## 3 Experimental Results and Discussion
### Comprehensive Discussion  
Traditionally, when we want to estimate the fractal dimensions of images, we use an estimator to compute. If necessary, we also can further use the fractal dimensions to classify their classes or types. In the past, the box-counting method was often used to estimate because of its simplicity; however, its accuracy is considerably low, and hence it is often unreliable, thereby resulting in a low recognition rate. Figure 10: The complete confusion matrix (**a**) and its lower right part (**b**) for clarity of the first fold of the five-fold cross-validation of GoogleNet 1 for 21 classes. Recently, [PERSON] [22] proposed an efficient MLE for 2D FBM, which is the most accurate estimator and, so far, the most efficient estimator among the MLEs. However, its computational time is still high for applications with larger image sizes and/or database sizes, and hence it is not appropriate for tasks of quick evaluation and analysis, especially for real-time systems. Therefore, when we want to get the feature map of an original image by transforming the sub-images of the image to a matrix with the corresponding Hurst exponent as an entry or element, it is essential to find out an alternative estimation tool for a wide range of use. This is the second purpose of the paper. In the paper, we have experimentally confirmed in the images of size 32 \(\times\) 32 \(\times\) 1 that the classification rates of suitably designed or chosen deep learning models can outperform those of machine learning models that receive the only feature--the estimated Hurst exponents from the efficient MLE. The computational cost of machine learning models can be neglected, but taking the only feature is terribly high. In general, directly estimating the Hurst exponent of images is enough to analyze the differences between images; the classification is practically not necessary. Performed on the estimated Hurst exponents, machine learning models aim to compare with deep learning models as well as to further evaluate the benefits of deep learning models. If workable, we can use well-trained deep learning models to replace the role of the efficient MLE as estimation or further feature transformation. At present, the comparison may not be considered fair because we use only one indicator as our feature for machine learning models, but deep learning models can implicitly and automatically extract or capture as more and more features as possible. However, the comparison tells us that deep learning models do work well even for the 2D FBM images from a 2D random process--various appearances coming from different realizations with the same parameter, as shown in Figures 3-6. In the future, we can manually select other promising indicators as our features, for example, entropy and spectrum, to raise the classification rate. Surprisingly, the chosen three pretrained networks--AlexNet, GoogleNet, and Xception--also work well for 2D FBM images (or FPIs) of size 32 \(\times\) 32 \(\times\) 1; in addition, the adjusted GoogleNet model is better than the adjusted AlexNet model and the adjusted Xception model in terms of the computational cost and classification rate. These pretrained networks, as we know, were designed for 1000 common object categories, such as keyboard, mouse, pencil, and many animals. They did not see the hidden features of FPIs and were not trained with them, and hence, intuitively, they should not perform well. However, they not only work better than machine learning models with the only feature of the estimated Hurst exponents, but they also give us a promising prospect for the future, especially GoogleNet 1. As for other image sizes, further experiments are needed in order to find out the most suitable deep-learning model for different cases or situations. In the experiments, we find that the models of Type 1--with the original images as the input images--can work better than machine learning models with the best estimates of the Hurst exponent as the only feature except Xception 1.