- Methodology
- Open Access
New parameter of roundness R: circularity corrected by aspect ratio
- Yasuhiro Takashimizu^{1}Email author and
- Maiko Iiyoshi^{2}
https://doi.org/10.1186/s40645-015-0078-x
© Takashimizu and Iiyoshi. 2016
- Received: 14 January 2015
- Accepted: 27 December 2015
- Published: 20 January 2016
Abstract
In this paper, we propose a new roundness parameter R, to denote circularity corrected by aspect ratio. The basic concept of this new roundness parameter is given by the following equation:
where Circularity_{perfect circle} is the maximum value of circularity and Circularity_{aspect ratio} is the circularity when only the aspect ratio varies from that of a perfect circle. Based on tests of digital circle and ellipse images using ImageJ software, the effective sizes and aspect ratios of such images for the calculation of R were found to range between 100 and 1024 pixels, and 10:1 to 10:10, respectively. R is thus given by
where C_{I} is the circularity measured using ImageJ software and C_{AR} is the sixth-degree function of the aspect ratio measured using the same software. The correlation coefficient between the new parameter R and Krumbein’s roundness is 0.937 (adjusted coefficient of determination = 0.874). Results from the application of R to modern beach and slope deposits showed that R is able to quantitatively separate both types of material in terms of roundness. Therefore, we believe that the new roundness parameter R will be useful for performing precise statistical analyses of the roundness of particles in the future.
Keywords
- Aspect ratio
- Circularity
- ImageJ software
- Krumbein’s visual roundness
- Roundness
Background
Many studies have investigated particle shape in the natural world, mostly based on the definitions of sphericity and roundness of rock particles proposed by Wadell (1932). Previous studies into particle shape have been discussed in a series of review articles (e.g., Barrett 1980; Clark 1981; Winkelmolen 1982; Diepenbroek et al. 1992; Blott and Pye 2008), and in general, such studies have mainly taken one of two approaches to understanding particle shape. The first is a simple method that involves the examination of visual images of particle grains (e.g., Krumbein 1941; Rittenhouse 1943; Powers 1953; Pettijohn 1957; Lees 1963). Determining roundness using the visual roundness chart proposed by Krumbein, which is further extended in this paper, is one of the most widely employed methods. However, such a method merely compares visual images, and therefore, the derived roundness values are not strictly quantitative. The second approach involves the quantitative determination of various shape parameters, and many evaluation methods have been designed to obtain relevant shape parameters (e.g., Schwarcz and Shane 1969; Orford and Whalley 1983; Diepenbroek et al. 1992; Yoshimura and Ogawa 1993; Vallejo and Zhou 1995; Bowman et al. 2001; Itabashi et al. 2004; Drevin 2007; Blott and Pye 2008; Lira and Pina 2009; Roussillon et al. 2009; Arasan et al. 2011; Suzuki et al. 2013, 2015). Both approaches, however, involve the analysis of each individual particle, and therefore, production of several thousand to several tens of thousands of shape parameters for reliable analysis is time consuming. Therefore, owing to the extensive time requirements and effort required for both approaches, neither is widely used, and there remains a need for an easy statistical method to derive parameters of particle shape.
In this study, we define a new roundness parameter, R, to denote the circularity corrected by aspect ratio, and present a case study of R calculation using ImageJ software (ver. 1.47q) released from the US National Institute of Health (Abramoff et al. 2004; Schneider et al. 2012). This represents a fairly simple method that helps to overcome the shortcomings of the previously published methods discussed above.
Basic concept of this study
where Circularity is the value defined by Eq. (1), Circularity_{perfect circle} is the maximum value of circularity, and Circularity_{aspect ratio} is the circularity when only the aspect ratio varies from that of a perfect circle.
Case study of parameter R calculation using ImageJ software
In this chapter, we present a specific case study, demonstrating the calculation of the parameter R defined in the previous chapter, using ImageJ software.
Methods
Test digital images were produced using Adobe Photoshop CS4 and Adobe Illustrator CS4. Shape parameters, including area, perimeter, circularity, aspect ratio, major axis length, and minor axis length, were measured from the test digital images using ImageJ software (ver. 1.47q). To validate the effectiveness of R defined in this paper, digital images of Krumbein’s original visual images (Krumbein 1941) were captured using a Fuji Xerox ApeosPort-IV C7780 scanner with a resolution of 600 × 600 dots per inch and a grayscale color profile; they were saved in a TIF image file format.
