嵌入在${\mathbb Z^n}$中的有偏于正方形矩阵的复杂性
Complexity of Shapes Embedded in ${\mathbb Z^n}$ with a Bias Towards Squares
摘要：
形状复杂性是一种难以量化的质量，主要是由于其相对性质。由于欧几里德思想的偏见，圆通常被认为是最简单的。然而，它们作为数字图像的构造仅仅是理想形式的近似。因此，参照圆计算的复杂度顺序是不稳定的。与数字图像中失去圆形的圆形不同，正方形保留了它们的特性。因此，我们认为正方形（$\mathbb Z^n$中的超立方体）是最简单的形状，相对于其复杂性顺序被构造。利用$L^\infty$范数和平方之间的联系，我们有效地编码平方适应简化，通过它我们获得多尺度复杂性度量，其中尺度决定了对边界的兴趣程度。边界特征（附属物）作用消失的突现尺度与附属物与主体接触宽度之比有关。我们讨论了零复杂性在信息重复性和可构造性方面意味着什么，以及除了正方形之外，什么样的形状具有零复杂性。
英文摘要：
Shape complexity is a hard-to-quantify quality, mainly due to its relative nature. Biased by Euclidean thinking, circles are commonly considered as the simplest. However, their constructions as digital images are only approximations to the ideal form. Consequently, complexity orders computed in reference to circle are unstable. Unlike circles which lose their circleness in digital images, squares retain their qualities. Hence, we consider squares (hypercubes in $\mathbb Z^n$) to be the simplest shapes relative to which complexity orders are constructed. Using the connection between $L^\infty$ norm and squares we effectively encode squareness-adapted simplification through which we obtain multi-scale complexity measure, where scale determines the level of interest to the boundary. The emergent scale above which the effect of a boundary feature (appendage) disappears is related to the ratio of the contacting width of the appendage to that of the main body. We discuss what zero complexity implies in terms of information repetition and constructibility and what kind of shapes in addition to squares have zero complexity.
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