Evaluating the Fractal Pattern of the Von Koch Island Using Richardson’s Method

Abstract

The principles of fractal geometry have revolutionized the characterization of complex geometric objects since Benoit Mandelbrot’s groundbreaking work. Richardson’s method for determining the fractal dimension of boundaries laid the groundwork for Mandelbrot’s later developments in fractal theory. Despite extensive research, challenges remain in accurately calculating fractal dimensions, particularly when dealing with digital images and their inherent limitations. This study examines the numerical artifacts introduced by Richardson’s method when applied to the Von Koch Island, a classic fractal curve, and proposes a novel approach for computing fractal dimensions in image analysis. The Koch snowflake serves as a key example in this analysis; it serves to assess the algorithm of fractal dimension calculation as his theoretical one is known. However, there is a fundamental difference between the theoretical calculation of fractal dimension and the actual calculation of the fractal dimension from digital images with a given resolution undergoing discretization. We propose eight different calculation methods based on Richardson’s area–perimeter relationship: the Self-Convolution Patterns Research (SCPR) method accurately estimates the fractal dimension, as the 95% confidence interval includes the theoretical dimension.

1. Introduction

1.1. About Fractal Geometry

In the field of image analysis and pattern recognition, the study of fractal geometry has emerged as a critical approach for characterizing complex structures. Since Benoît Mandelbrot’s seminal work “The Fractal Geometry of Nature” [1], fractal geometry has provided a framework for analyzing natural and artificial patterns that exhibit self-similarity across multiple scales. The fractal dimension, a key parameter within this framework, captures the complexity of these structures by quantifying how details change with scale. This makes it an invaluable tool for applications such as image analysis [2] and feature extraction [3]. Pattern recognition, as a discipline, relies on the accurate identification and classification of features within images [4]. Fractals are used in the target separation branch to enhance the distinction between the object and the background as presented in Zhu et Guo [5]. By constructing a self-similar fractal structure, the model can analyze details at multiple scales, enabling precise capture of complex boundaries. These saliency features with expanded boundaries amplify differences at the edges, thereby improving the complete and accurate separation of the object from the background. The fractal method used by most pattern recognition studies is a variant of the box-counting method, also known as the Minkowski–Bouligand dimension [6]. This variant estimates the fractal dimension by dilating an object at different scales and measuring the area covered at each step. The measurements are obtained using a cost map calculated with the IFT (Image Foresting Transform) algorithm [7]. A logarithmic curve is plotted to show how the area varies with dilation, and the slope of this curve allows the calculation of the fractal dimension. This method is effective for analyzing real and digital objects with partial or complex fractality such as plant leaf structures [8], shoeprint [9], video images [10], and others. Another method was proposed by Plotze et al., stating that a single non-integer number is insufficient to capture the full complexity of an object. The Multi-Scale Fractal Dimension (MSFD) overcomes this by using the derivative of the log–log curve, linking changes in object complexity to visualization scale changes. Unlike traditional fractal dimension, which relies on linear interpolation, MSFD offers more effective object discrimination. It accounts for irregular growth in the degree curve, often due to shape peculiarities, by applying the derivative of the log of the degree with respect to the distance. The Fourier Transform is used for derivative calculation, and a Gaussian low-pass filter is applied to reduce noise and high-frequency information [11].

However, an interesting example appears in the article by Torres [12], using a fractal dimension calculation based on the modified Minkowski–Bouligand dimension mentioned earlier, on a Koch snowflake. The Koch snowflake is a repetition of the Koch curve, which appeared in a 1904 paper titled “On a Continuous Curve Without Tangents, Constructible from Elementary Geometry” [13] by the Swedish mathematician Helge Von Koch. The fractal dimension of the snowflake given by the algorithm is 1.23, while the theoretical fractal dimension is 1.26 [12]. With its known theoretical fractal dimension, the Koch snowflake is used to assess the robustness of a fractal dimension calculation algorithm [14]. The shape of the Koch curve has also been applied to model the shape of fractal antenna [15].

A well-known challenge in calculating fractal dimensions for real structures is the strong dependency of the estimated dimension on the chosen algorithm [16,17]. As a result, each estimate is only directly comparable to those obtained using the same method. Additionally, different algorithms exhibit varying sensitivities to the fractal dimension they aim to estimate ideally, an algorithm should accurately determine the fractal dimension across the entire range. Moreover, the estimation process can be influenced by the resolution of the image and the number of pixels. These issues have been known by mathematicians since the 1990s. Numerous studies have examined the reliability of fractal dimension estimation by applying different algorithms to various fractal functions. These comparisons reveal that different estimation methods produce biased results, complicating their interpretation [18,19]. A key reason for these discrepancies is that real-world data sets are finite, whereas true fractals exhibit infinite resolution. Although mathematical studies have established error bounds for fractal dimension estimation [20,21] they often overlook practical issues, such as the inconsistencies between different estimators.

In this study, we propose eight new different methods to more accurately calculate the fractal dimension of a complex object such as the Koch snowflake. These methods are not based on those presented in the introduction but rather on an approach that is rarely used in pattern recognition yet widely applied in surface topography characterization [22]. This is the Yardstick method, theorized by the mathematician Richardson and later revisited by Mandelbrot in the article “How long is the coast of Britain” [23]. It involves using a fixed-length segment (the “yardstick”) to measure the curve. As the size of the yardstick decreases, the measured length increases, as it captures more details. By plotting the relationship between the measured length and the yardstick size on a log–log graph, the resulting slope allows for the calculation of the fractal dimension [24].

1.2. The Definition of the Koch Curve

The Von Koch snowflake is constructed using an initiator and a generator. The initiator is a triangle (with internal angles of 60°) and side length ${\mathrm{L}}_0$. At each step of the construction, every side of the triangle is replaced by the generator, a segment with a length one-third of the original segment. We are taking the construction formula from the paper from Bigerelle and Iost. [25]. This iterative process is repeated infinitely. The length of the Von Koch Island (perimeter P) is first determined by the following procedure: The Von Koch snowflake is a fractal constructed mathematically by starting with a triangle of side length ${\mathrm{L}}_0$. At each iteration, each side of the triangle is divided into three segments of equal length ${\mathrm{L}}_0/3^{\mathrm{n}}$, where $\mathrm{n}$ is the iteration number. The middle segment is replaced by two segments forming a new triangle “peak”, thus increasing the number of segments by a factor of 4 in each iteration. After n iterations, the curve consists of $3\times 4^{\mathrm{n}}$ segments. The fractal dimension D of the Von Koch snowflake is given by $\mathrm{D}={\displaystyle \frac{\mathrm{l}\mathrm{n}\left(4\right)}{\mathrm{l}\mathrm{n}\left(3\right)}}\approx 1.26$, indicating that it is more complex than a line (dimension 1) but does not fully occupy the plane (dimension 2). The resulting shape has an infinite perimeter yet encloses a finite area, a hallmark of fractal geometry. By eliminating n from the above equations, Equation (1) is obtained.

$${\mathrm{P}}_{\mathrm{n}}={\mathrm{P}}_0^{\mathrm{D}}({\epsilon }_{\mathrm{n}}{)}^{1-\mathrm{D}}$$

(1)

where D = ln 4/ln 3 is the self-similarity dimension.

