# An Approximated Box Height for Differential-Box-Counting Method to Estimate Fractal Dimensions of Gray-Scale Images

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Databases

#### 2.2. Fractal Dimension

#### 2.3. The DBC Method

#### 2.4. Box Height

**B:**Chen et al. considered $h=s$ in their work [15] to compute FD. Sometimes it may happen that the box height is greater than the number of gray-levels in an image ($h\ge G$); in that case only one box is required to represent the rough surface of a particular grid. Consequently, it leads to an inaccurate FD.**C:**Li et al. proposed $h=\frac{s}{0.5\times ({I}_{max}-{I}_{min})}$ to be their box height in [16], where ${I}_{max}$, ${I}_{min}$ are the maximum and minimum gray-level intensities present in an image respectively. The box height will be very small when an image is having a large texture variation. On the other hand, when gray-level differences (${I}_{max}-{I}_{min}$) will be small then the box height will be high as a result inaccurate FD value will be produced.**D:**Same authors proposed another formula for box height, $h=\frac{s}{1+2a\sigma}$, where $\sigma $ is the standard deviation of an image and $a=3$ is a constant in [17]. But, the proposed box height is too small and it requires more memory space for storage. The value of this box height is almost same to the box height used in [16] and like**C**this box height is dependent on image gray-levels.

## 3. Experimental Results and Discussion

#### 3.1. Impact of Box Heights

**A:**$\frac{s\times G}{M}$,

**B:**$h=s$ and

**D:**$\frac{s}{1+2a\sigma}$ (

**B**>

**A**>

**D**) are selected for further experiment from

**A**,

**B**,

**C**,

**D**and

**E**described in Section 2.4 because box height

**A**is almost same to

**E**for texture images and box height

**C**is nearly equal to

**D**. These box heights have been considered separately with DBC method by keeping other parameters same. The FD values along with DE values are then computed on Brodatz Database and they have been depicted in Figure 5 and Figure 6. However, all the images are scaled down to standard $512\times 512$ pixels before computation. Because for $M=256$,

**A**=

**B**, hence these two box heights cannot be distinguished and simultaneously $M=640$ is not selected since it is not a standard image size i.e., not a power of 2. The list of parameters of DBC method are reported in Table 1. It is clear from both Figure 5 and Figure 6 that FD value increases when box height decreases. On the other hand, DE values increases while box height increases.

#### 3.2. A Novel Method to Estimate Box Height

#### Approximating R

- If ${\mathrm{log}}_{2}\left(M\right)<2\Rightarrow R<0\Rightarrow h<0$.
- If ${\mathrm{log}}_{2}\left(M\right)=2\Rightarrow R=0\Rightarrow h=\infty $.
- If $2<{\mathrm{log}}_{2}\left(M\right)<3\Rightarrow R<{\mathrm{log}}_{2}\left(M\right)\Rightarrow h$ becomes relatively higher.

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 5.**Different FD values for images from Brodatz Database using DBC method together with box heights

**A**,

**B**and

**D**while image size is $512\times 512$.

**Figure 6.**Different DE values for images from Brodatz Database using DBC method together with box heights

**A**,

**B**and

**D**while image size is $512\times 512$.

**Figure 7.**Different FD values for images from Brodatz Database using DBC method along with box height, h$(=\frac{s}{R})$, while image size is $512\times 512$.

**Figure 8.**Different DE values for images from Brodatz Database using DBC method along with box height, h$(=\frac{s}{R})$, while image size is $512\times 512$.

**Figure 9.**Computed and actual FD values for generated FBM images using DBC, SDBC, IDBC, ITDBC along with the original and the proposed box height.

**Figure 10.**Computed FD values for images of Brodatz Database as shown in Figure 1 using DBC, SDBC, IDBC, ITDBC along with the original and the proposed box height.

