Morphological Features of Mathematical and Real-World Fractals: A Survey

Abstract

The aim of this review paper is to survey the fractal morphology of scale-invariant patterns. We are particularly focusing on the scale and conformal invariance, as well as on the fractal non-uniformity (multifractality), inhomogeneity (lacunarity), and anisotropy (succolarity). We argue that these features can be properly quantified by the following six adimensional numbers: the fractal (e.g., similarity, box-counting, or Assouad) dimension, conformal dimension, degree of multifractal non-uniformity, coefficient of multifractal asymmetry, index of lacunarity, and index of fractal anisotropy. The difference between morphological properties of mathematical and real-world fractals is especially outlined in this review paper.

1. Introduction

Since the introduction of the notion of fractals by Mandelbrot in the 1970s [1], a plethora of research efforts have been conducted trying to expose the fractal features of real-world systems (see, for review, Refs. [2,3,4,5] and references therein). The ultimate goal of these efforts was to reveal the interplay between the geometry and properties of the system. In this way it was recognized that fractal attributes are relevant for many physical, chemical, and biological phenomena occurring in the real world (see, for review, Refs. [6,7,8,9,10] and references therein). Hence, a proper characterization of the fractal features is of tremendous importance from both fundamental and practical points of view.

Although there is no universal definition of fractals in practical applications, the fractal features are commonly associated with scale invariance. Accordingly, the fractal pattern is characterized by the similarity dimension $D_S$, which strictly exceeds its topological dimension $d$ [11,12,13]. We recall that the topological dimension is associated with the way the pattern can be divided into parts, whereas the similarity (fractal) dimension characterizes the pattern’s ability to occupy the embedding Euclidean space. The scale invariance implies that the shape of the pattern is independent of the scale at which it is observed. Even though unlike the mathematical fractals, the real-world patterns exhibit only a statistical scale invariance over a finite range of scales with lower and upper limits [1,2,3,4,5]. Even so, the scale invariance of real-world systems can also be quantified by an appropriately defined fractal dimension $D$ [6,7,8,9,10].

However, two dimension numbers $D$ and $d$ are often insufficient to properly characterize the peculiar features of a scale-invariant system [14]. From a methodological viewpoint, the fractal properties of a system can be linked to its topological, morphological, topographical, and metrological features [15]. Regarding this background, it has been argued that the topological features of fractals can be accurately quantified by six generally independent dimension numbers accounting for the fractal connectedness, ramification, connectivity, loopiness, knottedness, and embeddability [16]. Nonetheless, topologically equivalent systems can possess quite different morphological features. The fractal morphological features are associated with the scale invariance and possible invariance under non-linear conformal transformations, as well as with the non-uniformity, heterogeneity, and anisotropy of the studied pattern and its sub-structures [15]. The purpose of the fractal analysis of system morphology is to reveal a spatial order in the system organization [17,18,19,20].

This review paper is devoted to the morphological features of mathematical and real-world fractals. Specifically, we are focused on the quantitative characterization of the scale and conformal invariance, as well as on the fractal non-uniformity (multifractality), inhomogeneity (lacunarity), and anisotropy (succolarity). Accordingly, we surveyed the morphological features of mathematical and real-world fractals. The differences between mathematical and real-world fractals are specially outlined.

2. Scale and Conformal Invariance of Mathematical and Real-World Fractals

Scale invariance is a key issue in fractal morphology studies. Geometrically, the scale invariance can be expressed in terms of affine transformations that map a pattern into its components. We recall that the affine transformations include all similarity transformations (translations, rotations, and isotropic scaling) and, additionally, shearing and anisotropic scaling. An isotropic mapping leads to self-similarity, whereas the self-affinity occurs when the rescaling is anisotropic. Accordingly, the self-similar patterns are made up of smaller copies of themselves (see, for instance, Figure 1), whereas self-affine patterns are composed of scaled-down affine copies of themselves (see, for example, Figure 2). Statistically scale invariant patterns comprise scaled-down components with the same statistical distribution as a whole (see, for example, Figure 1c and Figure 2d). Consequently, the scale invariant pattern is indistinguishable (at least in a statistical sense) under proper contraction or magnification.

