Statistical Insights into Spatial Patterns: A Panorama About Lacunarity

Abstract

This overview is designed to illuminate the concept and utility of lacunarity. We first establish a strong foundation with a pedagogical introduction to the lacunarity measure applied to images, detailing analytical examples and a general approach. In the second part, we compare the available software for estimating the lacunarity of images. Related to this goal, we also provide an open-source code in R and Python. The third part then synthesizes these theoretical and computational aspects by presenting an analysis of the diverse applications of lacunarity across various scientific disciplines, utilizing VOSviewer networks to visually organize research topics into distinct clusters. We identify distinct thematic clusters in materials science, biological systems, and medical imaging.

1. Introduction

In his seminal article from 1967, ‘How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension’ [1], Mandelbrot drew attention to the self-similarity inherent in certain geographic curves. This insight led to the definition of fractal dimension and, consequently, enabled the scientific community to undertake a more detailed quantitative analysis of non-trivial structures. In 1982, in his book entitled “The Fractal Geometry of Nature” [2], Mandelbrot warned that in some cases the fractal dimension was not sufficient to distinguish fractals, so he proposed some alternative measures, such as the lacunarity.

A more systematic study of lacunarity was presented by Allain and Cloitre [3], Plotnick and collaborators [4] and Mandelbrot [5]. In more recent works, such as those presented by Vernon-Carter et al. [6], Braun et al. [7] and Mao et al. [8], generalized lacunarity approaches were investigated to evaluate multifractal structures and trajectory information in the phase space of a dynamical regime.

Thus, lacunarity has evolved into a versatile tool for pattern analysis across disciplines [6,7,8]. From its theoretical development [3,9] to applications distinguishing complex structures [4,5], lacunarity has been proven to be a useful spatial measure. While fractal dimension has been extensively reviewed, lacunarity has received comparatively little systematic attention despite its broader applicability. We address this gap in the literature by integrating theoretical, computational, and applied perspectives related to lacunarity.

The overview article is organized as follows: in Section 2 we focus on the theoretical aspects, in Section 3 we present the computational features, in Section 4 we analyze the real-world applications of lacunarity, and in Section 5 we provide the final remarks. As we will show, the core contributions of this paper are threefold:

• The development of a novel pedagogical introduction to lacunarity supported by analytical examples;
• The provision of new R and Python implementations for lacunarity estimation, addressing existing software gaps and enhancing accessibility;
• An evidence-based mapping (via VOSviewer) of the widespread utility of the lacunarity across diverse disciplines.

2. Part I: Theoretical Aspects

2.1. General Concepts

Lacunarity ($\mathrm{\Lambda }$) serves as a measure of how gappy a fractal structure appears across multiple scales. A high $\mathrm{\Lambda }$ suggests an irregular distribution of gaps [3,4,10,11].

First, we define the n-th statistical moment:

$$\begin{array}{c}\hfill M_n=〈s^n〉=\sum s^{n}Q(m,s),\end{array}$$

(1)

where $Q(m,s)$ gives the probability of encountering s gaps in a sub-region of size m.

The lacunarity $\mathrm{\Lambda }\left(m\right)$ is then expressed in terms of the first two moments [3,4]:

$$\mathrm{\Lambda }\left(m\right)={\displaystyle \frac{M_2}{M_1^2}}.$$

(2)

As ${\sigma }^2=M_2-M_1^2$ the above equation can be written as

$$\mathrm{\Lambda }\left(m\right)={\displaystyle \frac{{\sigma }^2}{M_1^2}}+1=C_V^2+1,$$

(3)

where $C_V=\sigma /M_1$ is the coefficient of variation. As $M_1$ is the mean of gap sizes and $\sigma$ is the standard deviation of gap sizes, it becomes clear now that the interpretation of $\mathrm{\Lambda }\left(m\right)$ as a measure of the heterogeneity of gaps in a structure is valid.

