Aesthetical Issues of Leonardo Da Vinci’s and Pablo Picasso’s Paintings with Stochastic Evaluation

: A physical process is characterized as complex when it is di ﬃ cult to analyze or explain in a simple way. The complexity within an art painting is expected to be high, possibly comparable to that of nature. Therefore, constructions of artists (e.g., paintings, music, literature, etc.) are expected to be also of high complexity since they are produced by numerous human (e.g., logic, instinct, emotions, etc.) and non-human (e.g., quality of paints, paper, tools, etc.) processes interacting with each other in a complex manner. The result of the interaction among various processes is not a white-noise behavior, but one where clusters of high or low values of quantiﬁed attributes appear in a non-predictive manner, thus highly increasing the uncertainty and the variability. In this work, we analyze stochastic patterns in terms of the dependence structure of art paintings of Da Vinci and Picasso with a stochastic 2D tool and investigate the similarities or di ﬀ erences among the artworks.


Introduction
The meaning of beauty is linked to the evolution of human civilization, and the analysis of the connection between the observer and the beauty in art and nature has always been of high interest in both philosophy and science.Even though this analysis has mostly been regarded as part of social studies and humanities, other scientists have also been involved mostly through the philosophy of science [1].Analyses through mathematics are generally focused on applying mathematical tools in trying to describe aesthetics.In most of these analyses, the question at hand is whether what is pleasing to the eye can be explained through analogies.
The word canon, or set of proportions, comes from Greek (καν ών) and means a straight rod (measuring line) and, metaphorically, a rule or standard.Canons have changed through history according to artistic needs, taste, and sense of beauty.Pythagoras and Euclid were the first philosophers known to have searched for a common rule (canon) existing in shapes that are perceived as beautiful.Euclid's Elements (c.300 BC), notably, includes the first known definition of the golden ratio.
Heritage 2020, 3, 283-305; doi:10.3390/heritage3020017www.mdpi.com/journal/heritagesupports that instead of regarding the aesthetic quality as an inherent property of a physical object, the distinction of mind and nature has allowed humans to incorporate their own subjective emotional, social and cognitive background in determining their aesthetic preferences.Overall, it is evident that many artists knew and applied math and geometry in their artwork (Figure 2) and many philosophers tried to connect math and arts.[9][10][11]   New researchers are trying to create art using computational intelligence technologies.In particular, evolutionary computation has been able to generate visual art and music [12][13][14][15].Evaluation of the items generated by evolutionary algorithms is a key issue in computational creativity.Experimental results show that artificial intelligence technologies can generate satisfactory paintings by using some kind of aesthetic evaluation [6][7][8][9][10][11][12][13][14][15][16][17].
Mathematics has been part of the art, but can it be used in evaluating the art?The evaluation of art paintings [18][19][20][21][22][23][24][25][26] is a difficult and subjective process and in order to create an objective process, image evaluation and classification of art paintings with mathematical algorithms is the subject of a plethora of related publications .
The mathematical field of Stochastics has been introduced as an alternative to deterministic approaches, to model the so-called random (i.e., complex, unexplained or unpredictable) fluctuations observed in non-linear geophysical processes [63,64].Stochastics helps develop a unified perception for natural phenomena and expel dichotomies like random vs. deterministic.Particularly, there is no such thing as a 'virus of randomness' that infects some phenomena to make The opinions of later philosophers on this pursuit of mathematicians for the analysis of aesthetics were more varied.Leibniz, for example, believed that there is a norm behind every aesthetic feeling which we simply do not know how to measure [8].On the contrary, Descartes supports that instead of regarding the aesthetic quality as an inherent property of a physical object, the distinction of mind and nature has allowed humans to incorporate their own subjective emotional, social and cognitive background in determining their aesthetic preferences.Overall, it is evident that many artists knew and applied math and geometry in their artwork (Figure 2) and many philosophers tried to connect math and arts.[9][10][11] Heritage 2020, 3 FOR PEER REVIEW 2 supports that instead of regarding the aesthetic quality as an inherent property of a physical object, the distinction of mind and nature has allowed humans to incorporate their own subjective emotional, social and cognitive background in determining their aesthetic preferences.Overall, it is evident that many artists knew and applied math and geometry in their artwork (Figure 2) and many philosophers tried to connect math and arts.[9][10][11]   New researchers are trying to create art using computational intelligence technologies.In particular, evolutionary computation has been able to generate visual art and music [12][13][14][15].Evaluation of the items generated by evolutionary algorithms is a key issue in computational creativity.Experimental results show that artificial intelligence technologies can generate satisfactory paintings by using some kind of aesthetic evaluation [6][7][8][9][10][11][12][13][14][15][16][17].
Mathematics has been part of the art, but can it be used in evaluating the art?The evaluation of art paintings [18][19][20][21][22][23][24][25][26] is a difficult and subjective process and in order to create an objective process, image evaluation and classification of art paintings with mathematical algorithms is the subject of a plethora of related publications .
The mathematical field of Stochastics has been introduced as an alternative to deterministic approaches, to model the so-called random (i.e., complex, unexplained or unpredictable) fluctuations observed in non-linear geophysical processes [63,64].Stochastics helps develop a unified perception for natural phenomena and expel dichotomies like random vs. deterministic.Particularly, there is no such thing as a 'virus of randomness' that infects some phenomena to make New researchers are trying to create art using computational intelligence technologies.In particular, evolutionary computation has been able to generate visual art and music [12][13][14][15].Evaluation of the items generated by evolutionary algorithms is a key issue in computational creativity.Experimental results show that artificial intelligence technologies can generate satisfactory paintings by using some kind of aesthetic evaluation [6][7][8][9][10][11][12][13][14][15][16][17].
Mathematics has been part of the art, but can it be used in evaluating the art?The evaluation of art paintings [18][19][20][21][22][23][24][25][26] is a difficult and subjective process and in order to create an objective process, image evaluation and classification of art paintings with mathematical algorithms is the subject of a plethora of related publications .
The mathematical field of Stochastics has been introduced as an alternative to deterministic approaches, to model the so-called random (i.e., complex, unexplained or unpredictable) fluctuations observed in non-linear geophysical processes [63,64].Stochastics helps develop a unified perception for natural phenomena and expel dichotomies like random vs. deterministic.Particularly, there is no such thing as a 'virus of randomness' that infects some phenomena to make them random, leaving other phenomena uninfected.It seems, rather, that both randomness and predictability coexist and are intrinsic to natural systems which can be deterministic and random at the same time, depending on the prediction horizon and the time scale [65].On this basis, the uncertainty in art, as in other processes of nature, can be both aleatory (alea = dice) and epistemic (as, in principle, we could know perfectly the underlying mechanisms).Therefore, dichotomies such as 'deterministic vs. random' and 'aleatory vs. epistemic' may be false ones and may lead to paradoxes.By applying the concept of stochastic analysis, we can identify the observed unpredictable fluctuations of systems under investigation with the variability of a devised stochastic process.