The von Koch curve is a fractal where the apex angle “α” of the peaks added at each iteration determines its shape and complexity (Figure 1). In fact, when α is different from 60°, similar reasoning leads to Equation (2).

$$\mathrm{D}=\ln 4/\ln [2\left(1 + \cos \alpha \right)]$$

(2)

For α = 60°, representing the classic von Koch curve, D ≈ 1.2619. When α = 90°, the curve is highly complex with D = 2, while for α = 0°, the curve becomes a straight line with D = 1. As α decreases, the curve becomes smoother and less complex, whereas the α values increase the sharpness of the peaks and the fractal complexity of the curve.

Figure 1. The fluctuations observed on the Richardson curve are related to geometrical relations between the yardstick and generator lengths: we obtain α’ = α/2 and then ${\eta }_2 ={ \eta }_1/2\cos (\alpha /2)$, and recursively we obtain ${\eta }_{\mathrm{n}} ={ \eta }_{\mathrm{n}-1}/2\cos (\alpha /2)$.

Figure 1. The fluctuations observed on the Richardson curve are related to geometrical relations between the yardstick and generator lengths: we obtain α’ = α/2 and then ${\eta }_2 ={ \eta }_1/2\cos (\alpha /2)$, and recursively we obtain ${\eta }_{\mathrm{n}} ={ \eta }_{\mathrm{n}-1}/2\cos (\alpha /2)$.

2. Materials and Methods

2.1. Richardson’s Method and Perimeter-Based Fractal Dimension Estimation

The estimation of fractal dimension from geometric contours often relies on the principle originally described by L.F. Richardson [26] as the Richardson–Mandelbrot scaling law. This approach, commonly referred to as Richardson’s method or the compass method, is based on the observation that the measured length $\mathrm{P}(\eta )$ of a complex curve depends on the yardstick size η used to measure it. Specifically, for a self-similar curve, the perimeter scales as in Equation (3) where D is the fractal dimension. Taking logarithms yields a linear relation as in Equation (4) from which D can be estimated as the slope of a linear regression in a log–log plot.

$$\mathrm{P}(\eta )\propto \eta 1-\mathrm{D}$$

(3)

$$\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{P}(\eta )=(1-\mathrm{D})\mathrm{l}\mathrm{o}\mathrm{g} \eta +\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$$

(4)

In the present study, several of the eight tested methods are either direct implementations or digital adaptations of this principle. These include variations in how the measuring yardsticks are applied (fixed vs. sliding), how curvature is handled (straight-line vs. adaptive compass), and how regression is performed (global vs. piecewise). Other methods, such as box-counting or Fourier analysis, use different underlying models but often converge with similar estimates of boundary complexity. To maintain terminological clarity throughout the manuscript, we reserve the term “Richardson’s method” to denote this classical perimeter-scaling approach and explicitly indicate which of our tested algorithms are derived from it. A summary of the computational steps is also provided as a schematic flowchart in Appendix B.

2.2. Computer Software and Statistical Estimation

To investigate Richardson’s method, two original computer programs were especially developed to construct the fractal curves and to calculate their related fractal dimension. The reasons for creating our own computer programs are as follows:

• Images as large as possible are needed to analyze the properties of the fractal curves. The application we developed allows creating images without size limits (except for the RAM memory of the computer).
• Some of the fractal’s properties require a special implementation that will not be found in the usual software.
• As reported in the literature, some results could depend on the implementation such as error discretization or statistical methods. No doubt must remain about numerical implementation to analyze the efficiency of Richardson’s method, and therefore all parts of the software must be controlled without any assumptions about the implementation or the algorithm used.

2.3. Fractal Curve Generation Software (FCGS)

The Iterated Function System (IFS) is the system allowing us to create fractal curves. This process needs two steps to create a curve: 1- defining the coordinates of the initiator and the generator; 2- defining the number of iterations to generate the curve. The Von Koch curve is then created using vector notation. Thanks to the vector representation, the curve is constructed without errors of discretization (except for numerical representation). The total number of coordinates for the Von Koch snowflake, based on a p-sided polygon initiator, is given by $\mathrm{p}\times 4^{\mathrm{i}}$, where i is the number of iterations. A resolution of export must be selected for the software to connect coordinates with line segments, during this process the discretization errors can appear. The images are then saved in the PCX version 5 graphic format (as a widely used and versatile format).

2.4. Fractal Analysis System Software

The program created for the study decodes all PCX images. In our case, we shall analyze only the simulated curves given by the F.C.G.S. The binary images are shown in two colors (white and black), and the black one is considered as matter we shall call Islands. First, for each island, the perimeter is detected, and each point of the perimeter is numerated depending on the connexity used C8D or C4D. We then construct the polygon of pixels and calculate the coordinates (x, y) of each center of the pixels. In this step, the fractal island is defined as a polygon described by a list of pointers. This list points onto the properties of the considered island. A measuring unit of size η then recovers the polygon. Let P₀ be the arbitrary origin (Figure 2) of the covering files, P₁ and P₂, the first successive points of data list such that ${\mathrm{d}}_1 < \eta < {\mathrm{d}}_2$ with ${\mathrm{d}}_1 = \overline{{\mathrm{P}}_0{\mathrm{P}}_1}$ and ${\mathrm{d}}_2 = \overline{{\mathrm{P}}_0{\mathrm{P}}_2}$, and let P_x be the final point such that $\eta = \overline{{\mathrm{P}}_0{\mathrm{P}}_{\mathrm{x}}}$. The coordinates of P_x are the intersections of the circle with center P₀ and radius η with the segment $\overline{{\mathrm{P}}_1{\mathrm{P}}_2}$. This operation is repeated until ${\mathrm{P}}_{\mathrm{x}} = {\mathrm{P}}_0$ and we can then count the number of yardsticks of size η that allows us to recover the perimeter. This operation is then repeated for a yardstick of size $\eta + \delta \eta$ and so on. To construct the data bank of yardsticks [size-perimeters], noted $\left[\eta , {\mathrm{P}}_{(\eta )}\right]$, we choose a minimal yardstick size (in pixel), a maximal one and an increment. However, many algorithm parameters will lead to different estimations of $\mathrm{P}(\eta )$ and statistical artifacts can lead to different estimations of the fractal dimension that will be discussed in the Section 3.3. As regards all the parameters that will change the calculated fractal dimension, it will clearly appear that Richardson’s method must be processed on Personal Software.

