# Representativity of 2D Shape Parameters for Mineral Particles in Quantitative Petrography

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Overview

^{2}) and has been applied to shape-fabric analysis of deformed grains in a rock [16]. A variation of this parameter is the perimeter over area normalized ratio (PoA) (¥) [17]. This parameter is the inverse of the square root of the shape factor and has been used to assess pore space in sandstone [18].

#### 1.2. Shape Parameter Quantification

^{®}, Aphelion) with morphological functions and a programming module (e.g., Visual basic) to automate the procedure [40], and (iii) free and open source image processing programs (e.g., ImageJ: [35,41]). Due to precision image scales and high data density, observations and measurements are capable of combining roundness and roughness.

#### 1.3. Objectives

## 2. Materials and Methods

#### 2.1. Design and Development of Type Particles

^{®}CS6. Particles of an average diameter of 2.54 cm (1 inch) were taken as a base. The objects were scanned with resolutions of 900, 750, 600, 300, 150, and 75 DPI and saved in a raster format (tif, 8 bits, B/W).

#### 2.2. Parameters and Measurement Techniques

^{®}(v. 1.52), which is compatible with Microsoft

^{®}OS with Windows 10 and OS El Capitan

^{®}. The fractal dimension was calculated using Image-Pro Plus

^{®}(v. 7.0) software, in Microsoft OS with Windows 10, by applying the Richardson Method [31].

#### 2.3. Determination of Analysis Conditions

#### 2.4. Data Acquisition and Evaluation

^{®}), with each particle labelled with a specific identifier. Due to the use of three different programs (ImageJ, Image-Pro Plus, and Roussillon Toolbox), it was necessary to generate a file that would group all the measurements obtained for each particle. In order to facilitate the interpretation of the numerical measurements, type particles were arranged into graphic charts organized by increasing parameter values (e.g., particle template according to Rw and S). These charts were supplemented with qualitative information extracted from visual charts used in particle shape descriptions [6,43,48]. Through mathematical analysis of the measured parameters of the designed, theoretical particles, correlation curves and functions were obtained (e.g., Rw vs. FD, Rw vs. S, FD vs. S, etc.). These analyses were conducted in a previous study [49]. In order to limit the influence of S on other parameters, a correction was applied, based on the methodology proposed by [44].

#### 2.5. Estimation of a Regularity Indicator (RBC)

#### 2.6. Application to A Case Study and Validation

## 3. Results

#### 3.1. Description of Type Particles

#### 3.2. Optimal Analysis Conditions of Shape Parameters

#### 3.3. Shape Parameter Evaluation and A New Regularity Indicator (RBC)

#### 3.3.1. Shape Parameter Evaluation

#### 3.3.2. New Regularity Indicator (RBC)

_{1}) to each of the 18 particles is shown in Table 4. The multiple linear regression (Table 4) was based on the Equation (7):

^{2}= 0.82, and the equation presents a standard deviation of σ = 0.09. The absolute values of the standardized coefficients, a

_{2}= −0.692 and c

_{2}= −0.252 (Table 4), indicate that the independent variables with higher relative importance in the RBC equation are FD and ¥c (roughness and roundness, respectively). The RBC values of each of the 18 particles calculated from Equation (8) are presented in Table 4 as RBC. Figure 8 shows the 18 types of particles ordered according to the RBC parameter (obtained from the linear regression).

#### 3.4. Application and Validation of Shape Parameters

## 4. Discussion

#### 4.1. Representativity of the Shape Parameters Studied

#### 4.2. Minimum Particle Size for Shape Analysis

#### 4.3. Application of the Proposed Indicator of Regularity

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Criteria for the development of the 18 type particles, according to the evolution of different morphological parameters: (

**a**) decrease in sphericity. (

**b**) decrease in roundness, and (

**c**) increase in roughness.

**Figure 2.**Type particles drawn and digitized as a basis for quantifying morphological parameters (Rw, S, So, FD, and ¥) by means of image processing tools. The size of the particles is represented in the graphic scale and corresponds to the size of magnified particles.

**Figure 3.**(

**a**) Relationship between FD values of type particles (1, 5, and 10) vs. optical scanning resolution (75, 100, 150, 300, 600, and 900 DPI). (

**b**) Relationship between values of Rw of type particles (1, 6, and 12) vs. optical scanning resolution (75, 100, 150, 300, 600, 750, and 900 DPI). Calculations after [36].