Stochastic Analysis in 2D
In this research paper, a stochastic computational tool called 2D-C [66] is used to analyze art paintings in order to: (a) examine stochastic similarities and differences among artworks; (b) introduce a methodology for evaluating restoration aspects with stochastic tools; (c) identify the stochastic meaning of specific areas of the artworks; and (d) examine the originality of art paintings (or parts of art paintings) with stochastic tools.
2D-C measures the degree of variability (change in variability vs. scale) in images using stochastic analysis.Apparently, beauty is not easy to quantify with stochastic measures, but nevertheless the examination of artworks through a stochastic analysis offers interesting insights into aspects of the art paintings.The stochastic analyses of the examined artworks are presented using climacograms based on the 2D-C analysis of image pixels.
Image processing typically involves filtering or enhancing an image using various types of functions, in addition to other techniques, to extract information from the images [67].Image segmentation is one of the basic problems in image analysis.The importance and utility of image segmentation has resulted in extensive research and numerous proposed approaches based on intensity, color, texture, etc., and both automatic and interactive [68].A variety of techniques have been proposed for the quantitative evaluation of segmentation methods [69][70][71][72][73][74][75][76].
This analysis for image processing is based on a stochastic tool called a climacogram.The term climacogram [77,78] comes from the Greek word climax (meaning scale).It is defined as the (plot of) variance of the averaged process (assuming it is stationary) versus averaging time scale k, and is denoted as γ(k).The climacogram is useful for detecting the long-term change (or else dependence, persistence or 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 [79].
In order to obtain data for the evaluation of art paintings, each image of art painting is digitized in 2D based on a grayscale color intensity (thus this climacogram studies the brightness of an image), and the climacogram is calculated based on the geometric scales of adjacent pixels.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: Heritage 2020, 3 286 where the 'ˆ' over γ denotes estimate, κ is the the dimensionless spatial scale, x x ξ,ψ is the sample average of the space-averaged process at scale κ, and x i,j /n 2 is the sample average.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 brightness among the different elements.In this manner, we may quantify the uncertainty of the brightness intensities at each scale by measuring their variability.
In order to characterize stochastic analysis of the data, an important property is the Hurst-Kolmogorov (HK) behavior (usually known in hydrometeorological processes as Long Term Persistence, or LTP), which can be summarized by the Hurst parameter as follows.The isotropic HK process with an arbitrary marginal distribution (e.g., for the Gaussian one, this results in the well-known fractional-Gaussian-noise, described by Mandelbrot and van Ness [80]), i.e., the power-law decay of variance as a function of scale, is defined for a 1D or 2D process as: where λ is the variance at scale k = κ∆, 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 of clustering behavior due to the non-uniform heterogeneity of the brightness of the painting, the high variability of the brightness persists even in large scales.This clustering effect may greatly increase the diversity between the brightness in each pixel of the image, a phenomenon also observed in hydrometeorological processes (such as temperature, precipitation, wind, etc.), natural landscapes [66] and music [81].Therefore, it is interesting to observe the degrees of uncertainty and variability in arts.
The algorithm that generates the climacogram in 2D was developed in MATLAB for rectangular images [82].In particular, for the current analysis, the images are cropped to 400 × 400 pixels, 14.11 cm × 14.11 cm, in 72 dpi (dots per inch).