In this study, all the analyzed curves are defined by a resolution of 2048 × 2048 pixels, matching the current resolution of the CCD camera used to capture material morphology through optical microscopy.

A resolution of 2048 × 2048 pixels was defined for all the generated curves, as it is like the resolution of the CCD sensor equipped to most of the optical microscopy apparatus.

2.5. Statistical Estimation of the Fractal Dimension

To calculate the fractal dimension, we first choose the range of variation and one of the following incremental laws for the yardstick:

• The yardstick linear variation (YLV): ${\eta }_{\mathrm{n}} ={ \eta }_{\mathrm{n}-1} + \Delta \eta$, where $\Delta \eta$ is the linear increment,
• The yardstick geometrical variation (YGV): ${\eta }_{\mathrm{n}} = \mathrm{q}{\eta }_{\mathrm{n}-1}$, where q is a geometrical increment.

As the fractal dimension is calculated by the slope of the linear range obtained by the least square method plotting values of $\mathrm{F} = \left\{\log \mathrm{P}\left({\eta }_1\right), \log \mathrm{P}\left({\eta }_2\right),\dots , \log \mathrm{P}({\eta }_{\mathrm{n}})\right\}$ versus $\mathrm{E} = \left\{\log { \eta }_1, \log {\eta }_2,\dots , \log {\eta }_{\mathrm{n}}\right\}$, the distribution of the values of the E-set modifies the estimation of the fractal dimension. If the YLV is used, then the E-set will present a lognormal distribution since ${\eta }_1, {\eta }_2,\dots , {\eta }_{\mathrm{n}}$ are regularly spaced. The lower the yardstick, the higher the number of $\log {\eta }_{\mathrm{i}}$ terms. This means that the calculated fractal dimension is more influenced by the perimeter calculated by the lower yardstick than by the higher ones. Consequently, the evaluation of the fractal dimension will include image errors in discretization. In other cases, by using the YGV method then the E-set presents a uniform distribution, and the yardstick range does not influence ∆. To calculate analytically the error made on the determination of the fractal dimension, we suppose that the noise in the determination of the perimeter is independent of the yardstick size (no discretization error) and therefore the standard deviation of the fractal dimension calculated by both YGV and YLV is given, respectively, by Equations (5) and (6).

$${\sigma }_{\mathrm{YGV}}={\displaystyle \frac{\mathrm{s}\sqrt{12}}{\delta \eta \sqrt{\left(\mathrm{n}-1\right)}(\mathrm{n}-1)}}$$

(5)

$${\sigma }_{\mathrm{YLV}}={\displaystyle \frac{\mathrm{s}}{\sqrt{\left(\mathrm{n}-1\right)\left(\frac{{\eta }_{\max }\left({ \ln }^2 {\eta }_{\max }-2\ln {\eta }_{\max }+2\right)-{\eta }_{\min }\left({ \ln }^2 {\eta }_{\min }-2\ln {\eta }_{\min }+2\right)}{{\eta }_{\max }-{\eta }_{\min }}-{\left(\frac{{\eta }_{\max }\left(\ln {\eta }_{\max }-1\right)-{\eta }_{\min }\left(\ln {\eta }_{\min }-1\right)}{{\eta }_{\max }-{\eta }_{\min }}\right)}^2\right)}}}$$

(6)

where n is the number of yardsticks, $\delta \eta = \log { \eta }_{\mathrm{i}}-\log { \eta }_{\mathrm{i}-1}$, ${\eta }_{\max }-{\eta }_{\min } = (\mathrm{n}-1)\delta \eta \prime$ (with $\delta {\eta }^{\prime } = {\eta }_{\mathrm{i}}-{\eta }_{\mathrm{i}-1}$) the yardstick increment for, respectively, the YGV and the YLV model, and s the standard deviation for the regression residuals.

At this stage, three remarks can be stated:

• It can be proved than ${\sigma }_{\mathrm{YLV}} > {\sigma }_{\mathrm{YGV}}$, then the YGV method is always the more appropriate to calculate ∆ with a good accuracy.
• Using the YLV method, the experimental weight is not uniform and ∆ is more influenced by the estimated perimeter for large yardstick rather than for smaller one. The higher ${\eta }_{\max }-{\eta }_{\min }$, the higher the perimeter for large yardsticks.
• As we shall see in the next paragraph, discretization errors can lead to an erroneous measurement of the perimeter for large yardsticks, consequently the YLV method could overestimate or underestimate the fractal dimension of the image.

3. Curve Analyses

We shall then test our methods on Von Koch Island and the stochastic Von Koch Island.

3.1. Analyses on the Von Koch Flake

The fractal dimension of the Von Koch flake is calculated by the following method:

• A 1.2 fractal dimension Von Koch flake with α = 54° instead of 60° is computed with a resolution of 2048 × 2048 pixels (we use this dimension on purpose to compare with the Von Koch flake since it is impossible to construct a Stochastic flake defined in Section 4 without recovering). Seven iterations are carried out to construct the flake.
• The origin of the yardstick is chosen at random.
• The fractal dimension is calculated by the YGV method and the perimeter’s length is computed by Method 4 (floating number of yardstick).

Figure 3 represents the variation in the perimeter versus the yardstick length in log–log coordinates. The upper line corresponds to the true perimeter (same origin for the yardstick and the initiator), and the broken line is the result of the calculation. It is shown that the calculated perimeter’s length is shorter than the true one except if the origin of the yardstick corresponds to the origin of the initiator and if the yardstick’s length is L₀/3ⁱ. From simple geometrical relation (Figure 1), it is obvious that the distance between the maxima is $\log \left[2\cos (\alpha /2)\right] = 0.25$. An interesting discovery can be stated in relation with the serrated variation since a systematic departure from linearity is always observed in real microstructural features such as surface rupture of Titanium alloys and steel [3,4], rock mechanics [5], or wear processes [6]. Such variations may be related to grain diameter or other microstructural parameters to give information on the physical process involved.

