# Entropy of Shortest Distance (ESD) as Pore Detector and Pore-Shape Classifier

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## Abstract

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## 1. Introduction

#### 1.1. Motivation

## 2. Methodology

#### 2.1. Mathematical Model

^{th}possible event be ${p}_{i},i=1,\cdots ,N$ and suppose the forecaster gets a payoff $f\left({p}_{i}\right),i=1,\cdots ,N$ if he predicts this event, that is his expected payoff is $\sum {p}_{i}f\left({p}_{i}\right)$. If we want to keep the forecaster honest, we must select a function $f\left({p}_{i}\right)$ such that for any other probability distribution ${q}_{i},i=1,\cdots ,N$ one has:

**Figure 1.**Ludwig Boltzmann’s grave in the Central Vienna Cemetery, with his famous equation, S=k log W.

#### 2.2. Entropy of the Shortest Distance

**Figure 2.**A model representing the case of strong correlation between the placement of the mineral occurrences (yellow dots), and lineaments.

**Figure 3.**Spatial relation between three shapes ("granite outcrops" blue, "mineral occurrences" (red), and "lineaments", black). Scaled down by a factor ${10}^{5}$, the model might represent an outcrop of a vuggy, fractured limestone (see Figure 7), reducing it by ${10}^{8}$ it will resemble an optical micrograph of a triple porosity carbonate (Figure 8, Figure 9). Our entropy technique remains applicable through this enormous range of scales.

#### 2.3. Sliding Window Entropy Filtering for Bore Boundary Enhancement

**Figure 5.**Illustration of the sliding window entropy technique for a better definition of the boundary of the pore ${A}_{0}$. The sliding window W, which moves out of ${A}_{0}$, has a size less than half the distance to the nearest pore. The sequence ${A}_{0}\subset {A}_{1}\subset \cdots \subset {A}_{N}$ is strictly increasing, the difference sets ${\rho}_{k}={A}_{k}\backslash {A}_{k-1}$ $(k=1,\cdots ,N)$ form one pixel wide “rings” or “halos” around ${A}_{0}$.

## 3. Examples, Discussion, and Outlook

#### 3.1. PROGNOZ Application to Pore Boundary Detection

^{rd}image of Figure 7, the entropy cutoff $H\le 2$ reliably defines the “pores” (more exactly, vugs and caves in this case, as the picture represents the outcrop scale). The inset in Figure 7 shows the histogram of distances from randomly selected points to the nearest pore. To compute a histogram such as this, it is not necessary to move a sliding window W all over the image, we only need to randomly generate a large number of Poisson distributed points and compute the entropy of the probability distribution of their distances from the nearest pore. The mathematical treatment of the Poisson-distributed points approach is very challenging, and we have not attempted it in this paper. Mark Berman [25] derived the distribution of the distances of a fixed point from Poisson-distributed objects of random sizes and directions, as well as the distribution of distances between a fixed object and random Poisson-distributed points. We think that his results, combined with Tomiczková’s [26] Equation (7) for the area $\mu \left\{S(r;A)\right\}$ will form the foundations upon which the theory of ESD of random Poisson-distributed points from the nearest pore will be developed.

**Figure 7.**Entropy of shortest distance (ESD) processing of a carbonate outcrop photo. The second image in the sequence shows the entropy map over the whole image, as discussed in the text, the cutoff $H\le 2$ defines the pores (3rd image). The inset shows the histogram of distances from randomly selected points to the nearest pore.

**Figure 8.**10 × magnification of a rock sample, taken from the outcrop in Figure 6. The position of the section is perpendicular to the face of the rock wall.

**Figure 9.**Entropy of shortest distance (ESD) isolines of the micrograph on Figure 7. The ranges of entropy values are different for the various objects: Large vugs (H = 0.2–0.7), small vugs and pores (H = 1–1.7), solid matrix (H = 1.9–2.4).

#### 3.2. Concluding Remarks and Outlook

## Acknowledgments

## Conflict of Interest

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**MDPI and ACS Style**

Korvin, G.; Sterligov, B.; Oleschko, K.; Cherkasov, S.
Entropy of Shortest Distance (ESD) as Pore Detector and Pore-Shape Classifier. *Entropy* **2013**, *15*, 2384-2397.
https://doi.org/10.3390/e15062384

**AMA Style**

Korvin G, Sterligov B, Oleschko K, Cherkasov S.
Entropy of Shortest Distance (ESD) as Pore Detector and Pore-Shape Classifier. *Entropy*. 2013; 15(6):2384-2397.
https://doi.org/10.3390/e15062384

**Chicago/Turabian Style**

Korvin, Gabor, Boris Sterligov, Klaudia Oleschko, and Sergey Cherkasov.
2013. "Entropy of Shortest Distance (ESD) as Pore Detector and Pore-Shape Classifier" *Entropy* 15, no. 6: 2384-2397.
https://doi.org/10.3390/e15062384