DBC [13] | SDBC [15] | IDBC [18] | ITDBC [22] | |
---|---|---|---|---|

${s}_{min}$ | 2 | 2 | 2 | 2 |

${s}_{max}$ | $\frac{M}{2}$ | $\frac{M}{2}$ | $\frac{M}{2}$ | $\frac{M}{2}$ |

Allowed Grid sizes, s | ${s}_{max}\le s\le {s}_{max}$ | ${s}_{max}\le s\le {s}_{max}$ | 2^{i} i = 1, 2, .., log_{2} M − 1 | ${s}_{max}\le s\le {s}_{max}$ |

Grid shifting mechanism | No shifting | No shifting | Each grid is shifted by one pixel along South-East direction | No shifting |

Grid partitioning mechanism | No grid partitioning | No grid partitioning | No grid partitioning | Each grid is partitioned into unequal triangular grids in four different ways |

${n}_{r}$ | $\u2308{\displaystyle \frac{{g}_{max}}{h}}\u2309-\u2308{\displaystyle \frac{{g}_{min}}{h}}\u2309+1$ | $\u2308\frac{{g}_{max}-{g}_{min}+1}{h}\u2309$ | $\left\{\begin{array}{c}\u2308\frac{{g}_{max}-{g}_{min}+1}{h}\u2309,\mathrm{if}\phantom{\rule{4.pt}{0ex}}{g}_{max}\ne {g}_{min}\hfill \\ 1,\mathrm{if}\phantom{\rule{4.pt}{0ex}}{g}_{max}\ne {g}_{min}\hfill \end{array}\right.$ | $\left(\u2308\frac{t{g}_{max}}{h}\u2309-\u2308\frac{t{g}_{min}}{h}\u2309+1\right)\times \left(\frac{{A}_{t}}{s\times s}\right)$ ${A}_{t}$ is the number of image pixels in the tth triangular partition |

$xy$-plane partitioning | Partition image plane into non overlapping square grids. | Partition image plane into non overlapping square grids. | Partition image plane into non overlapping square grids. | Partition image plane into non overlapping square grids. |

Box Height, h | $\frac{s\times G}{M}$ | s | $\frac{s\times G}{M}$ | $\frac{s\times G}{M}$ |

**Table 2.**Finding out the smallest values of R that satisfy Equation (7) for different image sizes of Brodatz Database.

Image Size (M) | ${\mathit{R}}_{\mathit{max}}^{\mathit{M}}$ | R | ||
---|---|---|---|---|

$0.0589\mathit{M}+19.667$ | $\lceil {\mathbf{log}}_{2}\mathit{M}\times ({\mathbf{log}}_{2}\left(\mathit{M}\right)-3)\rceil $ | $\lceil {\mathbf{log}}_{2}\mathit{M}\times ({\mathbf{log}}_{2}\left(\mathit{M}\right)-2)\rceil $ | ||

128 | 28 | 27.21 (✕) | 28 (✓) | 35 (✓) |

256 | 32 | 34.75 (✓) | 40 (✓) | 48 (✓) |

512 | 49 | 49.82 (✓) | 54 (✓) | 63 (✓) |

640 | 62 | 57.36 (✕) | 59 (✕) | 69 (✓) |

1024 | 78 | 79.98 (✓) | 70 (✕) | 80 (✓) |

FBM Images with | Fractal Dimension (FD) | Distance Error (DE) | |||||||
---|---|---|---|---|---|---|---|---|---|

Hurst Parameter (H) | Actual FD (3-H) (P) | DBC (Q) | DBC + Proposed Box Height (R) | Absolute Difference in FD Values | Change in FD Values (R-Q) | DBC (X) | DBC + Proposed Box Height (Y) | Difference in DE Values (Y-X) | |

$\left|\mathit{P}-\mathit{Q}\right|$ | $\left|\mathit{P}-\mathit{R}\right|$ | ||||||||

1.0 | 2.0 | 1.9098 | 1.9857 | 0.0902 | 0.0143 | 0.0759 | 0.0045 | 0.0047 | 0.0002 |

0.9 | 2.1 | 1.9374 | 2.0248 | 0.1626 | 0.0752 | 0.0874 | 0.0042 | 0.0042 | 0.0000 |

