Takashimizu and Iiyoshi Progress in Earth and Planetary Science (2016) 3:2
DOI 10.1186/s40645-015-0078-x

METHODOLOGYOpen Access

New parameter of roundness R: circularity
corrected by aspect ratio
Yasuhiro Takashimizu1* and Maiko Iiyoshi2

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:
R = Circularity + (Circularity perfect circle Circularity aspect ratio)
where Circularityperfect circle is the maximum value of circularity and Circularityaspect 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 ImageJsoftware, 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
R = CI + (0.913−CAR)
where CI is the circularity measured using ImageJ software and CAR 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 ofparticles in the future.
Keywords: Aspect ratio, Circularity, ImageJ software, Krumbein’s visual roundness, Roundness

Background
Many studies have investigated particle shape in thenatural world, mostly based on the definitions ofsphericity and roundness of rock particles proposedby 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 oneof 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 visualroundness chart proposed by Krumbein, which is further
extended in this paper, is one of the most widely employedmethods. However, such a method merely compares visual
images, and therefore, the derived roundness values are* Correspondence: takashimi@ed.niigata-u.ac.jp
1
Faculty of Education, Niigata University, Niigata 950-2181, Japan
Full list of author information is available at the end of the article

not strictly quantitative. The second approach involves
the quantitative determination of various shape parameters, and many evaluation methods have been designed toobtain relevant shape parameters (e.g., Schwarcz andShane 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; Drevin2007; 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 eachindividual 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 forboth approaches, neither is widely used, and there remains
a need for an easy statistical method to derive parametersof 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 ImageJsoftware (ver. 1.47q) released from the US National© 2016 Takashimizu and Iiyoshi. Open Access This 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, andreproduction 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.

Takashimizu and Iiyoshi Progress in Earth and Planetary Science (2016) 3:2

Institute of Health (Abramoff et al. 2004; Schneider
et al. 2012). This represents a fairly simple method thathelps to overcome the shortcomings of the previously
published methods discussed above.

Basic concept of this study
The proposed concept for the new roundness parameter is quite simple: It is a correction of circularity using the aspect ratio of particles. The
definition of circularity, corresponding to that ofparameter K defined by Cox (1927), is given as
follows:
Circularity ¼ 4π⋅

Area
Perimeter2

ð1ÞThis indicates that circularity can be altered in two
ways: by changes in area and by changes in the perimeter of a particle. To consider this, an ideal perfect circle(true circle) is assumed. If the area and perimeter do notchange, then circularity is constant. However, if only theperimeter increases and the area does not change(Fig. 1a, towards the right), then circularity decreases. An increase in the perimeter length therefore represents a decrease in the roundness of theparticle. It should be noted that in this paper, we
use the term “roundness” to refer to the presence orabsence of surface irregularities. Therefore, with aaPage 2 of 16

decrease in roundness, circularity also decreases. In
comparison, in the case of the ellipses skewed froma circle, if only the area decreases and the perimeter
does not change, then circularity can also be seen
to decrease (Fig. 1b). A decrease in area in this way
represents an increase in the aspect ratio of the particle image. Therefore, with a increase in the aspectratio, the circularity also decreases. In Fig. 1b, however, the transformed images in the center and on
the right still show a high degree of roundness.
Consequently, it should be possible to determine
roundness if circularity can be corrected using aspectratio. In other words, if it is possible to combine thedifference in circularity value due to aspect ratio with thecircularity itself, then the aspect ratio-corrected circularity can be used to represent roundness. Thus, ourroundness parameter R can be defined by the following
equation:
R ¼ Circularity þCircularity perfect circle −Circularity aspect ratio

where Circularity is the value defined by Eq. (1),Circularityperfect circle is the maximum value of circularity, and Circularityaspect ratio is the circularity when
only the aspect ratio varies from that of a perfectcircle.