In order to gain further insights, let us analyze some extreme cases:

• At the smallest scale, each window corresponds to a single pixel, $m=1$. Letting $f_0$ denote the proportion of gap pixels in the matrix, the statistical moments simplify to $M_1=f_0$, $M_2=f_0$, then $\mathrm{\Lambda }\left(1\right)={\displaystyle \frac{1}{f_0}}$. This inverse relation reveals that at the pixel scale, lacunarity depends solely on gap density.
• When the window size matches the image dimensions, $m=L$, the entire matrix becomes a single observation. Then the variance is null, $\sigma =0$. Consequently, $\mathrm{\Lambda }\left(L\right)=1$.

2.2. Simple Analytical Examples

From the literature on lacunarity [3,4,11,12], we define $Q(m,s)$ as the probability of finding s lacunae (zeros) in a sliding window of size $m\times m$ within a binary image.

Consider a 3 × 3 matrix with variable elements:

$$I_v=\left[\begin{array}{ccc}1& 0& v\\ 0& 1& 0\\ v& 0& 1\end{array}\right]$$

(4)

where $v\in \{0,1\}$. We analyze both cases separately:

• Case 1: $v=1$ (Chessboard pattern)

For $m=2$ (4 possible positions):

• Top-left: $\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$ (2 zeros)
• Top-right: $\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$ (2 zeros)
• Bottom-left: $\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$ (2 zeros)
• Bottom-right: $\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$ (2 zeros)

Lacunarity distribution:

• $Q(2,0)=0/4=0$
• $Q(2,1)=0/4=0$
• $Q(2,2)=4/4=1$
• $Q(2,3)=0/4=0$
• $Q(2,4)=0/4=0$

• Case 2: $v=0$ (Diagonal pattern)

For $m=2$:

• Top-left: $\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$ (2 zeros)
• Top-right: $\left[\begin{array}{cc}0& 0\\ 1& 0\end{array}\right]$ (3 zeros)
• Bottom-left: $\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right]$ (3 zeros)
• Bottom-right: $\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$ (2 zeros)

Lacunarity distribution:

• $Q(2,0)=0/4=0$
• $Q(2,1)=0/4=0$
• $Q(2,2)=2/4=0.5$
• $Q(2,3)=2/4=0.5$
• $Q(2,4)=0/4=0$

• Measures of lacunarity

Using the lacunar distributions Q from the previous analysis, we compute the statistical moments and lacunarity measure $\mathrm{\Lambda }$ for each case.

• Case 1: $v=1$

Statistical moments:

$$M_1=\sum _{s=0}^{4}s·Q(2,s)=2$$

(5)

$$M_2=\sum _{s=0}^4{s}^2·Q(2,s)=4$$

(6)

Lacunarity measure:

$$\mathrm{\Lambda }\left(2\right)={\displaystyle \frac{M_2}{M_1^2}}={\displaystyle \frac{4}{2^2}}=1$$

(7)

• Case 2: $v=0$

Statistical moments:

$$M_1=\sum _{s=0}^{4}s·Q(2,s)=2.5$$

(8)

$$M_2=\sum _{s=0}^4{s}^2·Q(2,s)=6.5$$

(9)

Lacunarity measure:

$$\mathrm{\Lambda }\left(2\right)={\displaystyle \frac{M_2}{M_1^2}}={\displaystyle \frac{6.5}{2.5^2}}\approx 1.04$$

(10)

• Which Case Has Higher Lacunarity?

For window size $m=2$:

• Case $v=1$: $\mathrm{\Lambda }\left(2\right)=1$
• Case $v=0$: $\mathrm{\Lambda }\left(2\right)\approx 1.04$

The case with $v=0$ has higher lacunarity because its distribution is more spread out (between $s=2$ and $s=3$) compared to the concentrated distribution of the $v=1$ case.

