The equivalent area disc diameter approximates grain size by using the diameter of a disc with the same area as the grain (Figure 2A; Ren et al., 2021; Wang et al., 2008). It is easily computable but should only be used when the grain form is very similar for the entire data set, as it assumes that all the grains are perfectly circular (Pirard, 2004).

The maximum inscribed circle of a grain is the largest circle that can be drawn inside the grain without extending beyond its boundaries. It is used as a measure of the short axis. The minimum circumscribed circle is the smallest circle that can enclose the grain, and it is used as a measurement of the grain’s long axis (Figure 2C; Cho et al., 2006; Wadell, 1932).

The Feret diameter is defined as the maximum distance between two parallel tangential lines that can be drawn in a specific direction across the grain (Han et al., 2023; Persson, 1998). For measurements, the orientation of the object can be changed at regular intervals, e.g., every 5°, and the distance is computed. The result of the method is a list of distances describing the discrete version of the grain outline. The most commonly used Feret diameters are the maximum, minimum, and mean (Figure 2B). However, as it is an incremental method, this approach could miss the longest or shortest distance depending on the measurement step.

The minimum circumscribed circle diameter and the maximum Feret diameter measure the same axis, which is the longest distance between two points placed on the grain outline. In the case of the maximum inscribed circle diameter, if the center of the circle is the same as the grain’s center of rotation, the measure will be precisely the same as the minimum Feret diameter. The only difference between these methods is the precision induced by the analysis step (Figure 2B).

The equivalent-moment ellipse is defined as an ellipse having the same moments of inertia for each axis of the grain. It approximates the grain by an ellipsoid that is dynamically equivalent to it (Medalia, 1971; Figure 2D). This method has the advantage of obtaining a biaxial approximation of the grain and its orientation, and it can also weight orientation by area (Higgins, 2006), as it preserves the same perimeter and area (Rodieck, 2007).

The minimum enclosing rectangle is a rectangle that surrounds the grain with a specific direction to obtain the smallest possible area (Figure 2E; Maerz, 2004; Xing et al., 2023). It can be seen as the Feret measurements performed along the principal orientation and its perpendicular (Wang, 2006). This biaxial approximation of the grain has the advantage of imitating its passage through the smallest sieve possible.

The orientation of the long axis measures the angle between the long axis of the minimum enclosing rectangle and the horizontal axis of the image, ranging from 0° to 180°. The second-moment orientation provides the orientation of the least second-moment axis of the object binary image, which has the advantage of measuring the orientation even for complex contours (Higgins, 2006; Medalia, 1971).

The earliest use of mathematics to classify grain form involved comparing axis ratios or parameters to an ideal form (Wadell, 1933; Zingg, 1935). The axis ratio method uses 3D measurements to classify, for example, grain forms such as oblate to prolate. As discussed in the previous section on size measurement methods, there are various methods for measuring axes. Depending on the size measurement method being used, different formulas have been developed. However, they can be simplified by assuming that in 3D, grain size has three measurements: the long ( L ), intermediate ( I ), and short ( S ) axes (Equation 1) or the length, width, and thickness. In this article, the term “long axis” will be used because it is unclear in the second term which axis represents the longest between width and thickness. The simplification is S L , I L , S I  or  L S , L I , I S   (1)

The choice of formula depends on the user’s intention to represent numbers between 0 and 1 or greater than 1, but they describe the same grain aspect. In 2D, the only available axis ratio is elongation, the first ratio in the two groups representing the ratio between the long and short axis (Equation 1). Some authors have attempted to use mean diameter as a measure for the intermediate axis (Wettimuny and Penumadu, 2004).