Angularperimeter
increasing

equal area

equal area

Roundness = Constantarea
decreasing



ð2Þ

Round
perimeter
increasing

b



area
decreasing

equal perimeter
Fig. 1 Basic concept of transformation from a perfect circle. Narrow solid lines denote perfect circles before transformation. a Only the perimeterincreases; the area does not change. b Only the area decreases; the perimeter does not change

Takashimizu and Iiyoshi Progress in Earth and Planetary Science (2016) 3:2

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 theprevious 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 fromthe 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 beachsediments 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), andthird quartile (seventy-fifth percentile) of grain size wereobtained for each from the cumulative curves.
To obtain the new roundness parameter R value forthese modern slope and beach sediments, we used an
Olympus TG-1 digital camera to acquire digital imagesof particles in each grain-size class. Image analysis wasPage 3 of 16

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 themeasurement 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 ofthe 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 tothe c-axis, which included the major and intermediatea- 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 morethan 100 pixels. The digital images taken by Olympus
TG-1 were transferred to the ImageJ software andprocessed into binary images. The circularity andaspect 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 classwere integrated into a total R distribution for the individual samples using the weight percent of each grain sizeclass. The calculated R distributions thus ranged from0.400 to 0.925 with 0.025 intervals. The first quartile,

1448 pixels
1024 pixels
724 pixels

128

512

362

256

Fig. 2 Test circle images with diameters of 1 to 1448 pixels

181

64
91

32
45

16

8

23 11 64

2
3
1 pixelWidth of test circle
image (pixel)
1

AI

PI

2·r

A

Area measured by
ImageJ (pixel)

Perimeter measured
by ImageJ (pixel)

Major and minor axis length
measured by ImageJ

Area calculated by p·r

1

2.828

1.128

0.999328

P
2

ΔA (%)ΔP (%)

0.067260−20.196760

Perimeter calculated by 2·p·r
3.543716

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

41211.3143.909

12.001102

12.280483−0.009184−7.870074

632

19.3146.383

31.999226

20.052782

0.002417−3.684186

852

24.9718.137

52.001806

25.563134−0.003472−2.316360

11

9734.627

11.11396.995686

34.912512

0.004447−0.817792

16

208

52.28416.274208.007222

51.126268−0.003472

2.264456

23421

75.59823.152420.985191

72.734138

0.003518

3.93743932

804

104.569

31.995803.996244

100.515236

0.000467

4.032985

45

1581

147.196

44.866

1580.973153

140.950667

0.001698

4.430865

64

3196

211.4863.7913196.013548

200.405295−0.000424

5.526154

90

6320

296.735

89.7046319.946608

281.813369

0.000845

5.294863

128

12,796422.96

127.64212796.081336

400.999086−0.000636

5.476550

181

25,565597.47180.41725564.935720

566.796604

0.000251

5.411711

254

51,104842.607255.084

51104.156942

801.369854−0.000307

5.145832360

102,252

1193.97360.82102251.807881

1133.549225

0.000188

5.330229510

204,520

1690.871

510.297204520.017549

1603.144973−0.000009

5.472121

722

408,932

2391.253

721.574408932.420696

2266.891106−0.000103

5.486011

1020

818,1963381.743

1020.666

818195.500707

3206.516140

0.000061

5.464712

1442

1,636,0244893.536

1443.278

1636024.391899

4534.190619−0.000024

7.925238

Takashimizu and Iiyoshi Progress in Earth and Planetary Science (2016) 3:2

Table 1 Results of error of area and perimeter for various sizes of test circle images

p = 3.141592

Page 4 of 16

Takashimizu and Iiyoshi Progress in Earth and Planetary Science (2016) 3:2

Page 5 of 16

100
10Error (%)

1
0.1

64 pixels

1024 pixels

0.01

-1

0.001

-1

0.00010.00001
0.000001
1

10
100
1000
Width of digital circle image (pixel)

10000

Fig. 3 Plot of the diameter of a digital circle image (pixels) and error (%) in the area and perimeter estimations

1448 pixels

AR =

10
10

AR =

10
6

AR =

10
5

AR =

10
4

AR =

10
3

AR =

10
2

AR =

10
1

1 pixel

AR =

10
9

AR =

10
8

AR =

10
7

Fig. 4 Test circle/ellipse images with diameters/widths of 1 to 1448 pixels and aspect ratios of 10/10 to 10/1

Takashimizu and Iiyoshi Progress in Earth and Planetary Science (2016) 3:2

Page 6 of 16

True aspect ratio

10

ARI

10 : 1
10 : 2
10 : 3
10 : 4
10 : 5
10 : 6
10 : 7
10 : 8
10 : 9
10 : 10

1
1

10
100
Width of test ellipse image (pixel)

1000

Fig. 5 Plot of the diameter of a digital circle image (pixels) and the aspect ratio calculated using ImageJ (ARI)

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 theminimum units of the image. Hence, there should be anerror in shape parameter values between the geometrically obtained true values and those calculated from thedigital image. Therefore, to obtain the most effective sizeof 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

