#### 2.1. 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

nΔ ×

nΔ, where

n is the number of intervals (e.g., pixels) along each spatial direction, and Δ 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:

where the ‘^’ over

γ denotes estimate,

$\kappa $ is the the dimensionless spatial scale,

${\underset{\_}{x}}_{i,j}^{\left(\kappa \right)}=\frac{1}{{\kappa}^{2}}{\displaystyle \sum}_{\psi =\kappa \left(j-1\right)+1}^{\kappa j}{\displaystyle \sum}_{\xi =\kappa \left(i-1\right)+1}^{\kappa i}{\underset{\_}{x}}_{\xi ,\psi}$ is the sample average of the space-averaged process at scale

$\kappa $, and

$\overline{\underset{\_}{x}}={\displaystyle \sum}_{i,j=1}^{n}{\underset{\_}{x}}_{i,j}/{n}^{2}$ 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

$\lambda $ 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 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).

#### 2.2. 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

$\gamma \left(k\right)/\gamma \left(1\right)$ 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 (

Figure 6,

Figure 7,

Figure 8,

Figure 9,

Figure 10,

Figure 11,

Figure 12,

Figure 13,

Figure 14,

Figure 15,

Figure 16,

Figure 17 and

Figure 18) and

Section 4 the discussion of the results.