From Figure 3, the regression line is parallel to the theoretical one and gives $\Delta = {1.188}_{\pm 0.002}$. If we consider that when $\eta$ < 30 pixels the peak does not match with the upper line due the errors in discretization and when $\eta$ < 10 pixels the undulations are lost in the discretization noise, a new regression performed for 30 < $\eta$ < 300 pixels gives $\Delta = {1.198}_{\pm 0.002}$. By analyzing the computed perimeter, we can observe that this perimeter is more and more underestimated when the size of the yardstick increases. This can be explained away as follows: if the size of the yardstick matches with the length of the initiator, the perimeter will be underestimated except if we choose as origin the origin of the Koch construction. When the size of the yardstick increases, the probability that the yardstick has the same origin as the initiator decreases dramatically. Let us note this probability $\mathrm{P}\mathrm{r}(\eta )$ defined by Equation (7).

$$\mathrm{Pr}(\eta )=\delta \eta /\eta$$

(7)

where $\delta \eta$ is the constant size of the elementary pixel.

Then, Equation (10) shows that the probability to estimate the true perimeter decreases logarithmically with the size of the yardstick. Moreover, for large yardsticks, the curves are noisier and noisier since the variance of the measure of the perimeter increases with the size of the yardstick. Postulating that the variance decreases with the probability that the yardstick has the same origin as the Von Koch initiator becomes obvious.

We can also remark that the graph itself achieves a self-affinity multifractal structure meaning that the fractal dimension depends on the scale [7,8]. Figure 3 shows that some parts of the plot present successive segments with slope 1 meaning that the local fractal dimension is null and not 1.2.

To minimize all these artifacts, the result can be improved by the following averaged method:

• The starting point for the first yardstick is chosen at random.
• A second iteration is carried out taking the previous origin + 1 pixel.
• The operation is repeated for δ varying from 1 to 500 pixels.
• Then the following statistics are built:
  • The computed perimeter is the mean of these 500 perimeters.
  • The computed perimeter is the maximum of these 500 perimeters.
  • The standard deviation is computed.

Figure 4 represents the evolution of mean and maximal perimeters versus the yardstick length. For a long yardstick, a set of computed perimeters is equal to the theoretical one using the maximal value, but using the mean value, the perimeter will always be underestimated. This simulation confirms the hypothesis we state about the underestimation of the perimeter that could then be avoided using the maximal perimeter that approaches the true fractal dimension $\left(\Delta ={ 1.19965}_{\pm 0.00035}\right)$ with much accuracy although the mean perimeter gives $\Delta ={ 1.22}_{\pm 0.02}$. The standard deviation of the perimeter versus the yardstick’s length follows a linear relation $\sigma \left[\mathrm{P}\right(\epsilon \left)\right] = 0.8\epsilon$ and confirms the hypothesis that the measurement of the perimeter is less precise as the size of the yardstick increases.

Series of curves from dimensions 1.1 to 1.9 are now created and discretized at a resolution of 2048 × 2048 pixels with a stick from 1 to 800 pixels. Figure 5 shows the variability of length for the perimeter depending on the stick size for fractal dimensions of 1.1, 1.5 and 1.9. We can draw some insight about those graphs:

• For smaller sticks, the perimeter is increasingly underestimated as the fractal dimension grows and the underestimating range increases critically. This phenomenon is related to the length of the initiator that makes the fractal dimension grows with the smallest size of ${\epsilon }_{\mathrm{n}}$ in the Koch construction. Let ${\epsilon }_{\mathrm{n}}$ represent the length of the last stick after n iterations of the Koch construction. With similar principles as in Equation (1), we can lead eventually to Equation (7).

  $${\epsilon }_{\mathrm{n}}={\mathrm{L}}_0/{10}^{{\displaystyle \frac{\mathrm{n}\log 4}{\Delta }}}$$

  (8)
• The critical value ${\epsilon }_{\mathrm{c}}$, defined by the first significant variation in the log–log representation is plotted versus ${\epsilon }_5$ in Figure 6 (n = 5 means that five iterations are performed to construct the Von Koch Island). The very good correlation shows that the underestimation of the perimeter is a consequence of the size of the lowest initiator met in the construction of the Von Koch flake.

The maxima observed in Figure 5 exist with a path depending on the geometry of the generator. The errors of discretization imply that log P does not vary linearly versus log η and presents a cross-over at a critical value, t. To estimate the influence of this cross over when computing the fractal dimension, ∆ is calculated with t < η < 300 pixels for all the Von Koch islands. Figure 7 shows that the fractal dimensions vary and present oscillations whose maxima fit well with the theoretical value. According to Equations (2) and (7), if we noted d, the distance between two adjacent peaks, as $\cos \alpha /2 = \sqrt{\left(1 +\cos \alpha \right)/2}$, we finally obtain Equation (8).

$$\Delta ={\displaystyle \frac{\log 2}{\mathrm{d}}}$$

(9)

3.

The variance in the estimation of the perimeter rises with the fractal dimension.

4.

For a given yardstick, the perimeter is undervalued as the dimension increase.

From all these remarks, seven methods were developed to calculate the slope of the log–log plot.

Figure 5. Variation in the perimeter length versus the yardstick size (in log–log coordinates) for a triadic 3 Von Koch Island with a self-similarity dimension D = 1.1, 1.5 and 1.9. The island is defined on a 2048 × 2048 pixel grid shown on the graph. The continued line represents the theoretical perimeter and the doted one the regression line [25].

Figure 5. Variation in the perimeter length versus the yardstick size (in log–log coordinates) for a triadic 3 Von Koch Island with a self-similarity dimension D = 1.1, 1.5 and 1.9. The island is defined on a 2048 × 2048 pixel grid shown on the graph. The continued line represents the theoretical perimeter and the doted one the regression line [25].

Figure 6. Correlation between ${\epsilon }_5$ from Equation (10) and ${\epsilon }_{\mathrm{c}}$ corresponding to the first significant undulation of the log–log plot of Figure 5.

Figure 6. Correlation between ${\epsilon }_5$ from Equation (10) and ${\epsilon }_{\mathrm{c}}$ corresponding to the first significant undulation of the log–log plot of Figure 5.

Figure 7. Values ${\mathrm{r}}_{\mathrm{j}}$ and ${\Delta }_{\mathrm{i}}$ versus the minimal size of ${\eta }_{\mathrm{i}}$ (in pixel) for the Von Koch of theoretical fractal dimension of 1.8. We obtain ${\mathrm{r}}_{\max } = 27.5$ that gives a fractal dimension of ${\Delta }_{\max } = 1.74$.

Figure 7. Values ${\mathrm{r}}_{\mathrm{j}}$ and ${\Delta }_{\mathrm{i}}$ versus the minimal size of ${\eta }_{\mathrm{i}}$ (in pixel) for the Von Koch of theoretical fractal dimension of 1.8. We obtain ${\mathrm{r}}_{\max } = 27.5$ that gives a fractal dimension of ${\Delta }_{\max } = 1.74$.

