Deep learning approach to fractal surfaces: application to satellite images of urban areas

Abstract A Convolutional Neural Network (CNN) approach is proposed to quantify the Hurst exponent H of two-dimensional correlated random matrices. The approach (shortly referred to as RegFractNet ) has been tested on deterministic and stochastic fractals and trained on large datasets of artificially generated two-dimensional Fractional Brownian fields. As real-world data, WorldView-2 (WV-2) high-resolution satellite images are considered. RegFractNet yields accurate estimates of the Hurst exponent H . As a further assessment of the RegFractNet robustness and accuracy, the outcomes are compared with those yield by standard methods as Detrending Moving Average (DMA), Box-Counting (BC), Variogram (VA) and Climacogram (CL) algorithms. A specific feature of the proposed RegFractNet approach, compared to standard approaches, is the ability to operate to $128 \times 128$ 128 × 128 pixels size, essential to the implementation at small urban scales. As real world case study, the Hurst exponent H , estimated from WV-2 satellite imagery of urban areas, are fed in the relationship $D_{f}=2 -H$ D f = 2 − H . Fractal dimensions $D_{f}$ D f are found to vary in the range $1.52 \div 1.93$ 1.52 ÷ 1.93 corresponding to less and high urbanized areas, thus confirming earlier studies.