We can conclude from

Section 2.4 that the standard expression for box height is as follows:

where,

R is any real number (

$R\in \mathbb{R}$) and

R must be greater than or equal to one (

$R\ge 1$) because if the value of

R is less than one and the value of

s is large then

h will be more than

G in that case the obtained value of FD will be inaccurate. Most of the authors have considered

R as a positive natural number for their works. Thus, a series of natural numbers 1 to 100 have been chosen for the value of

R to compute FD value in order to find out a

R value for an image, which satisfies Equation (

7) in this work. Bicubic interpolation method is applied on each of these images of size

$640\times 640$ pixels to resize them into

$128\times 128$,

$256\times 256$,

$512\times 512$ and

$1024\times 1024$ pixels for further experiments. FD of ideal fractal sets are invariant to size of the set i.e., for any value of

s in Equation (

1), the value of

$F{D}_{s}$ will remain same. Since texture images are not ideal fractals, their FD value changes with the size of the image though the difference in FD values are less but not equal to zero. Hence we have used different image sizes in-order to relate their FD value and using their relation a new box height is proposed in this paper.

The Equation (

6) is applied as box height with other parameters of DBC described in

Table 1 on 112 textured images from Brodatz Database at different image sizes

$128\times 128$,

$256\times 256$,

$512\times 512$,

$640\times 640$ and

$1024\times 1024$ pixels.

Figure 7 and

Figure 8 show different FD values and DE values for different values of

R respectively, which has started from 1 to 100 with step size 1.

Figure 7 contains the FD values of images, which are shown in

Figure 2. The similar experiments have been performed on 112 images with different image sizes

$128\times 128$,

$256\times 256$,

$640\times 640$ and

$1024\times 1024$ pixels. It is observed that the same trend persist between these experiments as shown in

Figure 7. Moreover, it is clear from

Figure 7 that for all images when box height decreases i.e.,

R in Equation (

6) increases then the computed FD value increases, but after some higher value of

R the estimated FD value almost remains the same as

R increases. Similarly

Figure 8 shows that as box height decreases, incurred error i.e., DE, decreases. Though decreasing of DE values are not fully monotonic but eventually after some lower box height, i.e., for higher

R values, the DE values remain almost the same. So, we have to find out a proper value of

R for a particular image size in such a way that there is no significant change in the difference of two successive FD values for two successive values of

R.

#### Approximating R

Let,

$FD$${}_{R}^{{p}_{M}}$ be a FD value of an image numbered as

p from Brodatz Database having

$M\times M$ pixels with a particular value of

R where,

R starts from one and

$M\in [128,256,512,640,1024]$. Now, smallest integer value,

R, has to be found that satisfies Equation (

7) for each image

p at a particular image size

$M\times M$.

It is observed from the experiments that the value of FD up-to four decimal places is enough to distinguish textures. Moreover, floating point arithmetic is very expensive. So, a threshold value 0.0001 is chosen for our work.

Table 2 shows the number of images having the same

R value for different image size that satisfy Equation (

7). Here,

${R}_{max}^{M}$ is marked using red color in

Table 2, which represents the smallest

R value for an

M, after which FD value does not change significantly for all the images.

Then all the points

$(M,{R}_{max}^{M})$ are plotted into a 2D plane to get a line using linear least square, where

x-axis represents

M i.e., image size and

y-axis denotes

${R}_{max}^{M}$ of a particular image size

M. Here,

$(M,{R}_{max}^{M})$ corresponds to

$(128,28)$,

$(256,32)$,

$(512,49)$,

$(640,62)$ and

$(1024,78)$ that have been obtained from

Table 2. The expression of the plotted line is represented by Equation (

8).

Now, this line is validated by putting different values of

M (128, 256, 512, 640 and 1024) and the obtained R values are shown in

Table 3. It is clear from

Table 3 that the obtained

R value is not always greater than or equal to

${R}_{max}^{M}$. So, the next task is to find out an exact equation, which can replace Equation (

8). In other words, an exact equation is expected in this work, which can find out

R after putting

M, where the value of R will be always greater than or equal to

${R}_{max}^{M}$ without performing the above analysis. However, this analysis helps us to propose a novel box height, which depends on image size. The expression of

R in Equation (

8) is approximated by

${\mathrm{log}}_{2}\left(M\right)$, which is shown in Equation (

9). Initially we started with

$R=\lceil {\mathrm{log}}_{2}M\times ({\mathrm{log}}_{2}\left(M\right)-3)\rceil $ because it is the nearest multiple of

${\mathrm{log}}_{2}\left(M\right)$ to satisfy for least image size

M = 128 and its

${R}_{max}^{128}=28$.