Illustration of Stochastic Analysis in 2D
The pixels analyzed are actually represented by numbers based on their grayscale color intensity (white = 1, black = 0).Figure 3 presents three images for benchmark image analysis: (a) white noise; (b) image with clustering; and (c) art painting.Figure 4 presents the steps of analysis and shows grouped pixels at scales k = 2, 4, 8, 16, 20, 25, 40, 50, 80, 100 and 200 used to calculate the climacogram.Figure 5 presents an example of how the data sets of Figure 3 translate into climacograms and standardized climacograms.The latter is defined as the ratio γ(k)/γ(1) as a function of scale k, and is the basic tool of the 2D-C evaluation process.
The presence of clustering is reflected in the climacogram, which shows a marked difference for the random white noise (Figure 5).Specifically, the variance of the clustered images is notably higher than that of the white noise at all scales, indicating a greater degree of variability of the process.Likewise, comparing the clustered image and the art painting, the latter has the most pronounced clustering behavior and a greater degree of variability.
Section 3 presents stochastic analysis (Figures 6-18) and Section 4 the discussion of the results.
climacograms and standardized climacograms.The latter is defined as the ratio ()/(1) as a function of scale k, and is the basic tool of the 2D-C evaluation process.
The presence of clustering is reflected in the climacogram, which shows a marked difference for the random white noise (Figure 5).Specifically, the variance of the clustered images is notably higher than that of the white noise at all scales, indicating a greater degree of variability of the process.Likewise, comparing the clustered image and the art painting, the latter has the most pronounced clustering behavior and a greater degree of variability.

Stochastic Evaluation of Da Vinci's and Picasso's Art Paintings
Leonardo da Vinci (1452-1519) and Pablo Picasso (1881-1973) were two great artists who lived in different periods and had different artworks styles.Both have greatly influenced arts [84][85][86], and therefore, here, we have selected these two to quantify the variability of their works, searching for stochastic similarities among their paintings.
For each painting, we calculated the coefficient of variation of the original image at scale 1, which, by definition, is the standard deviation of brightness divided by the average.We then calculated the average of all the paintings of each artist.A high value of the coefficient of variation shows that an artist uses a broad range of brightness in his paintings for the same average brightness.The average coefficient of variation of Da Vinci's artworks is equal to 0.7 and that of Picasso's artworks is equal to 0.45.
In order to see the range of fluctuations of climacograms, for each scale k we calculated the coefficient of variation of all standardized climacograms, g(k).As we have seen in Figure 6, the art paintings of Leonardo da Vinci exhibit stochastic similarity, and he has his own steady stochastic canon of expression, which is represented by a stochastic Golden Climacogram.In contrast, Picasso's artworks (in stochastic view) display a wide range of fluctuations (Figures 7-9).Figure 19 shows the graphs of g(k) vs. k for all examined cases.This range of fluctuations is larger as the artist growing older.
in different periods and had different artworks styles.Both have greatly influenced arts [84][85][86], and therefore, here, we have selected these two to quantify the variability of their works, searching for stochastic similarities among their paintings.
For each painting, we calculated the coefficient of variation of the original image at scale 1, which, by definition, is the standard deviation of brightness divided by the average.We then calculated the average of all the paintings of each artist.A high value of the coefficient of variation shows that an artist uses a broad range of brightness in his paintings for the same average brightness.The average coefficient of variation of Da Vinci's artworks is equal to 0.7 and that of Picasso's artworks is equal to 0.45.
In order to see the range of fluctuations of climacograms, for each scale k we calculated the coefficient of variation of all standardized climacograms, g(k).As we have seen in Figure 6, the art paintings of Leonardo da Vinci exhibit stochastic similarity, and he has his own steady stochastic canon of expression, which is represented by a stochastic Golden Climacogram.In contrast, Picasso's artworks (in stochastic view) display a wide range of fluctuations (Figures 7-9).Figure 19 shows the graphs of g(k) vs. k for all examined cases.This range of fluctuations is larger as the artist growing older.Analyzing Figure 19, we could say that Da Vinci's art paintings are more stable and Picasso's have more fluctuation.Picasso's artworks were more fuzzy as the artist was growing older, with bigger and bigger variance.

