Introduction

The discrete element method (DEM) provides great convenience to model granular materials (such as sand, soil, rockfill, and debris flow) in geotechnical field (Bu et al. 2022; Xu et al. 2021; Xu and Dong 2021). However, the conventional DEM models use disks to replace the irregular particles (Gao et al. 2021). In fact, it has been proved by many studies that the morphological signatures of sand particles have a considerable impact on the mechanical behavior (such as friction, strength, dilation, compressibility, and crushability) (Guo and Su 2007; Tsomokos and Georgiannou 2010; Altuhafi and COOP 2011; Yang and Luo 2015). For example, the interlocking phenomena observed in sand have been generally attributed to the irregularity and angularity and considered to be closely related to the strength and dilatancy of sand particles (Mair et al. 2002; Suh et al. 2017). Thus, accurate shape analysis is pivotal to obtain deep understanding towards the complicated mechanical behavior of sand particles.

In order to quantify the particle shape, a set of shape descriptors (such as elongation, circularity, roundness, roughness, and convexity index) were adopted (Suhr et al. 2020; Suhr and Six 2020; Das 2007). Each of these shape descriptors only quantifies a single aspect of the geometry features of the particle and has limitation to provide comprehensive quantitative information about the particle morphology. The Fourier descriptors have been proved to be simple and efficient in describing and recognizing the object shape (Wang et al. 2021; Sokic and Konjicija 2016). Meloy applied the fast Fourier transform (FFT) to analyze the shape of particle silhouettes and postulated that the particle signatures were dependent on the Fourier descriptors and not on the phase angles (Meloy 1977). Mollon illustrated the qualitative and quantitative relationship between the Fourier descriptors and the particle morphological features and also pointed out that the variation in the phase angles resulted in changes of particle shape (Mollon and Zhao 2012).

The specific objectives of this study were to examine the effects of using Fourier descriptors to analyze and reconstruct sand particle boundary and to investigate the relationship between regular shape descriptors (such as elongation, circularity, convexity, and roughness) and Fourier descriptors. The study was conducted in the form of a case study of 600 grains of sand particle collected from the Hutuo River, the main water resource of Shijiazhuang County, Hebei Province. Firstly, shape descriptors of sand samples, including elongation, circularity, convexity, and roughness, were calculated, and statistical features were analyzed. Secondly, Fourier descriptors were obtained, and the Andrews plots of Fourier descriptors were used to discriminate the sand samples. And then, a method to reconstruct sand particles using Fourier descriptors was proposed. Lastly, a set of Fourier-descriptor-controlled experiments was conducted to research the relationship between shape descriptors and Fourier descriptors qualitatively. Furthermore, several 3D-scatter plots and Pearson’s coefficient were used to describe the correlation between shape descriptors and Fourier descriptors quantitively.

All the algorithms including image processing and data analysis were written specifically for this study and were executed on Matlab 2022-a.

Descriptors of particle shape

Shape descriptors

Although a variety of shape descriptors have been proposed in the literature, elongation, circularity, convexity, and roughness are most often adopted in related research. The elongation (Fig. 1a) is defined below in Eq. (1), denoting by I and L, the shortest and longest axes of the particle’s minimum bounding box (Suhr and Six 2020).

Fig. 1
figure 1

Shape descriptors of an illustrate particle a elongation = 0.6448, b circularity = 0.7959, c convexity = 0.9485, d roughness = 0.9935

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$$\mathrm{Elongation}=I/L$$
(1)

The particle circularity (Fig. 1b) may be defined as the ratio of the circumference of a circle of the same area as the particle to the actual circumference of the particle (Das 2007). The standard equation to calculate circularity is

$$\mathrm{Circularity}= 4\pi A/{P}^{2}$$
(2)

where A is the area of the particle, and P is the perimeter of the particle. The convexity (Fig. 1c) follows the definition:

$$\mathrm{Convexity}=A/{A}_{c}$$
(3)

where A is the area of the particle, and Ac is the area of the convex hull (Yang et al. 2019). The roughness (Fig. 1d) of a particle is defined as

$$\mathrm{Roughness}=P/{P}_{c}$$
(4)

which is the ratio of the particle perimeter to the convex perimeter (Janoo 1998).