0.8 | 2.2 | 1.9690 | 2.0757 | 0.2310 | 0.1243 | 0.1067 | 0.0042 | 0.0039 | -0.0003 |

0.7 | 2.3 | 2.0141 | 2.1369 | 0.2859 | 0.1631 | 0.1228 | 0.0042 | 0.0036 | -0.0006 |

0.6 | 2.4 | 2.0666 | 2.2056 | 0.3334 | 0.1944 | 0.1390 | 0.0041 | 0.0034 | -0.0007 |

0.5 | 2.5 | 2.1237 | 2.2779 | 0.3763 | 0.2221 | 0.1542 | 0.0042 | 0.0033 | -0.0009 |

0.4 | 2.6 | 2.1917 | 2.3514 | 0.4083 | 0.2486 | 0.1597 | 0.0041 | 0.0032 | -0.0009 |

0.3 | 2.7 | 2.2575 | 2.4172 | 0.4425 | 0.2828 | 0.1597 | 0.0040 | 0.0032 | -0.0008 |

0.2 | 2.8 | 2.3205 | 2.4742 | 0.4795 | 0.3258 | 0.1537 | 0.0038 | 0.0032 | -0.0006 |

0.1 | 2.9 | 2.3785 | 2.5233 | 0.5215 | 0.3767 | 0.1448 | 0.0036 | 0.0032 | -0.0004 |

0.0 | 3.0 | 2.4192 | 2.5612 | 0.5808 | 0.4388 | 0.1420 | 0.0037 | 0.0032 | -0.0005 |

**Table 5.**Computed FD and DE values using DBC and the proposed box height for images from Brodatz Database as shown in Figure 1.

Images from Brodatz Database with Image Name | Fractal Dimension (FD) | Distance Error (DE) | ||||
---|---|---|---|---|---|---|

DBC (P) | DBC + Proposed Box Height (Q) | Difference in FD Values (Q-P) | DBC (X) | DBC + Proposed Box Height (Y) | Difference in DE Values (Y-X) | |

D1 | 2.5281 | 2.6903 | 0.1622 | 0.0046 | 0.0035 | −0.0011 |

D5 | 2.5740 | 2.7375 | 0.1635 | 0.0048 | 0.0036 | −0.0012 |

D15 | 2.6188 | 2.7823 | 0.1635 | 0.0049 | 0.0034 | −0.0015 |

D20 | 2.6154 | 2.7793 | 0.1639 | 0.0050 | 0.0035 | −0.0015 |

D31 | 2.4877 | 2.6486 | 0.1609 | 0.0045 | 0.0034 | −0.0011 |

D37 | 2.4882 | 2.6521 | 0.1639 | 0.0040 | 0.0031 | −0.0009 |

D48 | 2.3346 | 2.4874 | 0.1528 | 0.0045 | 0.0033 | −0.0012 |

D54 | 2.5839 | 2.7471 | 0.1632 | 0.0049 | 0.0035 | −0.0014 |

D59 | 2.3496 | 2.5028 | 0.1532 | 0.0037 | 0.0025 | −0.0012 |

D62 | 2.5164 | 2.6773 | 0.1609 | 0.0047 | 0.0034 | −0.0013 |

D67 | 2.6172 | 2.7810 | 0.1638 | 0.0050 | 0.0035 | −0.0015 |

D74 | 2.5878 | 2.7508 | 0.1630 | 0.0049 | 0.0036 | −0.0013 |

D86 | 2.5868 | 2.7496 | 0.1628 | 0.0048 | 0.0034 | −0.0014 |

D92 | 2.5929 | 2.7569 | 0.1640 | 0.0049 | 0.0034 | −0.0015 |

D102 | 2.6090 | 2.7728 | 0.1638 | 0.0050 | 0.0035 | −0.0015 |

D112 | 2.5554 | 2.7185 | 0.1631 | 0.0047 | 0.0035 | −0.0012 |

FBM Images with | Fractal Dimension (FD) | Distance Error (DE) | |||||||
---|---|---|---|---|---|---|---|---|---|