Grain-size distributions of modern slope and beach sediments were measured using a sieve ranging from −5.0 to 4.0 phi, with 0.5 phi intervals. The first quartile (twenty-fifth percentile), second quartile (median), and third quartile (seventy-fifth percentile) of grain size were obtained for each from the cumulative curves.
To obtain the new roundness parameter R value for these modern slope and beach sediments, we used an Olympus TG-1 digital camera to acquire digital images of particles in each grain-size class. Image analysis was then conducted separately for each grain-size fraction. The measurement of grain sizes in this analysis ranged from values equal to or coarser than 1.0 phi, with 0.5 phi intervals, with the finer limit (1.0 phi) of the measurement range determined by the limitations of the Olympus TG-1. This range was sufficient for comparing the sediments in this study, because of the coarseness of the material. For imaging, the particles were laid out on a transparent board. As the minor c-axes of the particles in this layout were nearly perpendicular to the board, we assumed that an imaginary plane perpendicular to the c-axis, which included the major and intermediate a- and b-axes, respectively, was parallel to the board. To obtain sharp silhouettes of particles, the light source was placed on the opposite side of the digital camera, allowing intentional capture of backlit images. The major lengths of the silhouettes were then adjusted to be more than 100 pixels. The digital images taken by Olympus TG-1 were transferred to the ImageJ software and processed into binary images. The circularity and aspect ratio of the silhouettes in the digital binary images were then measured using the ImageJ software, and the R values were obtained using Eq. (8) described below. The obtained R values for each grain-size class were integrated into a total R distribution for the individual samples using the weight percent of each grain size class. The calculated R distributions thus ranged from 0.400 to 0.925 with 0.025 intervals. The first quartile, median, and third quartile of R were calculated using Microsoft Excel 2007 software.
Validation of the effective resolution of digital images using ImageJ software
A digital image is an aggregate of pixels, which are the minimum units of the image. Hence, there should be an error in shape parameter values between the geometrically obtained true values and those calculated from the digital image. Therefore, to obtain the most effective size of digital images for shape analysis, we examined the errors in basic shape parameters, including area, perimeter, and aspect ratio, using ImageJ software.
Area and perimeter
Results of error of area and perimeter for various sizes of test circle images
Width of test circle image (pixel) | A _{I} | P _{I} | 2·r | A | P | ΔA (%) | ΔP (%) |
---|---|---|---|---|---|---|---|
Area measured by ImageJ (pixel) | Perimeter measured by ImageJ (pixel) | Major and minor axis length measured by ImageJ | Area calculated by p·r ^{2} | Perimeter calculated by 2·p·r | |||
1 | 1 | 2.828 | 1.128 | 0.999328 | 3.543716 | 0.067260 | −20.196760 |
2 | 4 | 5.657 | 2.257 | 4.000856 | 7.090573 | −0.021393 | −20.218015 |
3 | 9 | 9.657 | 3.385 | 8.999267 | 10.634289 | 0.008145 | −9.189979 |
4 | 12 | 11.314 | 3.909 | 12.001102 | 12.280483 | −0.009184 | −7.870074 |
6 | 32 | 19.314 | 6.383 | 31.999226 | 20.052782 | 0.002417 | −3.684186 |
8 | 52 | 24.971 | 8.137 | 52.001806 | 25.563134 | −0.003472 | −2.316360 |
11 | 97 | 34.627 | 11.113 | 96.995686 | 34.912512 | 0.004447 | −0.817792 |