**Figure 4.**Log-log relationship of percentage variation of area (A) and perimeter (P) of the type particle 1 vs. particle size (in pixels) with (

**a**) 900 DPI resolution, (

**b**) 600 DPI resolution, and (

**c**) 150 DPI resolution. Calculations after [44].

**Figure 5.**(

**a**) Values of the parameters measured for the 18 type particles. (

**b**–

**k**) Schematic diagrams of correlation between the different parameters (Based on [49]).

**Figure 6.**Representation of the 18 particles according to the obtained values (

**a**) of Rw vs. S. (

**b**) of ¥ vs. S. (

**c**) of FD vs. S. (

**d**) of So vs S.

**Figure 7.**Schematic correlation diagrams between the parameters So, FD, Rwc, and ¥c to verify their independence (based on [49]): (

**a**) FD vs. So, (

**b**) FD vs. ¥c, (

**c**) FD vs. Rwc, (

**d**) So vs. ¥c, (

**e**) So vs. Rwc, (

**f**) ¥ c vs. Rwc.

**Figure 9.**Microphotographs of the sandstones studied. Main mineralogy in plane polarized light conditions of: (

**a**) heterogeneous sandstone (SI) and (

**b**) homogeneous sandstone (SB). Segmented quartz particles of (

**c**) SI sandstone and (

**d**) SB sandstone. Images of the SB sample are amplified (3×) to improve visualization.

**Figure 10.**Grain size classes (µm) vs. grain area percentages. The average values of RBC (black) and SF (average and standard deviation, in red) are defined for each grain size class: (

**a**) Sample SI with a minimum size of 80 µm, (

**b**) sample SB with a minimum size of 80 µm, (

**c**) Sample SI with a minimum size of 240 µm, and (

**d**) sample SB with a minimum size of 240 µm.

**Figure 12.**(

**a**) Graphical representation of the values of shape parameters of particles 1, 2, and 18. (

**b**) Graphical representation of the values of shape parameters of particles 3, 4, and 5. X-axis on a logarithmic and dimensionless scale.

Properties | Shape Parameters | References |
---|---|---|

Sphericity (form) | Wadell’s circularity | [11,23] |

Sphericity | [6] | |

Ellipticity-aspect ratio-elongation | [16,24] | |

Rittenhouse’s sphericity | [25] | |

Roundness | Wentworth’s roundness | [10] |

Wadell’s roundness | [11] | |

Angularity | [12] | |

Shape Factor | [13] | |

Angularity Factor | ||

Circularity of Cox/ Shape factor | [14,15,16] | |

Perimeter over area normalized ratio | [17] | |

Solidity-convexity | [19,20] | |

Krumbein’s roundness number | [43] | |

Roundness | [44] | |

Roughness (surface texture) | Fractal Dimension | [26,27,28,29,30] |

**Table 2.**Morphological parameters together with their definition, mathematical expression, and the software tools used.

Morphological Parameter | Mathematical Expression | Software |
---|---|---|

Roundness (Rw) [11]. where r _{i} is the radius of curvature of each corner, N is the number of corners, and r_{ins} is the radius of the maximum circle inscribed in the particle. | $Rw=\frac{\raisebox{1ex}{${\Sigma}_{i=1}^{N}{\mathrm{r}}_{i}$}\!\left/ \!\raisebox{-1ex}{$N$}\right.}{{r}_{ins}}$ | Custom application in DOS OS Roussillon Toolbox [24] |

Sphericity (S) [6]. where d _{1} is the length of the particle and d_{2} is the width. | $S=\frac{{d}_{2}}{{d}_{1}}$ | ImageJ (v. 1.52) |

Solidity-convexity (So) [19,20]. where A _{T} is the area of the particle, and A_{convex} is the area defined by the convexity produced by the irregularity of the edge of the particle. | $So=\frac{{A}_{T}}{{A}_{T}+{A}_{convex}}$ | ImageJ (v. 1.52) |

Fractal Dimension (FD) [26]. | Richardson Method [31] | Image-Pro Plus^{®} (v.7) |

¥ (normalized PoA) [17]. where P _{i} and A_{i} are, respectively, the perimeter and the area measured for a particle. | $\mathsf{\yen}=\frac{{P}_{i}}{2\sqrt{\pi {A}_{i}}}$ | ImageJ (v. 1.52) |

Sf (Shape factor) [14]. where P _{i} and A_{i} are, respectively, the perimeter and the area measured for a particle. | $\mathrm{Sf}=\frac{4{\mathsf{\pi}\mathrm{A}}_{\mathrm{i}}\text{}}{{\mathrm{P}}_{\mathrm{i}}^{2}}$ | ImageJ (v. 1.52) |

**Table 3.**Minimum particle size (case a: 150 pixels and case b: 50 pixels) in metric units on which particle shape analysis is possible. The data was obtained in reference to the objectives and characteristics of a Leica 6000 M microscope.

Objective | Camera Lenses | Total | Geometric Calibration µm/pixel | Minimum Ø (µm) Case a | Minimum Ø (µm) Case b |
---|---|---|---|---|---|

40 | 0.63 | 25.32 | 0.156 | 23.4 | 7.80 |

20 | 0.63 | 12.66 | 0.312 | 46.8 | 15.60 |

10 | 0.63 | 6.33 | 0.624 | 93.6 | 31.20 |

5 | 0.63 | 3.17 | 1.248 | 187.2 | 62.40 |

4 | 0.63 | 2.52 | 1.564 | 234.6 | 78.20 |

2 | 0.63 | 1.26 | 3.120 | 468 | 156.00 |

**Table 4.**Values of parameters FD, So, ¥c, Rwc, and RBC

_{1}(1: estimated by expert criteria). Values of factors a, b, c, and d in the applied lineal regression (2: normalized value and σ: standard deviation). Values of RBC (a regression function applied).