Another way to describe form is by comparing the grain to an ideal shape such as a circle, rectangle, or ellipse. The parameters used are areas or perimeters, represented as a ratio using both the grain and ideal shape parameters (Arasan et al., 2011; Cox and Budhu, 2008; Li et al., 2021; Ulusoy et al., 2003). The result addresses the question, “How closely does this shape resemble the ideal shape?” This concept gives rise to circularity (often referred to as sphericity, even in 2D studies) and encompasses the variety of formulas that can exist for a single descriptor (Kuo and Freeman, 2000). It also demonstrates the potential differences in usage for the same descriptor. Circularly has been widely used to describe roundness and grain form, highlighting the lack of consensus in regard to specific formulas stemming from the varying scientific approaches and interpretations (Arasan et al., 2011; Maroof et al., 2020b).

A modern method for mathematically describing grains involves applying Fourier analysis to the grain contour curve. There are two techniques for obtaining the grain contour curve: the Rθ method and the elliptic method. The Rθ method, the oldest, plots the ratio of the mean diameter to the diameter as a function of θ, ranging from 0° to 360° (Bui et al., 1989). However, this method is only applicable to convex shapes. The elliptic method obtains separate curves for the x and y coordinates as functions of θ, which allows for characterizing concave grains. Fourier analysis is then performed on the combined curves (Caple et al., 2017). In this article, this method will be referred to as the xy method to indicate contour information extraction.

Fourier analysis begins by applying the discrete Fourier transform to the grain contour curve to convert spatial information into frequency domain information. Then, the power spectral density is computed to provide frequency distribution information of the boundary shape. Low-frequency components correspond to form characteristics. Analyzing the low harmonics and their amplitude reveals the harmonics that affects the most the grain form. Combined, these two parameters create the shape descriptor. For instance, in the case of a square, the fourth harmonic will be present, and its amplitude will be high, whereas for an octagon, it will be the eighth harmonic. Consequently, it is essential to specify the harmonics used, as they could change the meaning of the form descriptors entirely.

To address the need for a systematic approach to analyzing over 1000 papers on quantitative shape descriptors in various sciences, we adopted a statistical approach. The goal of this extensive bibliographic research was to identify a wide range of methods for quantifying grain shape in fields such as geology (Barrett, 1980; Yang et al., 2022), agronomy and crop sciences (Igathinathane et al., 2008; Vaezi et al., 2013), chemical engineering (Haffar et al., 2021; Huo et al., 2016; Lu et al., 2010), and civil engineering (Al-Raoush, 2007; Chandan et al., 2004; Maroof et al., 2020a). We selected articles using the following steps:

1. A broad query, including necessary keywords and their synonyms, was initiated and subsequently refined to obtain a substantial number (100 s–1,000 s) of articles matching the research topic. This initial selection was based on a quick data set review, including brief readings of the abstracts.

The first step in the approach involved refining the query multiple times to obtain a reasonable number of articles. The number of articles per query ranged from 2044 to 42,262. Tens of thousands of articles were found for cases where physics-related articles do not use image-based size measurement methods, which was not desired for this study. The selected query produced 2127 articles, which were condensed to 274 using VOSviewer by removing articles lacking cross-referencing. These 274 articles were then grouped into 24 clusters, resulting in the selection of 82 articles for further study. Moreover, when an article referenced a formula used, the reference article was read, increasing the number of articles read to 109 (see the Supplementary Material for the list). However, of the 359 shape descriptors identified during the literature review, only 66% of the descriptors in the papers had a cited reference to indicate their source (when the article proposed a new method, it was considered as referenced). This absence of sourcing indicated a lack of rigor in the use of these descriptors and measurement methods.

The second step involved testing shape descriptors from the literature survey using images that were generated with known shape parameters for grain image simulation. The objective was to measure and compare size and orientation using different methods. This step was important because shape descriptors often use size measurements in their formulas. As seen earlier, there are several methods for size measurements, and the chosen method may not match what is needed for the form descriptors. For example, rectangularity should be computed with the minimum enclosing rectangle parameters, not the Feret maximum and minimum measurements, as the two axes of the rectangle are perpendicular, which may not be the case for Feret’s size. In addition to size and orientation, few also evaluated form descriptors and the dependencies between form, roundness, and roughness. Finally, we applied our method to an actual grain population to evaluate its effectiveness in a real-life scenario.

