Modeling of Geometric Size Distribution of Almond

Abstract

The purpose of this study was modeling the mass and size distribution of three varieties of almond and its kernel (seed) using the Weibull distribution function. Furthermore, some physical properties of seeds were measured using an image processing technique. A two-parameter Weibull distribution function was chosen for modeling the size and mass distributions. The Weibull distribution of width was better modeled than the other sizes and the distribution of sizes mainly were better modeled than the mass. The mass distribution of almonds was approached to the normal probability density function, whereas the dimension distributions of seeds had negatively skewed.

Keywords:

• Almond
• Image processing technique
• Modeling
• Physical properties
• Weibull distribution

INTRODUCTION

Almonds are one of the most popular tree nuts worldwide. Almond (Amygdalus communis L.) is a perennial plant growing in the Mediterranean and cold climates of Iran. The almond and its kernel play an important role as a source of protein in the human diet. In 2004, the annual production of Iran was 110 × 10⁶ tons as the third producing country in the world after the USA and Syria. Proper post-harvest handling is critical for producing high-quality nuts. Some physical properties, such as size, surface area, and volume, are required to predict the transport properties, different handling, processing operations, and drying rates of grains through simulation models.[1,2] Furthermore, fracture mechanics depend not only on the parameters of the compression test, but also on the shape, size, and structure of the samples.[3]

A large number of different distribution functions have been utilized to model diameter distributions of plant and fruit, including the Beta, Lognormal, Johnson's Sb, and Weibull. The Weibull function has been widely used due to its flexibility in modeling reverse-J, skewed, and unimodal shapes.[4] Furthermore, the Weibull function is not required to estimate frequencies because of integration[5]; in this respect, the Weibull function was chosen to model size and mass of the almond in this research.

The primary form of the probability density function (PDF) has three parameters: shape (β), scale (η), and location (γ) (Eq. 1). These parameters must be calculated to predict Weibull distribution.[6] The three-parameter Weibull distribution is suitable for situations in which an extreme value cannot take values less than γ. For independent materials of location, the parameter γ is zero so the three-parameter Weibull function is converted to the two-parameter Weibull function.

Riquelme et al. studied the physical properties of different almond cultivars grown under homogeneous conditions and arranged varieties as a function of their sphericity.[7] Kalyoncu measured the size and rupture strength of ten almond varieties and determined the relationship between the size and rupture strength of their kernels.[8] Then Aydin evaluated several physical properties of almond and its kernel as functions of moisture content.[9]

Some statistical distributions, such as normal, lognormal, and Weibull, were used to determine the nonlinear models of dimensions of forage particles and vegetables.[10,11] The statistical distribution that best described the pattern in all cases was found to be the two-parameter Weibull function. Sharan et al. modeled weight and length of the Himachal tomato using the two-parameter Weibull function and it was found that weight and longitudinal axis length of the Himachal tomato are both described satisfactorily by the Weibull distribution function.[12]

Bullock and Boone derived tree diameter distributions using the Weibull function[5] and Smith and Bullock compared parameter estimation techniques to calculate the Weibull parameter for diameter distributions of loblolly pine in a genotype and environment (GxE) study. The Weibull probability density function was used for this modeling and the analyses indicated that the Weibull provides a more accurate canopy description for environmental and physiological modeling.[12] Fragment size of glass particles under impact loading conditions have been modeled using Weibull distribution in several researches.[4,13–18] The two-parameter Weibull function has been investigated by several researchers.[4,5,11,15,16,19]

The purpose of this study was modeling the mass and size distribution of three varieties of almond and its kernel using the Weibull distribution function. Finally, the Weibull distribution function (WDF) values were compared to measure values. The performance of this model was investigated by comparing the determination of coefficients R ² and root mean square error (RMSE) between the measured and predicted distributions.

Theoretical Consideration

The primary form of the PDF has three parameters (β, η, γ):

(1)

where X is the independent variable, γ is the location parameter, η is the scale parameter, and β is the shape parameter. Note that when γ = 0 the two-parameter Weibull is produced. For almond the parameter γ was zero because the physical properties of almond are independent of location:

(2)

The cumulative density function (CDF), as F(X) of Weibull distribution function (WDF) is calculated by:

(3)

(4)

The Weibull CDF is the probability of X that takes on a value in the interval [a,b], and it is the area under the Weibull PDF in this interval. The X is the independent variable representing either length, width, thickness, mass, and so on, of food in this research. In the context of this study, F(X), represents the probability that the size and mass are equal to or less than X. The parameters η and β of the distribution function F(X) are estimated from observations. The methods usually employed in the estimation of these parameters are method of linear regression, method of maximum likelihood, and method of moments.[20–22]

The parameter β was a pure number (i.e., it was dimensionless). According to the literature[23] of Weibull distribution, one can see that the shape of the Weibull PDF can take on a variety of forms based on the value of β. For 0 < β ≤ 1, as X → 0, f(X) → ∞ and as X → ∞, f(X) → 0. The f(X) decreases monotonically and is convex as X increases beyond the value of γ. For β > 1, f(X) = 0 at X = 0 that f(X) increases as (the mode) and decreases thereafter. For β < 2.6, the Weibull PDF is positively skewed (has a right tail), for 2.6 < β < 3.7, its coefficient of skewness approaches zero (no tail). Consequently, it may approximate the normal PDF, and for β > 3.7, it is negatively skewed (left tail). If η is increased while β is kept the same, the distribution gets stretched out to the right and its height decreases, while maintaining its shape, and if η is decreased while β is kept the same, the distribution gets pushed in towards the left (i.e., towards its beginning or towards 0), and its height increases.

