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.

中文翻译：

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分形和非分形轮廓的粗糙度缩放特性的连续变化

在本研究中，均方根粗糙度的尺度特征（Rq) 通过使用我们之前工作中提出的粗糙度缩放提取 (RSE) 方法对分形和非分形轮廓进行了研究。分析了通过 Weierstrass-Mandelbrot (W-M) 函数生成的人工轮廓和实际轮廓，包括银薄膜的表面轮廓和脑电图信号。基于之间的关系曲线Rq和尺度，发现用RSE方法计算的尺寸值存在连续变化（DRSE) 跨越分形和非分形剖面。在分形区域范围内，DRSE可以准确匹配理想的分形维数（FD) W-M 函数的输入。在非分形区域，DRSE值可以表征配置文件的复杂性，类似于FD分形剖面的价值，从而能够检测信号中的某些事件，例如癫痫发作。此外，传统方法（Box-Counting 和 Higuchi）FD计算未能反映非分形剖面的复杂性变化，因为它们FD一般为 1。通过对 W-M 函数性质的分析，讨论了 W-M 函数异常实现的可行性和 RSE 方法的能力，这将有助于进一步理解这两种理论的非线性行为。和实用的功能。