Fourier descriptors

Fourier descriptors are the Fourier transform coefficients, actually the amplitudes of spectrum computation of particle silhouettes. Image processing is necessary to obtain the coordinates of the points on the particle silhouettes before Fourier descriptors are calculated, which consists of image denoising, image binarizing, edge detection, and lastly returning x–y coordinates of contour points (see Fig. 2).

Fig. 2
figure 2

Image processing a original particle image, b binarization, c edge detection

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The x–y coordinates of N contour points can be treated as a set of complex numbers so that

$$s\left(n\right)=x\left(n\right)+iy\left(n\right), n= 0, 1, 2, \dots , N-1$$
(5)

That is, the x-axis is treated as the real axis and the y-axis as the imaginary axis of a sequence of complex numbers. Although the interpretation of the sequence is restated, the nature of the boundary itself is not changed. Of course, this representation has one great advantage: it reduces a 2-D to a 1-D description problem (Gonzalez 2009). The discrete Fourier transform (DFT) and the inverse Fourier transform (IFFT) are

$$\left\{\begin{array}{c}z\left(k\right)= \sum_{n=0}^{N-1}\left(x\left(n\right)+iy\left(n\right)\right){e}^{-i\frac{2\pi }{N}kn}\\ s\left(n\right)= \frac{1}{N}\sum_{k=0}^{N-1}z\left(k\right){e}^{i\frac{2\pi }{N}kn}\end{array}\right.$$
(6)

where k = 0, 1, 2, …, N − 1; n = 0, 1, 2, …, N − 1. The Fourier descriptors are defined as

$$d\left(k\right)=\left|\left|z\left(k\right)\right|\right|, k= 0, 1, 2, \dots , N-1$$
(7)

in which ||·|| means calculating the absolute value of a complex number. The Fourier descriptors should be as insensitive as possible to translation, rotation, and scale changes. Hence, the normalized Fourier descriptors Dk are proposed in order to remove the influence of rotation, translation, and scale changes of the particle silhouettes on Fourier descriptors. From basic mathematical analysis, rotation can be considered by an angle φ, translation by a displacement {Δx0, Δy0}, and scale changes by r times, and the new Fourier coefficients should be

$$\left\{\begin{array}{c}{z}^{^{\prime}}\left(0\right)=r{e}^{i\varphi }z\left(0\right)+F\left({x}_{0}+i{y}_{0}\right), k=0 \\ {z}^{^{\prime}}\left(k\right)=r{e}^{i\varphi }{e}^{i\frac{2\pi }{N}ka}z\left(k\right), k=1, 2,\cdots ,N-1\end{array}\right.$$
(8)

and

$$\frac{\left|\left|{z}^{^{\prime}}\left(k\right)\right|\right|}{\left|\left|{z}^{^{\prime}}\left(1\right)\right|\right|}= \frac{r\left|\left|{e}^{i\varphi }{e}^{i\frac{2\pi }{N}ka}z\left(k\right)\right|\right|}{r\left|\left|{e}^{i\varphi }{e}^{i\frac{2\pi }{N}ka}z\left(1\right)\right|\right|} \equiv \frac{\left|\left|z\left(k\right)\right|\right|}{\left|\left|z\left(1\right)\right|\right|}$$
(9)

Thus, rotation and scale changes simply affect all coefficients equally by a multiplicative constant term \(r{e}^{i\varphi }\). Note that the translation only affects D0 and has no effect on the other descriptors Dk for k > 0, so that the first descriptor D0 can be set to zero. Finally, the normalized Fourier descriptors are defined as

$$D\left(k\right)=\frac{\left|\left|z\left(k\right)\right|\right|}{\left|\left|z\left(k\right)\right|\right|} , k= 1, 2, \dots , N-1$$
(10)

An illustration of Fourier descriptors of the particle in Fig. 2 is shown in Fig. 3. According to the Fourier transform, low-frequency components determine the overall shape of particles, and high-frequency components account for fine detail (Gonzalez 2009). As Fig. 3 shows, the descriptors Dk when k > 18 are almost equal to zero. Moreover, the particle boundary can be reconstructed using Fourier descriptors by the IFFT in Eq. (8).

