Morphological analysis of textiles by fractal geometry

各种被归类为纤维的材料都是细的、线状的、柔韧的。纤维的一般定义注重其形状，即实用纤维的最小长宽比为100。纤维的另一个重要特性是柔韧性。有些是自然卷曲的，有些是人造波浪。缠绕纤维形状的随机性难以描述，但希望能够像纵横比那样用数值表示纤维形状的复杂性。Mandelbrot的分形几何概念是估计不规则集合时最方便的数学方法之一。分形几何的特点是自相似（精确和统计）和非积分维数。将单纤维、多孔膜等图像视为随机分形集，是一种统计上自相似的对象。粗略地说，分形维数是一个集合中不规则性显著性的判据。Mandelbrot在定义分形时，特别提到了Hausdorff维数，它是通过覆盖来定义的，可以应用于任何集合。豪斯多夫维数的优点是数学上的严格性和清晰性。另一方面，其主要缺点是难以通过计算方法进行估计，在寻找最有效的覆盖条件时存在模糊性，并且取极限很麻烦。在许多非积分覆盖的实际尺寸中，选择计数维数来描述纺织产品的复杂性。确定卷曲纤维分形维数的盒计数法，多孔聚氨酯混合膜，和修改随机科赫曲线。尼龙6卷曲长丝的盒数维分布在1.00-1.65之间，聚氨酯共混膜的盒数维分布在1.8660-1.8793之间，修正后的随机Koch曲线分布在1.033-1.310之间。

Various kinds of materials classified as fibers are thin, thread-like, and flexible. A general definition of fibers pays attention to their shape, that is, the least aspect ratio of practical fibers is 100. Another important characteristic of fibers is their flexibility. Some are naturally crimped and others have artificial waves added to them. The randomness of the twisted fiber shapes is difficult to describe, but it is desirable to be able to express the complexity of fiber shapes numerically, like the aspect ratio.Mandelbrot's concept of fractal geometry is one of the most mathematically convenient when estimating irregular sets. Fractal geometry is characterized by self-similarity (both exact and statistical) and non-integral dimensions. The images of single fibers, multi-pored films, and so on, are regarded as a random fractal set, a kind of statistically self-similar object. Roughly speaking, the fractal dimension is a criterion of the prominence of irregularities in a set.When Mandelbrot defined fractal, he singled out Hausdorff's dimension, which is defined by covering and can be applied to any set. The advantages of Hausdorff's dimension are mathematical strictness and clarity. On the other hand, the major disadvantages are that it is hard to estimate by computational methods, it is ambiguous in finding the most efficient covering condition, and it is troublesome to take a limit. In many nonintegral covering practical dimensions, the box-counting dimension has been chosen to describe the complexity of the textile products.The box-counting method for determining the fractal dimensions of crimped fibers, porous polyurethane-blend films, and modified random Koch curves. The box-counting dimension of nylon 6 crimped filament has a distribution of 1.00-1.65, polyurethane-blend films has 1.8660-1.8793,and modified random Koch curves has 1.033-1.310.