To verify the applicability of this method to real-life situations, we acquired images of galena grains from glacial sediments, using the protocol of Back et al. (2023), with the acquisition parameters held constant, except for the change in light source to transmitted light and an exposure time of 364.2 µs.

Transmitted light images of opaque minerals provide clear grain outlines and simplify the segmentation process. The segmentation procedure involved 1) converting the image from RGB color space to grayscale; 2) applying a pixel intensity threshold set at 30; 3) implementing opening and closing operations to eliminate anomalous pixels in both the object and background; and 4) extracting each grain contour using the find Contours method from the Open CV library. Finally, the contours were filled with white (pixel intensity at 255), and the background was colored black (pixel intensity at 0), saving the result as a binary image. Some contours were manually removed because of acquisition artifacts caused by the stitching method of the Olympus Stream® software and touching grains. We used 580 galena grain images for training and 4 images for testing.

The PCA methodology used for the form analysis was replicated, starting with an initial PCA on all computed features. Then, using the loadings and explained variances, we selected the most discriminant descriptors (eight in this case), and a second PCA was performed exclusively on these descriptors. To ensure method reproducibility, four characteristic grains were removed from the PCA training data set and transformed into the PCA domain after the learning process.

The size measurement methods exhibited significant differences between the short- and long-axis measurements (Figure 5). The short axis (Figure 5A) showed low reference deviation for the maximum inscribed circle, minimum Feret diameter, and bounding rectangle height; however, it showed a high reference deviation for the mean Feret diameter, equivalent-moment ellipse short axis, and equivalent area disc diameter. The long axis (Figure 5B) generally had higher values of reference deviation for all methods, and these differences can be attributed to the disparity between the measured axis in the measurement method and the reference axis of the shape (Figure 5).

For the short-axis measurement, the reference deviation was −0.08 ± 0.10 (mean ± standard deviation) for the maximum inscribed circle diameter, −0.06 ± 0.06 for the minimum Feret diameter, 0.06 ± 0.17 for the mean Feret diameter, −0.05 ± 0.05 for the bounding rectangle height, −0.03 ± 0.11 for the equivalent-moment ellipse short axis, and −0.01 ± 0.12 for the equivalent area disc diameter. Given these results and the boxplot information (Figure 5A), the method having the lowest reference deviation and highest consistency was the bounding rectangle height.

For the long-axis measurements, the reference deviation was 0.07 ± 0.14 for the minimum circumscribed circle, 0.06 ± 0.15 for the maximum Feret diameter, 0.00 ± 0.10 for the mean Feret diameter, −0.01 ± 0.06 for the bounding rectangle width, −0.01 ± 0.08 for the equivalent-moment ellipse short axis, and −0.06 ± 0.10 for the equivalent area disc diameter. Despite its higher mean reference deviation and standard deviation, the minimum circumscribed circle diameter was the most consistent method, as highlighted by the boxplot graph (Figure 5B).

The long-axis orientation showed great precision for shapes having a clear long axis and a diameter elongation greater than 1.2 (Figure 6A). Below this value, the precision was highly variable. The second-moment orientation was less precise (Figure 6B), but all points remained within a 10% error margin, even for those with a very low elongation close to 1. This method was more accurate than the long-axis orientation for shapes with low elongation.

The first PCA loadings indicated the following parameters were being discriminant for the 13 ideal shapes: minimum circumscribed circle, mean Feret diameter, convex hull perimeter, maximum inscribed circle, rectangularity perimeter, and Fourier amplitude for both methods. The selected amplitudes were the 1st, 3rd, 4th, 6th, and 8th for the Rθ method and the 1st, 3rd, 5th, 7th, 9th, and 11th for the elliptic method (xy; Table 3). When two methods described the same aspects, e.g., minimum circumscribed circle and maximum Feret diameter, we selected the one with the highest loading value.