Fig. 3
figure 3

Illustrative normalized Fourier descriptors

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Materials

The sand particles collected for this study encompass natural sands from the Hutuo River, the main water resource of Shijiazhuang County, Hebei Province, and manufactured crushed sands. The river sands are divided into 2 groups according to particle size: RS-I (particle size in 2–3 mm, see Fig. 4) and RSII (particle size in 3–4 mm, see Fig. 5). The particle size of the manufactured sands, marked as MS (see Fig. 6), varies from 3 to 10 mm. RSI, RSII, and MS consist of 200 particles, respectively.

Fig. 4
figure 4

Sand sample of RSI

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Fig. 5
figure 5

Sand sample of RSII

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Fig. 6
figure 6

Sand sample of MS

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Results and discussion

Analysis of shape descriptors

The shape descriptors, elongation, circularity, convexity, and roughness, have been obtained using Eqs. (1) to (4), respectively, from 600 sand particles of RSI, RSII, and MS. Then, frequency distribution histograms for the four shape descriptors were plotted in Fig. 7. As can be seen from the figure, frequency distribution histograms of RSI and RSII are both skewed, while irregular distributions occurred for MS. Moreover, probability plots for each shape descriptor of RSI and RSII were plotted in Fig. 8. It is clearly showed that both elongation and circularity obey normal distribution for RSI and RSII, while the convexity obeys Weibull distribution and the roughness subjects to Rayleigh distribution. It can be seen from Figs. 7 and 8 that the elongation, circularity, convexity, and roughness were found out to display a statistically similar pattern for both RSI and RSII, while those for MS were found out to obey a statistically different pattern. The same statistical distribution models are also found in related literatures (Blott and Pye 2008; Yang et al. 2018), which mainly result from different formation processes. The Andrews plot of the four shape descriptors also showed that the Andrews curves of RSI and RSII blend into each other (Fig. 9). Thus, it is clear that the RSI, RSII, and MS can be divided into two types of sands: natural sand for RSI and RSII and manufactured sand for MS by elongation, circularity, convexity, and roughness.

Fig. 7
figure 7

Frequency distribution histograms of elongation circularity convexity and roughness for RSI, RSII, and MS

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Fig. 8
figure 8

Probability plots of elongation circularity convexity and roughness for RSI and RSII

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Fig. 9
figure 9

Andrews plot of elongation circularity convexity and roughness for RSI, RSII, and MS

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Analysis of Fourier descriptors

The Fourier descriptors of RSI, RSII, and MS were calculated and then normalized based on the programs written by ourself. D0 was set to 0 because it only depends on the initial position. Then, the Andrews plot of the mode Fourier descriptors for D1 to D18 was plotted in Fig. 10. As can be seen, there are very few intersection points between the three Andrews curves of RSI, RSII, and MS. It leads to a conclusion that the Fourier descriptors can adequately discriminate the sand samples.

Fig. 10
figure 10

Andrews plot of Fourier descriptors for D1 to D18

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On the other hand, it has been known that Fourier descriptors are quite convenient for shape retrieval and reconstruction of the particles (Yang and Yu 2019; Mollon and Zhao 2013). The boundary of a sand particle from RSI, as shown in Fig. 11a, consists of 356 points. The corresponding 356 Fourier descriptors were obtained using Eq. (8). Figure 11b shows the boundary reconstructed using one-half of the 356 Fourier descriptors by the IFFT in Eq. (8). It is clear that there is no significant difference between this boundary and the original. Figures 11c through f show the boundaries reconstructed with the number of Fourier descriptors being 36, 18, 12, and 10, respectively. Figure 12 shows the shape descriptors of particles in Fig. 11. The more Fourier descriptors used in reconstruction, the generated particles are more similar to the original ones, which is also proved by the shape descriptors quantitatively. An algorithm to reconstruct particle boundary was proposed using all of the Fourier descriptors of a sand particle (Mollon and Zhao 2013). However, it turns out to be very time-consuming when simulating realistic geotechnical materials, which generally consist of thousands of particles. This study indicates that there is no need to use all of the Fourier descriptors when simulating realistic materials. However, it is not suggested to adopt less than 18 Fourier descriptors. Figure 13 shows 36 particles reconstructed using one quarter of the Fourier descriptors.