Conversely, the conformal transformations locally preserve angles but not necessarily interval lengths [21]. In particular, the linear similarity transformations preserving the shape of pattern but not necessarily its size belong to the class of conformal linear transformations. Accordingly, there is a class of self-conformal patterns that do not possess the scale invariance [22,23,24]. The self-conformal fractals are made up of conformal images of themselves, as this is illustrated in Figure 3.

Quantitatively, the self-similarity can be characterized by the similarity dimension $D_S$ defined via the Hutchinson–Moran formula.

$${{\displaystyle \sum }}_i^m{c}_i^{D_S}=1,$$

(1)

where $0<c_i<1$ are the contraction ratios and m is the number of contractions at each iteration step [25]. In contrast to this, the dimension numbers characterizing the self-affine invariance need not vary continuously under the affine transformations. The anisotropy of self-affine patterns is often quantified by the Hurst exponent [1], which is defined as follows:

$$H=\ln c_x/\ln {\mathrm{c}}_{\mathrm{y}}<1,$$

(2)

where $c_x$ and $c_y$ are the contraction ratios in the orthogonal directions x and y, such that $c_y<c_x<1$ (see, for illustration, Figure 2b). The scale invariant measure of a self-affine curve can be characterized by the divider dimension D_D [26], which is defined as follows:

$$D_D{=\dim }_{D}F=1-\underset{\mathcal{l}\to 0}{\lim }\left\{\ln L\left(\mathcal{l}\right)/\ln \mathcal{l}\right\},$$

(3)

where $L\left(\mathcal{l}\right)$ is the curve length measured by means of a yardstick (divider) of length $\mathcal{l}$. It is a straightforward matter to verify that for self-affine curves hold the following relations. Verification of self-affine curves is as follows:

$$D_D=\left\{\begin{array}{c}1/H, \mathrm{if} H\ge 1/2 \hfill \\ 2, \mathrm{if} H\le 1/2\hfill \end{array}\right.,$$

(4)

whereas for self-similar curves $D_D=D_S$, while H = 1 [27]. Notice that in the literature, the divider dimension D_D is also called the compass or ruler dimension [1,6,8]. In a three-dimensional space, one can define three different scaling exponents $H_{zx}\le H_{zy}\le H_{yx}\le 1$ associated with the contraction ratios in three mutually orthogonal directions From the definition of scaling exponents it follows that $H_{zx}=H_{zy}\times H_{yx}$. Therefore, we can define the Hurst exponent as the geometric mean of three scaling exponents, such that for self-affine patterns we get

$$H=\sqrt{H_{zx}{H}_{zy}{H}_{yx}}=H_{zy}{H}_{yx}=H_{zx}<1,$$

(5)

whereas for self-similar patterns $H=H_{zx}=H_{zy}=H_{yx}=1$.

The similarity and divider dimensions both can be linked to a dimension number associated with a suitable defined covering measure. Three other most common covering measures are the Hausdorff, the box-counting, and the Assouad measures. The Hausdorff measure is defined as follows:

$$H_{ϵ}^s\left(F\right)=\inf \left\{{{\displaystyle \sum }}_{i=1}^{\infty }{\left|U_i\right|}^s:\left\{U_i\right\} \mathrm{is} ϵ-\mathrm{cover} \mathrm{of} F\right\},$$

(6)

where s is a non-negative number and the infimum is taken over all countable n-dimensional balls with diameters $r\le ϵ$ (see Figure 4a). The Hausdorff dimension is equal to

$$D_H{=\dim }_{H}F=s, \mathrm{such} \mathrm{that} \underset{ϵ\to 0}{\lim }{H}_{ϵ}^s\left(F\right)=\left\{\begin{array}{c}0, \mathrm{if} s>D\\ \infty , \mathrm{if} s<D\end{array}\right.,$$