3.2. Method 1: All Range of the Yardstick Variation (ARYV)

The ARYV and MSMV methods are the same as those used in the publication by Bigerelle and Iost, [25] showing similar results. The aim of this study is to include these methods in the general comparison of seven methods. We provide a summary of the ARYV and MSMV methods, as too much detail on previous results would automatically result in plagiarism. This method involves plotting the relationship between area and yardstick length on a log–log scale. It uses the area covered by the curve as a function of yardstick size to estimate the fractal dimension. Calculations show that smaller yardsticks tend to underestimate the fractal dimension, as they do not capture fine details of the curves, leading to a reduced slope in the log–log plot and thus indicating a lower dimension estimate. As presented in

Table 1. Fractal dimension values using the different methods of computation (ARYV to TRROFD). ${∆}_{\mathrm{t}}$ is the theoretical dimension as ${∆}_{\mathrm{c}}$ is the dimension calculated using the range of sticks size (η_min and η_max).

	1 ARYV		2 IEMMV		3 MSMV		4 TMSMV		5 FTPR	6 SCPR	7 RROFD	8 TRROFD
Δt	Δc	ηmin ηmax	Δc	ηmin ηmax	Δc	ηmin ηmax	Δc	ηmin ηmax	Δc	Δc	ηmin ηmax	Δc	ηmin ηmax
1.1	1.102	1 800	1.101	4 580	1.099	1 580	1.102	1 800	1.141	1.096	27 187	1.098	37 172
1.2	1.181	1 800	1.188	6 573	1.183	2 573	1.189	2 800	1.244	1.197	69 712	1.187	37 172
1.26	1.224	1 800	1.241	8 609	1.231	2 609	1.243	3 800	1.244	1.255	53 724	1.260	37 172
1.3	1.248	1 800	1.275	9 619	1.267	4 619	1.285	9 800	1.244	1.295	38 259	1.318	37 172
1.4	1.306	1 800	1.368	13 645	1.352	5 645	1.359	5 800	1.348	1.396	44 362	1.367	37 172
1.5	1.352	1 800	1.459	17 672	1.438	7 672	1.457	10 800	1.452	1.480	59 343	1.444	37 172
1.6	1.389	1 800	1.555	22 681	1.537	14 691	1.543	13 800	1.555	1.581	64 380	1.583	37 172
1.7	1.417	1 800	1.647	28 717	1.644	24 717	1.652	24 800	1.659	1.671	75 408	1.692	37 172
1.8	1.439	1 800	1.738	34 737	1.737	30 737	1.743	30 800	1.763	1.786	40 179	1.792	37 172
1.9	1.462	1 800	1.816	41 755	1.826	35 755	1.825	35 800	1.866	1.901	36 187	1.893	37 172

, the method increasingly underestimates the fractal dimension as it grows. This occurs because ε₅ increases with the fractal dimension, which in turn lowers the slope in the log–log plot due to perimeter underestimation. The result of the calculation of the fractal dimension (1.5) shows that the method is ineffective for accurately determining higher fractal dimensions.

3.3. Method 2: Initiator Epsilon Min-Max Variation (IEMMV)

The slope is calculated as follows. The yardstick varies between the size of the initiator ε₁ (between 580 and 755 pixels) and the smallest size of the iterator after five iterations, i.e., ε₅ (between 4 and 41 pixels). When choosing the yardstick range, we obtain (Table 1) a better estimation of the fractal dimension, and errors vary between 0.01 and 0.09. The best results obtained by Method 2 mean that

• The yardstick’s size must be higher than a critical value corresponding to the beginning of the fractal regime.
• The yardstick’s size must be lower than a critical size depending on the support of the fractal.

In our case, these critical sizes are determined from the values ε₁ and ε₅ related to the Von Koch flake construction. However, the fractal dimension is not well evaluated by this method because the perimeter is underestimated for short yardsticks even if the yardstick size is higher than ε₅. This is since the fractal curve is discretized in a highly anisotropic matrix and then it is not possible for ε₅ to be confounded for each segment with the Von Koch flake even with a high number of randomly distributed origins. This effect is amplified for high fractal dimensions, which increase the anisotropy of the discretized image.

This method which requires the knowledge of both ε₁ and ε₅ (the way to construct the fractal curve and therefore its fractal dimension) is not relevant to calculate the fractal dimension of unknown curves.

3.4. Method 3: Maximal Slope with Minimal Variation (MSMV)

This method, also presented in detail in Bigerelle and Iost [25], measures the mean square variation in the fractal dimension as a function of yardstick size. To evaluate the set of perimeters that is significantly less than the expected one, A linear method on a log–log plot is used to estimate the fractal dimension ${∆}_{\mathrm{j}}$. To avoid errors related to small scales, an estimator ${\mathrm{r}}_{\mathrm{j}}={\displaystyle \frac{{∆}_{\mathrm{j}}}{{\mathrm{S}}_{\mathrm{j}}}}$ is defined, where ${\mathrm{S}}_{\mathrm{j}}$ is the standard deviation of the residuals.

The best estimate of the fractal dimension ${∆}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ is obtained at the highest value of ${\mathrm{r}}_{\mathrm{j}}$. This method allows estimation of the smallest meaningful scale (${\epsilon }_5$) without knowing the construction parameters, but it requires prior knowledge of the largest scale (${\epsilon }_1$).

3.5. Method 4: Total Maximal Slope with Minimal Variation (TMSMV)

This method is the same as the MSMV method except for the largest yardstick which is not given by ε₁ but is equal to 800 (the maximal yardstick size used in our simulations). The results are nearer than those obtained by Methods 3 and 2 in which parameters of the fractal were completely or partially known. Moreover, this method does not require any hypothesis on the flake constructor and can then be applied to any fractal curve and then to any experimental image.

