The differential box-counting (DBC) method is useful for determining the fractal dimension of grayscale images. It is simple to learn and implement and has been extensively utilized. However, this approach has several problems, such as over- or undercounting the number of boxes due to inappropriate parameter choices, limiting the calculation accuracy. Many studies have been conducted to increase the algorithm's computational accuracy by improving the calculating parameters of the differential box-counting method. The grid size is a crucial parameter for the DBC method. Generally, there are two typical ways for selecting the grid size in relevant studies: consecutive integer and divisors of image size. However, both methods for grid size selection are problematic. The consecutive integer method cannot partition the image entirely and will result in the undercounting of boxes; the divisors of image size can partition the image completely. However, this method uses fewer grid sizes to compute fractal dimensions and has a relatively huge distance error (DE). To address the shortcomings of the above-mentioned two approaches, this research presents an improved grid size selection strategy. The improved method enhances computational accuracy by computing the discarded image edge areas in the consecutive integer method, allowing the original image information to be used as thoroughly as the divisor strategy. Based on fractional Brownian motion (FBM), Brodatz, and Aerials image sets, the accuracy of the three grid size selection techniques (consecutive integer method, divisors of image size method, and the improved algorithm) to compute the fractal dimension is then compared. The results reveal that, compared to the two prior techniques, the revised algorithm described in this study minimizes the distance error and increases the accuracy of the fractal dimension computation.
Keywords: fractal dimension; differential box-counting; grid size selection; fractional Brownian motion; Brodatz database; Aerials database
An image captured by a camera is a projection of an actual three-dimensional shape on a plane. The image textures can reflect the morphological features of the physical object. As a result, objects can be identified by evaluating the grayscale variations of an image. Textured surfaces, on the other hand, are inherently complex in natural scenes. Irregular and complex objects cannot be described using Euclidean geometry. Mandelbrot proposed utilizing fractal geometry to describe this type of phenomenon in 1983 [[
Researchers have proposed many calculating methods to precisely estimate the fractal dimension. Mandelbrot first proposed calculating the fractal dimension when determining the length of the British coastline [[
In 1994, Sarkar et al. [[
For the DBC method, the fractal dimension is calculated by fitting a series of points (log1/r, logN
Jin et al. [[
Chen et al. [[
Long et al. [[
Liu et al. [[
According to the studies mentioned above, each algorithm overcomes the shortcomings of the original DBC method by optimizing the calculation parameters to enhance the computing accuracy and minimize distance error. Various parameters affect the computation of the fractal dimension; different approaches have different optimization focuses. Nevertheless, they all achieved better results than the DBC method under their corresponding validation method. However, some studies failed to provide more persuasive evidence on the enhanced accuracy because their validations were not performed using images with known theoretical fractal dimensions, such as synthesized FBM Database [[
Among the various parameters utilized in DBC calculations, the grid size s is crucial. The choice of grid size for a square image impacts how the plane is partitioned during computing. The original DBC approach was based on consecutive integer partitioning, which discards areas that cannot be partitioned in an integer manner [[
These two approaches for calculating s values are frequently used in DBC algorithms. Both strategies, however, have limitations. The consecutive integer partitioning method discards the boundary region; hence, the original image is not fully utilized, which will lead to the problem of undercounting boxes. Moreover, as noted previously, the curve of the consecutive integer method is shaped like "steps" at large s values in the log–log plots, which will also affect the calculation accuracy [[
The weight approach has been utilized to compute n
In order to further investigate the impact of the grid size selection on fractal dimension calculation, this study first introduces the principles of two typical grid size selection methods and assesses their advantages and disadvantages. Then, by processing the discarded edge regions of an image with the weight method, an improved strategy is proposed to tackle the undercounting problem of the original consecutive integer approach and the result distortion problem of the divisor method. Subsequently, based on three image sets (synthesized FBM database, Brodatz database, and Aerials database), the impacts of the three grid size selection methods (original consecutive integer, divisors of image size, and the improved method) were then analyzed, evaluated, and compared. The results reveal that, compared with the original consecutive integer method and the divisor method, the DBC technique based on the improved grid size selection strategy can better estimate fractal dimension values, produce lower DE values, and obtain more consistent goodness of fit values. The various notations and abbreviated terms used in this work are summarized in Abbreviations.