In the second PCA, using the 10 and 11 selected descriptors for Rθ and xy, respectively, PC1 to PC3 collectively accounted for 87.0% and 83.8% of the total explained variance, with PC1 alone accounting for 43.9% and 49.4% for the Rθ and xy PCA, respectively.

Using Table 3, which presents the first three principal components of both methods, we could rebuild the principal component’s linear combinations (Equation 5) as P C = P C l o a d i n g s 1 × z d e s c r i p t o r s 1 + P C l o a d i n g s 2 × z d e s c r i p t o r s 2 + P C l o a d i n g s 3 × z d e s c r i p t o r s 3 + … + P C l o a d i n g s n × z d e s c r i p t o r s n (5) where P C is the selected principal components, P C   l o a d i n g s n is the associated loadings and its related descriptors ( d e s c r i p t o r s n ), n is the row number in the table, and z is the scaling function described in Equation 4. An example using the first principal component of Rθ is presented in Equation 6. P C 1 R θ = 0.465 × z d c i r + 0.427 × z d F m e a n + 0.418 × z P c h + 0.165 × z d i n s + 0.309 × z R p + 0.191 × z R θ 1 + 0.191 × z R θ 3 + 0.371 × z R θ 4 + 0.205 × R θ 6 + 0.228 × z R θ 10 (6)

Figure 7 presents the PC2 and PC3 axes, as most differences were easily visible using this axis pair. The scatterplots of Figures 7A,B depict ideal shapes (without roughness and roundness), whereas those of Figures 7C,D represent shapes having maximum roughness. In the PCAs of Rθ (Figures 7A,C) and xy (Figures 7B,D), the Rθ clusters were more tightly defined and grouped than their xy counterparts. This divergence was greatest for the cases of maximum roughness (Figures 7C,D), making form discrimination challenging when using xy amplitudes. Nonetheless, even under these conditions, the Rθ PCA remained capable of differentiating triangular, circular (hexagons, ellipses, and circles), and rectangular (squares, rectangles) shapes, thereby highlighting the robustness of the Rθ method. Given the negligible contribution of concave segments of the grain perimeter to its form description, coupled with the Rθ method’s ability to retain discrimination capability even in noisy conditions, Rθ amplitudes emerged as a preferred method for describing shapes.

Figure 8 illustrates the effect of roughness on the Rθ PCA. Roughness had a significant effect on PC1 and a smaller effect on PC2 and PC3 (Figures 8A,C). Moreover, the Rθ method retained its ability to distinguish triangular, circular, and rectangular forms despite elevated roughness.

Figure 9 illustrates the effect of roundness on the Rθ PCAs. All graphs (Figures 9A–C) demonstrated that form recognition using Rθ amplitudes was greatly affected by roundness, and no PC showed any apparent reduced effect from roundness. All the shape clusters tended to be attracted toward the circle cluster, making initial form recognition more difficult as roundness increased.

In Figure 10, we divided the 13 forms into two columns for better visualization. The left-hand column zooms in slightly on the PCA axes to distinguish differences between forms approximating a circle (high number of sides). The right-hand column shows the remaining forms. The delineated zones represent the convex hull of the test 1 and test 8 data sets. Tests 1 and 8 are distinct data sets of ideal shape images characterized by identical parameters (R and N were set at 0), demonstrating excellent reproducibility. Pentagons, hexagons, and decagons were clearly distinguished in Figures 10A,E, whereas shapes with fewer than four sides were easily discriminated in Figures 10B,D,F. The scatterplot PC2 vs PC3 (Figure 10D) best discriminated these forms. The circle and square are particular cases of the ellipse and rectangle, respectively, having a constant side length. Both form clusters appeared as a straight line at the longest extremity of their respective form cluster (Figures 7B,9B,D,F).