(7)

providing that the limit exists [28]. Even though the Hausdorff dimension is often used in theoretical research, it is very difficult to estimate in practice since it involves covers with balls of widely different diameters. In view of that, in practical applications, it is more convenient to use the box-counting dimension defined as follows: The pattern F is covered by n-dimensional boxes of the same size $ϵ$ (see Figure 4b). If $N_B\left(ϵ\right)$ is the smallest number needed to cover the pattern F, then the box-counting dimension is equal to

$$D_B{=\dim }_{B}F=-\underset{ϵ\to 0}{\lim }\left\{\ln N_B\left(ϵ\right)/\ln ϵ\right\},$$

(8)

if the limit exists [1]. Otherwise, one needs to define the upper and the lover box counting dimensions by the substitution of ${\lim }_{ϵ\to 0}\left\{\dots \right\}$ in Equation (8) on ${\lim }_{ϵ\to 0}\inf \left\{\dots \right\}$ and ${\lim }_{ϵ\to 0}\sup \left\{\dots \right\}$, respectively. For the self-affine fractal, the box-counting dimension is related to the Hurst exponent by the following relation:

$$D_B=n-H<D_D$$

(9)

where n is the embedding Euclidean dimension [23], whereas for self-similar fractals H = 1 and $D_B=D_D$.

The Hausdorff and box-counting dimensions are both associated with the measures of the whole pattern, giving rise to an “average” dimension number. In order to capture more fine information about the thickest and thinnest parts of the pattern, the Assouad dimension was introduced [29]. The Assouad dimension is defined as follows:

$$D_A{=\dim }_{A}F=\left\{\begin{array}{c}s>0: \mathrm{there} \mathrm{exists} C>0 \mathrm{such} \mathrm{that} N_r\left[B\left(x,R\right)\cap F\right]\le C{\left(R/r\right)}^s\\ \mathrm{for} \mathrm{all} x\in F \mathrm{and} 0<r<R\end{array}\right\},$$

(10)

where $B\left(x,R\right)$ denote the covering balls (see, for illustration, Figure 4c). The key point in this definition is that one does not seek the covers of a whole pattern but only a small ball, and the covering number is properly normalized. Although, generally dimension numbers satisfy the following inequalities

$$D_H\le D_B\le D_A,$$

(11)

for many kinds of (statistically) self-similar fractals, the three following dimension numbers are equal between each other [28].

The Hausdorff, box-counting, and Assouad dimensions are bi-Lipschitz invariants, whereas the topological dimension is invariant under homeomorphisms. An intermediate class of transformation between the homeomorphic and bi-Lipschitz invariant maps is the quasisymmetric maps [30]. In order to classify patterns up to quasisymmetric equivalence it was introduced the notion of conformal dimension [31] defined as follows

$$D_{Cf}={\dim }_{Cf}F=\inf \left\{{\dim }_{H}f\left(F\right):f \mathrm{is} \mathrm{quasisimmetric}\right\}.$$

(12)

such that $D_{Cf}\le D_H$ [32]. The conformal dimension is invariant under quasisymmetric maps, and so it is also invariant under bi-Lipschitz transformations but not under homeomorphisms [33].

In mathematics, the scale invariance refers to the absence of characteristic scales in the system. This, for instance, is the case of deterministic self-similar and self-affine fractals. Consequently, the fractal dimensions defined in the limit of $ϵ\to 0$ coincide with the similarity dimension defined on any scale. Unlike mathematical fractals, real-world patterns (see, for example, Figure 5) exhibit statistical scale invariance only over a finite range of length scales.

