Structures are abundant in both natural and human-made environments and usually studied in the form of images or scattering patterns. To characterize structures a huge variety of descriptors is available spanning from porosity to radial and correlation functions. In addition to morphological structural analysis, such descriptors are necessary for stochastic reconstructions, stationarity and representativity analysis. The most important characteristic of any such descriptor is its information content - or its ability to describe the structure at hand. For example, from crystallography it is well known that experimentally measurable $S_2$ correlation function lacks necessary information content to describe majority of structures. The information content of this function can be assessed using Monte-Carlo methods only for very small 2D images due to computational expenses. Some indirect quantitative approaches for this and other correlation function were also proposed. Yet, to date no methodology to obtain information content for arbitrary 2D or 3D image is available. In this work, we make a step toward developing a general framework to perform such computations analytically. We show, that one can assess the entropy of a perturbed random field and that stochastic perturbation of fields correlation function decreases its information content. In addition to analytical expression, we demonstrate that different regions of correlation function are in different extent informative and sensitive for perturbation. Proposed model bridges the gap between descriptor-based heterogeneous media reconstruction and information theory and opens way for computationally effective way to compute information content of any descriptor as applied to arbitrary structure.

翻译：结构在自然和人造环境中广泛存在，通常以图像或散射图案的形式进行研究。为表征这些结构，存在从孔隙度到径向函数和关联函数的大量描述符。除了形态结构分析外，这些描述符对于随机重建、平稳性和代表性分析也至关重要。任何此类描述符的最重要特征是其信息内容——即其描述当前结构的能力。例如，从晶体学中已知，实验可测量的$S_2$关联函数缺乏描述多数结构所需的信息内容。由于计算开销，该函数的信息内容只能通过蒙特卡洛方法对非常小的二维图像进行评估。针对该函数及其他关联函数，也提出了一些间接的定量方法。然而，迄今尚无方法可用于任意二维或三维图像的信息内容获取。本文向开发通用分析框架迈出了一步，以解析方式执行此类计算。我们表明，可以评估扰动随机场的熵，且场的关联函数的随机扰动会降低其信息内容。除了解析表达式，我们还展示了关联函数的不同区域在信息量和扰动敏感性方面存在差异。所提出的模型弥合了基于描述符的非均质介质重建与信息理论之间的鸿沟，并为以计算高效的方式计算任意描述符应用于任意结构时的信息内容开辟了道路。