## 2 Materials and Methods
### Disorder of Layer Structure (DStr) and Size (DSize)  
Let us first review the meaning of the DStr and DSize parameters and the basic elements of their calculation. The parameters of DStr and DSize are introduced to describe the morphological features of arbitrary patterns, with respect to the reference image, which are made up of layers that have an isotropic size and structure. Thus:  
* The DStr serves as a measure of deviation of an arbitrary pattern that has an anisotropic structure from a pattern that has an isotropic structure;
* The DSize serves as a measure of deviation of an arbitrary pattern of an anisotropic size from a pattern of an isotropic size. An isotropic structure implies that lines that comprise an image have no breaks and confluences. An isotropic size suggests that the distance between lines remains constant along R\({}_{1}\),..., R\({}_{N}\) for any angles of an image rotation. Thus, an image with an isotropic size and structure consists of straight parallel lines that are at a constant distance from any nearby lines. The notions of isotropic and anisotropic structure and size come from the formalization of growth layers (i.e., the growth increments) of living systems such as fish scales [4; 5] and lamellar bone of humans and animals [2; 6; 7]. If the growth rate remains constant over a period of time, then the thickness of the growth increment remains constant across a 2-D area of growth, and growth increments contain no breaks and confluences in a layer's structure. Thus, a constant growth rate results in the isotropic size and structure of growth layers. If the growth rate varies, then the thickness of growth lines also varies, and the structure usually exhibits breaks and confluences. Consequently, the values of DStr and DSize signal a deviation of the growth rate from the constant speed of layer formation. Overall, the EM provides tools to quantify the anisotropic growth rate across 2-D growth increments [1,5]. Figure 1: Test images with high edge gradient. Earth’s surface. Credit: Remote Sensing of Environment [13] and Google Earth Pro. Mars’ surface. Credit: Jet Propulsion Laboratory [14]. Tidal sand ripples. Credit: Reproduced with permission from [PERSON] [15]; [[https://natureinfocus.blog/2016/09/03/inch-2//](https://natureinfocus.blog/2016/09/03/inch-2//)]([https://natureinfocus.blog/2016/09/03/inch-2//](https://natureinfocus.blog/2016/09/03/inch-2//)) accessed on 17 January 2022. It has been shown [2,3] that breaks and confluences are inherent in various images formed in nature and beyond. Hence, it is possible to use the EM to create a morphological description of these images. Moreover, the breaks and confluences within a layer are the building blocks of any (i.e., arbitrary) binary patterns [3]. This means that the EM can be used to create a morphological description of an arbitrary binary image. Let us consider the basic steps of calculating DStr and DSize. First, DStr and DSize are functions of GAI alignment. This is contrary to commonly used morphological characteristics of images, such as circularity, curvature, perimeter, convexity, and compactness. Thus, when the image under study rotates, DStr and DSize are calculated for each image position. Second, the concept of isotropic and anisotropic edges/vertices (i.e., isotropic/anisotropic connectivity) of G(N) was introduced [2,3] in order to calculate DStr; the measure of DStr is the ratio between anisotropic vertices and all G(N) vertices. Third, the set of transects R\({}_{1}\),..., R\({}_{\rm i}\),..., R\({}_{\rm N}\) is what is used to calculate DStr and DSize. Each transect is a straight line, and the distance between R\({}_{\rm j-1}\) and R\({}_{\rm j}\) remains constant across R\({}_{1}\),..., R\({}_{\rm N}\). The value of both DStr and DSize depends on the number of transects (i.e., sampling density) used in calculating them. The general principle used when choosing the number of transects is based on the fact that we do not know _a priori_ how many transects will best describe the particular layered pattern.