$${\xi }_0<ϵ<{\xi }_C,$$

(13)

where ${\xi }_0$ is the characteristic size of structural elements from which the fractal pattern is composed and ${\xi }_C$ is the correlation length beyond which the pattern appears homogeneous [34,35,36]. So, the scale-invariant real-world patterns are pre-fractals rather than true fractals. In this regard, it was noted that the scale invariance implies a power law behavior of certain properties of the studied pattern. Accordingly, the fractal dimension of real-world patterns is commonly estimated using the finite-size scaling over a statistically identified region of scale invariance [37]. Specifically, the covering measures of the statistically scale-invariant pattern scale with the ratio of the pattern size L to the box size $ϵ$ are calculated using the following expression:

$$M\left(L,ϵ\right)\propto {\left(L/ϵ\right)}^D,$$

(14)

where D is the fractal dimension associated with the covering (e.g., Hausdorff or box-counting) measure, while ${\xi }_0<ϵ\ll L<{\xi }_C$. Consequently, the fractal dimension of a real world fractal can be defined as the slope of the linear part of the log-log plot of the covering measure M versus the ratio $\left(L/ϵ\right)$. The values of $\left(L/ϵ\right)$ at which the numerical data deviate from the power-law scaling allow the estimation of the values of ${\xi }_0$ and ${\xi }_C$.

In particular, in the box-counting method, a two- or three-dimensional image can be covered by a sequence of grids with boxes of decreasing size $ϵ$, as shown in Figure 6a. The minimum number of boxes needed to cover a statistically scale-invariant pattern scales with the box size as $N_B\propto {ϵ}^{-D_B}$, and so the box-counting dimension can be determined from the slope of the fitting line in the log-log coordinates [38]. An alternative algorithm for estimating the box-counting dimension is the sliding box-counting procedure [39], in which the box of size $ϵ$ is slid over the pattern so that it overlaps itself, as shown in Figure 6b. A practical implementation of any box-counting technique demands specifying certain requirements, including the minimum and maximum sizes of boxes, the scaling box sizing (linear or exponential), and box orientation. A detailed discussion of practical implementations of the box-counting method can be found in Refs. [40,41,42,43].

A modification of the box-counting technique is the sandbox scheme [44,45,46]. In the sandbox method, each point (pixel) of the pattern’s image is surrounded by a sequence of boxes with increasing sizes ${\epsilon }_{i+1}>{\epsilon }_i$, as it is shown in Figure 6b. The mean number of points that belong to the pattern $N_{sb}$ inside the boxes of size ${ϵ}_i$ is calculated by averaging over the boxes of this size around all pattern points. For scale-invariant patterns, $N_{sb}$ obeys the power law behavior $N_{sb}\propto {ϵ}_i^{D_C}$, where the scaling exponent $D_C$ is the correlation dimension. Accordingly, the correlation dimension can be estimated as the slope of a linear fit in the double-logarithmic coordinates. Other techniques used to measure the fractal dimension of real-world patterns include the slit-island method based on the perimeter area analysis [47], the wavelet method [48], the Higuchi method [49], and the power spectrum analysis [50], among other procedures [51,52,53,54,55,56,57].

3. Fractal Non-Uniformity and Multifractality

Many real-world fractals are created by a series of complex processes that operate at different scales. Linear regression methods allow to distinguish between patterns with one scaling region (12) and those with various scaling regions [58]. The pattern with various scaling regions can be characterized by different apparent fractal dimensions at different length scales. More frequently, the log-log plot of the number of covering boxes versus the box size has no linear region. Nonetheless, if $N_B\left(ϵ\right)$ is a smooth function of the measurement scale, the pattern can be characterized by the local box-counting dimension $D_b$. The local box-counting dimension is defined as follows [59]:

$$D_b\left(ϵ\right)=-d\ln N_B\left(ϵ\right)/d\ln ϵ ,$$

(15)

and so it depends on the scale of observation. Consequently, the number of boxes needed to cover the non-uniform fractal of size L complies with the following relation:

$$N_b\left(ϵ/L\right)\propto \exp \left\{{{\displaystyle \int }}_{ϵ}^L{D}_b\left(\epsilon \right)d\ln \epsilon \right\},$$