First, the area (AI), perimeter (PI), and major and minor
axis lengths of the fit ellipses (2·r; both lengths are equal)of twelve test circle images with diameter lengths of 1 to21 pixels, produced using Adobe Photoshop CS4, were
calculated using ImageJ software (Fig. 2). To calculate
the area and perimeter, the measuring algorithms of the
ImageJ software were sought from the manual. However,there were no detailed descriptions of the algorithms, so
we investigated the value determinations ourselves. The
obtained algorithms are therefore as follows. AI is equal to
the total number of pixels in a grain. For test circle images
with a diameter length of 1 to 2 pixels, PI was calculated
by determining the geometric mean of the numbers of
pixels in the images and the numbers of circumscribedpixels in the images. In contrast, for test circle images withdiameter lengths greater than 2 pixels, PI was the sum of

the marginal pixels, in which the sizes of pixels located at
the corners are assumed to be 20.5, while those of otherpixels is 1. The major axis length of the test circle images
was 2·r, because all of the test images comprised perfectcircles. The error (ΔA) between the area measured using
ImageJ software (AI) and the area calculated from radius
length (A), and the error (ΔP) between PI and the perimeter obtained by the geometric procedure (P), were therefore determined as follows (Table 1):

ΔA ¼


AI −A
 100
A


ΔP ¼


P I −P
 100
P

ð3Þ
ð4Þ

From our calculations, the absolute error for the area(|ΔA|) was found to decrease with an increase in the
diameter of the test circle images (Table 1 and Fig. 3).
All 21 |ΔA| values obtained in this test were below 0.1 %,which is sufficiently small to assume accuracy. These
values are therefore sufficiently reliable for use in shapeanalysis. In contrast, the absolute perimeter error |ΔP|
was approximately 20 % for diameters of 1 to 2 pixels, but
decreased with an increase in the diameter of the test
circle images. |ΔP| attained a minimum of approximately
0.8 % at a diameter of 11 pixels; however, the error
increased again to approximately 8 % for diameters of 16
to 1442 pixels. For diameters of 64 to 1024 pixels, the|ΔP| value remained constant at around 5 % (Table 1 andFig. 3).

Takashimizu and Iiyoshi Progress in Earth and Planetary Science (2016) 3:2

Aspect ratio

In this section, we examine the relationship between the
aspect ratio measured using ImageJ software (ARI) andthe widths of test ellipse images (Fig. 4), which were obtained by subjecting the test circle images in Fig. 2 to
10 % aspect ratio deformation intervals in Adobe Photoshop CS4. The ImageJ software defines the aspect ratio
as follows:

Page 7 of 16

ARI ¼

major axis length of approximate ellipse
minor axis length of approximate ellipse

ð5Þ

Therefore, the aspect ratio is equal to one for a perfectcircle and increases with an increase in deformation. To
validate this relationship for different image sizes, weprepared test circle images with diameters of 20–10.5 (1 to1448) pixels, and test ellipse images were obtained from

10:10

10:9.1100
pixels
10:1
Fig. 6 Test circle/ellipse images with aspect ratios of 10/10 to 10/1 and diameters/widths of 100 pixels

aspect ratio of test ellipse image

CI

ARI

aspect ratio of test ellipse image

CI

ARI

aspect ratio of test ellipse image

CI

10:10.0

0.911

1.000

10:6.9

0.871

1.449

10:3.8

0.668

ARI
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

12:1.40.31

7.15

10:7.50.89

1.33

10:4.40.72

2.27

13:1.30.29

7.64

10:7.40.88

1.35

10:4.30.72

2.32

14:1.2

0.27

8.34

10:7.30.88

1.37

10:4.20.7

2.38

15:1.1

0.25

8.98

16:1.0

0.23

10.05

0.88

1.39

10:4.1

0.7

2.44

0.88

1.41

10:4.00.68

2.5

10:7.00.87

1.43

10:3.90.68

2.57

Page 8 of 16

10:7.2
10:7.1

Takashimizu and Iiyoshi Progress in Earth and Planetary Science (2016) 3:2

Table 2 CI and ARI values calculated using ImageJ applied to the test images of Fig. 6

Takashimizu and Iiyoshi Progress in Earth and Planetary Science (2016) 3:2

Page 9 of 16

Fig. 7 Plot of ARI and CI values of the test images, calculated using ImageJ software. Solid line represents a sixth-degree polynomial

the test circle images through 10 % deformation intervals
in height (Fig. 4). As such, ARI showed a range of values,
controlled by the widths of the test circle/ellipse images(Fig. 5). Notably, the test circle/ellipse images with diameters or widths of less than 100 pixels are unstable, as
shown in Fig. 5. However, the widths of the test circle/
ellipse images greater than or equal to 100 pixels canbe seen to remain constant.