3.6. Method 5: Fourier Transform Patterns Research (FTPR)

A periodicity in the representation of log(P) versus log(η) related to the self-similarity dimension of the Von Koch construction by $\mathrm{D} =\log 2/\mathrm{d}$ is shown in Figure 4 and Figure 5. To estimate d with this relation, we first calculated the fractal dimension by Method 1. Secondly the signal is straightening up on by keeping only the residuals of the least square regression line to avoid lower frequencies related to the slope of the log–log plot which leaves unchanged the d value. Thirdly we processed a Discrete Fourier Transform. Figure 8 represents the spectra of the log–log plot for ∆ = 1.2. As it can be observed, the fundamental peak appears on d = 0.242 and two harmonic peaks at d/2 = 0.126 and d/4 = 0.063. However, even if the border effects are neglected (in our cases we used the Hamming window to process to the weighted moving average transformation), the lowest frequencies are not defined very precisely. In our experiment, the two adjacent periods for d = 0.242 are 0.223 and 0.263. This lower resolution is intrinsic to the Fourier transform and can only be increased by taking a higher number of points, which means increasing dramatically the image size and then the calculation time. However, according to Equation (11) and a fractal dimension varying between one and two, the d-range lies in the [log √2, log 2] interval, i.e., [0.1505, 0.3010]. Consequently, using the Fourier analysis, the precision on ∆ cannot exceed 0.2. In Figure 8, we obtain ∆ = 1.24 which lies in the interval [1.14, 1.35]. This procedure is repeated for the different fractal dimensions and the results with their confidence intervals are shown in

Table 2. Results of the Discrete Fourier Transform. d_theo represents the period of the log–log plot of the Richardson graph according to Equation (11). d_mean, d_inf and d_sup represents the statistics of the adjacent points of the higher harmonics (period) (see Figure 8). The fractal dimension could be calculated from these D values and give an interval of variation ${\Delta }_{\mathrm{c} }\in \left[{\Delta }_{\min },{ \Delta }_{\max }\right]$.

∆	dtheo	dmean	dinf	dsup	∆c	∆inf	∆sup
1.10	0.274	0.264	0.242	0.290	1.14	1.04	1.24
1.20	0.251	0.242	0.223	0.264	1.24	1.14	1.35
1.26	0.239	0.242	0.223	0.264	1.24	1.14	1.35
1.30	0.232	0.242	0.223	0.264	1.24	1.14	1.35
1.40	0.215	0.223	0.207	0.242	1.35	1.24	1.45
1.50	0.201	0.207	0.193	0.223	1.45	1.35	1.56
1.60	0.188	0.194	0.181	0.207	1.56	1.45	1.66
1.70	0.177	0.181	0.171	0.194	1.66	1.56	1.76
1.80	0.167	0.171	0.161	0.181	1.76	1.66	1.87
1.90	0.158	0.161	0.153	0.171	1.87	1.76	1.97

. As can be observed, the uncertainty on the determination of the fractal dimension is around 0.2, which represents an error of 20%. However, the theoretical fractal dimension always lies in the confidence interval determined by this spectrum method, proving that the research pattern method is adequate but that its precision must be improved. The next method will present an original technique to better estimate d.

3.7. Method 6: Self-Convolution Patterns Research (SCPR)

This method aims at determining d precisely and then, according to Equation (11), the fractal dimension. As could be proved for the FTPR method, the resolution of all methods based on the mathematical basis of the Fourier transform is insufficient to calculate the fractal dimension. For this reason, we use an autocorrelation function by calculating the coefficient of regression R(δη) between the points log P(η) and log P(η + δη). By taking the mean of all values for different profiles, we obtain Equation (10).

$$\mathrm{R}(\delta \eta )={\displaystyle \frac{1}{{\sigma }^2}}{\displaystyle \frac{1}{{\eta }_{\max }-{\eta }_{\min }-\delta \eta }}\underset{{\eta }_{\min }}{\overset{{\eta }_{\max }-\delta \eta }{\int }}\log \left({\mathrm{P}}^{\ast }\left(\eta \right)\right)\log \left({\mathrm{P}}^{\ast }\left(\eta +\delta \eta \right)\right)\mathrm{d}\eta$$

(10)

where σ² is the variance of $\log \left({\mathrm{P}}^{\ast }\left(\eta \right)\right)$ with $\log \left({\mathrm{P}}^{\ast }\left(\eta \right)\right)=\log \left(\mathrm{P}\left(\eta \right)\right)-\mathsf{\mu }$ and μ the mean of log P(η).

As we have used the yardstick geometrical variation (YGV) method, each distance between discretized adjacent points is constant and the integral can be discretized by Equation (11).

$$\mathrm{R}\left(\mathrm{i}\right)={\displaystyle \frac{1}{{\sigma }^2(\mathrm{N}-\mathrm{i})}}\sum _{\mathrm{j}=1}^{\mathrm{N}-\mathrm{i}}\log \left({\mathrm{P}}^{\ast }\left(\mathrm{j}\right)\right)\log \left({\mathrm{P}}^{\ast }\left(\mathrm{j}+\mathrm{i}\right)\right)$$

(11)

where N is the number of discretized points $\log \left({\mathrm{P}}^{\ast }\left(\mathrm{j}\right)\right)$.

To find d, δη is calculated such that the functional R(δη) is maximal with $\delta \eta \in [\log \sqrt{2},\log 2]$, R(δη) is plotted versus $\Delta (\delta \mathrm{n}) = {\displaystyle \frac{\log 2}{\delta \eta }}$ (Figure 9) for all the Von Koch curves whose fractal dimension varies from 1.1 to 1.9. The maximum of each curve gives the fractal dimension calculated by the SCPR method with an error lower than 0.03 (Table 1). This method can estimate the highest fractal dimension and is more precise than the previous ones. Moreover, this original method allows us to calculate the angle of the constructor α (with the relation $\mathrm{d} =\log \left[2\cos (\alpha /2)\right]$) without estimation of the regression slope. Figure 5 shows that if m is the number of maxima met in the log–log plot and i the number of iterations for the snowflake, the relation $\mathrm{m} = 2\mathrm{i}-1$ is obtained. Consequently, all the odd peaks represent the value of ε_n used in the construction of the flake.

As the relation ${\epsilon }_{\mathrm{n}} = {\mathrm{L}}_0/{[2\left(1+\cos \alpha \right)]}^{\mathrm{n}}$ holds, by plotting log ε_n versus log n and by the least square linear regression, the values of the ordinate give the values of L₀. Then it becomes very simple to reconstruct all the parameters of the initial fractal curve without restriction. The SCPR method becomes an inverse method that could be used to find the beginning of a fractal process and its origin. However, it can only be used to calculate the fractal dimension of fractal curves constructed by Linear Iterative Function System (LIFS) and cannot be applied to other forms.

3.8. Method 7: Range Research of Optimized Fractal Dimension (RROFD)

RROFD is not a method to calculate the fractal dimension, but it allows us to find the optimal range of [η_min, η_max] on which the calculated fractal dimension corresponds to the theoretical one (Table 1). To find this interval, we calculate the fractal dimension from the set of points

$${\mathrm{E}}_{\mathrm{i},\mathrm{j}}=\left\{\left(\log {\eta }_{\mathrm{i}},\log \mathrm{P}\left({\eta }_{\mathrm{i}}\right)\right),\left(\log {\eta }_{\mathrm{i}+1},\log \mathrm{P}\left({\eta }_{\mathrm{i}+1}\right)\right),\dots ,\left(\log {\eta }_{\mathrm{j}-1},\log \mathrm{P}\left({\eta }_{\mathrm{j}-1}\right)\right),\left(\log {\eta }_{\mathrm{j}},\log \mathrm{P}\left({\eta }_{\mathrm{j}}\right)\right)\right\}$$

noted ${\Delta }_{\mathrm{i},\mathrm{j}}$.