(16)

where scale dependence of $D_b\left(ϵ\right)$ can be attributed to the fractal non-uniformity [60,61,62]. Accordingly, the local box-counting dimension can be expressed as follows, in terms of the statistical distribution of empty covering boxes $p\left(\epsilon \right)$:

$$D_b\left(ϵ\right)=n-ϵ{{\displaystyle \int }}_{ϵ}^{L}p\left(\epsilon \right)d\epsilon /{{\displaystyle \int }}_0^{ϵ}\left[{{\displaystyle \int }}_{\zeta }^{L}p\left(\epsilon \right)d\epsilon \right]d\zeta ,$$

(17)

where n is the Euclidean dimension of covering boxes. Notice that the integral in the denominator of the right-hand side of Equation (17) is equal to

$${{\displaystyle \int }}_0^{ϵ}\left[{{\displaystyle \int }}_{\zeta }^{L}p\left(\epsilon \right)d\epsilon \right]d\zeta =ϵ{{\displaystyle \int }}_{ϵ}^{L}p\left(\epsilon \right)d\epsilon +{{\displaystyle \int }}_0^{ϵ}\epsilon p\left(\epsilon \right)d\epsilon .$$

(18)

For the uniform fractals, like that shown in Figure 1, the empty box size distribution obeys power law behavior $p\propto {ϵ}^{-\beta }$, and so we get

$$D_b\left(ϵ\right)=D_B=\beta -1,$$

(19)

whereas the completely random patterns obey the exponential distribution $p=\left(ϵ/\overline{ϵ}\right)\exp \left(-ϵ/\overline{ϵ}\right)$ and so

$$D_b\left(ϵ\right)=n-\left(ϵ/\overline{ϵ}\right)\exp \left(ϵ/\overline{ϵ}\right)/\left[1-\exp \left(ϵ/\overline{ϵ}\right)\right],$$

(20)

where the overline denotes the mean value. For non-uniform real-world fractals, the scale dependence of $D_b\left(ϵ\right)$ can be obtained with the help of Equations (17) and (18). It is pertinent to note that in the literature the scale-dependent dimension $D_b\left(\epsilon \right)$ is also called the effective fractal dimension [62], the coverage-dimension function [63], and the geometric dimension [64].

An alternative characteristic of the fractal non-uniformity is the singularity spectrum curve $f\left(\alpha \right)$ defined via the following scaling relation:

$$N_{\alpha }\left(ϵ\right)\propto {ϵ}^{-f\left(\alpha \right)},$$

(21)

where $N_{\alpha }\left(ϵ\right)$ is the numbers of boxes in which the Hölder exponent $\alpha$ has a value within the range $\left[\alpha ,\alpha +d\alpha \right]$ [28]. Hence, the function $f\left(\alpha \right)$ signifies the box-counting dimension of the subset, which has singularity $p_i\propto {ϵ}^{{\alpha }_i}$ around any point $i$ of the pattern. For uniform fractals $f\left(\alpha =D_B\right)=D_B$ and $f\left(\alpha \ne D_B\right)=0$. For multifractals, $f\left(\alpha \right)$ varies in the range of $\left[{\alpha }_{min},{\alpha }_{max}\right]$ (see, for example, Figure 7). Hence, the singularity spectrum characterizes the abundance of boxes with Hölder exponent $\alpha$. Equivalently, fractal non-uniformity can be characterized by the spectrum of Rényi dimensions which is defined as follows:

$$D_q=-\underset{ϵ\to 0}{\lim }\left[I_q\left(ϵ\right)/\ln ϵ\right], \mathrm{with}-\infty <q<\infty ,$$

(22)

where $I_q$ is the Rényi information. The Rényi information is defined as follows

$$I_q\left(ϵ\right)={\left(1-q\right)}^{-1}{{\displaystyle \sum }}_{i=1}^{N_B\left(ϵ\right)}{p}_i^q\left(ϵ\right),$$

(23)

where $p_i\left(ϵ\right)$ is the probability that a point of pattern lies in the box number $i$ of the covering grid with boxes of size $ϵ$ such that

$${{\displaystyle \sum }}_{i=1}^{N_B\left(ϵ\right)}{p}_i=1,$$

(24)

and so $Z_q\left(ϵ\right)={{\displaystyle \sum }}_{i=1}^{N_B\left(ϵ\right)}{p}_i^q\left(ϵ\right)$ is the partition function. In the case of real-world fractals, the limit $ϵ\to 0$ in Equation (22) should be replaced by the limit $ϵ\to {\xi }_0$.