16 | 208 | 52.284 | 16.274 | 208.007222 | 51.126268 | −0.003472 | 2.264456 |
23 | 421 | 75.598 | 23.152 | 420.985191 | 72.734138 | 0.003518 | 3.937439 |
32 | 804 | 104.569 | 31.995 | 803.996244 | 100.515236 | 0.000467 | 4.032985 |
45 | 1581 | 147.196 | 44.866 | 1580.973153 | 140.950667 | 0.001698 | 4.430865 |
64 | 3196 | 211.48 | 63.791 | 3196.013548 | 200.405295 | −0.000424 | 5.526154 |
90 | 6320 | 296.735 | 89.704 | 6319.946608 | 281.813369 | 0.000845 | 5.294863 |
128 | 12,796 | 422.96 | 127.642 | 12796.081336 | 400.999086 | −0.000636 | 5.476550 |
181 | 25,565 | 597.47 | 180.417 | 25564.935720 | 566.796604 | 0.000251 | 5.411711 |
254 | 51,104 | 842.607 | 255.084 | 51104.156942 | 801.369854 | −0.000307 | 5.145832 |
360 | 102,252 | 1193.97 | 360.82 | 102251.807881 | 1133.549225 | 0.000188 | 5.330229 |
510 | 204,520 | 1690.871 | 510.297 | 204520.017549 | 1603.144973 | −0.000009 | 5.472121 |
722 | 408,932 | 2391.253 | 721.574 | 408932.420696 | 2266.891106 | −0.000103 | 5.486011 |
1020 | 818,196 | 3381.743 | 1020.666 | 818195.500707 | 3206.516140 | 0.000061 | 5.464712 |
1442 | 1,636,024 | 4893.536 | 1443.278 | 1636024.391899 | 4534.190619 | −0.000024 | 7.925238 |
Aspect ratio
Effective size range of digital images in this study
The above results can be summarized as follows: (1) The |ΔA| values are sufficiently small to assume accuracy for all test circle images with diameters ranging from 1 to 1442 pixels; (2) The |ΔP| value is constant, at approximately 5 %, for test circle images with diameters of 64 to 1024 pixels; and (3) The AR_{I} value for the test circle/ellipse images with widths greater than or equal to 100 pixels, and a true aspect ratio greater than or equal to 10:1, remains constant. Consequently, in this paper, we consider the effective size and aspect ratios of digital images for shape analysis using ImageJ software, to be 100 to 1024 pixels and 10:1 to 10:10, respectively.
Relationship between aspect ratio and circularity in ImageJ software
Circularity in ImageJ software
where A _{I} and P _{I} are the area and perimeter measured using ImageJ (ImageJ User 2012), respectively. This therefore implies that C _{I} is directly determined by A _{I} and P _{I}. For instance, if their two different P _{I} values are provided for digital images with the same A _{I} values, the image showing high circularity will have a shorter perimeter than that of the other image.
Relationship between circularity and aspect ratio in ImageJ software
When considering roundness as a shape parameter, the degree of roundness of a deformed circle (ellipse) image should be the same as that of a perfect circle image. Different values of C _{I} will therefore correspond to changes in aspect ratio, and C _{I} alone should not be used as a roundness shape parameter. For this reason, the relationship between C _{I} and AR_{I} is examined in this section, and we attempt to correct C _{I} using AR_{I}.
CI and ARI values calculated using ImageJ applied to the test images of Fig. 6
aspect ratio of test ellipse image | CI | ARI | aspect ratio of test ellipse image | CI | ARI | aspect ratio of test ellipse image | CI | ARI |
---|---|---|---|---|---|---|---|---|
10:10.0 | 0.911 | 1.000 | 10:6.9 | 0.871 | 1.449 | 10:3.8 | 0.668 | 2.626 |
10:9.9 | 0.905 | 1.010 | 10:6.8 | 0.867 | 1.471 | 10:3.7 | 0.654 | 2.711 |
10:9.8 | 0.912 | 1.020 | 10:6.7 | 0.860 | 1.493 | 10:3.6 | 0.646 | 2.775 |
10:9.7 | 0.913 | 1.031 | 10:6.6 | 0.856 | 1.515 | 10:3.5 | 0.634 | 2.860 |
10:9.6 | 0.912 | 1.042 | 10:6.5 | 0.855 | 1.539 | 10:3.4 | 0.621 | 2.936 |
10:9.5 | 0.913 | 1.054 | 10:6.4 | 0.851 | 1.560 | 10:3.3 | 0.609 | 3.038 |