2.3. Random Binary Surface

Another pedagogical case for which analytical results can be obtained has to do with the problem of a perfectly homogeneous surface on which one creates a fraction p of uniformly distributed lacunae; herein, the probability of finding s lacunae in an $m\times m$ matrix reads,

$$Q(m,s)={\displaystyle \frac{m^2!}{(m^2-s)!s!}}{p}^s{(1-p)}^{m^2-s},$$

(11)

where the combinatorial terms define the number of ways of having s lacunae in $m^2$ sites. In furtherance of the final result, it is key to heed that in being homogeneous—i.e., there is no established deterministic structure—$Q(m,s)$ is valid whatever the box chosen within $L\times L$. Thence, from an analytical perspective, we need to compute the moments $M_2$ and $M_1$ considering an $m\times m$ box; nonetheless, in numerical implementations we must screen the entire $L\times L$ image using an $m\times m$ gliding box and average over them for correct results. Plugging this expression into Equation (1) and subsequently Equation (2), one obtains, after some algebra,

$$\mathrm{\Lambda }\left(m\right)=1-{\displaystyle \frac{1}{m^2}}\left(1-{\displaystyle \frac{1}{p}}\right).$$

(12)

In Figure 1, we plot the dependence of the lacunarity on the fraction of lacunae, p, for the case $m=2$ and the overall dependence of $\mathrm{\Lambda }$ on m and the parameter p.

2.4. Sierpinski-like Fractals

Figure 2 depicts a flowchart in which we boil down the key difference between fractality and lacunarity using a random pattern and two other distinct patterns within the family of Sierpinski carpets—the ‘Quadrilux’ and ‘Unilux’ fractals—which are generated from a set of iterative replacement rules. Both fractals have the same dimension $D={\displaystyle \frac{log\left(N_f\right)}{log\left(S\right)}}={\displaystyle \frac{log\left(32\right)}{log\left(6\right)}}\approx 1.934$, where $N_f=32$ is the number of filled sub-squares in the generating rule, and $S=6$ is the scaling factor.

The fractal dimension (D) is determined by the slope of the linear relationship in a log-log plot of the number of occupied boxes (N) versus the inverse of the box size ($1/m$), described by the equation $log\left(N\right(m\left)\right)\propto Dlog(1/m)$. Note that the random pattern scales with the Euclidean dimension $D=2$. We see clear differences with respect to the visual structure and texture after four generations for the Quadrilux and Unilux, as shown in the plot on the right-hand side of the flowchart;

The quantitative distinction between the two patterns is drawn resorting to the computation of the lacunarity as visible in the plot on the left-hand side. For a given scale $m>1$, the Unilux has higher lacunarity compared to the Quadrilux fractal. This analysis exemplifies how lacunarity supplements the fractal dimension by characterizing patterns.

3. Part II: Computational Aspects

3.1. Algorithm to Compute the Lacunarity for Images: General Case

The most widely used algorithm to measure the lacunarity of images is the “gliding box” method proposed by Allain and Cloitre [3], which was explored in detail in Ref. [4]. This method is intuitively clear and computationally simple, as can be seen in the following description:

• Step 1: The pixel matrix of the original image, composed of $z(x,y)$ elements distributed in the $XY$-plane, must be binarized using a suitable threshold according to Equation (13). The choice of threshold generally depends on the characteristics of the image. Nonetheless, there are automated methods for identifying thresholds, e.g., the Otsu method [11].

  $$g(x,y)=\left\{\begin{array}{cc}1,& z(x,y)\ge threshold\\ 0,& otherwise\end{array}\right.$$

  (13)
• Step 2: A square box of size m must slide over the binarized image, from the top right corner to the bottom left corner, advancing one column and one row per scan, while the number of zeros s (lacunar pixels) inside the box is counted.
• Step 3: From the distribution of s values found in the previous step, the lacunarity can be calculated using Equation (2).

The “gliding box” method can be applied to any type of digital image, from radiographs [13] to scanning microscopy [14]; yet, the interpretation of the results depends on the physical nature of the data and the binarization threshold. Conversely, lacunarity is strongly dependent on image resolution [11,15], thus comparisons should be made between images with the same resolution. Comparative studies between the gliding box algorithm and the box-counting [3] and sandbox [15] algorithms can be found in the literature.