The partition function scales with the box size as $Z_q\left(ϵ\right)\propto {ϵ}^{\tau \left(q\right)}$, where

$$\tau \left(q\right)=\left(q-1\right)D_q$$

(25)

is the mass exponent function [65]. The mass exponent function can be linked to the singularity spectrum $f\left(\alpha \right)$ via the Legendre transform (Equation (26)).

$$f\left(\alpha \right)=q\alpha -\tau \left(q\right), \alpha \left(q\right)=d\tau \left(q\right)/dq, q=d\left\{f\left[\alpha \left(q\right)\right]\right\}/d\alpha$$

(26)

Legendre transform allows for the determination of the spectrum of Rényi dimensions (22) from the singularity spectrum $f\left(\alpha \right)$, as illustrated in Figure 7. In particular, from Equations (22)–(26), it immediately follows that

$$D_{-\infty }={\alpha }_{min}\le D_0=D_B\le D_1=D_I\le D_2=D_C\le D_{\infty }={\alpha }_{max},$$

(27)

where $D_B$, $D_I$, and $D_C$ are the box-counting, information, and correlation dimensions, respectively [1]. Accordingly, the degree of fractal non-uniformity can be characterized by either the difference $\mathsf{\Delta }D=\left(D_{-\infty }-D_{\infty }\right)$ or the degree of multifractal non-uniformity $\mathfrak{O}$, which is defined in [8] as follows:

$$\mathfrak{O}=\left(D_B-D_C\right)/D_I$$

(28)

For the uniform fractals $\mathfrak{O}=\mathsf{\Delta }D=0$. The asymmetry of the multifractal spectrum can be quantified by the coefficient of multifractal asymmetry, defined as

$$\mathfrak{K}=\left(D_B-D_{\infty }\right)/\left(D_{-\infty }-D_B\right),$$

(29)

such that if the $f\left(\alpha \right)$ spectrum is symmetric, then $\mathfrak{K}=1$ (see Figure 7). A thorough discussion of practical implementations of multifractal analysis in studies of real world systems can be read in refs [66,67,68,69,70,71].

4. Fractal Inhomogeneity and Lacunarity

Spatially homogeneous patterns are invariant under translations as well as under affine transformations. The hierarchical arrangement leads to breaking the translational symmetry in fractal patterns. The deviation from the translational invariance is associated with the spatial distribution of the pattern density [1]. The last can be characterized by the statistical moments of the mass distribution, as seen below.

$$Z_q\left(ϵ/L\right)={{\displaystyle \sum }}_{m=1}^{N\left(ϵ/L\right)}{m}^{q}P\left(m,ϵ/L\right),$$

(30)

where $P(m,ϵ/L)$ is the probability density function of mass distribution, L is the size of the pattern, and $ϵ$ is the size of covering boxes. Accordingly, the departure from the translational invariance is commonly quantified by the lacunarity, which is defined as follows:

$$\mathcal{L}=Z_2/Z_1^2=\langle \mathsf{\Delta }{\mu }^2\rangle /{\langle \mu \rangle }^2+1,$$

(31)

where $\langle \mu \left(ϵ/L\right)\rangle$ is the mean density within the box of size $ϵ$ and $\langle \mathsf{\Delta }{\mu }^2\rangle$ is the statistical variation. The lacunarity describes the pattern’s heterogeneity whether or not it is fractal [72,73,74]. Homogeneous patterns have $\mathcal{L}=1$, whereas heterogeneous patterns are characterized by $\mathcal{L}>1$. For the self-similar patterns $Z_2\propto Z_1\propto {\left(L/ϵ\right)}^{n-D_B}$, and so the lacunarity scales are expressed as follows:

$$\mathcal{L}=c_{\mathcal{L}}{\left(L/ϵ\right)}^{n-D_B},$$

(32)

where $c_{\mathcal{L}}$ is the geometric constant, while $D_B<n.$ The lacunarity is a scale-dependent measure of the fractal heterogeneity [71,72,73,74]. Notice that for the true mathematical fractals $\mathcal{L}\to \infty$ as $ϵ\to 0$, whereas the lacunarity of pre-fractals and real world patterns varies in the following range:

$$1\le \mathcal{L}\left(L/ϵ\right)\le {\rho }_0/{\rho }_C,$$

(33)

where ${\rho }_C$ is the mean density of the pattern of size $L\ge {\xi }_C$, while ${\rho }_0$ is the density of structural elements having the size ${\xi }_0$. For multifractal patterns, the following expression is used:

$$\mathcal{L}\propto {\left(L/ϵ\right)}^{n-D_2},$$

(34)

where $D_2$ is the correlation dimension [74]. In the case of black-and-white images, the following expression is used:

$${\rho }_C={\rho }_0\left(1-p\right),$$

(35)

where p is the pattern porosity, if the matrix is pre-fractal, or p is the relative density, while ${\rho }_C/{\rho }_0$ is the porosity, if the pore space is pre-fractal. In both cases, the geometric coefficient is equal to the following:

$$\tilde{\mathcal{L}}=\left(1-p\right){\left({\xi }_C/{\xi }_0\right)}^{n-D_2}$$

(36)

and can be used as the scale-independent index of lacunarity. Low lacunarity indexes correspond to relatively homogeneous patterns, whereas high lacunarity indexes correspond to heterogeneous fractals. Notice that for deterministic pre-fractals ${\xi }_C=L$. Useful discussions of the lacunarity analysis performance in studies of real-world fractals can be found in Refs. [75,76,77,78,79,80,81].

5. Fractal Anisotropy and Succolarity

Isotropic patterns are invariant under rotations and reflections. Although many mathematical and real-world fractals are isotropic, at least in a statistical sense, the connectivity of most fractals is different in different directions. In order to describe anisotropic fractals, Mandelbrot has suggested the notion of succolarity [1]. The succolarity characterizes intercommunications and can be defined as the percolation degree of a fractal pattern. Thereby, a virtual fluid will percolate the image in the chosen direction, crossing the voids and bypassing the white obstacles. Hydrostatic pressure in a point immersed in the fluid at depth h is determined by the weight of the fluid column above that point. Therefore, the pressure difference between two points is proportional to the difference in depth $\mathsf{\Delta }h$ between these points. The succolarity determines how much a virtual fluid can flow through the pattern in a chosen direction $\overrightarrow{e}$. Accordingly, the value of succolarity can be computed with the help of an adaptation of the box-counting algorithm that simulates the percolation of a virtual fluid using the pressure notion. In order to calculate the succolarity, the studied pattern is binarized to obtain a square black-and-white image of size $L^n$, where is the embedding Euclidean dimension. The black areas (volumes in 3D) correspond to the matrix, whereas the white areas (volumes) represent the voids (pores, channels, cracks, etc.). The binary image of the studied fractal is partitioned into a grid of boxes of varying sizes r. The value of succolarity $\mathcal{S}\left(\overrightarrow{e}\right)$ is equal to the ratio of the summation of $\mathcal{O}\left(r\right)\times \mathcal{P}\left(r\right)$ to the summation of $\mathcal{P}\left(r\right)$ for all possible values of r, where $\mathcal{O}$ denotes the spatial occupation and $\mathcal{P}$ is the fluid pressure.