The remaining parts of the paper are as follows: Section 2 describes the classic DBC method, the typical grid size selection strategies, and the drawbacks of each strategy. In the third section, an improved grid size selection strategy based on the consecutive integer method is proposed. Section 4 discusses the three image databases (FBM, Brodatz, Aerials) utilized for validation and evaluation metrics. Section 5 describes and discusses the results of the two typical grid size selection strategies and the improved one based on three image databases. Section 6 presents the conclusions.
In 1994, Sarkar et al. [[
(
(
G is the total number of grayscale orders in the 8-bit grayscale image, which is 256; g
(
Finally, the fractal dimension D is computed by fitting the points (log1/r, logN
Graph: Figure 1 Sketch of determination of the number of boxes (n r) by DBC method (here M = 12, s = 3, n r is 3 for the grid).
Many modified DBC methods have emerged based on Sarkar's DBC method. The grid size s is an essential parameter in these methods. It determines how to partition the xy plane. In general, there are two types of grid size selection methods: consecutive integer and divisors of image size. However, each of these solutions has its shortcomings.
The original DBC method uses all successive integers between s
We take a 512 × 512 image as an example. For the consecutive integer method, s
The results in Figure 3 can then be obtained based on the above analysis. The relationship between the actual area for calculation (percentage of the image area) and the gird size s is depicted in Figure 3. The amplitude of the curve increases with increasing s, as shown in the figure. The minimum value of the curve is even less than 50%. When consecutive integers are used as s values, the area used for calculation differs significantly from the actual image area. This deviation becomes more pronounced as the s value increases. Because of the discarded areas, the number of boxes is undercounted, resulting in distorted results.
In theory, a smaller s value should result in a larger N
Biswas [[
However, the reduced number of s values introduces other issues. Images of natural settings are frequently not ideal fractal objects [[
As mentioned above, the divisor method can thoroughly partition the image, but the number of s values is minimal, which can easily lead to distorted results. Although more s values are used in the consecutive integer technique, most of the s values, which cannot entirely partition the image, are prone to the undercounting problem of boxes, especially at large grid size s. Thus, we expect an improved method that can entirely partition the image like the divisor method, but also has numerous s values involved in FD calculation like the consecutive integer method.
To achieve this, we can improve the consecutive integer method by supplementing it with the undercounting boxes. If a given value of s does not completely partition the image, a zone with a width less than s is generated at the boundaries. This is depicted in the blue region of Figure 4. Although the area of each grid in the blue region is less than s × s, it also corresponds to a part of a two-dimensional surface with grayscale variations inside, and the n
The weight method would assign the corresponding weights according to the actual area of the grids [[
(
(
The code of the DBC method based on two consecutive integer methods is shown in Figure 5. The improved method's specifics are as follows:
Partition an
Graph: Figure 5 Algorithm code of the DBC method with different consecutive integer methods (Left: the original one; Right: the improved one).
By replacing Equation (
(
Finally, linear least squares regression (LLS) is used to fit the obtained points (log1/r, logN
To compare the performance of the consecutive integer technique, the divisor method, and the improved method, we validate these three approaches using the synthesized FBM images, Brodatz database, and Aerials database. The FBM images can be synthesized based on the theoretical fractal dimension, and the produced grayscale images can be used to validate the accuracy of the three algorithms. The Brodatz and the Aerials databases are typical for validating the fractal dimension algorithm, both obtained in natural scenes. With the help of the Brodatz and the Aerials databases, the algorithms' performance for texture images and aerial photographs could be evaluated, respectively.