Some spatial patterns and their respective lacunarity curves on the log-log scale are presented in Figure 3. The behavior of the curves is very characteristic of each evaluated pattern, which confirms the discriminatory power of $\mathrm{\Lambda }$. We can see the change in the concavity of the $\mathrm{\Lambda }\left(m\right)$ curve filtering out organized heterogeneity from disorganized heterogeneity. These two extreme cases shed light on the effectiveness of lacunarity even within a self-similar context. As discussed in Ref. [12], the Sierpinski carpet can be extended to other similar patterns; for example, instead of grouping the 9 lacunae at the center of the generator, one can distribute them (homogeneously) on the 49 sites of the initiator. Although the fractal dimension is still $D_F=log40/log7$, one does not have the central large gap any longer; i.e., this new structure is perceived as having fewer discontinuities, and thus its lacunarity is smaller.

In view of this, several studies have recently emerged with approaches to represent the information contained within a lacunarity curve as a single number, which would facilitate comparison between multiple images. Among these proposals, we highlight the lacunarity parameter $\lambda$ [3], the lacunarity exponent $\beta$ [11,16], and the spatial heterogeneity index h [17]. These parameters can be calculated using the following relationships:

$$\mathrm{\Lambda }\left(m\right)=\lambda {\left[{\displaystyle \frac{m}{L}}\right]}^{D_F-D_E}$$

(14)

$$\mathrm{\Lambda }\left(m\right)=\alpha ·m^{-\beta }$$

(15)

$$h=1-{\displaystyle \frac{2}{(1+m_{max})N_{max}}}\sum _{i=1}^{N_{max}}{\displaystyle \frac{m_i\mathrm{\Lambda }\left(m_i\right)}{\mathrm{\Lambda }\left(1\right)}}$$

(16)

where L is the order of the matrix of pixels, $D_F$ is the fractal dimension, $D_E$ is the Euclidean dimension, $\alpha$ is a constant, $N_{max}$ is the number of m values required to obtain $\mathrm{\Lambda }=1$, and $m_{max}$ is the m value that corresponds to the first time $\mathrm{\Lambda }=1$ appears in the lacunarity curve. Besides being dimensionless, this index is bounded within the interval $[0,1]$, which allows comparisons between patterns with unlike scale properties. Explicitly, $h=0$ corresponds to a perfectly homogeneous case, whereas the opposite limit, $h=1$, signals extreme heterogeneity.

Alternatively, the parameters $\lambda$ and $\beta$ are restricted to fractal structures, as their definitions stem from the power laws derived from scale invariance. On the other hand, the parameter h, although it can be applied to non-fractals, is limited to small matrices due to computational cost. More general approaches based on the identification of the optimal value of m [7,18] are still being explored. With respect to $\beta$, previous work indicates that it was unable to provide a complete analysis of surface topography on its own. This has been fixed by a complementary analysis of the Moran index, which is a measure of spatial autocorrelation [11].

3.2. Computational Tools: A Comparison

There are some computational tools for the fractal analysis of images offered by companies or research groups, such as software (Table 1) and open codes in languages such as Fortran [19], MATLAB [17], Python [20], C++ [21], and R [11]. However, most computational tools with user interfaces have implemented only the fractal dimension, probably contributing to the less expressive presence of lacunarity in the literature compared to $D_F$.

To our knowledge, no commercial image analysis software offers the lacunarity measure, and its implementation in user interfaces is restricted to plugins of the free software FIJI/ImageJ2. FIJI/ImageJ2 is a version of the free software ImageJ that includes several useful plugins provided by the scientific community. ImageJ is written in Java and can run on Linux, macOS, and Windows, both in 32-bit and 64-bit. Since its earliest versions, ImageJ has already offered the fractal dimension measure. In contrast, lacunarity was only implemented in plugins such as Fraclac in 2013 [22], Multifrac in 2020 [23] and, more recently, ComsystanJ in 2023 [24]. We highlight the latter, which includes several other measures of complexity and heterogeneity in 1D, 2D, and 3D.

All of these plugins only allow one to evaluate images in grayscale or RGB format. Although useful, they are very limited for analyzing Atomic Force Microscopy (AFM) data, for example. Furthermore, free software such as WSxM [25] and Gwyddion [26], widely used in AFM studies, does not offer lacunarity analysis. Consequently, programming in various languages is commonly employed for more robust analysis.