The most popular approach for calculating succolarity was suggested by Melo and Conci [82]. The image is flooded in a given direction, ensuring that all voids in the first column are detected. Then, all the four connected voids are selected until an impenetrable mass of black matrix is encountered. The succolarity is calculated as follows:

$${\mathcal{S}}_{\overrightarrow{e}}=\left[{{\displaystyle \sum }}_r^N\mathcal{O}\left(r\right)\mathcal{P}\left(r\right)\right]/{{\displaystyle \sum }}_r^N\mathcal{P}\left(r\right)$$

(37)

in one direction and then in the opposite direction, e.g., from left to right (${\mathcal{S}}_{\to }$) and from right to left (${\mathcal{S}}_{\leftarrow }$), from top to bottom $\left({\mathcal{S}}_{\downarrow }\right)$ and from bottom to top (${\mathcal{S}}_{\uparrow }$). This allows to determine the succolarity in the vertical and (one or two) horizontal directions as the arithmetic means, e.g., ${\mathcal{S}}_V\left({\overrightarrow{e}}_i\right)=0.5\left[{\mathcal{S}}_{\downarrow }\left({\overrightarrow{e}}_i\right)+{\mathcal{S}}_{\uparrow }\left({\overrightarrow{e}}_i\right)\right]$. Accordingly, the two-dimensional fractal anisotropy index is defined as follows:

$$\mathcal{A}=\max \left\{{\mathcal{S}}_H,{\mathcal{S}}_V\right\}/\min \left\{{\mathcal{S}}_H,{\mathcal{S}}_V\right\}$$

(38)

where ${\mathcal{S}}_V$ and ${\mathcal{S}}_H$ are the succolarities in the vertical and horizontal directions, respectively. In the case of a 3D pattern, one can define two independent two-dimensional fractal anisotropy indices [83]. The use of succolarity to characterize real-world fractals was discussed in Refs. [82,83,84,85].

Morphological attributes of some mathematical and real world fractals are summarized in

Table 1. Morphological attributes of the pre-fractal Sierpinski carpet (see Refs. [86,87]), Menger sponge (see Ref. [88]), phosphate alumina gel (see Ref. [51]), balls folded from randomly crumpled paper sheets (see Ref. [35]), retinal vessel networks (see Refs. [89,90,91,92,93,94,95,96]), and human brains (see Refs. [97,98,99,100]).

Pre-Fractal		${\mathit{D}}_{\mathit{B}}$	${\mathit{D}}_{\mathit{C}\mathit{f}}$	$\mathcal{D}$	$\mathfrak{K}$	$\tilde{\mathcal{L}}$	$\mathcal{A}$
Sierpinski carpet after six iteration steps		1.893	1.797	0	-	1.03	1
Menger sponge after six iteration steps		2.727	2.262	0	-	5.05	1
Phosphate alumina gel		1.84 ± 0.02	1.4 ± 0.1	0	-	1.2	0
Balls folded from randomly crumpled paper sheets		2.66 ± 0.03	2	0.00 ± 0.02	-	4.8	1.0 ± 0.1
Retinal vessel network images	Normal	1.697 ± 0.003	1	0.065 ± 0.007	0.96 ± 0.02	2.7	1.3 ± 0.3
	Hypertension	1.41 ± 0.01	1	0.067 ± 0.007	0.92 ± 0.02	12.1	1.4 ± 0.3
	Glaucoma	1.39 ± 0.02	1	0.07 ± 0.01	0.89 ± 0.03	13.3	1.4 ± 0.3
Human brains (tomography)		2.69 ± 0.07	-	0.05 ± 0.01	0.87 ± 0.03	3.0	1.5 ± 0.3

.

6. Conclusions

The aim of morphological studies is to reveal how parts of the system are linked together. In this study, we survey the attributes characterizing fractal morphology. Different algorithms and techniques used to estimate the fractal dimension quantifying the scale invariance are highlighted. The use of conformal dimension for characterizing fractal morphology is sketched. The methods for quantifying the fractal non-uniformity are also discussed. In this regard, we argue that fractal non-uniformity can be properly categorized with the help of two attributes, i.e., degree of multifractal non-uniformity and coefficient of multifractal asymmetry. In this way, fractal inhomogeneity and anisotropy can be specified by the index of lacunarity and the index of fractal anisotropy associated with the succolarity. The difference between morphological properties of mathematical and real-world fractals is especially highlighted.