In this research, FBM images are synthesized based on the random midpoint displacement (RMD) method. [[
Nine images with a resolution of
The Brodatz database [[
As illustrated in Figure 8, twelve satellite images from the SIPI image database [[
Gaylevel = 0.2989R + 0.5870G + 0.1140B (
Generally, LLS fit to a set of points in a log–log plot is required to calculate the fractal dimension. The goodness of fit (R
The goodness of fit is used to evaluate the fit of the regression line to the observed values. The maximum value of R
Then, calculate the total sum of squares (TSS):
(
Calculate the residual sum of squares (RSS):
(
Then, we can get:
(
The fractal dimension is calculated using LLS fitting in the various DBC techniques. The distance error (DE) is defined as the root-mean-square distance between the scatter points and the fitted line in a log–log plot. DE is a crucial metric for evaluating DBC methods and has been utilized in many studies [[
(
Correlation coefficients are used to show the linear relationship between two sets of data. The algorithms provide a number between −1 and 1, where 1 represents a strong positive relationship, −1 represents a strong negative association, and zero represents no relationship at all. For two sets of variables, X(x
(
(
Combining the above content, we can get the experimental process of this study, as shown in Figure 9. The experimental results in the fifth section are obtained according to this procedure.
The RMD method was used to generate FBM images with theoretical fractal dimensions ranging from 2.1 to 2.9. The following methods were used to calculate grid size: divisor of image size (ER), original consecutive integer method (CI), and optimized consecutive integer method (OCI). Table 2 displays the fractal dimension calculation, the goodness of fit, and the distance error corresponding to the three methods. Figure 10, Figure 11 and Figure 12 illustrate the data in Table 2. The correlation coefficients and slope
Figure 10 shows that the fractal dimension value calculated by the three methods rises as the TFD value increases. The consecutive integer method achieves the largest value from the fractal dimension value calculation, the divisor method produces the smallest value, and the computed value of the enhanced consecutive integer method is between the two. According to the data in Table 3, the improved approach has the largest correlation coefficient, the original consecutive integer method has the smallest, and the divisor method has a correlation coefficient between the two. This demonstrates that the fractal dimension values calculated by the improved consecutive integer method correlate better with TFD values than those of the other two methods. In terms of slope, the improved consecutive integer method has a somewhat lower slope than the original method. However, both have higher slope values than the divisor method. The above results demonstrated that, compared to the consecutive integer methods, the computed fractal dimensions derived by the divisor method have a relatively good correlation with the TFD values but the trend deviation is large. The calculated results of the original consecutive integer method can better match the trend of the TFD values, but the correlation is the worst. The improved method outperforms the other two in terms of linear correlation between calculated FD and TFD values, although it has slightly lower slope values than the original consecutive integer method.
When comparing the goodness of fit results in Table 2 and Figure 11, it can be observed that the improved method has the highest goodness of fit, followed by the divisor method, and the original consecutive integer method has the lowest goodness of fit. When the DE values are compared (Figure 12), the divisor method has the largest distance error; the consecutive integer method has a relatively moderate distance error, and the enhanced method has the smallest distance error.
To investigate the causes of this phenomenon, we generated Figure 13 based on the calculation of the FBM image with a theoretical fractal dimension of 2.9. Figure 13a–c depict the log–log plot curves of the three approaches, respectively. We produced Figure 13d to compare the difference in the number of boxes between the improved consecutive integer method and the original method.
In this work, eight s values are employed for the divisor method and 255 for the consecutive integer method. These 255 s values contain the divisor method's eight s values. This is equivalent to the consecutive integer technique having 247 more data points than the divisor method for LLS fitting. The absence of these 247 points is also responsible for the distinction between the divisor method and the consecutive integer method.
According to Figure 13c, the divisor method requires fewer scatter points for the calculation, which is related to the lower number of s values needed for the calculation. Fewer points reduce the computing work accordingly. However, the relatively small number of grid size values leads to the insufficient utilization of the original image, which can easily distort the fractal dimension result. As a result, although the divisor method has a higher R
For the consecutive integer methods, more s values are used, and more corresponding N
The improved method considerably improves the number of computed boxes compared to the original consecutive integer method, as shown in Figure 13d. The higher the s value, the greater the percentage of improvement. This curve resembles the trend in Figure 3. It can be interpreted as the improved approach compensating exactly for the original method's discarded boxes, and the larger the s value, the larger the percentage of compensation. Compared to the original consecutive integer method, the improved method makes better use of the image's grayscale information. The compensating for the discarded boxes, on the one hand, raises the N
On the other hand, the "steps" in the curve of the original consecutive integer method are deleted, the incorrect box-counting is revised, and the computational accuracy is enhanced. Thus, compared to the original consecutive integer method, the improved method achieves larger goodness of fit and a smaller distance error, and the computed fractal dimension has a better correlation with the TFD value.