Particle | Shape Parameters | Linear Multiple Regression | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

FD | So | ¥c | Rwc | RBC_{1} | a, a_{2}, σa | b, b_{2}, σb | c, c_{2}, σc | d, d_{2}, σd | e, σe | RBC | |

1 | 1.0180 | 1.00 | 1.06 | 0.94 | 1.00 | −3.230 −0.692 0.787 | 0.003 0.001 0.339 | −0.161 −0.252 0.15 | 0.037 0.043 0.129 | 4.314 0.82 | 0.90 |

2 | 1.0137 | 0.99 | 1.08 | 0.67 | 0.95 | 0.90 | |||||

3 | 1.0209 | 0.88 | 1.31 | 0.42 | 0.90 | 0.82 | |||||

4 | 1.0358 | 0.86 | 1.57 | 0.38 | 0.70 | 0.73 | |||||

5 | 1.0657 | 0.83 | 1.83 | 0.26 | 0.55 | 0.59 | |||||

6 | 1.1608 | 0.80 | 1.85 | 0.62 | 0.40 | 0.29 | |||||

7 | 1.0801 | 0.90 | 1.32 | 0.55 | 0.50 | 0.63 | |||||

8 | 1.1339 | 0.78 | 1.63 | 0.49 | 0.45 | 0.41 | |||||

9 | 1.0317 | 0.97 | 1.10 | 0.91 | 0.85 | 0.84 | |||||

10 | 1.0335 | 0.92 | 1.19 | 0.90 | 0.80 | 0.82 | |||||

11 | 1.0459 | 0.85 | 1.41 | 0.82 | 0.65 | 0.74 | |||||

12 | 1.0812 | 0.81 | 1.97 | 0.78 | 0.55 | 0.54 | |||||

13 | 1.0333 | 0.96 | 1.06 | 0.94 | 0.80 | 0.84 | |||||

14 | 1.0389 | 0.88 | 1.15 | 0.91 | 0.65 | 0.81 | |||||

15 | 1.0544 | 0.78 | 1.55 | 0.82 | 0.60 | 0.69 | |||||

16 | 1.0174 | 0.81 | 1.09 | 0.94 | 0.95 | 0.90 | |||||

17 | 1.0208 | 0.66 | 1.43 | 0.91 | 0.90 | 0.82 | |||||

18 | 1.0130 | 1.00 | 1.12 | 0.73 | 0.95 | 0.90 |

**Table 5.**Average values and standard deviations of shape parameters of the quartz particles studied. All particles in the SI sample were larger than 50 pixels (80 microns). Only 68% of the particles in SB were larger than 50 pixels (80 microns).

Samples | Statistics | FD | So | Sf | ¥c | Rwc | S | RBC |
---|---|---|---|---|---|---|---|---|

SI sample 105 particles > 50 pixels | Average | 1.1353 | 0.82 | 0.40 | 1.65 | 0.57 | 0.61 | 0.40 |

Standard deviation | 0.0557 | 0.09 | 0.16 | 0.36 | 0.14 | 0.16 | 0.23 | |

SB sample 99 particles | Average | 1.1213 | 0.83 | 0.37 | 1.50 | 0.77 | 0.61 | 0.48 |

Standard deviation | 0.0465 | 0.06 | 0.13 | 0.24 | 0.14 | 0.17 | 0.18 | |

SB sample 68 particles > 50 pixels | Average | 1.1124 | 0.85 | 0.43 | 1.53 | 0.70 | 0.62 | 0.50 |

Standard deviation | 0.0045 | 0.05 | 0.12 | 0.26 | 0.18 | 0.16 | 0.18 |

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**MDPI and ACS Style**

Berrezueta, E.; Cuervas-Mons, J.; Rodríguez-Rey, Á.; Ordóñez-Casado, B.
Representativity of 2D Shape Parameters for Mineral Particles in Quantitative Petrography. *Minerals* **2019**, *9*, 768.
https://doi.org/10.3390/min9120768

**AMA Style**

Berrezueta E, Cuervas-Mons J, Rodríguez-Rey Á, Ordóñez-Casado B.
Representativity of 2D Shape Parameters for Mineral Particles in Quantitative Petrography. *Minerals*. 2019; 9(12):768.
https://doi.org/10.3390/min9120768

**Chicago/Turabian Style**

Berrezueta, Edgar, José Cuervas-Mons, Ángel Rodríguez-Rey, and Berta Ordóñez-Casado.
2019. "Representativity of 2D Shape Parameters for Mineral Particles in Quantitative Petrography" *Minerals* 9, no. 12: 768.
https://doi.org/10.3390/min9120768