We retain the values (i, j) that give the functional $\left|{\Delta }_{\mathrm{i},\mathrm{j}}-{\Delta }_{\mathrm{theoretical}}\right|$ minimal. It is then possible to plot the surface response of the error $\left|{\Delta }_{{\eta }_{\min },{\eta }_{\max }}-{\Delta }_{\mathrm{theoretical}}\right|$ versus the minimal and maximal sizes of the pixel η_min and η_max used to estimate the fractal dimension. The main problem is that the surface on which optima are found does not consist of valleys, making the optima not very stable, particularly for higher dimensions. Moreover, each optimum depends on the fractal dimension and is quite different. Therefore, all methods used to calculate the fractal dimension must be tested on curves with different fractal dimensions and must lead to robust minima.

3.9. Method 8: Total Range Research of Optimized Fractal Dimension (TRROFD)

This method is close to Method 7 (RROFD) but its aim is to find the unique optimal range $\left[{\eta }_{\mathrm{m}\mathrm{i}\mathrm{n}},{\eta }_{\mathrm{m}\mathrm{a}\mathrm{x}}\right]$ on which all calculated fractal dimensions correspond to the theoretical ones. The research algorithm described in Method 6 is used to retain the shortest size of the pixel η_min and the largest one η_max used to estimate the fractal dimension which minimizes the functional ${\displaystyle \frac{1}{\mathrm{n}}}\sum _{\mathrm{i}=1}^{\mathrm{n}}\left|{}^{\mathrm{i}}{\Delta }_{{\eta }_{\min ,}{\eta }_{\max }}-{}^{\mathrm{i}}{\Delta }_{\mathrm{theoretical}}\right|$ where 1.1 < ${}^{\mathrm{i}}∆$ < 1.9 represents the fractal dimension for the ith Koch flakes between n curves. The surface response of the mean error for η_min = 37 and η_max = 172 gives a fractal dimension with an error lower than 0.016 (Table 1). Note that changing the previous functional by $\sqrt{{\displaystyle \frac{1}{\mathrm{n}}}\sum _{\mathrm{i}=1}^{\mathrm{n}}{\left({}^{\mathrm{i}}{\Delta }_{{\eta }_{\min ,}{\eta }_{\max }}-{}^{\mathrm{i}}{\Delta }_{\mathrm{theoretical}}\right)}^2}$ does not affect ${\mathsf{\eta }}_{\mathrm{m}\mathrm{i}\mathrm{n}}$ and ${\mathsf{\eta }}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ proving that the optimization problem is well stated.

3.10. Influence and Optimization of Yardstick Ranges

The accuracy of fractal dimension estimation using Richardson’s method strongly depends on the selection of the yardstick range, defined by ${\mathsf{\eta }}_{\mathrm{m}\mathrm{i}\mathrm{n}}$ and ${\mathsf{\eta }}_{\mathrm{m}\mathrm{a}\mathrm{x}}.$ Each of the eight methods described in this study adopts a different strategy for choosing or optimizing this interval.

Table 3. Definition of $\eta \mathrm{m}\mathrm{i}\mathrm{n} \mathbf{a}\mathbf{n}\mathbf{d} \eta \mathrm{m}\mathrm{a}\mathrm{x}$in the eight methods.

Method	Full Name	$${\mathsf{\eta }}_{\mathbf{m}\mathbf{i}\mathbf{n}}$$	$${\mathsf{\eta }}_{\mathbf{m}\mathbf{a}\mathbf{x}}$$	Description
1. ARYV	All Range of the Yardstick Variation	Fixed (e.g., 1 pixel)	Fixed (e.g., 800 pixels)	Full range arbitrarily chosen; prone to errors due to discretization (low η) or oversmoothing (high η).
2. IEMMV	Initiator Epsilon Min-Max Variation	ε5 = smallest segment of the generator (iteration 5)	ε1 = initial segment of the initiator	Requires knowledge of the fractal’s construction parameters; not usable on unknown curves.
3. MSMV	Maximal Slope with Minimal Variation	Selected to minimize slope variance	Selected to minimize slope variance	Searches for stable intervals with low slope fluctuation, without relying on explicit geometric knowledge.
4. TMSMV	Total Maximal Slope with Minimal Variation	Like MSMV	Fixed at 800 pixels	More general version of MSMV that can be applied to unknown or experimental fractals.
5. FTPR	Fourier Transform Pattern Research	Implicit (derived from residuals)	Implicit (derived from residuals)	No explicit η range; analysis is based on the periodicity of residuals in the log–log perimeter plot.
6. SCPR	Self-Convolution Patterns Research	Determined automatically via autocorrelation	Determined automatically via autocorrelation	Peak lag ${\mathrm{r}}_{\mathrm{m}\mathrm{a}\mathrm{x}} \mathrm{defines} \mathrm{the} \mathrm{periodicity}; {\mathsf{\eta }}_{\mathrm{m}\mathrm{i}\mathrm{n}} \mathrm{and} {\mathsf{\eta }}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ are implicit from the correlated signal.
7. RROFD	Range Research of Optimized Fractal Dimension	Explored by grid search	Explored by grid search	Optimized per curve to find the best $[{\mathsf{\eta }}_{\mathrm{m}\mathrm{i}\mathrm{n}}, {\mathsf{\eta }}_{\mathrm{m}\mathrm{a}\mathrm{x}}]$ interval with minimal deviation from theoretical D.
8. TRROFD	Total Range Research of Optimized Fractal Dimension	${\mathsf{\eta }}_{\mathrm{m}\mathrm{i}\mathrm{n}}$ = 37 pixels	${\mathsf{\eta }}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ = 172 pixels	Globally optimized range that minimizes average error across all tested fractal dimensions; provides stable and general recommendation.

summarizes how these values are defined or computed in each method, whether they are fixed, derived from the fractal geometry, statistically optimized, or indirectly determined through signal analysis.

