Deep-Learning Estimators for the Hurst Exponent of Two-Dimensional Fractional Brownian Motion

The fractal dimension (D) is a very useful indicator for recognizing images. The fractal dimension increases as the pattern of an image becomes rougher. Therefore, images are frequently described as certain models of fractal geometry. Among the models, two-dimensional fractional Brownian motion (2D FBM) is commonly used because it has specific physical meaning and only contains the finite-valued parameter (a real value from 0 to 1) of the Hurst exponent (H). More usefully, H and D possess the relation of D = 3 − H. The accuracy of the maximum likelihood estimator (MLE) is the best among estimators, but its efficiency is appreciably low. Lately, an efficient MLE for the Hurst exponent was produced to greatly improve its efficiency, but it still incurs much higher computational costs. Therefore, in the paper, we put forward a deep-learning estimator through classification models. The trained deep-learning models for images of 2D FBM not only incur smaller computational costs but also provide smaller mean-squared errors than the efficient MLE, except for size 32 × 32 × 1. In particular, the computational times of the efficient MLE are up to 129, 3090, and 156248 times those of our proposed simple model for sizes 32 × 32 × 1, 64 × 64 × 1, and 128 × 128 × 1.