After validating our approach using generated images, we then applied our process to actual images of galena grains. The first PCA loadings indicated that particle perimeter, polygon to circle area, diameter elongation, minimum circumscribed circle diameter, maximum inscribed circle diameter, circularity, and the first and third amplitudes of the elliptic Fourier method were discriminant form descriptors for the galena grain population. A second PCA that applied the eight selected descriptors had PC1 to PC3 collectively accounting for 94.4% of the total explained variance, with PC1 alone accounting for 53.2%. The other PCs were omitted, as they did not significantly contribute to shape discrimination. Loadings of the first three principal components for the eight descriptors are presented in Table 4.

The PCA output for the galena binary images demonstrated how the first principal component sorted the galena grains by size (Figures 11A,C). The scatterplots were consistent with the grain data set image (Figure 11E), and clearly showed a large number of small grains around 180 pixels (e.g., grains 263, 391, and 494) and a smaller number of larger grains, e.g., grain 101 (Figure 11D). PC2 distinguished rounded and elongated rectangular grains, as indicated by its high positive or negative loadings for the polygon to circle area, circularity, and diameter elongation. These loadings values are in accordance with the grain position (Figure 11D) along PC2 (Figures 11A,B), ranging from the most rectangular and elongated grain 391 to the less elongated subangular grain 494 to the sub-round grain 101 and finally to rounded grain 263 having a low diameter elongation of 1.19. PC2 accurately reflected the shapes observed in the grain data set image (Figure 11E), with a small number of well-rounded and very elongated rectangular grains, most being rectangular grains similar to that of grain 494. Finally, PC3 described both grain roughness and roundness, with smooth and rounded particles (e.g., grains 263, 494, and 101) having low values and grains with angular and complex outlines (e.g., grain 391) having higher values (Figures 11B–D).

In conclusion, the PCAs indicated that the galena grain population consisted mainly of small stubby rectangular subangular grains, such as grain 494, with a smaller proportion of grains that were either well rounded (263) or very elongated (391). This quantitative description of the grains aligned perfectly with the grain data set image showing the same patterns (Figure 11E). An interactive graph displaying each point within the PCA plots, along with the corresponding grain images, can be accessed at https://pca-grain-shapes.onrender.com. The loading of the webpage may take a while.

The objects used in this framework are binary images obtained from an image generation pipeline or a segmented transmitted light microscope image. This technique can be applied to any image as long as a precise contour of the object of study is obtained and transformed into a 2D-binary image. However, the image resolution must exceed a certain threshold to obtain unbiased results. This threshold can vary depending on the applied descriptor; for the tested form descriptors, a higher threshold of 130 pixels is set for object length (Sun et al., 2019). The segmentation technique must also be coherent with the image and mineral properties to ensure proper segmentation. Segmentation inaccuracies can produce significant errors in the shape descriptor results (Zheng and Hryciw, 2016).

Regarding the generated images, there are newer tools available for simulating more realistic grain images (Mollon and Zhao, 2013). However, evaluating the Fourier amplitudes on generated images using the same techniques and tools would be inappropriate, given their reliance on Fourier’s methods.

The most commonly used formula (see Appendix B) is used to compute circularity. However, there exists a wide range of formulas for circularity, differing in whether the circularity is squared or square-rooted. These equations describe the same concept, but the distribution of the function changes. This change highlights differences within certain intervals of x. For example, the square root exaggerates x values between 0 and 1, whereas the square exaggerates high values of x. Using PCA, we can determine which of those functions has a greater effect, providing valuable insights into a data set. If the square root of circularity has a more significant effect on the variance, it indicates that the differences in our data set lie in very low circularity values, illustrating the degree of roundness of the grains. Emphasizing these differences will assist with clustering in a later step.

Form discrimination can be improved in several ways. First, the orientation of objects influences the results of the Fourier method because of the finite number of pixels (Wettimuny and Penumadu, 2004). Aligning all objects in the same direction would reduce this influence. Additionally, using the convex hull rather than the actual grain outline for rough shapes can minimize the roughness effect and improve form recognition. The form of the grain convex hull can be considered as the grain form. Finally, size difference affects PCA output. Scaling all images to the same height or width while maintaining the aspect ratio should, therefore, enhance form discrimination.