In summary, the improved consecutive integer method outperforms the divisor method and the original consecutive integer method in computing accuracy for the fractal dimension calculation of FBM images.
The Brodatz texture database, a commonly used database in DBC investigations, is utilized for validation to examine the three methods' performance in fractal dimension calculation of texture images. This database is composed of grayscale images with various textures. The calculation in this section is based on the consecutive integer method, the divisor method, and the improved consecutive integer method for 16 images from the Brodatz database. Table 4 displays the acquired results for fractal dimension, the goodness of fit, and distance error, and Figure 14, Figure 15 and Figure 16 illustrate the data in the table.
Figure 14 shows that the three methods for calculating the fractal dimension have a similar variation trend. The trend depicted in Figure 14 is generally consistent with the prior study [[
The Brodatz texture database, unlike the FBM images, is derived from the natural scene. As discussed in Section 2.2.2, images of natural scenes are not always ideal fractal patterns [[
The preceding section mentioned that using fewer s values can result in distorted computing results for the divisor method. For the Brodatz database, the distortion of the divisor method results in small fractal dimensions and large distance errors. This result agrees with the FBM-based study described above.
In summary, the improved consecutive integer method outperforms the original method for texture image calculation. Because of distortion, the divisor method does not produce satisfactory results when calculating texture images. In 2021, Liu et al. [[
To compare the performance of the three methods, twelve images from the Aerials database with a resolution of 512 × 512 were chosen. As these are high-altitude aerial images, they are suitable for remote sensing applications. Figure 17, Figure 18 and Figure 19 and Table 5 illustrate the fractal dimension, goodness of fit, and DE values obtained by three different methods.
According to Figure 17, the estimated fractal dimensions are identical to those of Brodatz and FBM. The original consecutive integer method produces the highest fractal dimension, the divisor method produces the smallest fractal dimension, and the improved consecutive integer method produces values in between. In terms of goodness of fit, the improved method outperforms the original consecutive integer method. Unlike Brodatz and FBM, the divisor technique's goodness of fit for Aerials images is almost equivalent to that of the improved consecutive integer approach. The distance error results are similar to the FBM and Brodatz results, with the divisor method generating the highest error, followed by the original consecutive integer method; the improved consecutive integer method produces the lowest error.
The Aerials images are pictures of natural scenes, which are also not ideal fractals. Compared to Figure 14 and Figure 16 in the preceding section, the fractal dimension and distance error results of the Aerials images (Figure 17 and Figure 19) are extremely similar to the computed results of Brodatz images. This also validates the conclusion of the previous section. It indicates that, for Aerials images, the improved method still achieves lower distance errors than the divisor method and the original consecutive integer method.
However, unlike the results for Brodatz images, the goodness of fit values obtained by the divisor method are not intermediate between the two consecutive integer methods. We indicated in Section 5.1 that the divisor method requires eight points to fit an LLS line, whereas the consecutive integer method takes 255 points, including the 8 points of the divisor method and the other 247 data points. The absence of 247 points in the divisor method makes the calculation results "distorted". The effect of the "distortion" varies depending on the image type. Comparing Figure 7 and Figure 8, it can be seen that the aerial image has different characteristics from the texture image. The Brodatz texture image is more homogeneous and has basically the same features in all image regions, but the Aerials image usually appears with some conspicuous objects.
Because there are fewer s values in the divisor technique, the fitting results will be considerably influenced if the image's conspicuous objects influence certain s values. Because the consecutive integer method uses a large number of s values, some abnormal scatter points do not affect the results. As a result, in Figure 18, the divisor method's goodness of fit curve seems to oscillate around the curve of the improved consecutive integer method. Of course, further research is required to confirm this. In summary, the improved consecutive integer method outperforms the divisor method and the original continuous integer method for computing the fractal dimension of Aerials images.
