Multiscale multifractal detrended-fluctuation analysis of two-dimensional surfaces.

Area of Science:

• Fractal Geometry
• Image Analysis
• Data Science

Background:

• Traditional 2D multifractal detrended fluctuation analysis (MF-DFA) can yield biased generalized Hurst exponents due to scale crossovers.
• Multiscale multifractal analysis (MMA) was developed for 1D data to address scale-dependent fractal properties.

Purpose of the Study:

• To adapt and apply the multiscale multifractal analysis (MMA) method for analyzing two-dimensional (2D) surfaces.
• To identify key parameters influencing MMA results on 2D surfaces and assess its robustness to noise.

Main Methods:

• Synthesized 2D surfaces were generated to test the MMA method and identify scale crossovers.
• MMA was applied to synthesized and natural/real-world 2D images to estimate generalized Hurst exponents.
• The influence of moving window length (WL) and slide length (SL) parameters was investigated.

Main Results:

• Crossovers were consistently observed in synthesized surfaces, indicating scale-dependent fractal properties.
• Optimal MMA parameters for 2D surfaces were identified as WL=4 and SL=4.
• The generalized Hurst exponent (h(2,s)) showed high robustness to noise at large scales but varied at small scales.
• Image noise, particularly Gaussian and salt-and-pepper, weakened long-term correlations.

Conclusions:

• MMA effectively reveals scale-dependent fractal characteristics in 2D surfaces, overcoming limitations of traditional MF-DFA.
• The study provides optimal parameter settings and noise robustness insights for MMA application on 2D data.
• Findings significantly enhance the performance and applicability of MMA for analyzing complex 2D surfaces and textures.