Our PCA output demonstrated that roundness and roughness significantly influence form descriptors, and the reverse should also be true. Therefore, including roundness and roughness descriptors in the PCA analysis will enhance form discrimination.

Regarding the analysis of real grain images, we focus on galena grains extracted from glacial sediments. Within this population, only a few grains are large (exceeding a value of 4 on PC1), and none exhibit a high degree of roundness. Additionally, many grains, such as grain 494, are medium-sized and exhibit a rounded side along with an angular one, leading to their classification as sub-rounded. Given galena’s perfect cleavage properties, it is plausible that large grains often fracture into smaller pieces, resulting in grains showing an angular and rounded side. These shape analysis results are consistent with a short distance of glacial transport for such minerals (Paulen et al., 2011). With extended transport, sub-rounded grains would likely become smaller, as they would result from the separation of medium-rounded grains and the number of medium-rounded grains would increase.

Our results on real grain images (Figure 11) demonstrated that combining PCA with form descriptors can reveal shape patterns within the data set. The next step is to use clustering algorithms on the captured PCA data to automatically identify and quantify the number of distinct grain shape clusters and the number of grains in each category. However, a greater distance between points is required to cluster the data set more accurately and efficiently. In a future study, we will enhance these differences by including the roundness and roughness descriptors.

More advanced techniques, such as machine-learning algorithms (Li and Iskander, 2022), can be applied to the data generated by all form descriptors. Nevertheless, PCA retains the advantage of interpretability. The loadings indicate the role of each descriptor in shape discrimination, and the related formulas help understand the physical parameters underlying these variations.

The quantitative framework for grain shape analysis proposed in this study has significant potential for applications in various real-world scenarios. The proposed method can be directly applied to geological problems where object shape is a critical factor. It has the potential to advance research applications, such as developing a quantitative zircon classification algorithm inspired by Pupin (1980), or addressing exploration challenges, like gold grain classification in till sediments, which relies on morphological characteristics (Girard et al., 2021).

This approach could also help in planetary science, where chondrule shape and size analysis provides valuable insights into the relationships among meteorite groups, the classification of ungrouped chondrites, and the temporal and spatial variability within the solar nebula (Ebel et al., 2024; Floyd et al., 2024). Moreover, the shape (or texture) of the corundum-bearing Ca–Al-rich inclusions found within chondrites could indicate different origins for these inclusions: condensation (rough and irregular) or crystallization from a melt (rounded inclusion with radiated corundum; Needham et al., 2017).

The method also holds potential for industrial applications. For instance, geometallurgy relies on a quantitative understanding of the characteristics of primary resources (Frenzel et al., 2023), with shape being a key factor. Improved measurement of roughness, and by extension, surface area could enhance understanding of grain flotation capacity (Wang and Zhang, 2020), aiding in the optimization of flotation processes. Additionally, more consistent size measurements would contribute to the improved design of gravimetric separation processes, where size plays a critical role (Andò, 2020).

The shape of soil grains plays a crucial role in determining various soil properties, significantly impacting geotechnical behavior. Studies have shown that grain shape influences factors such as compaction, permeability, and shear strength (Altuhafi et al., 2016; Lu et al., 2019). For instance, rough grains tend to interlock more effectively than rounded grains, resulting in higher shear strength (Lu et al., 2019) and better stability in soil structures. Additionally, the arrangement and packing of different grain shapes can affect the void ratio and porosity of the soil (Lu et al., 2019), which in turn influences water retention and drainage capabilities. As a result, understanding and analyzing grain shape is essential for predicting soil behavior in engineering applications, ensuring the reliability and safety of geotechnical projects.

Finally, by offering a reproducible and adaptable tool, this approach could enhance precision and standardize grain analysis practices across disciplines. Future studies should further explore these applications. It includes integrating the methodology with different imaging techniques and assessing its impact on existing practices in related fields. Such efforts will help fully realize the broader utility and influence of this framework.