From the above analysis based on the original DBC method, it is found that the improved grid selection strategy, i.e., the improved consecutive integer method, can obtain better accuracy of fractal dimension calculation. For further validation, we tested the effectiveness of the improved strategy based on the three methods of Long'2013DBC [[
From the fractal dimension results in Table 6, we can find that the Long'2013 method with the improved strategy obtains a higher fractal dimension than the original method. FDs of Lai'2016 and Liu'2021 methods show a fractal dimension decrease after applying the improved strategy. From the results in Table 7 and Table 8, it can be found that for all three methods, the improved strategy improves the goodness of fit and reduces the distance error compared to the original methods.
The strategy of integer r used in the Long'2013 method leads to fewer values of s involved in the calculation. This is very similar to the divisor method. Therefore, it is easy to produce the problem of small fractal dimension, large goodness of fit, and large distance error. However, because the weight method is used in Long's method, the distance error obtained is not much different from that of the consecutive integer method even though the above problems exist. After applying the "consecutive integer method + weighting method" strategy, Long's method could involve more s values and fully use the grayscale variation of the original image, resulting in a larger fractal dimension, higher goodness of fit, and smaller distance error.
In both Lai'2016 and Liu'2021 methods, the original consecutive integer method is used as the grid selection strategy. Thus, even though various optimizations have been performed in their methods, the problems caused by the original consecutive integer method still exist. Based on the analysis of previous sections, the original consecutive integer method has the problem of undercounting boxes. The larger the value of s, the higher the percentage of undercounting boxes. After applying the improved strategy, it is equivalent to raising the vertical coordinates of the points corresponding to large s values. This change reduces the excessive fractal dimension while increasing the goodness of fit and reducing the distance error. The fractal dimension, goodness of fit, and distance error results for Lai'2016 and Liu'2021 (Table 6, Table 7 and Table 8) confirm this conclusion.
In summary, the grid selection strategy based on the "consecutive integer method + weight method" can improve the goodness of fit, reduce the distance error, and obtain more accurate fractal dimension results for DBC methods. However, from the results in this section, although the grid selection strategy improves the computational accuracy compared to the original method, the calculated results of the three methods are very different. This is because many parameters affect the fractal dimension, and each method uses different optimization strategies for other parameters, resulting in many deviations in the results.
This study proposes an improved consecutive integer method to address the current difficulties of distorted calculation results and huge distance errors caused by the inappropriate choice of grid size s in the differential box-counting method. The synthetic FBM images, Brodatz database, and Aerials database are then used to evaluate and compare the effects of three grid size selection methods, namely the consecutive integer method, the divisor method, and the improved consecutive integer method, on the accuracy of fractal dimension calculation. Except for the different grid selections, all other parameters are identical to the DBC approaches. The results indicate that the original consecutive integer method ignores the boxes along the edges, resulting in fewer boxes to calculate. This reduces the goodness of fit and increases the distance error, distorting the estimated fractal dimension.
Although the divisor method has a small computation and can partition the whole image completely, the number of s values is too small. Then, a large amount of effective information is ignored compared with the consecutive integer method, resulting in a severe distortion of fractal dimension results and more significant distance error, and the fitting goodness is not stable. The improved strategy solves the undercounting problem of the original consecutive integer method by retaining the edge regions of images. Furthermore, the method can partition the image completely like the divisor method. Thus, the improved consecutive integer method obtains a smaller distance error than the original consecutive integer method and the divisor method, improving the accuracy of fractal dimension calculation.
Graph: Figure 2 The orange area is the calculated area, and the blue area is the ignored area. Image size is 512 × 512. (a) s = 128, (b) s = 160, (c) s = 220, (d) s = 240, (e) s = 256.
Graph: Figure 3 The relationship between the grid size and the percentage of the calculated area to the actual area of the image.
Graph: Figure 4 A part of the M×M picture is partitioned by square grids (
Graph: Figure 6 Nine synthesized FBM images with their TFD values.
Graph: Figure 7 Sixteen sample natural grayscale images of the Brodatz database. The numbering sequence starts from the first row, left to right, top to bottom: D1, D11, D21, D30, D41, D47, D53, D61, D65, D74, D81, D85, D91, D99, D104, D112.
Graph: Figure 8 Twelve images of Aerials database. The numbering sequence starts from the first row, left to right, top to bottom: 2.1.01~2.1.12.
Graph: Figure 9 Computational flow chart for experimental research.
