Complexity-Entropy Causality Plane as a Complexity Measure for Two-Dimensional Patterns

Abstract

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 fractal landscapes generated numerically where we compare our measures with the Hurst exponent; liquid crystal textures where nematic-isotropic-nematic phase transitions were properly identified; 12 characteristic textures of liquid crystals where the different values show that the method can distinguish different phases; and Ising surfaces where our method identified the critical temperature and also proved to be stable.

Estimating physical properties from liquid crystal textures via machine learning and complexity-entropy methods

“…Due to this underlying simplicity and also because our approach is very fast and scalable from the computational point of view, we believe it can be easily implemented and adapted for other more complex experimental situations involving the study of liquid crystals and perhaps for probing physical properties of different materials. The normalized permutation entropy H [25] and statistical complexity C [33] are two complexity measures originally proposed for characterizing time series [26], and that were more recently generalized for considering higher dimensional data such as images [27,28]. We refer the more detail-oriented reader to the previouslycited references, where a complete description of these techniques can be found.…”

“…Thus, the k-nearest neighbors algorithm achieves a remarkable accuracy of ≈ 99.2% in the regression task of predicting the order parameter p solely based on the values of H and C. We further observe that pratically the same accuracy is obtained when using only the values of H or only the values of C. This happens because the values of H and C are strongly correlated to each other for these textures. However, in general, the values of C are not a trivial function of H and usually care additional information related to the "structural" complexity of images [27][28][29]. The performance of this algorithm is much higher than those obtained from simple baseline regressors that always predict the expected values or the median (accuracy of ≈ 0%).…”

“…Our goal in this case is to identify the pitch of a cholesteric liquid crystal based on the values of H and C obtained from the textures. To do so, we create a dataset of textures composed of one hundred replicas for each pitch value η ∈ (15,17,19,21,23,25,27,29,40) nm, where η = 40 nm is large enough to mimic a nematic texture. The optical textures are numerically obtained by solving the model described in Appendix D, which is based on the Landau-de Gennes theory [32].…”

Discriminating image textures with the multiscale two-dimensional complexity-entropy causality plane

“…Despite all these significant efforts, the development of a robust methodology to detect and quantify spatial structures in images still represents an open and subtle problem. Along this research direction, we have previously introduced an extension of the complexity-entropy causality plane to more than one dimension [13]. It has been shown that the two-dimensional version of this information-theory-derived tool is very promising for distinguishing between two-dimensional patterns.…”

Quantifying ecoacoustic activity in the Amazon rainforest through Information Theory quantifiers

“…is the maximum possible value of Q[P, P e ], obtained when only one component of P is 493 equal to one and all the others become zero [30]. This means λ 1 = 1 and λ 2:τmax = 0 in 494 our definition.…”