基于单幅形态学图像的分形维数精确计算
Accurate evaluation of the fractal dimension based on a single morphological image
摘要：
分形维数（D）是表征物理气相沉积薄膜中常见的自仿射表面不规则性和破碎性的有效参数。D可以通过使用原子力显微镜（AFM）测量表面粗糙度的缩放性能来评估，但需要大量不同尺度（L）的AFM图像。在这项研究中，表面粗糙度预测（SRP）的方法，提出了一个单一的AFM图像的D值，其中在较小的L的粗糙度估计与平坦化修改的图像分割。首先，利用Weierstrass-Mandelbrot（W-M）函数生成了一系列理想维数（Di）在2.1 ~ 2.9之间的人工分形曲面，并将SRP方法与传统的盒计数法、功率谱密度法等方法进行了比较。SRP法计算的尺寸Dc比其他方法更接近Di，平均相对误差仅为0.64%。其次，利用SRP方法对L为1-70 μm的非晶氧化铝薄膜的AFM图像进行处理。在10 μm以上的L处，SRP方法得到的Dc值与我们以前的多图像分析结果也很接近，而在较小的L处得到的较大的Dc值与实际表面特征相吻合。讨论了SRP方法的有效性和真实曲面的物理本质，有助于加深对分形几何的理解。
英文摘要：
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 μm. 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 μm, 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.
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