Graph: Figure 10 Calculated fractal dimension of images with different theoretical fractal dimension values (FBM).
Graph: Figure 11 Goodness of fit of images with different theoretical fractal dimension values (FBM).
Graph: Figure 12 Distance error (DE) of images with different theoretical fractal dimension values (FBM).
Graph: Figure 13 LLS fitting lines of FBM images with a theoretical fractal dimension of 2.9 by three methods. (a) Consecutive integer; (b) improved consecutive integer method; (c) image size divisor; (d) compared with the original consecutive integer method, the improved method has an increase in the number of boxes under different s values.
Graph: Figure 14 Calculated fractal dimension of different images (Brodatz).
Graph: Figure 15 Goodness of fit of different images (Brodatz).
Graph: Figure 16 Distance error (DE) of different images (Brodatz).
Graph: Figure 17 Calculated fractal dimension of different images (Aerials).
Graph: Figure 18 Goodness of fit of different images (Aerials).
Graph: Figure 19 Distance error (DE) of different images (Aerials).
Table 1 Parameters for synthesizing FBM images.
Item Value Maxlevel 9 Sigma 4 H 0.1~0.9 Addition 1 Seed 1
Table 2 Fractal dimension, the goodness of fit, and distance error of three methods based on FBM images.
TFD FD R2 DE ER CI OCI ER CI OCI ER CI OCI 2.1 2.0653 2.2097 2.1255 0.9979 0.9925 0.9992 0.0232 0.0045 0.0015 2.2 2.1116 2.2490 2.1654 0.9980 0.9928 0.9995 0.0230 0.0045 0.0012 2.3 2.1512 2.3030 2.2182 0.9986 0.9930 0.9995 0.0189 0.0044 0.0011 2.4 2.2171 2.3733 2.2857 0.9986 0.9931 0.9995 0.0192 0.0044 0.0011 2.5 2.2820 2.4490 2.3551 0.9985 0.9930 0.9995 0.0201 0.0045 0.0012 2.6 2.3489 2.5231 2.4243 0.9985 0.9925 0.9995 0.0203 0.0046 0.0012 2.7 2.4131 2.5939 2.4918 0.9984 0.9929 0.9995 0.0211 0.0045 0.0012 2.8 2.4727 2.6586 2.5575 0.9982 0.9940 0.9994 0.0220 0.0042 0.0013 2.9 2.5238 2.7169 2.6132 0.9980 0.9944 0.9994 0.0233 0.0040 0.0013
Table 3 The correlation coefficient between the calculated fractal dimension and the theoretical fractal dimension; the slope of the fitted line.
ER CI OCI γ 0.9981 0.9979 0.9983 0.5955 0.6648 0.6355
Table 4 Fractal dimension, goodness of fit, and distance error of three methods based on Brodatz images.
No. ITEM FD R2 DE ER CI OCI ER CI OCI ER CI OCI 1 D1 2.5490 2.8369 2.7290 0.9933 0.9915 0.9973 0.0429 0.0050 0.0028 2 D11 2.7029 2.9103 2.8015 0.9963 0.9920 0.9988 0.0322 0.0049 0.0018 3 D21 2.8238 2.9919 2.8823 0.9976 0.9933 0.9988 0.0262 0.0045 0.0018 4 D30 2.4027 2.7106 2.6001 0.9907 0.9864 0.9954 0.0502 0.0063 0.0036 5 D41 2.6880 2.9076 2.7982 0.9957 0.9912 0.9985 0.0345 0.0051 0.0021 6 D47 2.5025 2.8229 2.7130 0.9926 0.9919 0.9962 0.0450 0.0049 0.0033 7 D53 2.7945 2.9720 2.8624 0.9967 0.9935 0.9990 0.0305 0.0044 0.0017 8 D61 2.5250 2.7589 2.6503 0.9949 0.9903 0.9982 0.0373 0.0053 0.0023 9 D65 2.6220 2.8856 2.7758 0.9931 0.9901 0.9977 0.0436 0.0054 0.0026 10 D74 2.6461 2.9308 2.8196 0.9928 0.9925 0.9975 0.0448 0.0047 0.0027 11 D81 2.7382 2.9468 2.8397 0.9968 0.9943 0.9989 0.0298 0.0041 0.0018 12 D85 2.7160 2.8882 2.7774 0.9978 0.9943 0.9993 0.0245 0.0041 0.0014 13 D91 2.3144 2.5885 2.4793 0.9928 0.9879 0.9960 0.0439 0.0059 0.0034 14 D99 2.4858 2.7552 2.6470 0.9931 0.9895 0.9974 0.0434 0.0056 0.0028 15 D104 2.8115 2.9987 2.8893 0.9966 0.9937 0.9986 0.0308 0.0043 0.0020 16 D112 2.6276 2.8853 2.7759 0.9948 0.9929 0.9982 0.0378 0.0046 0.0023
Table 5 Fractal dimension, goodness of fit, and distance error of three methods based on Aerials images.