We conducted a comparative study applying various computational tools to the Sierpinski Carpet. The Fraclac plugin does not allow free selection of box sizes; the user chooses the number of size values and the minimum number of pixels in the box. The plugin then generates box sizes based on the percentage of image coverage up to $100\%$. Additionally, the measurement is performed on the image file. On the other hand, the ComsystanJ plugin allows selection of box sizes identical to the R and Python codes. Nevertheless, the measurement is also performed on the image file, which potentially generates fluctuations in the lacunarity values at some scales, as shown in Figure 4. The Multifrac plugin is not compatible with the current version of FIJI/imageJ2.

Table 1. Computational tools with a user interface for fractal analysis.

Name/Type	Availability	Measures Offered	Reference
MountainsMap® Premium/Software	Paid	Fractal Dimension ($2D$)	—
XEI 4.3.0®/Software	Paid	Fractal Dimension ($2D$)	—
Gwyddion 2.69/Software	Free	Fractal Dimension ($2D$)	[26]
ImageJ 1.54p/Software	Free	Fractal Dimension ($2D$)	[27]
Fraclac/Plugin of FIJI-ImageJ2	Free	Fractal Dimension and Lacunarity ($2D$ and $3D$)	[22]
Multifrac/Plugin of FIJI-ImageJ2	Free	Fractal Dimension and Lacunarity ($2D$ and $3D$)	[23]
ComsystanJ/Plugin of FIJI-ImageJ2	Free	Fractal Dimension and Lacunarity ($1D$, $2D$, and $3D$)	[24]
Codes/Python and R languages	Free	Lacunarity ($2D$)	[28]

Table 1. Computational tools with a user interface for fractal analysis.

Name/Type	Availability	Measures Offered	Reference
MountainsMap® Premium/Software	Paid	Fractal Dimension ($2D$)	—
XEI 4.3.0®/Software	Paid	Fractal Dimension ($2D$)	—
Gwyddion 2.69/Software	Free	Fractal Dimension ($2D$)	[26]
ImageJ 1.54p/Software	Free	Fractal Dimension ($2D$)	[27]
Fraclac/Plugin of FIJI-ImageJ2	Free	Fractal Dimension and Lacunarity ($2D$ and $3D$)	[22]
Multifrac/Plugin of FIJI-ImageJ2	Free	Fractal Dimension and Lacunarity ($2D$ and $3D$)	[23]
ComsystanJ/Plugin of FIJI-ImageJ2	Free	Fractal Dimension and Lacunarity ($1D$, $2D$, and $3D$)	[24]
Codes/Python and R languages	Free	Lacunarity ($2D$)	[28]

The codes in R and Python perform the measurement directly on the pixel matrix, which, for AFM data, for example, corresponds to the direct spatial evaluation of the physical properties measured in the experiment.

4. Part III: Applications

4.1. Time Evolution

Figure 5 shows the temporal evolution of the number of scientific publications related to the fractal dimension and lacunarity from two major bibliographic databases (Scopus and Web of Science). The limited engagement with fractality and lacunarity in the academic literature prior to 1970, notwithstanding Mandelbrot’s pioneering work in 1967 [1], led us to establish 1970 as the starting point for our chronological investigation. Note that the number of publications related to the fractal dimension is much larger than the corresponding number of publications associated with the lacunarity. This trend is consistent between both databases, despite some differences that might reflect variations in database coverage.

4.2. Applications of Lacunarity: An Overview with VOSviewer

To analyze the applications of lacunarity, we employed VOSviewer [29], a widely used software tool for constructing and visualizing bibliometric networks. The primary purpose of using VOSviewer was to identify and map the landscape of lacunarity research. The networks generated by VOSviewer are based on co-occurrence data of keywords extracted from a dataset from Scopus.

We present our results in Figure 6, which displays distinct clusters of lacunarity-related research, each represented by a given color. Items that frequently co-occur are placed closer together, forming clusters that represent related research themes or application areas. The clustering algorithm within VOSviewer was utilized to identify distinct groups of keywords. Each cluster was then assigned a unique color for visual differentiation, allowing for an intuitive understanding of the thematic subdivisions within the broader field of lacunarity applications.

