## 5 Conclusions  
Recently, an efficient MLE for 2D FBM has been proposed to estimate Hurst exponents or their corresponding fractal dimensions of fractal surfaces or images. The MLE for 2D FBM is an unbiased estimator and the best in mean-squared error, but it is inevitable to contain estimation error. That is, the exact value of any estimated Hurst exponent is not absolutely reliable. Accordingly, in the paper, we replace estimated Hurst exponents with  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Models** & **Mean** & **Ratio** \\ \hline Our model & 0.0034 & 1.00 \\ GoogleNet 1 & 0.0138 & 4.09 \\ MLE & 0.4996 & 148.36 \\ \hline \hline \end{tabular}
\end{table}
Table 13: Classifying or estimating time (in seconds) and its ratio.  
the corresponding Hurst exponents of their classes or equivalent Hurst exponents. For the Hurst exponent, three ranges or cases are meaningful: \(H\) < 0.5, \(H\) = 0.5, and \(H\) > 0.5, and different values tell different levels of seriousness. The exact \(H\) = 0.5, however, cannot be evaluated. Therefore, we considered two cases of 11 classes and 21 classes of the Hurst exponent to contain the three ranges mentioned above.  
Experimental results show that the best classification rate of 11 classes through four machine learning models run on the estimated Hurst exponents is 81.4%; that of 21 classes is 53.2%. Therefore, the exact value of any estimated Hurst exponent has lost its true meaning under the resolution of 1/21. On the other hand, when deep learning models were run on the same images, the best classification rate of 11 classes through three deep learning models was 96.00%; that of 21 classes was 93.70%.  
Since deep learning models, if well designed or chosen, can learn more from various hidden features, the results of deep learning models are better than those of machine learning models learning from the only feature--the estimated Hurst exponents. In the future, except for the estimated Hurst exponents, we would like to find out other useful features to raise the classification rate of machine learning models and further compare the results to those of deep learning models by further adjusting or choosing hyperparameters. With the success of deep learning models, more inputs will also be developed to improve the whole performance in the upcoming future.  
Based on their high classification accuracies, these well-designed or well-chosen pre-trained deep learning models can be used to \"equivalently\" estimate the Hurst exponent or fractal dimension--first, determine the class of an image and then calculate its corresponding Hurst exponent. Consequently, the time saving can be up to 148 times that of the best estimator--the MLE. In addition, different models will learn various hidden features; hence, when they are applied to \"equivalently\" estimate the Hurst exponents of practical images--modeled as 2D FBM, but not the exact one, they can find out various appearances that contribute to explaining diverse details from various perspectives.  
More importantly, the concept of feature maps--in our case, transforming an original image to a feature map of the Hurst exponent--can be widely used to develop other feature maps, such as spectrum or entropy maps, in order to raise the whole recognition rate.  
Conceptualization, Y.-C.C. and [PERSON]-[PERSON]; methodology, Y.-C.C. and [PERSON]-T.J.; programming, Y.-C.C.; writing--original draft preparation, Y.-C.C. and [PERSON]-T.J.; writing--review and editing, Y.-C.C. and [PERSON]-T.J.; funding acquisition, Y.-C.C. All authors have read and agreed to the published version of the manuscript.  
This work was supported by the Ministry of Science and Technology, Taiwan, Republic of China, under Grant MOST 111-2221-E-040-009.  
Not applicable for studies not involving humans or animals.  
Not applicable for studies not involving humans.  
Data sharing is not applicable.  
The authors declare no conflict of interest.