Fig. 11
figure 11

Particle boundary reconstructed using different number of Fourier descriptors. a Original boundary. b–f Boundaries reconstructed using 178, 36, 18, 12, and 10 Fourier descriptors, respectively

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Fig. 12
figure 12

Shape descriptors of particles in Fig. 11

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Fig. 13
figure 13

Particles reconstructed using Fourier descriptors

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Furthermore, Meloy proposed that there is a functional relationship between Fourier descriptors Dk and the frequency of the kth component (Meloy 1977). The sand particle in Fig. 11 was used to calculate the Fourier descriptors, and then the logarithms of the obtained Dk and k were taken to the base 2. And Fig. 14 shows a log–log plot of Dk and k; it is clear that there is a linear relationship between log2(Dk) and log2(k).

Fig. 14
figure 14

Log–Log plot of Dk versus k

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Shape descriptors versus Fourier descriptors

In order to assess the relationship between shape descriptors and Fourier descriptors, Pearson’s correlation coefficients were calculated for each kind of shape descriptors and Fourier descriptors of D1 through D4. The correlation coefficient matrix was plotted in Fig. 15, where Pearson correlation coefficients are presented as different color gradients, from red (absolute positive correlation: correlation coefficient 1) to blue (absolute negative correlation: correlation coefficient − 1). Note that elongation and circularity have a moderate negative correlation with D1 of RSI and RSII, coefficient r ranging from − 0.63 to − 0.65. As for MS, circularity and convexity show a moderate negative correlation with D2, D3, and D4, coefficient r ranging from − 0.62 to − 0.65, and roughness shows a moderate positive correlation with D2, D3, and D4, coefficient r ranging from 0.60 to 0.66.

Fig. 15
figure 15

Correlation coefficients matrix plots between shape descriptors and Fourier descriptors

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To further find out the relationship between shape descriptors and Fourier descriptors, a set of Fourier-descriptors-controlled experiments was conducted. A grain of sand was taken to calculate its Fourier descriptors. First of all, D1, D2, and D3 were set to 0. Then, two of the three Fourier descriptors were kept constant, and the remaining Fourier descriptor increased from 0 to 0.5, then to 1.0. Hence, 9 grains of sand were obtained, and then their shape descriptors, elongation, circularity, convexity, and roughness, were calculated, which can be seen in Fig. 16. It can be seen that the elongation decreased linearly as D1 increased, while the circularity and the convexity decreased in a nonlinearly manner as D1, D2, and D3 increased. But there is no significant change in roughness. Figure 17 furtherly proved that there exhibits a moderate negative linear correlation between D1 and elongation. Figure 18 indicates the statistic distribution of Fourier descriptors for D2, D3, and D4 and their relationship with elongation, circularity, convexity, and roughness. It can be seen that D2 varies from 0 to 0.6, D3 varies from 0 to 0.6, and D4 varies from 0 to 0.3. The elongation, circularity, convexity, and roughness were comparative evenly distributed.

Fig. 16
figure 16

Relationship between shape descriptors and Fourier descriptors

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Fig. 17
figure 17

Relationship between elongation and D1

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Fig. 18
figure 18

3D-scatter-diagram of D2, D3, D4, and shape descriptors

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Conclusion

Based on the programs written by ourselves, this paper studies shape descriptors, Fourier descriptors, and their relationship of both natural and manufactured sand particles, as well as the method to simulate realistic particles using Fourier descriptors. The main conclusions could be drawn:

  1. (a)

    Elongation and circularity obey normal distribution for natural sands, while the convexity obeys Weibull distribution and roughness subjects to Rayleigh distribution. And irregular distribution occurs to manufactured sands.

  2. (b)

    Fourier descriptors provide comprehensive quantitative information about sand particles. And an algorithm is proposed to simulate sand particles using Fourier descriptors.

  3. (c)

    Elongation shows a moderate negative correlation with Fourier descriptors of D1 for both natural and manufactured sands, and roughness shows a moderate positive correlation with Fourier descriptors of D2, D3, and D4 for manufactured sands.

Despite the above findings, this paper raises several questions that need to be explored:

  1. (a)

    The Fourier descriptors were used to divide sand samples adopted in this paper very well. But its universality needs further verification.

  2. (b)

    Although Fourier descriptors provide comprehensive information about sand particles, whether they contain more physical meanings needs further study to verify.

  3. (c)

    A method is proposed to reconstruct sand particles using the Fourier descriptors. However, how to simulate the complex structure of geotechnic materials deserves more study.