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 to1442 pixels; (2) The |ΔP| value is constant, at approximately 5 %, for test circle images with diameters of 64 to
1024 pixels; and (3) The ARI 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, weconsider 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

Circularity can be calculated as a shape parameter index
in the ImageJ software. The definition of circularity (CI)in the ImageJ software is as follows:
C I ¼ 4π⋅

AI
PI 2

ð6Þ

where AI and PI are the area and perimeter measured
using ImageJ (ImageJ User 2012), respectively. This therefore implies that CI is directly determined by AI and PI.
For instance, if their two different PI values are provided

Fig. 8 Plot of RK and R values derived in this study. Solid line represents the quadratic regression

Takashimizu and Iiyoshi Progress in Earth and Planetary Science (2016) 3:2

for digital images with the same AI values, the image
showing high circularity will have a shorter perimeter thanthat of the other image.

Relationship between circularity and aspect ratio in ImageJsoftware

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 CI will therefore correspondto changes in aspect ratio, and CI alone should not
be used as a roundness shape parameter. For thisreason, the relationship between CI and ARI is examined in this section, and we attempt to correct CI
using ARI.
First, a test circle image with a diameter of 100 pixels
was produced using Adobe Illustrator CS4. Then, 91 test
ellipse images were obtained by deforming a test circle
image, in 0.1 % height intervals (Fig. 6). From the shapeanalysis of circularity and aspect ratio using ImageJ software, the triadic relationship between the true aspectratio, CI, and ARI could be obtained (Table 2). ARI
versus CI is plotted in Fig. 7. It should be noted that the
maximum CI value is 0.913 (Table 2), as ImageJ is
unable to output 1.0 as a maximum value for these test
images, because the digital images comprise an aggregate of pixels and include errors. We used a polynomial
regression to analyze the relationships. From thisanalysis, a sixth-degree polynomial was obtained for the
relationship, with a strong correlation (r = 0.999836119,
p < 0.005, adjusted coefficient of determination =Page 10 of 16

0.999648855; solid curve in Fig. 7). Thus, the estimatedregression equation for the survey line isC I ¼ 0:826261 þ 0:337479  ARI −0:335455  ARI 2
þ0:103642  ARI 3 −0:0155562  ARI 4
þ 0:00114582  ARI 5 −0:0000330834  ARI 6
ð7Þ
where CI is the circularity calculated using ImageJ software and ARI is the aspect ratio calculated using the same
software. This indicates that CI is a sixth-degree polynomial of ARI when test ellipse images are made from thedeformation of a perfect circle image with 0.1 % intervals
in height. We therefore refer to the CI value newly derived
from this regression equation as CAR.
Calculating roundness parameter R using ImageJ software

Equation (7) implies that the circularity of a perfectcircle changes with varying aspect ratio. This meansthat if a shape’s perimeter is highly rounded, then the
degree of roundness should also be close to 0.913.
Therefore, after the correction of circularity by aspectratio, which represents the addition of the difference
between 0.913 and CAR to CI, it is possible to calculate the circularity without considering aspect ratio.
The corrected circularity is therefore considered the
new roundness parameter (R) and can be presented
as follows:
R ¼ C I þ ð0:913 − C AR Þ

ð8Þ

When using R in particle shape analysis, the roundness values can be easily handled as numerical data.

Fig. 9 Location of the study area. a Large-scale map of study area in Northeastern Japan and facing the Pacific Ocean. b Detailed map with sampling
location in the Masaki area of Iwate Prefecture, Northeastern Japan. Location map is based on “Taro,” the 1:25,000 topographic map from theGeographical Survey Institute (GSI) of Japan. Solid circle and solid square indicate beach and slope deposits, respectively

Takashimizu and Iiyoshi Progress in Earth and Planetary Science (2016) 3:2

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.