4. Analyses of the Stochastic Von Koch Flake

We shall now introduce the stochastic construction of the Von Koch flake (size 2048 × 2048 with five iterations) (Figure 10). All the steps of the iterative construction are the same as in the deterministic case, but the direction of the initiator is chosen at random, with a probability ½, (these operations leave unchanged the theoretical fractal dimension). These fractal curves presenting a stochastic structure met in nature are more appropriate to test Richardson’s method (Figure 11) and introduce a stochastic measure for the fractal dimension that was impossible for deterministic curves. However, during the construction, some recovering may appear that diminishes the fractal dimension of the curves leaving unchanged the self-similarity dimension. We have shown that when using a fractal dimension lower than 1.2, no recovering appears and then the fractal dimension of the flake equals its self-similarity dimension (∆ = D). It is to be noticed that some analytical models applied to calculate the fractal dimension of ruptured surfaces do not take this fact into account and give overestimated values of ∆.

To estimate the fractal dimension of the stochastic Von Koch flake, the following scheme is applied:

• 100 stochastic Von Koch flakes are constructed with five iterations and a resolution of 2048 × 2048 pixels.
• The fractal dimensions are calculated by the six different methods (ARYV, IEMMV, MSMV, TMSMV, SCPR, and TRROFD (taking the range obtained by the nearer optimization, i.e., η ∈ [37, 172]) as indicated by Method 8.
• Statistics on the 100 fractal dimensions are performed (mean, standard deviation, standard error, min, max, median, 95% confidence level for the mean) (

  Table 4. Descriptive statistics on the calculation of the fractal dimension (theoretical fractal dimension 1.2) shown in Figure 10 for 100 Stochastic Von Koch flakes built with five iterations on a resolution of 2048 × 2048. Fractal dimension is calculated by the six different methods: ARYV, IEMMV, MSMV, TMSMV, SCPR, and TRROFD. IC 95% represents the 95% confidence interval of means, and Std. dev the standard deviation.

  Method	Mean	−IC 95%	+ IC 95%	Median	Minimum	Maximum	Std Dev
  ARYV	1.186945	1.186567	1.187322	1.186793	1.182414	1.192838	0.001892
  TMSMV	1.197857	1.197272	1.198442	1.197949	1.191600	1.207049	0.002932
  SCPR	1.199980	1.198943	1.201017	1.198000	1.191000	1.212000	0.005200
  TRROFD	1.197210	1.195674	1.198745	1.196271	1.180984	1.214840	0.007698
  IEMMV	1.196716	1.196080	1.197351	1.196651	1.190029	1.207746	0.003184
  MSMV	1.180796	1.180464	1.181128	1.180775	1.177221	1.185031	0.001665

  ).

The SCPR Method perfectly estimates the fractal dimension because the 95% confidence interval includes the theoretical dimension. On stochastic curves, the results obtained by this original method are also very accurate. The TRROFD method also speaks for itself, confirming the fact that the range on which the yardstick is chosen plays an important role in giving a precise determination of the fractal dimension. Moreover, this method helps to determine the yardstick range. The methods ARYV and IEMMV underestimate Δ, proving once more that the yardstick range cannot be chosen arbitrarily. Because of the lack of precision on the measure of the fractal dimension, the MSMV method is not appropriate for stochastic curves. Finally, the TMSMV method is accurate and can be applied both to stochastic and deterministic curves.

A promising avenue for future work is the extension of the He–Liu formulation [27], originally developed for porous materials, to partially fractal geometries such as the stochastic Von Koch curve as shown in Appendix A. By treating the probability p of recursive refinement as a proxy for geometric porosity, we derive in Appendix A, a heuristic dimension α(p,n) that captures the progressive loss of complexity. This model enables a continuous transition between linear and fully fractal regimes and defines a critical threshold p_transition where the structure reaches the classical Koch dimension. These results suggest that the He–Liu approach could bridge geometric and physical perspectives in fractal analysis.

5. Conclusions

This study highlights both the strengths and limitations of using Richardson’s method for estimating the fractal dimension of self-similar structures. A key strength of our approach is the development and comparison of multiple computational methods, allowing for a more robust and precise estimation of fractal dimensions, particularly when applied to digital images. Methods such as the Self-Convolution Patterns Research (SCPR) technique demonstrated high accuracy, with confidence intervals including the theoretical fractal dimension. Additionally, our findings emphasize the importance of selecting an appropriate yardstick range, as arbitrary choices can significantly bias results. However, certain limitations persist. The dependency of the estimated fractal dimension on the chosen algorithm remains a challenge, making cross-method comparisons difficult. Discretization errors introduce artifacts, particularly for large yardstick sizes, affecting the precision of the perimeter estimation. Moreover, while our approach is effective for deterministic fractals such as the Koch snowflake, its applicability to more complex, naturally occurring fractals or stochastic fractal structures requires further investigation. Future research could extend these findings by integrating physical simulation models to explore how characteristic scales of interface modification emerge under specific physical mechanisms, such as erosion, diffusion, or mechanical stress. Additionally, refining computational techniques to mitigate discretization errors and improve cross-method comparability will be crucial for advancing fractal analysis in various scientific and engineering applications.

In conclusion, this study demonstrates the effectiveness of Richardson’s method for estimating the fractal dimension of self-similar structures. Our analyses underscore the emergence of artifacts when the condition of self-similarity is not strictly met, highlighting the critical role of selecting appropriate parameters for accurate analysis. We observed that the fractal dimension is influenced by various morphological factors in the Koch curve construction, suggesting a potential link between shape parameters and underlying physical processes.

Using an innovative approach, we analyze the perimeter as a function of yardstick size, enabling the extraction of key parameters for constructing fractal curves. This inverse method proves valuable in identifying critical experimental features. Furthermore, our findings confirm that specific methods, such as Self-Convolution Patterns Research (SCPR), provide accurate fractal dimension estimations, whereas methods like All Range of the Yardstick Variation (ARYV) and Initiator Epsilon Min-Max Variation (IEMMV) reveal the sensitivity of results to the chosen yardstick range. This study also highlights the suitability of certain methods for either stochastic or deterministic curves, offering robust tools for fractal analysis.

Looking ahead, this work opens promising avenues for leveraging physical simulation models on fractal constructions such as the Von Koch curve. These models could explore how characteristic scales of interface modification emerge under specific physical mechanisms. For instance, simulations of erosion, deposition, or diffusion along a fractal interface could help identify critical points of interaction, while models of mechanical stress or adhesion could elucidate the role of fractality in fracture propagation. Moreover, dynamic simulations involving fluid–structure interactions, growth processes, or thermal ablation could reveal how external forces alter the geometry of fractal structures. By combining these simulations with a quantification approach based on morphological indicators, such as fractal dimension or specific roughness parameters, it would be possible to systematically study the interplay between fractal geometries and physical phenomena. These methodologies may provide new insights into interface dynamics, enabling the identification of characteristic features and scales relevant to applications in materials science, surface engineering, and even biological systems.