A CONTINUOUS VARIATION OF ROUGHNESS SCALING CHARACTERISTICS ACROSS FRACTAL AND NON-FRACTAL PROFILES

In this study, the scaling characteristics of root-mean-squared roughness (Rq) was investigated for both fractal and non-fractal profiles by using roughness scaling extraction (RSE) method proposed in our previous work. The artificial profiles generated through Weierstrass-Mandelbrot (W-M) function and the actual profiles, including surface contours of silver thin films and electroencephalography signals, were analyzed. Based on the relationship curves between Rq and scale, it was found that there was a continuous variation of the dimension value calculated with RSE method (DRSE) across the fractal and non-fractal profiles. In the range of fractal region, DRSE could accurately match with the ideal fractal dimension (FD) input for W-M function. In the non-fractal region, DRSE values could characterize the complexity of the profiles, similar to the functionality of FD value for fractal profiles, thus enabling the detection of certain incidents in signals such as an epileptic seizure. Moreover, the traditional methods (Box-Counting and Higuchi) of FD calculation failed to reflect the complexity variation of non-fractal profiles, because their FD was generally 1. The feasibility of abnormal implementation of W-M function and the capability of RSE method were discussed according to the analysis on the properties of W-M function, which would be promising to make more understandings of the nonlinear behaviors of both theoretical and practical features.

Automated summary

This study investigated the scaling characteristics of root-mean-squared roughness for both fractal and non-fractal profiles, finding continuous variation of dimension values between the two types of profiles. It successfully matched the ideal fractal dimension in the fractal region and characterized the complexity of non-fractal profiles using the RSE method. The traditional FD calculation methods failed to reflect the complexity variation of non-fractal profiles, highlighting the feasibility of abnormal implementation of W-M function and the capability of RSE method in understanding nonlinear behaviors.