Departamento de Física and National Institute of Science and Technology for Complex Systems, Universidade Estadual de Maringá, Maringá, Brazil.
PLoS One. 2012;7(8):e40689. doi: 10.1371/journal.pone.0040689. Epub 2012 Aug 14.
Complexity measures are essential to understand complex systems and there are numerous definitions to analyze one-dimensional data. However, extensions of these approaches to two or higher-dimensional data, such as images, are much less common. Here, we reduce this gap by applying the ideas of the permutation entropy combined with a relative entropic index. We build up a numerical procedure that can be easily implemented to evaluate the complexity of two or higher-dimensional patterns. We work out this method in different scenarios where numerical experiments and empirical data were taken into account. Specifically, we have applied the method to [Formula: see text] fractal landscapes generated numerically where we compare our measures with the Hurst exponent; [Formula: see text] liquid crystal textures where nematic-isotropic-nematic phase transitions were properly identified; [Formula: see text] 12 characteristic textures of liquid crystals where the different values show that the method can distinguish different phases; [Formula: see text] and Ising surfaces where our method identified the critical temperature and also proved to be stable.
复杂性测度对于理解复杂系统至关重要，并且有许多定义可用于分析一维数据。然而，这些方法的扩展到二维或更高维数据，如图像，就不那么常见了。在这里，我们通过应用排列熵的思想并结合相对熵指数来缩小这一差距。我们建立了一个数值程序，可以很容易地实现来评估二维或更高维模式的复杂性。我们在不同的场景中应用了这种方法，考虑了数值实验和经验数据。具体来说，我们已经将该方法应用于[公式：见正文]分形景观的数值生成，我们将我们的度量与赫斯特指数进行了比较；[公式：见正文]向列型-各向同性-向列型相转变的液晶纹理；[公式：见正文]液晶的 12 种特征纹理，不同的数值表明该方法可以区分不同的相；[公式：见正文]和伊辛表面，我们的方法确定了临界温度，并且也被证明是稳定的。