The red and brown clusters highlight lacunarity’s relevance in materials science, physics, engineering, and geophysics. Terms like “surface roughness”, “surface cracks”, “fractal fissures,” and “texture” point to the use of lacunarity in characterizing the structural patterns in materials and geological formations.

The purple, blue, and green clusters delve into the biological and life sciences. “Morphology,” “tissue”, “animal tissue”, “ultrastructure”, “histology”, and “tomography” strongly suggest lacunarity’s role in characterizing the structural organization of biological systems at various scales.

Terms like “fractal” and “texture analysis” appear across clusters, emphasizing lacunarity’s utility in quantifying irregular patterns. The presence of “remote sensing” and “medical imaging” in independent clusters highlights parallel methodologies in vastly separate domains.

Terms related to numerical methods and algorithms are present across diverse clusters, indicating a strong computational foundation for calculating the lacunarity. This underscores that a significant portion of lacunarity analysis relies on extracting quantitative information from data that can be encapsulated as images.

4.3. Lacunarity and Fractal Dimension: A Comparison

This section details the findings of a bibliometric comparative analysis performed in June 2025. The academic database Dimensions AI served as the primary data source. A comprehensive search was executed using the precise terms “Fractal Dimension” and “Fractal Lacunarity,” with the “full data” filter applied to ensure the retrieval of all relevant indexed records. Our complete dataset is available at [30].

Figure 7 illustrates the percentage distribution of major knowledge areas employing fractal dimension (FD) and lacunarity (FL). The results show that lacunarity and fractal dimension have cross-disciplinary applications in multiple areas of knowledge, with an emphasis on engineering, mathematics, physical sciences, and computer sciences, but with notable prominence in biomedical and clinical sciences, biological sciences, and environmental sciences.

5. Conclusions

In contrast to the brief treatment given to the lacunarity in previous surveys [31,32,33,34,35,36,37,38,39,40,41], our review provides the first comprehensive and practical exposition of this measure, bridging the gap between its theoretical foundations and its applicability in the real world.

The initial part provided a new pedagogical guide to lacunarity in images, complemented with illustrative analytical examples and a generalized methodological framework. A key contribution of the second part was a novel comparative analysis of current software tools for lacunarity estimation, which was significantly augmented by our release of open-source implementations in R and Python. Finally, the third part integrated these theoretical and computational insights, employing VOSviewer networks to systematically explore diverse real-world applications. Our analysis revealed the widespread relevance of lacunarity across numerous scientific domains, including areas such as material science, biological and medical imaging analysis, as well as geology and geophysics.

Despite the potential for characterizing time series presented by lacunarity in Ref. [4], where the authors illustrate applications in geological and ecological data, the use of this parameter has focused mainly on two-dimensional data, such as medical images [13], Scanning Electron Microscopy (SEM) [14], and Atomic Force Microscopy (AFM) [11]. Additionally, studies with rainfall series can also be highlighted [16,42,43,44]. Furthermore, lacunarity has been used recently to find transition regimes in nonlinear dynamics by applying it to time series recurrence maps [7,8].Three-dimensional assessments of lacunarity are currently becoming popular in the field of geophysics as a means of using the micro-computed tomography (micro-CT) technology to characterize the spatial heterogeneity of different types of rock [45,46] and soil [47,48].

As a new trend we mention, for instance, that in Ref. [18] the lacunarity of binary sequences provided insights into the characteristics of switching protocols. By analyzing a lacunarity-persistence plane, the researchers gained a more nuanced understanding of what makes a switching strategy effective in achieving the Parrondo’s Paradox. The lacunarity of sequences is also an important topic in the mathematical community [49]. In this way, the development of new lacunarity-based measures for time series and sequences is a worthwhile future research direction.

Looking ahead, there are promising research avenues. Integrating lacunarity into machine learning workflows would allow more powerful and descriptive feature engineering. Furthermore, the development of cross-platform toolkits is essential to standardize the methodology and broaden its adoption across different fields. While our study provides a broad overview, a more focused investigation into the smaller clusters of the VOSviewer network could offer new insights into emerging research topics.