Hurst Parameter (H) | Actual FD (3-H) (P) | SDBC (Q) | SDBC + Proposed Box Height (R) | Absolute Difference in FD Values | Change in FD Values (R-Q) | SDBC (X) | SDBC + Proposed Box Height (Y) | Difference in DE Values (Y-X) | |

$\left|\mathit{P}-\mathit{Q}\right|$ | $\left|\mathit{P}-\mathit{R}\right|$ | ||||||||

1.0 | 2.0 | 1.9364 | 2.0272 | 0.0636 | 0.0272 | 0.0908 | 0.0038 | 0.0042 | 0.0004 |

0.9 | 2.1 | 1.9595 | 2.0605 | 0.1405 | 0.0395 | 0.1010 | 0.0038 | 0.0039 | 0.0001 |

0.8 | 2.2 | 2.0037 | 2.1056 | 0.1963 | 0.0944 | 0.1019 | 0.0039 | 0.0037 | −0.0002 |

0.7 | 2.3 | 2.0675 | 2.1611 | 0.2325 | 0.1389 | 0.0936 | 0.0041 | 0.0035 | −0.0006 |

0.6 | 2.4 | 2.1376 | 2.2251 | 0.2624 | 0.1749 | 0.0875 | 0.0040 | 0.0033 | −0.0007 |

0.5 | 2.5 | 2.2078 | 2.2935 | 0.2922 | 0.2065 | 0.0857 | 0.0038 | 0.0032 | −0.0006 |

0.4 | 2.6 | 2.2767 | 2.3640 | 0.3233 | 0.2360 | 0.0873 | 0.0036 | 0.0031 | −0.0005 |

0.3 | 2.7 | 2.3408 | 2.4273 | 0.3592 | 0.2727 | 0.0865 | 0.0037 | 0.0031 | −0.0006 |

0.2 | 2.8 | 2.3951 | 2.4828 | 0.4049 | 0.3172 | 0.0877 | 0.0037 | 0.0032 | −0.0005 |

0.1 | 2.9 | 2.4347 | 2.5306 | 0.4653 | 0.3694 | 0.0959 | 0.0037 | 0.0032 | −0.0005 |

0.0 | 3.0 | 2.4679 | 2.5678 | 0.5321 | 0.4322 | 0.0999 | 0.0034 | 0.0032 | −0.0002 |

**Table 7.**Computed FD and DE values using SDBC and the proposed box height for images from Brodatz Database as shown in Figure 1.

Images from Brodatz Database with Image Name | Fractal Dimension (FD) | Distance Error (DE) | ||||
---|---|---|---|---|---|---|

SDBC (P) | SDBC + Proposed Box Height (Q) | Difference in FD Values (Q-P) | SDBC (X) | SDBC + Proposed Box Height (Y) | Difference in DE Values (Y-X) | |

