## 1 Introduction  
Fractal geometry plays a very important role in explaining irregular images (or surfaces) and signals, for example, a classification for pathological prostate images [1], a classification for solitary pulmonary nodules [2], an online detection method for dry coal screening [3], an evaluation of changes in the structure of modified cement composite [4], a temperature fluctuation prediction [5], and a stock prediction [6]. For fractal images and signals, we often use fractal dimension (FD) to describe irregular structures--a smoother image/signal is from a smaller FD and a rougher one from a larger FD. For example, the intensities of some medical images, especially for pathological tissues, can be modeled as fractal images. In general, the textures of malignant tissues, such as nodules, are rougher than those of benign tissues, and more serious ones appear much rougher. Therefore, we can use the FD as an indicator to distinguish between benign and malignant tissues. Among models to describe fractal images or irregular ones with self-similarity, two-dimensional fractional Brownian motion (2D FBM) [7, 8] is often adopted because it can give physical meanings to explain the characteristics of images. Furthermore, two-dimensional fractional Gaussian noise (2D FGN)--the corresponding two-dimensional derivative process of 2D FBM--can be used to describe much rougher images. These two models can also be appropriate to describe some natural scenes [9, 10, 11]. These two models contain only one parameter--the Hurst exponent (\(H\)), a real value between 0 and 1--to distinguish between characteristics of various images; it is directly related to the FD (\(D\)) via \(D=3-H\)[7]. The Hurst exponent is limited, and hence it suitably serves as an effective feature; in addition, its value between 0 and 1 is particularly suitable for mapping or transforming an original image to a feature image or map. For machine learning or deep learning on certain images, we can provide this feature as an extra input to improve the classification rate. Generally, the efficacy of the extra feature--the Hurst exponent--depends on the estimation accuracy because the higher the estimation accuracy is, the better the classification rate. Therefore, it is very important to find out the best estimator of the Hurst exponent as our auxiliary tool. Related estimators are calculated through the fractal Brownian surface [12], the Fourier power spectrum [13], the reticular cell counting [14], the differential box-counting [15; 16; 17; 18; 19; 20], and the blanket [21]. Recently, [PERSON] [22] proposed the maximum likelihood estimator (MLE)--the best estimator--for the 2D FBM images. It is an unbiased estimator and has the smallest mean-squared error. Therefore, we are pleased to adopt the MLE for 2D FBM images to estimate the Hurst exponent or its corresponding fractal dimension. On the one hand, we can directly use these estimates to differentiate between different images or tissues; on the other hand, we also can use these estimates as a feature to classify them through machine learning models. In the past, when surfaces or images were modeled as 2D FBM--the former is called 2D FBM surfaces and the latter 2D FBM images, we estimated their Hurst exponents or their corresponding fractal dimensions through estimators and then used these values or estimates as an index or indicator. However, any estimator, even the best estimator--the MLE--more or fewer results in error, and hence their estimates are not completely reliable. In addition, although the best estimator can obtain the most accurate estimate, its computational time is still high even though it has lowered much by an efficient MLE [22]. As deep learning models have continued to flourish over the years, another novel and practical point of view emerged that estimates might be calculated from deep learning models, not traditional estimators.