Page 11 of 16

Validation of R using Krumbein’s pebble images for visualroundness

In order to validate the effectiveness of R defined in thispaper, the R values of Krumbein’s visual images were calculated and examined. Figure 8 plots Krumbein’s roundness

a

b

c

d

e

f

Fig. 10 Photographs and binary images of sediment particles. These were obtained from beach and slope environments and grain size ranges
from −1.5 to 2.0 phi. Photographs of a beach sediments and b slope sediments (scoop length is 82 cm); photographs of prepared samples ofc beach sediments and d slope deposits; binary images processed by ImageJ of e beach sediments and f slope deposits

Takashimizu and Iiyoshi Progress in Earth and Planetary Science (2016) 3:2

Page 12 of 16

Fig. 11 Grain-size distributions and R distributions of beach deposits and slope deposits(RK) against the R values calculated using the method
defined in this study. From this analysis, a second-degree
polynomial was obtained for the relationship, with strong
correlation (r = 0.936801380, p < 0.005; adjusted coefficientof determination = 0.874458283; Fig. 8), and the estimatedregression equation for the survey line is therefore
RK ¼ 6:9940−20:575  R þ 15:349  R2

ð9Þ

This implies that R is an effective parameter for roundness.

Correlation analyses have been conducted, comparing
shape parameters and the roundness of Krumbein’svisual images, by many previous authors (Table 3). Thesepreviously published shape parameters were calculated
through various methods, including Fourier analysis(Mi: Itabashi et al. 2004), fractal analysis (D: Vallejo andZhou 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)

Fig. 12 Plot of median R versus median grain size for modern sedimentary materials. Solid circles and solid squares indicate beach and slope
deposits, respectively. Vertical and horizontal error bars represent the ranges between the first and third quartiles

ShapeCorrelation coefficient
Regression expressions
parameters
Adjusted coefficient of determination Mean values (n = 9) Adjusted coefficient of determination
Individual valuesa(n = 81)of individual values (n = 81)of mean values (n = 9)D––0.778b0.549b

References

D = 1.0541 − 0.0335·RK

Vallejo and Zhou (1995)

RK = 19.255 − 18.079·D

c

0.940

–

–

–

Mi = 28.38 − 46.18 RK + 21.71 RK 2

FD

0.939

–

–

–

FD = 1.03655 − 0.05799·RK + 0.02698·RK

FUd

0.857

–

–

–

FU = 0.0736 + 0.264·RK

Itabashi et al. (2004)–

Roussillon et al. (2009)

0.987

RK = 6.9940 − 20.575·R + 15.349·R2

This study

Mi

rW

0.919

–

0.992

R

0.937

0.874

0.995

Itabashi et al. (2004)2

Itabashi et al. (2004)

Takashimizu and Iiyoshi Progress in Earth and Planetary Science (2016) 3:2

Table 3 Correlation and regression expression between shape parameters values and the Krumbein’s visual imagesa

The shape factor for each group is the average of nine shape factors corresponding to the nine particles forming each group in Krumbein’s pebble images
This parameter is recalculated values to the third decimal place by the author
This expression is recalculated by the authordThis parameter is the same definition as circularity on ImageJbcPage 13 of 16

Takashimizu and Iiyoshi Progress in Earth and Planetary Science (2016) 3:2

roundness. In particular, Mi, FD, and rW all had highcorrelation coefficients of more than 0.9 (0.940, 0.939,
and 0.919, respectively). Similar to these studies, theshape parameter R defined in this study also exhibits ahigh 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 definedparameters 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 canbe expected to have a significant effect on future statisticalanalyses 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 casefor 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

In this section, we apply our shape parameter to digital
images of samples of sedimentary materials collected frommodern beaches and slopes in the Masaki area of EasternJapan (Figs. 9 and 10). The lithology of this region is mainly
Cretaceous rhyolite, dacite, sandy siltstone, sandstone, andconglomerate (Shimazu et al. 1970). The beach deposits consist primarily of coarse-grained sands to granules with pebbles, which are well-abraded by wave action on the beach(Fig. 10a). The slope deposits in this area comprise heavily
weathered Cretaceous basement rocks, which occur as angular granule- to pebble-sized clasts (Fig. 10b). Thus, the slope
deposits can be considered immature clastics, while thebeach deposits represent more mature clastic material. We
selected these two types of materials for roundness analysis
because they differ markedly in particle roundness, but bothhave a fairly coarse grain-size distribution.
Grain size distribution

