Accurate evaluation of the fractal dimension based on a single morphological image

TL;DR

This paper introduces a novel surface roughness prediction method to accurately evaluate the fractal dimension from a single AFM image, reducing the need for multiple scale images and improving precision over traditional techniques.

Contribution

A new SRP method for estimating fractal dimension from a single image, validated with artificial and real surfaces, showing higher accuracy than traditional methods.

Findings

SRP method achieves mean relative error of 0.64% on artificial surfaces.

SRP estimates of real surface fractal dimension are consistent with multi-image analysis.

The method enhances understanding of surface fractal properties from limited data.

Abstract

Fractal dimension (D) is an effective parameter to represent the irregularity and fragmental property of a self-affine surface, which is common in physical vapor deposited thin films. D could be evaluated through the scaling performance of surface roughness by using atomic force microscopy (AFM) measurements, but lots of AFM images with different scales (L) are needed. In this study, a surface roughness prediction (SRP) method was proposed to evaluate D values of a single AFM image, in which the roughness at smaller L was estimated by image segmentation with flatten modification. Firstly, a series of artificial fractal surfaces with ideal dimension (Di) values ranging from 2.1 to 2.9 were generated through Weierstrass-Mandelbrot (W-M) function, in order to compare SRP method with traditional methods such as box counting method and power spectral density method. The calculated dimension (Dc) by SRP method was much closer to Di than the other methods, with a mean relative error of only 0.64%. Secondly, SRP method was utilized to deal with real surfaces, which were AFM images of amorphous alumina thin films with L of 1-70 {\mu}m. Dc obtained by SRP method based on a single AFM image was also close to the result in our previous study by multi-image analysis at L above 10 {\mu}m, while the larger Dc at smaller L was consisted with the actual surface feature. The validity of SRP method and the physics nature of real surfaces were discussed, which might be helpful to obtain more understandings of fractal geometry.