10:9.4 | 0.909 | 1.064 | 10:6.3 | 0.845 | 1.588 | 10:3.2 | 0.599 | 3.116 |
10:9.3 | 0.910 | 1.075 | 10:6.2 | 0.842 | 1.612 | 10:3.1 | 0.583 | 3.230 |
10:9.2 | 0.909 | 1.087 | 10:6.1 | 0.836 | 1.641 | 10:3.0 | 0.570 | 3.335 |
10:9.1 | 0.910 | 1.099 | 10:6.0 | 0.832 | 1.665 | 10:2.9 | 0.558 | 3.439 |
10:9.0 | 0.904 | 1.110 | 10:5.9 | 0.826 | 1.694 | 10:2.8 | 0.545 | 3.573 |
10:8.9 | 0.906 | 1.123 | 10:5.8 | 0.822 | 1.724 | 10:2.7 | 0.530 | 3.696 |
10:8.8 | 0.904 | 1.136 | 10:5.7 | 0.815 | 1.756 | 10:2.6 | 0.516 | 3.847 |
10:8.7 | 0.909 | 1.149 | 10:5.6 | 0.810 | 1.785 | 10:2.5 | 0.503 | 3.984 |
10:8.6 | 0.902 | 1.161 | 10:5.5 | 0.807 | 1.816 | 10:2.4 | 0.486 | 4.172 |
10:8.5 | 0.905 | 1.175 | 10:5.4 | 0.798 | 1.851 | 10:2.3 | 0.470 | 4.338 |
10:8.4 | 0.898 | 1.190 | 10:5.3 | 0.791 | 1.887 | 10:2.2 | 0.453 | 4.549 |
10:8.3 | 0.901 | 1.205 | 10:5.2 | 0.777 | 1.924 | 10:2.1 | 0.438 | 4.755 |
10:8.2 | 0.897 | 1.219 | 10:5.1 | 0.780 | 1.959 | 10:2.0 | 0.422 | 5.009 |
10:8.1 | 0.898 | 1.234 | 10:5.0 | 0.773 | 2.002 | 10:1.9 | 0.405 | 5.236 |
10:8.0 | 0.891 | 1.250 | 10:4.9 | 0.761 | 2.039 | 10:1.8 | 0.386 | 5.570 |
10:7.9 | 0.893 | 1.265 | 10:4.8 | 0.758 | 2.082 | 10:1.7 | 0.369 | 5.860 |
10:7.8 | 0.886 | 1.281 | 10:4.7 | 0.748 | 2.127 | 10:1.6 | 0.350 | 6.260 |
10:7.7 | 0.892 | 1.298 | 10:4.6 | 0.741 | 2.173 | 11:1.5 | 0.334 | 6.636 |
10:7.6 | 0.886 | 1.315 | 10:4.5 | 0.733 | 2.221 | 12:1.4 | 0.31 | 7.15 |
10:7.5 | 0.89 | 1.33 | 10:4.4 | 0.72 | 2.27 | 13:1.3 | 0.29 | 7.64 |
10:7.4 | 0.88 | 1.35 | 10:4.3 | 0.72 | 2.32 | 14:1.2 | 0.27 | 8.34 |
10:7.3 | 0.88 | 1.37 | 10:4.2 | 0.7 | 2.38 | 15:1.1 | 0.25 | 8.98 |
10:7.2 | 0.88 | 1.39 | 10:4.1 | 0.7 | 2.44 | 16:1.0 | 0.23 | 10.05 |
10:7.1 | 0.88 | 1.41 | 10:4.0 | 0.68 | 2.5 | |||
10:7.0 | 0.87 | 1.43 | 10:3.9 | 0.68 | 2.57 |
where C _{I} is the circularity calculated using ImageJ software and AR_{I} is the aspect ratio calculated using the same software. This indicates that C _{I} is a sixth-degree polynomial of AR_{I} when test ellipse images are made from the deformation of a perfect circle image with 0.1 % intervals in height. We therefore refer to the C _{I} value newly derived from this regression equation as C _{AR}.
Calculating roundness parameter R using ImageJ software
When using R in particle shape analysis, the roundness values can be easily handled as numerical data. For instance, the first quartile (twenty-fifth percentile), second percentile (median), and third quartile (seventy-fifth percentile) of the roundness of a large number of particle grains can be easily and quickly examined.
Validation of R using Krumbein’s pebble images for visual roundness
This implies that R is an effective parameter for roundness.
Correlation analyses have been conducted, comparing shape parameters and the roundness of Krumbein’s visual images, by many previous authors (Table 3). These previously published shape parameters were calculated through various methods, including Fourier analysis (Mi: Itabashi et al. 2004), fractal analysis (D: Vallejo and Zhou 1995; FD: Itabashi et al. 2004), and computer-assisted geometrical analysis (FU: Itabashi et al. 2004; rW: Roussillon et al. 2009). Strong correlation coefficients were obtained between these shape parameters and the Krumbein’s (1941) roundness. In particular, Mi, FD, and rW all had high correlation coefficients of more than 0.9 (0.940, 0.939, and 0.919, respectively). Similar to these studies, the shape parameter R defined in this study also exhibits a high correlation coefficient (0.937). This demonstrates that R is a suitable parameter for discussing the roundness of particle grains as Mi, FD, and rW.