No. ITEM FD R2 DE ER CI OCI ER CI OCI ER CI OCI 1 2.1.01 2.5653 2.7998 2.6939 0.9973 0.9951 0.9983 0.0272 0.0038 0.0022 2 2.1.02 2.6274 2.8440 2.7351 0.9975 0.9951 0.9982 0.0263 0.0038 0.0023 3 2.1.03 2.3861 2.5221 2.4275 0.9992 0.9968 0.9985 0.0148 0.0031 0.0021 4 2.1.04 2.5664 2.7880 2.6770 0.9981 0.9952 0.9985 0.0226 0.0038 0.0021 5 2.1.05 2.4971 2.7461 2.6369 0.9962 0.9940 0.9977 0.0323 0.0042 0.0026 6 2.1.06 2.5538 2.7413 2.6338 0.9976 0.9952 0.9982 0.0254 0.0037 0.0023 7 2.1.07 2.4586 2.6197 2.5094 0.9993 0.9929 0.9985 0.0134 0.0045 0.0021 8 2.1.08 2.4564 2.7155 2.6068 0.9977 0.9933 0.9955 0.0248 0.0044 0.0036 9 2.1.09 2.3832 2.5449 2.4448 0.9994 0.9914 0.9982 0.0132 0.0050 0.0023 10 2.1.10 2.5125 2.7052 2.5996 0.9986 0.9921 0.9976 0.0197 0.0048 0.0026 11 2.1.11 2.4268 2.6870 2.5769 0.9966 0.9943 0.9975 0.0303 0.0041 0.0027 12 2.1.12 2.4979 2.7237 2.6204 0.9984 0.9920 0.9968 0.0211 0.0048 0.0030
Table 6 Fractal dimension of three methods (with/without the improved strategy) based on FBM images.
TFD Long'2013DBC with CW Improvement Lai'2016DBC with CW Improvement Liu'2021DBC with CW Improvement 2.1 2.0501 2.0935 0.0434 2.2388 2.1521 −0.0867 2.2875 2.2003 −0.0872 2.2 2.0959 2.1361 0.0402 2.2900 2.2010 −0.0889 2.3376 2.2483 −0.0893 2.3 2.1522 2.1881 0.0360 2.3288 2.2383 −0.0905 2.3722 2.2821 −0.0901 2.4 2.2161 2.2485 0.0324 2.3866 2.2944 −0.0922 2.4295 2.3366 −0.0928 2.5 2.2831 2.3141 0.0309 2.4629 2.3664 −0.0964 2.5019 2.4061 −0.0958 2.6 2.3491 2.3807 0.0316 2.5373 2.4402 −0.0971 2.5721 2.4762 −0.0959 2.7 2.4113 2.4459 0.0346 2.6084 2.5099 −0.0984 2.6420 2.5441 −0.0979 2.8 2.4669 2.5056 0.0387 2.6726 2.5739 −0.0987 2.7063 2.6080 −0.0984 2.9 2.5148 2.5575 0.0426 2.7290 2.6284 −0.1006 2.7621 2.6608 −0.1013
Table 7 Goodness of fit of three methods (with/without the improved strategy) based on FBM images.
