Quantifying multifractal anisotropy in two dimensional objects

Area of Science:

• Fractal analysis
• Image processing
• Data science

Background:

• Understanding anisotropy in fractal properties is crucial for characterizing complex 2D surfaces and images.
• Existing methods may not fully capture directional variations in fractal dimensions.

Purpose of the Study:

• To develop and validate a direction-sensitive method for analyzing multifractal properties in 2D data.
• To explore the interdisciplinary applications of this novel technique.

Main Methods:

• A direction-sensitive generalization of multifractal detrended fluctuation analysis (MF-DFA) is proposed for 2D data.
• The method is tested on synthetic structures to confirm its effectiveness and consistency.

Main Results:

• The method successfully identified directional multifractality in synthetic data.
• Novel directional multifractal characteristics were discovered in Martian surface data and the Crab Nebula.
• The analysis of Jackson Pollock's paintings revealed evolving multifractal patterns over their creation time.

Conclusions:

• The proposed method offers an efficient way to explore anisotropy in fractal properties of 2D surfaces and images.
• This technique has broad interdisciplinary potential for analyzing diverse real-world datasets.
• The findings suggest new insights into the structural complexity of natural phenomena and artistic creations.