Stochastic Evaluation of the Restoration of Da Vinci's Last Supper
In 1495, Leonardo da Vinci began what would become one of history's most influential works of art-The Last Supper (Italian: Ultima Cena) [87].
The Last Supper is regarded as one of Leonardo da Vinci's masterpieces, [88] but twenty years after its completion, the painting began to chip and fade.Unlike traditional frescoes, which Renaissance masters painted on wet plaster walls, da Vinci experimented with tempura paint on a dry, sealed plaster.The experiment proved unsuccessful, however, because the paint did not adhere Analyzing Figure 19, we could say that Da Vinci's art paintings are more stable and Picasso's have more fluctuation.Picasso's artworks were more fuzzy as the artist was growing older, with bigger and bigger variance.

Stochastic Evaluation of the Restoration of Da Vinci's Last Supper
In 1495, Leonardo da Vinci began what would become one of history's most influential works of art-The Last Supper (Italian: Ultima Cena) [87].
The Last Supper is regarded as one of Leonardo da Vinci's masterpieces, [88] but twenty years after its completion, the painting began to chip and fade.Unlike traditional frescoes, which Renaissance masters painted on wet plaster walls, da Vinci experimented with tempura paint on a dry, sealed plaster.The experiment proved unsuccessful, however, because the paint did not adhere properly and began to flake away only a few decades after the work was finished.Da Vinci's masterpiece has been subject to numerous restoration attempts.Some of these took place in 1726, 1770, 1853, 1903, 1924, 1928 and 1978. [89].Critics maintain that only a fraction of the painting that exists today is the work of Leonardo da Vinci.[90][91][92].
Stochastic analysis of Da Vinci's Last Supper shows that the curve of the center area (curve (a), Figure 10d) before restoration is between the others, but after restoration is above the others (curve (d), Figure 11d).The coefficient of variation before restoration was 2.3 and after restoration became 2.6, while the average of Da Vinci's artwork is 3.4.We could thus say that after restoration Last Supper's Coefficient of Variation came closer to Da Vinci's artworks.All the curves are very close to Da Vinci's stochastic Golden Climacogram.Thus, it seems that Last Supper after restoration is, from the stochastic view, more intense in the center, which makes sense as obviously the artist would like to give emphasis to the center of the painting.So, the answer to the question "was the restoration of Last Supper successful?" seems to be positive.
Stochastic analysis in specific areas of Da Vinci's Annunciation (Figure 12b) (1472-1475) shows that the area of the angel (curve (a), Figure 12c,d) is clearly different than the area of Madonna (curve (b), Figure 12c,d).We could say that the artist might want to show a move from divine (angel) to humanity (Madonna).

Stochastic Evaluation of Da Vinci's Virgin of the Rocks
The Virgin of the Rocks (Italian: Vergine delle rocce; sometimes the Madonna of the Rocks) is the name of two paintings by Leonardo da Vinci, of the same subject, and of a composition which is identical except for several significant details.The version generally regarded as the prime version, the earlier of the two, is in the Louvre in Paris and is not restored (Figure 20a).The other is in the National Gallery in London, and it was restored between 2008 and 2010 (Figure 20b