This expression is again validated using different values of

M (128, 256, 512, 640 and 1024). Unfortunately, it fails for image size 640 and 1024 as shown in

Table 3. Equation (

9) has two parts and 1 has been added to the second part,

$({\mathrm{log}}_{2}\left(M\right)-3)$. New formula is shown in Equation (

10).

Again, Equation (

10) has been validated for all image size

M (128, 256, 512, 640 and 1024) and this time Equation (

10) works fine for all image sizes

M. The ceil function,

$\lceil .\rceil $ is used to deal with the image sizes,

M which are not a power of 2 like when

M = 640 i.e., to obtain an integer value. Hence, Equation (

10) is used to fix the value of

R. Therefore using Equations (

6) and (

10), the new box height proposed in this paper for DBC [

13] is as follows:

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.

Hence when

${\mathrm{log}}_{2}\left(M\right)<3$, we need to restrict

R. Hence if

${\mathrm{log}}_{2}\left(M\right)<3$ then

$R={\mathrm{log}}_{2}\left(M\right)$. Therefore the final proposed formula to compute box height is as follows:

Eleven generated FBM images as shown in

Figure 2 have been used for the first set of experiment to validate the proposed box height as denoted in Equation (

12). The FBM images are generated using random mid-point displacement algorithm with different Hurst components. The value of Hurst component varies from 0 to 1 with step size 0.1 in this work. So, we have altogether 11 Hurst values, which help to create eleven FBM images. Moreover, Hurst component defines the actual FD value of an image. The different Hurst values and their acutal FD values are shown in 1st column and 2nd column of

Table 4 respectively. The 3rd column of

Table 4 depicts the FD values obtained using original DBC method.

Table 1 shows different parameters of DBC, SDBC [

15], IDBC [

18] and ITDBC [

22]. Now, the proposed box height have been used and the achieved FD values for all the above mentioned images are shown in column number 4 of

Table 4. The 5th and 6th columns of

Table 4 show the difference of the FD values obtained by DBC method and new box height from actual FD values respectively. The DE values using both the methods are also reported in

Table 4. It is clear from the columns 5th and 6th of

Table 4 that the proposed box height allows to better estimate FD values, which are closer to actual FD values with less DE values. The same set of experiments using both the methods have been done on sixteen textured images from Brodatz Database as shown in

Figure 1 and the results are shown in

Table 5. We do not have actual FD values for these textured images. So, the actual FD values are missing in

Table 5. However, the obtained FD values using the proposed box height of DBC are larger than the achieved FD values by the original DBC method. Thus, we can conclude that new box height of DBC estimates better FD values than the original DBC method.

In the next set of experiment, original SDBC [

15] and SDBC with proposed box height have been considered on eleven generated FBM images and sixteen textured images from Brodatz Database and the obtained results are shown in

Table 6 and

Table 7 respectively.

Then IDBC [

18] method and the proposed box height have been used for the next set of experiments on eleven generated FBM images and sixteen textured images from Brodatz Database and the achieved results are shown in

Table 8 and

Table 9 respectively.

Again, similar experiments using ITDBC method [

22] and proposed box height have been considered on both set of images i.e., on generated FBM, textured images and the gained results are reported in

Table 10 and

Table 11 respectively.

To know the impact of the new box height for different methods such as DBC, SDBC, IDBC and ITDBC, we have also plotted the results of these methods obtained using their original parameters and the results obtained with the proposed box height as shown in

Figure 9. While plotting, the actual FD values are represented in

x-axis and the corresponding computed FD values in

y-axis. It is clear from

Figure 9 that for all the cases the proposed box height allows the four existing methods to compute better FD values compare to their original methods. Moreover, we can conclude from

Figure 9 that the proposed box height together with ITDBC estimate better FD values for maximum set of experiments.

We have also plotted a similar graph for Brodatz Database in

Figure 10. In

x-axis of

Figure 10 represents image number and

y-axis denotes computed FD values using all the above mentioned methods. We can easily determine that the proposed box height along with ITDBC outperforms than other methods shown in

Figure 10.

FD is the main factor that indicates image roughness. DE is merely a secondary component of FD values and if less DE is generated than it is good but slightly high DE is acceptable and that does not mean that the method is entirely bad. The main priority lies in better estimating FD values, Hence we relaxed DE value while defining proposed box height i.e., not all DE values generated by proposed box height for DBC, SDBC, IDBC and ITDBC are less when compared with their original box heights as shown in last columns of

Table 8,

Table 9,

Table 10 and

Table 11.