Fractal dimension (D) is a widely used quantity to represent the irregularity of surfaces or profiles, e.g. it is often applied together with surface roughness to evaluate the quality of machined surfaces objectively and precisely. There are some conventional algorithms to calculate D values through the morphological images of measured surfaces. However, the accuracies or efficiencies of these algorithms sometimes might be insufficient to satisfy the requirement of high-precision machining technology. In this paper, an artificial neural network (ANN) model is proposed to evaluate the D value based on a single morphological image. First, the artificial fractal surfaces with preset ideal D values are generated via Weierstrass–Mandelbrot (W–M) function. Then these surfaces are divided into a training dataset and a test dataset, which are used to train the ANN model and compare the model against the conventional algorithms (including box counting, power spectral density, autocorrelation function, structural function, and roughness scaling extraction with flatten order of 1), respectively. The accuracy and efficiency of D calculation by using the trained ANN model are much superior. The mean relative error of ANN model is just 0.25%, while those of conventional algorithms are in the range of 2.22–9.33%. The average time cost for D calculation of ANN model is 1.87ms, while those of conventional algorithms are in the range of 46ms–8s. Based on the advantages verified above, the trained ANN model is utilized to calculate the D values of machined surfaces and investigate the influences of different cutting parameters. It is found that the D values of machined surfaces could be influenced significantly by the feed rate, while the cutting speed and depth are relatively irrelevant.

中文翻译：

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基于人工神经网络的分形维数计算及其在机加工表面中的应用

分形维数 (D) 是一个广泛使用的量，用于表示表面或轮廓的不规则性，例如，它通常与表面粗糙度一起应用，以客观准确地评估加工表面的质量。有一些传统的算法可以计算D通过测量表面的形态图像获取值。然而，这些算法的准确性或效率有时可能不足以满足高精度加工技术的要求。在本文中，提出了一种人工神经网络（ANN）模型来评估D基于单个形态图像的值。一、预设理想的人工分形面D值是通过 Weierstrass-Mandelbrot (W-M) 函数生成的。然后将这些表面分为训练数据集和测试数据集，用于训练 ANN 模型并将模型与常规算法（包括框计数、功率谱密度、自相关函数、结构函数和粗糙度缩放提取分别展平 1) 的顺序。准确性和效率D使用经过训练的 ANN 模型进行的计算要优越得多。人工神经网络模型的平均相对误差仅为0.25%，而传统算法的平均相对误差在2.22-9.33%之间。平均时间成本为DANN模型的计算为1.87ms，而传统算法的在 46 范围内毫秒–8s。基于以上验证的优势，利用训练好的 ANN 模型计算D加工表面的值，并研究不同切削参数的影响。发现该D加工表面的值可能会受到进给速率的显着影响，而切削速度和深度相对无关紧要。