## 4 Applications  
For medical images, there often appear some fractal characteristics--especially in the small regions--and hence if we want to recognize the differences between tissues, in the past, we only used estimators to compute their fractal dimensions or their corresponding Hurst exponents, and then judged them by their estimates. Under the images of size 32 \(\times\) 32 \(\times\) 1, experimental results show that some pretrained or well-designed deep learning models can \"equivalently\" outperform the efficient MLE--the best estimator--in an efficient and effective way; therefore, in the future, we can adopt some promising deep learning models to classify--or equivalently estimate--the Hurst exponent. For comparison, we chose one medical image database--the chest X-ray database--from Kaggle [40]; they are grayscale images. First, we find out the smallest size of all images--127 \(\times\) 384--as our input size. According to the size, we then clipped the central parts of all images because other parts generally contain some unimportant and unrelated components. Finally, we chose one clipped image as our subject, shown in Figure 11.  
Among experimental models, our proposed model and GoogleNet 1 run on 21 classes suitably serve the role of transforming the original image to two feature maps of the Hurst exponent. For estimation or feature transformation, we first chose the size of each sub-image as 32 \(\times\) 32--the size of our model design. Second, we chose 1 as our stride or shift in the horizontal and vertical directions. For all sub-images, on the one hand, we classified them through our proposed model and GoogleNet 1 and then converted these classes to their corresponding Hurst exponents--for example, the first class is equivalent to \(H=1/42\) and the last (or 21 st) class equivalent to \(H=41/42\). On the other hand, we estimated them through the efficient MLE. Figure 12 shows their resultant images or feature maps (96 \(\times\) 353) (top) and their corresponding negatives (bottom). The darker points correspond \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & & \multicolumn{2}{c}{**MAE**} & \multicolumn{2}{c}{**MSE**} \\ \hline
**Methods** & **No. of Classes** & **Mean** & **Std.** & **Mean** & **Std.** \\ \hline Our model & 11 & \(3.64\times 10^{-3}\) & \(5.02\times 10^{-4}\) & \(3.33\times 10^{-4}\) & \(4.47\times 10^{-5}\) \\  & 21 & \(3.22\times 10^{-3}\) & \(3.42\times 10^{-4}\) & \(1.74\times 10^{-4}\) & \(2.15\times 10^{-5}\) \\ GoogleNet 1 & 11 & \(6.16\times 10^{-3}\) & \(2.55\times 10^{-4}\) & \(5.61\times 10^{-4}\) & \(2.60\times 10^{-5}\) \\  & 21 & \(3.68\times 10^{-3}\) & \(1.91\times 10^{-4}\) & \(1.93\times 10^{-4}\) & \(7.81\times 10^{-6}\) \\ MLE (FBSs) & 11 & \(2.60\times 10^{-2}\) & \(2.05\times 10^{-2}\) & \(1.09\times 10^{-3}\) & \(1.68\times 10^{-3}\) \\  & 21 & \(2.59\times 10^{-2}\) & \(2.05\times 10^{-2}\) & \(1.09\times 10^{-3}\) & \(1.69\times 10^{-3}\) \\ MLE (FBIs) & 11 & \(2.83\times 10^{-2}\) & \(2.45\times 10^{-2}\) & \(1.40\times 10^{-3}\) & \(2.73\times 10^{-3}\) \\  & 21 & \(2.93\times 10^{-2}\) & \(2.83\times 10^{-2}\) & \(1.66\times 10^{-3}\) & \(4.46\times 10^{-3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 12: Comparison among four methods in terms of the MAE and MSE. to the smaller Hurst exponents; the lighter points, the larger Hurst exponents. The black point is equivalent to \(H=0\), and the white point is equivalent to \(H=1\). Figure 11: A sample of clipped chest X-ray images (\(127\times 384\)). Figure 12: The feature image or map (\(96\times 353\)) (**top**) and its corresponding negative (**bottom**) of transforming a clipped chest X-ray image (\(127\times 384\)) to Hurst exponents through our proposed model (**a**), GoogleNet 1 (**b**), and MLE (**c**). It is obvious from Figure 12 that these three sets of feature maps are different from one another because all sub-images are not exactly images from 2D FBM, but they are just modeled as 2D FBM.