D1 | 2.6238 | 2.6958 | 0.0720 | 0.0046 | 0.0035 | −0.0011 |

D5 | 2.6693 | 2.7427 | 0.0734 | 0.0048 | 0.0035 | −0.0013 |

D15 | 2.7138 | 2.7871 | 0.0733 | 0.0047 | 0.0034 | −0.0013 |

D20 | 2.7100 | 2.7841 | 0.0741 | 0.0048 | 0.0034 | −0.0014 |

D31 | 2.5828 | 2.6544 | 0.0716 | 0.0046 | 0.0034 | −0.0012 |

D37 | 2.5896 | 2.6583 | 0.0687 | 0.0037 | 0.0031 | −0.0006 |

D48 | 2.4281 | 2.4944 | 0.0663 | 0.0043 | 0.0033 | −0.0010 |

D54 | 2.6790 | 2.7522 | 0.0732 | 0.0048 | 0.0035 | −0.0013 |

D59 | 2.4478 | 2.5101 | 0.0623 | 0.0033 | 0.0025 | −0.0008 |

D62 | 2.6102 | 2.6829 | 0.0727 | 0.0047 | 0.0034 | −0.0013 |

D67 | 2.7115 | 2.7859 | 0.0744 | 0.0048 | 0.0034 | −0.0014 |

D74 | 2.6822 | 2.7558 | 0.0736 | 0.0048 | 0.0036 | −0.0012 |

D86 | 2.6817 | 2.7547 | 0.0730 | 0.0047 | 0.0034 | −0.0013 |

D92 | 2.6880 | 2.7619 | 0.0739 | 0.0048 | 0.0034 | −0.0014 |

D102 | 2.7034 | 2.7777 | 0.0743 | 0.0048 | 0.0035 | −0.0013 |

D112 | 2.6516 | 2.7238 | 0.0722 | 0.0047 | 0.0035 | −0.0012 |

FBM Images with | Fractal Dimension (FD) | Distance Error (DE) | |||||||
---|---|---|---|---|---|---|---|---|---|

Hurst Parameter (H) | Actual FD (3-H) (P) | IDBC (Q) | IDBC + Proposed Box Height (R) | Absolute Difference in FD Values | Change in FD Values (R-Q) | IDBC (X) | IDBC + Proposed Box Height (Y) | Difference in DE Values (Y-X) | |

$\left|\mathit{P}-\mathit{Q}\right|$ | $\left|\mathit{P}-\mathit{R}\right|$ | ||||||||

1.0 | 2.0 | 2.0921 | 2.1543 | 0.0921 | 0.1543 | 0.0622 | 0.0114 | 0.0062 | −0.0052 |

0.9 | 2.1 | 2.1222 | 2.1891 | 0.0222 | 0.0891 | 0.0669 | 0.0155 | 0.0073 | −0.0082 |

0.8 | 2.2 | 2.1744 | 2.2352 | 0.0256 | 0.0352 | 0.0608 | 0.0163 | 0.0095 | −0.0068 |

0.7 | 2.3 | 2.2420 | 2.2896 | 0.0580 | 0.0104 | 0.0476 | 0.0174 | 0.0117 | −0.0057 |

0.6 | 2.4 | 2.2717 | 2.3496 | 0.1283 | 0.0504 | 0.0779 | 0.0111 | 0.0137 | 0.0026 |

0.5 | 2.5 | 2.3462 | 2.4099 | 0.1538 | 0.0901 | 0.0637 | 0.0134 | 0.0152 | 0.0018 |

0.4 | 2.6 | 2.3986 | 2.4711 | 0.2014 | 0.1289 | 0.0725 | 0.0133 | 0.0167 | 0.0034 |

0.3 | 2.7 | 2.4654 | 2.5232 | 0.2346 | 0.1768 | 0.0578 | 0.0147 | 0.0174 | 0.0027 |

0.2 | 2.8 | 2.5182 | 2.5699 | 0.2818 | 0.2301 | 0.0517 | 0.0151 | 0.0180 | 0.0029 |

0.1 | 2.9 | 2.5712 | 2.6080 | 0.3288 | 0.2920 | 0.0368 | 0.0165 | 0.0182 | 0.0017 |

0.0 | 3.0 | 2.6091 | 2.6392 | 0.3909 | 0.3608 | 0.0301 | 0.0171 | 0.0181 | 0.0010 |

**Table 9.**Computed FD and DE values using IDBC and the proposed box height for images from Brodatz Database as shown in Figure 1.

