Polar partitioning method for the multifractal characterization of complex heterogeneous structures

Abstract

Classical Euclidean geometry effectively characterizes regular-shaped structures but often falls short when attempting to describe the intricate, irregular-shaped objects, commonly found in biological systems. Multifractal spectrum, a hump function, captures the variability of scaling behavior across an object and provides an alternative approach for analyzing complex biological structures. It is increasingly being used to link structural disorganization of biological tissues to the risk for chronic diseases. Therefore, enhancing the accuracy of existing methods for multifractal analysis may further the understanding of diseases. The box counting method which is often used for computing the multifractal spectrum requires partitioning the object in order to compute the multifractal spectrum. This study proposes a polar partitioning method for the multifractal analysis of complex structures embedded in two-dimensional (2D) space. Its performance is compared with existing methods, including radial and rectangular partitioning. The accuracy of multifractal spectrum computed by polar, radial and rectangular partitioning methods was evaluated using theoretically-derived multifractal structures for which the analytical multifractal spectra exist. The results from this study demonstrate that the polar partitioning method greatly outperforms the rectangular and radial partitioning method when analyzing objects with circular support. The radial partitioning method performs well only when analyzing objects whose scaling rate vary solely along the radial direction. This study identifies the most suitable use cases for each partitioning method to minimize errors in the computations. Instead of the rectangular and radial partitioning methods, the polar partitioning method should be used for computing the multifractal spectrum for complex objects with circular support.