TFD Long'2013DBC with CW Improvement Lai'2016DBC with CW Improvement Liu'2021DBC with CW Improvement 2.1 0.99942 0.99969 0.00027 0.99102 0.99915 0.00813 0.99192 0.99944 0.00752 2.2 0.99940 0.99971 0.00031 0.99150 0.99898 0.00748 0.99271 0.99934 0.00663 2.3 0.99939 0.99974 0.00035 0.99275 0.99921 0.00646 0.99343 0.99938 0.00595 2.4 0.99937 0.99975 0.00039 0.99308 0.99950 0.00642 0.99374 0.99967 0.00593 2.5 0.99933 0.99975 0.00042 0.99344 0.99960 0.00617 0.99428 0.99977 0.00550 2.6 0.99927 0.99974 0.00047 0.99367 0.99955 0.00588 0.99457 0.99980 0.00523 2.7 0.99919 0.99973 0.00053 0.99373 0.99948 0.00575 0.99461 0.99982 0.00521 2.8 0.99912 0.99971 0.00059 0.99410 0.99932 0.00523 0.99492 0.99972 0.00479 2.9 0.99904 0.99969 0.00065 0.99475 0.99909 0.00434 0.99553 0.99954 0.00401
Table 8 Distance error of three methods (with/without the improved strategy) based on FBM images.
TFD Long'2013DBC with CW Improvement Lai'2016DBC with CW Improvement Liu'2021DBC with CW Improvement 2.1 0.00350 0.00092 −0.00258 0.00500 0.00152 −0.00348 0.00475 0.00124 −0.00352 2.2 0.00355 0.00089 −0.00267 0.00488 0.00167 −0.00321 0.00453 0.00135 −0.00318 2.3 0.00360 0.00084 −0.00276 0.00451 0.00147 −0.00304 0.00431 0.00131 −0.00300 2.4 0.00369 0.00082 −0.00287 0.00443 0.00118 −0.00325 0.00422 0.00096 −0.00325 2.5 0.00383 0.00083 −0.00299 0.00433 0.00105 −0.00328 0.00405 0.00080 −0.00325 2.6 0.00401 0.00085 −0.00315 0.00427 0.00112 −0.00315 0.00396 0.00075 −0.00320 2.7 0.00422 0.00088 −0.00334 0.00426 0.00122 −0.00304 0.00396 0.00072 −0.00324 2.8 0.00443 0.00091 −0.00352 0.00415 0.00140 −0.00275 0.00385 0.00091 −0.00295 2.9 0.00463 0.00095 −0.00368 0.00392 0.00162 −0.00230 0.00362 0.00116 −0.00246
Conceptualization, W.J., Y.L. and J.W.; data curation, W.J.; formal analysis, W.J.; funding acquisition, W.J. and J.W.; investigation, W.J., Y.L. and J.W.; methodology, W.J., Y.L. and X.L.; resources, X.L. and J.Z.; software, R.L.; supervision, R.L., X.L. and J.Z.; validation, W.J.; visualization, W.J.; writing—original draft, W.J.; writing—review and editing, R.L. and J.Z. All authors have read and agreed to the published version of the manuscript.
Not applicable.
Not applicable.
Not applicable.
The authors declare no conflict of interest.
s Size of a square grid smin The minimum value of s smax The maximum value of s M Image size for a square image G Total number of gray levels in an image gmax Maximum gray level over a grid gmin Minimum gray level over a grid nr Total number of boxes at scale r, required to cover the rough surface over (i, j)th grid Nr Total number of boxes at scale r, required to cover the rough surface of an image r The scale of a square grid for an image h Height of a box γ Correlation coefficient R2 Goodness of fit The slope of LLS fitted line (for evaluation of FBM images) ER Equal ratio. Indicates that s is a divisor of the image size CI Consecutive integer. Indicates that the value of s is all integer from smin to smax. CW The "consecutive integer + weight method" strategy OCI Optimized consecutive integer. Indicates that the division of the TFD Theoretical fractal dimension refers to the fractal dimension used to generate FBM images FD Fractal dimension DE Distance error FBM Fractional Brownian motion
The authors would also like to acknowledge the receipt of a research grant award as principal investigators from the National Natural Science Foundation of China under Grant No. 51479030.
By Wenxuan Jiang; Yujun Liu; Ji Wang; Rui Li; Xiao Liu and Jian Zhang
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