However, R has an advantage over previously defined parameters in that it can be used to easily obtain roundness values using widely available software (such as ImageJ software). Consequently, the new roundness parameter R can be expected to have a significant effect on future statistical analyses of roundness. The roundness parameter R is also advantageous as it can be applied as a part of simple new field studies into clastic grain shapes, which is not the case for other methods such as fractal dimensions or Fourier descriptors. This simple approach to calculating the circularity corrected by aspect ratio has a great potential that can advance research in a wide variety of scientific fields.
Applying R to modern deposits using ImageJ software
Grain size distribution
R distribution
Correlation and regression expression between shape parameters values and the Krumbein’s visual images
Shape parameters | Correlation coefficient | Regression expressions | References | |||
---|---|---|---|---|---|---|
Individual values^{a} (n = 81) | Adjusted coefficient of determination of individual values (n = 81) | Mean values (n = 9) | Adjusted coefficient of determination of mean values (n = 9) | |||
D | – | – | 0.778^{b} | 0.549^{b} | D = 1.0541 − 0.0335·R _{ K } | Vallejo and Zhou (1995) |
R _{ K } = 19.255 − 18.079·D ^{c} | ||||||
Mi | 0.940 | – | – | – | Mi = 28.38 − 46.18 R _{ K } + 21.71 R _{ K } ^{ 2 } | Itabashi et al. (2004) |
FD | 0.939 | – | – | – | FD = 1.03655 − 0.05799·R _{ K } + 0.02698·R _{ K } ^{2} | Itabashi et al. (2004) |
FU ^{d} | 0.857 | – | – | – | FU = 0.0736 + 0.264·R _{ K } | Itabashi et al. (2004) |
rW | 0.919 | – | 0.992 | – | Roussillon et al. (2009) | |
R | 0.937 | 0.874 | 0.995 | 0.987 | R _{ K } = 6.9940 − 20.575·R + 15.349·R ^{2} | This study |
Comparison between beach and slope deposits using R and grain-size distributions
Through the examination of both R and grain-size distributions (Fig. 12), distinctive differences between beach and slope deposits are revealed. In order to compare the characteristics of the different deposits, a plot of the mean values and ranges between the first and third quartiles is shown for all samples in Fig. 12. This diagram shows clearly that the areas in which the beach deposits and slope deposits plot are completely separate. The beach deposits had high R values and ranged from coarse-grained sands to granules. In contrast, the slope deposits had low R values and ranged in size from granules to pebbles. These distinct variations imply that there is a significant difference in cumulative energy between the two deposit types. The beach deposits comprise particles that are highly abraded by wave action and beach drift transport, whereas the slope deposits were comparatively unaffected by such physical abrasion. Therefore, the new roundness parameter R can be considered helpful for the study of sedimentary processes and the estimation of particle origins.
Conclusions
- 1.
The basic concept of the new roundness parameter R can be defined as:
- 2.
The effective diameter of a digital image suitable for R calculations ranges from 100 to 1024 pixels, based on shape analysis of test circle images of various sizes using ImageJ software.
- 3.
The effective aspect ratio of digital images for R calculations ranges from 10:1 to 10:10, based on shape analysis for various test circle and ellipse images in ImageJ software.
- 4.
Given that a digital image is of an appropriate size, circularity (C _{AR}) is given by a sixth-degree polynomial with respect to aspect ratio (AR_{I}):
- 5.
The new roundness parameter R is thus defined as:
- 6.
Validation of R using the pebble images for visual roundness provided by Krumbein (1941) reveals a strong correlation coefficient (r = 0.937) between Krumbein’s roundness and R.
- 7.
Based on the application of R to modern beach and slope deposits, we can confirm that the new roundness parameter R represents a useful new tool in the analysis of particle shape.
Declarations
Acknowledgements
We thank the two anonymous reviewers who provided constructive comments and helpful suggestions. This research was partly supported by Grants-in-Aid for Scientific Research from the Japan Society for the Promotion of Science (Y. Takashimizu, no. 24740341). We also acknowledge Dr. A Urabe (Niigata University) who assisted us in sampling.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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