Images from Brodatz Database with Image Name | Fractal Dimension (FD) | Distance Error (DE) | ||||
---|---|---|---|---|---|---|

IDBC (P) | IDBC + Proposed Box Height (Q) | Difference in FD Values (Q-P) | IDBC (X) | IDBC + Proposed Box Height (Y) | Difference in DE Values (Y-X) | |

D1 | 2.7045 | 2.7051 | 0.0006 | 0.0290 | 0.0282 | −0.0008 |

D5 | 2.7618 | 2.7625 | 0.0007 | 0.0301 | 0.0295 | −0.0006 |

D15 | 2.8487 | 2.8495 | 0.0008 | 0.0235 | 0.0231 | −0.0004 |

D20 | 2.8359 | 2.8370 | 0.0011 | 0.0271 | 0.0267 | −0.0004 |

D31 | 2.6590 | 2.6594 | 0.0004 | 0.0285 | 0.0275 | −0.0010 |

D37 | 2.6898 | 2.7124 | 0.0226 | 0.0209 | 0.0209 | 0.0000 |

D48 | 2.5668 | 2.5882 | 0.0214 | 0.0171 | 0.0148 | −0.0023 |

D54 | 2.7797 | 2.7804 | 0.0007 | 0.0291 | 0.0285 | −0.0006 |

D59 | 2.6008 | 2.6121 | 0.0113 | 0.0122 | 0.0088 | −0.0034 |

D62 | 2.6973 | 2.6979 | 0.0006 | 0.0289 | 0.0280 | −0.0009 |

D67 | 2.8378 | 2.8389 | 0.0011 | 0.0264 | 0.0261 | −0.0003 |

D74 | 2.8378 | 2.8389 | 0.0011 | 0.0264 | 0.0261 | −0.0003 |

D86 | 2.8378 | 2.8389 | 0.0011 | 0.0264 | 0.0261 | −0.0003 |

D92 | 2.8378 | 2.8389 | 0.0011 | 0.0264 | 0.0261 | −0.0003 |

D102 | 2.8378 | 2.8389 | 0.0011 | 0.0264 | 0.0261 | −0.0003 |

D112 | 2.8378 | 2.8389 | 0.0011 | 0.0264 | 0.0261 | −0.0003 |

**Table 10.**Computed FD and DE values using ITDBC and the proposed box height for generated FBM images.

FBM Images with | Fractal Dimension (FD) | Distance Error (DE) | |||||||
---|---|---|---|---|---|---|---|---|---|

Hurst Parameter (H) | Actual FD (3-H) (P) | ITDBC (Q) | ITDBC + Proposed Box Height (R) | Absolute Difference in FD Values | Change in FD Values (R-Q) | ITDBC (X) | ITDBC + Proposed Box Height (Y) | Difference in DE Values (Y-X) | |

$\left|\mathit{P}-\mathit{Q}\right|$ | $\left|\mathit{P}-\mathit{R}\right|$ | ||||||||

1.0 | 2.0 | 2.0811 | 2.1286 | 0.0811 | 0.1286 | 0.0475 | 0.0027 | 0.0022 | −0.0005 |

0.9 | 2.1 | 2.1232 | 2.1786 | 0.0232 | 0.0786 | 0.0554 | 0.0027 | 0.0021 | −0.0006 |

0.8 | 2.2 | 2.1659 | 2.2395 | 0.0341 | 0.0395 | 0.0736 | 0.0021 | 0.0019 | −0.0002 |

0.7 | 2.3 | 2.2094 | 2.3063 | 0.0906 | 0.0063 | 0.0969 | 0.0018 | 0.0018 | 0.0000 |

0.6 | 2.4 | 2.2457 | 2.3759 | 0.1543 | 0.0241 | 0.1302 | 0.0014 | 0.0017 | 0.0003 |

0.5 | 2.5 | 2.2988 | 2.4446 | 0.2012 | 0.0554 | 0.1458 | 0.0012 | 0.0017 | 0.0005 |

0.4 | 2.6 | 2.3582 | 2.5097 | 0.2418 | 0.0903 | 0.1515 | 0.0014 | 0.0017 | 0.0003 |

0.3 | 2.7 | 2.4257 | 2.5672 | 0.2743 | 0.1328 | 0.1415 | 0.0015 | 0.0017 | 0.0002 |

0.2 | 2.8 | 2.4890 | 2.6159 | 0.3110 | 0.1841 | 0.1269 | 0.0015 | 0.0017 | 0.0002 |

0.1 | 2.9 | 2.5494 | 2.6570 | 0.3506 | 0.2430 | 0.1076 | 0.0016 | 0.0017 | 0.0001 |

0.0 | 3.0 | 2.5903 | 2.6904 | 0.4097 | 0.3096 | 0.1001 | 0.0016 | 0.0017 | 0.0001 |

**Table 11.**Computed FD and DE values using ITDBC and the proposed box height for images from Brodatz Database as shown in Figure 1.