A total of eight sedimentary samples were selected frombeach and slope environments for this study, consistingof four beach deposits, referred to as B1, B2, B3, and B4,
and four slope deposits, referred to as S1, S2, S3, andS4. Before shape analysis, their grain-size distributions
were measured using a sieve ranging from −5.0 to
4.0 phi with 0.5 phi intervals (Fig. 11). Median grain
sizes of the deposits in each environment were 0.33
(B1), −0.50 (B2), −1.25 (B3), and −0.50 (B4) phi for thebeach deposits, and −3.18 (S1), −5.00 (S2), −4.77 (S3),and −2.47 (S4) phi for the slope deposits. The range between
the first and third quartiles in the grain size distribution canbe considered a proxy for the degree of sorting. Thus, theseranges in each environment were 0.821 to 0.869 (B1), −1.34
to 0.50 (B2), −1.93 to −0.40 (B3), and −1.70 to 0.16 (B4) phi

Page 14 of 16

for the beach deposits and −3.96 to −1.96 (S1), −5.00
to −4.94 (S2), −5.00 to −4.30 (S3), and −3.46 to −1.33(S4) phi for the slope deposits. Together, these data indicate that the beach deposits were finer than the slope
deposits but had similar degree of sorting.
R distribution

The R values were measured using ImageJ software, following the above-described methodology. The measured
R distributions in these two deposits are shown in Fig. 12.
The median R of deposits in each environment was
0.847 (B1), 0.860 (B2), 0.865 (B3), and 0.866 (B4) for thebeach deposits and 0.764 (S1), 0.778 (S2), 0.746 (S3),and 0.784 (S4) for the slope deposits. The range between
the first and third quartiles of R in the deposits of eachenvironment were 0.821 to 0.869 (B1), 0.831 to 0.860(B2), 0.835 to 0.889 (B3), and 0.838 to 0.889 (B4) forthe beach deposits and 0.703 to 0.804 (S1), 0.759 to
0.792 (S2), 0.726 to 0.776 (S3), and 0.741 to 0.815 (S4)for the slope deposits. Together, these data indicate thatthe beach deposits were more rounded than the slope
deposits.
Comparison between beach and slope deposits using R andgrain-size distributions

Through the examination of both R and grain-size distributions (Fig. 12), distinctive differences between beachand slope deposits are revealed. In order to compare thecharacteristics of the different deposits, a plot of the meanvalues and ranges between the first and third quartiles isshown for all samples in Fig. 12. This diagram shows
clearly that the areas in which the beach deposits andslope deposits plot are completely separate. The beach deposits had high R values and ranged from coarse-grainedsands 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 arehighly abraded by wave action and beach drift transport,
whereas the slope deposits were comparatively unaffectedby 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
In this study, we propose a new particle roundness parameter, R, which can be defined as the circularity corrected by the aspect ratio, and we demonstrate thecalculation of this parameter from particle images,
using ImageJ software. The results of this study can besummarized as follows:Takashimizu and Iiyoshi Progress in Earth and Planetary Science (2016) 3:2

1. The basic concept of the new roundness parameter
R can be definedas:


R ¼ Circularity þCircularity perfect circle −Circularity aspect ratio

Page 15 of 16

Author details
1
Faculty of Education, Niigata University, Niigata 950-2181, Japan. 2Oh-shima
Elementary School, Joetsu 942-1103, Japan.
Received: 14 January 2015 Accepted: 27 December 2015

where Circularityperfect circle is the maximum value ofcircularity and Circularityaspect ratio is the circularity when
only the aspect ratio varies from that of a perfect circle.
2. The effective diameter of a digital image suitable for
R calculations ranges from 100 to 1024 pixels, basedon 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 onshape analysis for various test circle and ellipse
images in ImageJ software.
4. Given that a digital image is of an appropriate size,
circularity (CAR) is given by a sixth-degree polynomialwith respect to aspect ratio (ARI):C AR ¼ 0:826261 þ 0:337479  ARI −0:335455  ARI 2
þ0:103642  ARI 3 −0:0155562  ARI 4
þ 0:00114582  ARI 5 −0:0000330834  ARI 6

5. The new roundness parameter R is thus defined as:
R ¼ C I þ ð0:913−C AR Þ
where CI is the circularity measured using ImageJsoftware.
6. Validation of R using the pebble images for visualroundness provided by Krumbein (1941) reveals astrong correlation coefficient (r = 0.937) between
Krumbein’s roundness and R.
7. Based on the application of R to modern beach andslope deposits, we can confirm that the new roundness
parameter R represents a useful new tool in theanalysis of particle shape.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
YT originally produced the ideas for R and designed the study. YT and MI
carried out the experimental study of modern sediment particles. All authors
read and approved the final manuscript.
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 Promotionof Science (Y. Takashimizu, no. 24740341). We also acknowledge Dr. A Urabe(Niigata University) who assisted us in sampling.

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