Towards effective information content assessment: analytical derivation of information loss in the reconstruction of random fields with model uncertainty

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.Comment: Keywords: correlation functions, structure characterization, structural descriptors, image analysis, information conten

Improving stochastic reconstructions by weighting correlation functions in an objective function

Spatial correlation functions (CFs) are prominent descriptors of any structure. In this letter, we show for the first time how proper weighting of CFs in an objective function can lead to significant improvements in reconstruction accuracy and in the likelihood of convergence. We develop a simple weighting scheme and display its effectiveness on two- and three-dimensional structures utilizing up to 27 CFs in one set. Proper weighting of the objective functions led to completely accurate reconstructions not achievable by conventional unweighted approaches. The proposed approach combining numerous CFs can potentially characterize and reconstruct structures of any complexity

The dynamic nature of crystal growth in pores

The kinetics of crystal growth in porous media controls a variety of natural processes such as ore genesis and crystallization induced fracturing that can trigger earthquakes and weathering, as well as, sequestration of CO2 and toxic metals into geological formations. Progress on understanding those processes has been limited by experimental difficulties of dynamically studying the reactive surface area and permeability during pore occlusion. Here, we show that these variables cause a time-dependency of barite growth rates in microporous silica. The rate is approximately constant and similar to that observed on free surfaces if fast flow velocities predominate and if the time-dependent reactive surface area is accounted for. As the narrower flow paths clog, local flow velocities decrease, which causes the progressive slowing of growth rates. We conclude that mineral growth in a microporous media can be estimated based on free surface studies when a) the growth rate is normalized to the time-dependent surface area of the growing crystals, and b) the local flow velocities are above the limit at which growth is transport-limited. Accounting for the dynamic relation between microstructure, flow velocity and growth rate is shown to be crucial towards understanding and predicting precipitation in porous rocks

Improving pattern reconstruction using directional correlation functions

In this letter we introduce a new method to calculate correlation functions in four principal directions (i.e. two orthogonal and two diagonal) and separately utilize them for image reconstruction. We show that this method is particularly suitable for anisotropic porous media but that it also improves image reconstruction for isotropic structures. Based on the analysis of numerous reconstructions of four binary patterns using different sets of two-point probability and linear (for both phases) correlation functions, we quantify the accuracy of each set. Averaging of correlation functions in all directions almost always results in poorer reconstructions. Addition of separate directions significantly improves the quality of replicas with only a minor increase in computational effort

Mimicking 3D food microstructure using limited statistical information from 2D cross-sectional image

We used statistical correlation functions (CFs) to describe food microstructure and to reconstruct their 3D complexity by using limited information coming from single 2D microtomographic images. Apple fleshy parenchyma tissue and muffin crumb were chosen to test the ability of the reconstructions to mimic structural diversities. Several metrics based on morphological measures and cluster functions were utilized to analyze the fidelity of reconstructions. For the apple, reconstructions are accurate enough proving that lineal, L2, and two-point, S2, functions sufficiently describe the complexity of apple tissue. Muffin structure is isotropic but statistically inhomogeneous showing at least two different porosity domains which reduced the fidelity of reconstructions. Further improvement could be obtained by using more CFs as input data and by implementation of the techniques dealing with statistical non-stationarity. Novel stochastic reconstruction and CF-based characterization methods could improve the fidelity of reconstruction and future advances of this technology will allow estimating macroscopic food properties based on (limited) 2/3D input information

Soil thin-section information.

<p>*according to Russian soil classification [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0126515#pone.0126515.ref083" target="_blank">83</a>]</p><p>Soil thin-section information.</p

Overall scheme of the reconstruction procedure.

<p>Illustrations are provided for each stage using reconstruction of circles as example.</p

All original eight soil type images (left column) with their best performing reconstructions based on a cluster function analysis (middle column) or pore morphological analysis (right column) (if reconstruction performance for both analyses is identical, then only one image is shown).

<p>Size of thin section = 2.1×2.1 cm^<i>2</i>. Blue shaded areas highlight pore features that were poorly reconstructed: type II) vertical pore; III) complex elongated pores; V) one connected pore dominating entire image; VI) one connected fissure-like pore; VII) numerous horizontal cracks; VIII) horizontal features in the upper-right marked region.</p

Main concepts of the morphological analysis.

<p>a) morphological parameters calculated for each pore element, and b) examples of pores extracted from original soil images and their shape classifications (all five shape classes are shown in roundness (<i>4πA/P</i>^<i>2</i>)—isometry (<i>D/L</i>) coordinates).</p