Images from Brodatz Database with Image Name | Fractal Dimension (FD) | Distance Error (DE) | ||||
---|---|---|---|---|---|---|

ITDBC (P) | ITDBC + Proposed Box Height (Q) | Difference in FD Values (Q-P) | ITDBC (X) | ITDBC + Proposed Box Height (Y) | Difference in DE Values (Y-X) | |

D1 | 2.6618 | 2.8231 | 0.1613 | 0.0024 | 0.0030 | 0.0006 |

D5 | 2.7075 | 2.8705 | 0.1630 | 0.0023 | 0.0026 | 0.0003 |

D15 | 2.7600 | 2.9236 | 0.1636 | 0.0021 | 0.0020 | −0.0001 |

D20 | 2.7602 | 2.9232 | 0.1630 | 0.0022 | 0.0022 | 0.0000 |

D31 | 2.6176 | 2.7765 | 0.1589 | 0.0024 | 0.0032 | 0.0008 |

D37 | 2.6186 | 2.7808 | 0.1622 | 0.0015 | 0.0020 | 0.0005 |

D48 | 2.4861 | 2.6272 | 0.1411 | 0.0019 | 0.0025 | 0.0006 |

D54 | 2.7183 | 2.8812 | 0.1629 | 0.0023 | 0.0025 | 0.0002 |

D59 | 2.4726 | 2.6228 | 0.1502 | 0.0008 | 0.0014 | 0.0006 |

D62 | 2.6520 | 2.8118 | 0.1598 | 0.0023 | 0.0028 | 0.0005 |

D67 | 2.7603 | 2.9234 | 0.1631 | 0.0022 | 0.0021 | −0.0001 |

D74 | 2.7603 | 2.9234 | 0.1631 | 0.0022 | 0.0021 | −0.0001 |

D86 | 2.7603 | 2.9234 | 0.1631 | 0.0022 | 0.0021 | −0.0001 |

D92 | 2.7603 | 2.9234 | 0.1631 | 0.0022 | 0.0021 | −0.0001 |

D102 | 2.7603 | 2.9234 | 0.1631 | 0.0022 | 0.0021 | −0.0001 |

D112 | 2.7603 | 2.9234 | 0.1631 | 0.0022 | 0.0021 | −0.0001 |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Panigrahy, C.; Garcia-Pedrero, A.; Seal, A.; Rodríguez-Esparragón, D.; Mahato, N.K.; Gonzalo-Martín, C.
An Approximated Box Height for Differential-Box-Counting Method to Estimate Fractal Dimensions of Gray-Scale Images. *Entropy* **2017**, *19*, 534.
https://doi.org/10.3390/e19100534

**AMA Style**

Panigrahy C, Garcia-Pedrero A, Seal A, Rodríguez-Esparragón D, Mahato NK, Gonzalo-Martín C.
An Approximated Box Height for Differential-Box-Counting Method to Estimate Fractal Dimensions of Gray-Scale Images. *Entropy*. 2017; 19(10):534.
https://doi.org/10.3390/e19100534

**Chicago/Turabian Style**

Panigrahy, Chinmaya, Angel Garcia-Pedrero, Ayan Seal, Dionisio Rodríguez-Esparragón, Nihar Kumar Mahato, and Consuelo Gonzalo-Martín.
2017. "An Approximated Box Height for Differential-Box-Counting Method to Estimate Fractal Dimensions of Gray-Scale Images" *Entropy* 19, no. 10: 534.
https://doi.org/10.3390/e19100534