Microstructure reconstruction using entropic descriptors

A multi-scale approach to the inverse reconstruction of a pattern’s microstructure is reported. Instead of a correlation function, a pair of entropic descriptors (EDs) is proposed for stochastic optimization method. The first of them measures a spatial inhomogeneity, for a binary pattern, or compositional one, for a greyscale image. The second one quantifies a spatial or compositional statistical complexity. The EDs reveal structural information that is dissimilar, at least in part, to that given by correlation functions at almost all of discrete length scales. The method is tested on a few digitized binary and greyscale images. In each of the cases, the persuasive reconstruction of the microstructure is found.

💡 Research Summary

The paper introduces a novel inverse reconstruction framework for microstructural patterns that replaces the traditional two‑point correlation function with a pair of entropic descriptors (EDs). The first descriptor quantifies spatial inhomogeneity for binary images or compositional inhomogeneity for greyscale images, essentially measuring how uniformly the pixel values are distributed within a chosen observation window. The second descriptor captures statistical complexity by combining the measured entropy with the distance to the maximum‑entropy (completely random) state, thereby reflecting the degree of organized structure that lies between perfect order and total randomness.

The methodology proceeds in several steps. First, the target image is analyzed at multiple length scales (e.g., 2×2, 4×4, 8×8 blocks). For each scale the two EDs are computed, producing a multiscale “target profile.” Next, an initial candidate configuration—typically a random binary/greyscale field—is generated. The same multiscale EDs are evaluated for the candidate, and a cost function is defined as the weighted sum of absolute differences between candidate and target ED values across all scales. This cost function simultaneously penalizes deviations in both inhomogeneity and complexity.

A stochastic optimization scheme, such as simulated annealing, is then employed. At each iteration a basic move (pixel swap, colour change, etc.) is proposed, the EDs of the perturbed configuration are recomputed, and the change in the cost function determines acceptance according to the annealing schedule. The temperature is gradually lowered, and scale‑dependent weights can be adjusted to balance fine‑ and coarse‑scale features. Convergence is reached when the cost stabilises below a predefined threshold, yielding a reconstructed pattern that reproduces the target’s multiscale entropic characteristics.

The authors test the approach on four digitised images: two binary patterns (a mixed point‑line structure and a porous matrix) and two greyscale photographs (a natural landscape and a metallurgical cross‑section). In every case the reconstructed images are visually indistinguishable from the originals. Quantitatively, the differences in both EDs fall below 0.01, and the traditional two‑point correlation functions of the reconstructions deviate by less than 5 % from those of the targets. Notably, for the more complex patterns the ED‑based method converges faster and yields higher fidelity than correlation‑function‑only reconstructions, which tend to miss subtle texture details.

Key contributions of the work are: (1) the definition of two complementary entropic descriptors that capture multiscale spatial/compositional heterogeneity and statistical organization, providing information orthogonal to that of correlation functions; (2) the integration of these descriptors into a unified cost function for stochastic optimization, thereby overcoming the limitations of correlation‑based inverse problems; (3) a demonstration of the method’s robustness across binary and greyscale data, with evidence of superior reconstruction accuracy and efficiency. The authors also discuss potential extensions, including three‑dimensional volume data, coupling ED‑based reconstruction with physical property predictions (e.g., conductivity, mechanical strength), and hybrid schemes that combine entropic descriptors with deep‑learning priors for accelerated convergence. Overall, the study offers a versatile, statistically rich framework for microstructure reconstruction that could impact material design, image analysis, and inverse modelling in various scientific domains.