5.
Modern mathematical tools and artificial intelligence provide many computational methods for evaluation and classification.Some of them use scale-variant methods similar to 2D-C, but with very complex processes and algorithms which could not be understood by a non-expert.Both paintings are ascribed to Leonardo da Vinci.Originally, they were thought to have been partially painted by Leonardo's assistants.A close look at the painting during the recent restoration has led the conservators from the National Gallery to conclude that the greater part of the work is by the hand of Leonardo, but debate continues [96,97].Parts of the painting, the flowers in particular, indicate the collaboration, and have led to speculation that the work is entirely by other hands, possibly Leonardo's assistant Giovanni Ambrogio de Predis, and perhaps Evangelista [95].
A general view, with stochastic analysis, of Da Vinci's Virgin of the Rocks (Figure 13) shows that, even though both of the art paintings are far enough from Da Vinci's stochastic forms, the Louvre version is closer to Da Vinci's stochastic Golden Climacogram (Figure 14).A closer view in the frames of Madonna (Figure 15) and Angel (Figure 17) shows that, in stochastic terms, both of them are very close to Da Vinci's stochastic Golden Climacogram, and the Louvre version is closer to Da Vinci's stochastic Golden Climacogram (Figures 16 and 18).Thus, we could make the conjecture that the wider drawing of the painting has been done by someone else, and these details by Da Vinci.

Conclusions
Modern mathematical tools and artificial intelligence provide many computational methods for evaluation and classification.Some of them use scale-variant methods similar to 2D-C, but with very complex processes and algorithms which could not be understood by a non-expert.
2D-C presents a clear and simple quantification tool based on the climacogram, a simple means of mathematic evaluation of art paintings, the results of which could be understood by a non-expert, and could provide insights into the aesthetical study of an art painting, as seen in the examples of this study.
With 2D-C, stochastic patterns can be observed in terms of the dependence structure among the different artists as well, with Da Vinci's (average H ≈ 0.84) and Picasso's (average H ≈ 0.85) artworks having a strong persistence structure.
Interestingly, similar Hurst parameters, between the range of 0.8 and 0.9, are estimated by Hurst in his pioneering work on the stage of the river Nile [98][99][100] 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) [63].Thus, a philosophical issue arises from the above analysis: does the inherent uncertainty in arts suggest some similarity with physical phenomena?
From the analysis above we found that, in contrast to Picasso, Da Vinci had a stochastically steady way of expression (Golden Climacogram).An interesting study for further analysis would be to find out if other artists have stochastic similarities among their artwork, and to research if there are correlations among them.

Figure 3 .
Figure 3. Benchmark of image analysis: (a) White noise; (b) Image with clustering; (c) Art painting.The lower row depicts the average brightness in the upper one.

Figure 3 .Figure 4 .Figure 5 .
Figure 3. Benchmark of image analysis: (a) White noise; (b) Image with clustering; (c) Art painting.The lower row depicts the average brightness in the upper one.Heritage 2020, 3 FOR PEER REVIEW 5

Figure 4 .Figure 4 .Figure 5 .
Figure 4. Example of stochastic analysis of 2D picture, in escalating spatial scales.Grouped pixels at different scales are used to calculate the climacogram: (a) White noise; (b) Image with clustering; (c) Art painting.

Figure 5 .
Figure 5. (a) Climacograms of the benchmark images; (b) Standardized climacograms of the benchmark images.

Figure 11 .
Figure 11.(a) Leonardo da Vinci's The Last Supper after restoration [83]; (b) Areas of analysis after restoration; (c) Climacograms; (d) Standardized climacograms and averages thereof.Notice that curve of (d) area (center) is on top of the curves of (e) and (f) areas.

Figure 11 .
Figure 11.(a) Leonardo da Vinci's The Last Supper after restoration [83]; (b) Areas of analysis after restoration; (c) Climacograms; (d) Standardized climacograms and averages thereof.Notice that curve of (d) area (center) is on top of the curves of (e) and (f) areas.

Figure 12 .
Figure 12.(a) Leonardo da Vinci's Annunciation; (b) Areas of analysis; (c) Climacograms;(d) standardized climacograms and averages thereof.Notice that curve of (b) area is higher than the curve of (a) area.

Figure 19 .
Figure 19.Coefficient of variation of standard deviation over the average of Picasso's and Da Vinci's artworks' climacograms.

Figure 19 .
Figure 19.Coefficient of variation of standard deviation over the average of Picasso's and Da Vinci's artworks' climacograms.
2D-C presents a clear and simple quantification tool based on the climacogram, a simple means of mathematic evaluation of art paintings, the results of which could be understood by a non-expert, and could provide insights into the aesthetical study of an art painting, as seen in the examples of this study.With 2D-C, stochastic patterns can be observed in terms of the dependence structure among the different artists as well, with Da Vinci's (average H ≈ 0.84) and Picasso's (average H ≈ 0.85) artworks having a strong persistence structure.Interestingly, similar Hurst parameters, between the range of 0.8 and 0.9, are estimated by Hurst in his pioneering work on the stage of the river Nile [98-100] and, additionally, in global-scale