these paintings depict religious figures being in divine states of mine,of often mine, praying often praying [93,94]. [The93,94 mean]. Theing meaning of pray ofin prayOrthodox in Orthodox dogma dogmais described is described in one in one of the mostof the 91.important most Sargentis, important theological G.-F. theological Aesthetic books Element of books Orthodox, ofin Orthodox,Wat Philokaliaer, Hydraulic Philokalia (Φιλοκαλία Works (Φιλ ando) καλ[95],Dams.ια´ )[ Master’s95], which Thesis, National Technical University of which containscontains the writings theAthens, writings Athens,of the of Greece,Eastern the Eastern 1998; Fathers Fathersdoi:10.5281/zenodo.3770806. from from the the fourth fourth to to the the fifteenth fifteenth centuries. In centuries. In thisthis work, work,92. 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Iwannis«”Μην«M ηνεπιδιώκεις o επιδι Theologosωκεις´ να να δεχτείς(Ιερόν δεχτε ις´Ησυχαστήριο την την ώραωρα´ της της «προσευχήςΕυαγγελιστής πρoσευχης´ με µε Ιωάννηςκανένα κανενα´ τρο óπΘεολόγοςo µoρϕη´»), Souroti. 2012. Available online: τρόπο μορφή η´ή σχσχήμαηµα´ ».».https://www.politeianet.gr/ekdotis/ieron-is The identified identified stochastic struct structureure corresponds correspondsuchastirion-euaggelistis-ioannis-theolo to to this this theology theology as thisgos-2852 ideal (accessed on 4 February 2021). 94. Holy Bible, New Testament, Mathew (Κατά Ματθαίον) 22,37; Mark (Κατά Μάρκον) 12,30; Luke (Κατά Λουκά) 10,27, Available as this ideal of religiousreligious figuresfigures experiencingexperiencing aa divine divine humbling humbling experience experience is is better better delivered with online: http://www.apostoliki-diakonia.gr/bible/bible.asp?contents=new_testament/contents.asp&main= (accessed on 4 weaker dependence structures. delivered with weaker Februarydependence 2021). structures. 95. Philocalia (Φιλοκαλία), edt. Ioannou Mavrogordatou (Ιωάννου Μαυρογορδάτου), Venice 1782, Greek translation. Available 5. Conclusions 5. Conclusions online: https://greekdownloads.wordpress.com/φιλοκαλία/ (accessed on 2 January 2021). In recent years,In artificial recent years, intelligence artificial processes intelligence and processes mathematical and mathematicalcomputational computational tools tools have been usedhave to develop been used methods to develop for classi methodsfication for and classification evaluation and of evaluationaesthetics. Scale- of aesthetics. Scale- variant methodsvariant similar methods to the similarpresented to thestochast presentedic analysis stochastic have analysisalso been have used, also but been they used, but they usually includeusually very includecomplex very processes complex and processes algorithms and algorithms not easily not understood easily understood by non- by non-experts. experts. With the presented 2D-C methodology quantifying the brightness’s variability over scales of pixels, we can observe stochastic patterns among different groups of art paintings. With the presented 2D-C methodology quantifying the brightness’s variability over In this work, we have focused on paintings and photos depicting the human face. We identi- scales of pixels, we can observe stochastic patterns among different groups of art fied dependence structures similar to that of natural processes, with an average H ≈ 0.9, in paintings. In this work, we have focused on paintings and photos depicting the human photographs of faces, as well as in portraits belonging to the Renaissance/Baroque period face. We identified dependence structures similar to that of natural processes, with an and Rembrandt’s self-portraits. Modern art portraits and Picasso’s portraits also have an average H≈0.9, in photographs of faces, as well as in portraits belonging to the average H ≈ 0.9 implying underlining strong dependence structures, but showing a wider Renaissance/Baroque period and Rembrandt’s self-portraits. Modern art portraits and range of fluctuations indicative of the pursuit of freer artistic expression during these years. Picasso’s portraits also have an average H≈0.9 implying underlining strong dependence On the contrary, the stricter dogma-inspired Byzantine figures exhibit an overall weaker structures, but showing a wider range of fluctuations indicative of the pursuit of freer dependence structure (average H ≈ 0.8) and the smaller coefficient of variation (0.379) artistic expression during these years. On the contrary, the stricter dogma-inspired among the other sets. Byzantine figures exhibitThese findings an overall suggest weaker that dependence the presented structure methodology (average can H capture≈ 0.8) and the interrelation the smaller coefficientbetween of the variation stochastic (0.379) expression among ofthe the other art paintingssets. and the philosophical issues they These findingswant tosuggest describe that (desideratum the presented). This methodology aspect is encouraging can capture for the the interrelation use of stochastics in the between the stochasticanalysis of expression art paintings of the and art their paintings relation and to wider the philosophical philosophical issues and cultural they issues. want to describe (desideratum). This aspect is encouraging for the use of stochastics in the analysis of artAuthor paintings Contributions: and their relationConceptualization, to wider philosophical G.-F.S.; methodology, and cultural G.-F.S. issues. and P.D.; software, P.D.; validation, G.-F.S.; formal analysis, G.-F.S.; investigation, G.-F.S.; resources, G.-F.S.; data curation, Author Contributions:G.-F.S.; writing—original Conceptualization, draft G.-F.S.; preparation, methodology, G.-F.S., G.-F.S. and P.D.; and writing—review P.D.; software, andP.D.; editing, T.I. and validation, G.-F.S.;D.K.; formal visualization, analysis, G.-F.S.; G.-F.S.; supervision, investigation, D.K.; G.-F.S.; project resources, administration, G.-F.S.; data G.-F.S. curation, All authors have read G.-F.S.; writing—originaland agreed draft to the preparation, published versionG.-F.S., ofand the P.D.; manuscript. writing—review and editing, T.I. and D.K.; visualization, G.-F.S.; supervision, D.K.; project administration, G.-F.S. All authors have read and agreed to theFunding: publishedThis version research of receivedthe manuscript. no external funding. Funding: This researchInstitutional received Review no external Board Statement: funding Not applicable. Institutional ReviewInformed Board Consent Statement: Statement: Not applicableNot applicable. Informed ConsentAcknowledgments: Statement: Not applicableWe thank the editors of Heritage-MDPI for the invitation of this paper and the processing of the paper, as well as one anonymous reviewer for comments that helped improve and Acknowledgments:enrich We the thank manuscript the editors and aof second Heritage-MDPI anonymous for reviewerthe invitation for the of enthusiasticthis paper and review. the processing of the paper, as well as one anonymous reviewer for comments that helped improve and enrich the manuscriptConflicts and of a Interest: second anonymousThe authors reviewer declare no for conflict the enthusiastic of interest. review.

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