MATERIALS AND METHODS

Modeling the Mass and Size Distributions

The almond varieties used in this study were Fransis, Mamaei, and Rabei. The almonds were collected from Research Centre (Shahrekord, Iran) during the spring season in 2007. After the almonds were cleaned of all foreign matter, the kernels were separated manually from the shell. To predict Weibull WDF, one hundred almonds were randomly selected from the samples to find the size and mass of each almond and its kernel of three varieties. The mass of each seed (almond and their kernel) was weighed by a balance (AB204-s/fact, Mettler Toledo, Bern, Switzerland) with ±0.1 mg accuracy, and the dimensions were measured using a digital micrometer with ±0.1 mm accuracy. The parameters η and β were calculated to predict Weibull distribution. The linear regression was used to calculate the parameters in this article. After that, 5–9) were used to calculate the parameters, they were replaced in Eq. (2), and the Weibull WDF was calculated.

The Weibull CDF, Eq. (3), can be written in linear form by taking the logarithm twice, which reduces to the following form (Eq. 5):

(5)

The parameters in Eq. (3) can be estimated by applying the linear regression using the least squares procedure, with β being equal to the slope and η equal to 2.718 raised to the power of the intercept divided by β. With the consideration of obtained β and η, the Weibull WDF can be predicted of the Weibull CDF and Weibull PDF distribution values for each class of either size or mass of almond and its kernel.

(6)

(7)

(8)

which results in the linear equations:

(6)

(9)

Some Physical Properties of Almonds and Kernels

In the second step, the modeled seeds of each variety were divided into three classifications (small, medium, big; 3 × 30) used to calculate some physical properties, such as length, width, thickness, area, geometric mean diameter, and degree of sphericity. To measure the physical properties, the image processing technique was used. The samples were transferred into the light box (Fig. 1) and an image from each side of almond was acquired and stored (Fig. 2). The length (L), width (W), and thickness (T) were defined according to Fig. 2. A digital camera (Model G7 Canon, Japan) with resolution of 1200 × 1600 pixels was used to record images. It was located vertically over the light box at a distance of 30 cm. The images were taken on a textile, which can be easily subtracted by standard segmentation routines because of the difference in color with the seeds using MATLAB software.[24]

Figure 1 Schematic of light box with adjustable light and changeable distance. The images were taken from each side of the almond, used to calculate the area of almonds.

Figure 2 The dimensions and areas of an almond: A_xy (a), A_xz (b). A_xy is the horizontal area and A_xz is the vertical area of the almond.

The segmentation algorithm is based on the similarity of gray-level values. The algorithm uses thresholding, region growing, region splitting, and then merging. Thresholding was an important part of image segmentation. In this research to obtain the physical properties, first the images were converted to gray-scale images then the binary images were obtained by random threshold (0.45, depending on images) and the holes of the seeds in the images were filled, the images were labeled, and the final threshold was calculated according to maximum object area, mainly the maximum object area equal to seed area for each image. Finally, the noises of the background of the binary images were removed and final binary images were obtained. The dimensions, horizontal area (A_xy ) and vertical area (A_xz ), were calculated of the seed images using MATLAB 7.2 software (Fig. 3). The calculated dimensions using an image processing technique and using a digital micrometer were compared. These calculated dimensions were used as an index for converting scale of pictures (pixel) to scale of seed dimensions (mm). The area was equal to total number of pixels in the object (almond) of images converted to mm².

Figure 3 Whole stages of image segmentation of each almond.

The geometric mean diameter (D_p ) of the seeds was calculated using the following relationship[25]:

(10)

The degree of sphericity () can be expressed as follows[25]:

(11)

Finally, the effectiveness of the Weibull WDF in describing both size and mass distribution was assessed. The performance of this model was investigated by comparing the determination of coefficients R ² and RMSE between the measured and predicted distributions.

RESULTS AND DISCUSSION

Some Physical Properties of Almonds and Kernels

Values of the physical properties measured for seeds were summarized in Table 1 (almonds) and Table 2 (kernels). The mean length, width, and thickness of three varieties were 35.95 ± 0.66, 21.54 ± 0.39, and 15.46 ± 0.33 mm for almonds and 26.37 ± 0.47, 13.07 ± 0.26, and 8.31 ± 0.24 mm for their kernels. The geometric mean diameter of almonds of Mamaei was the highest among the varieties and ranged from 19.87 to 28.81. The corresponding value of kernels of Rabei was the highest among the varieties. The mean area of seeds of Fransis variety was the least value in the varieties, whereas the Rabei variety had the most value of area in the varieties. The average of sphericity of almonds and their kernels were 64.76 and 54.28%, 63.19 and 52.72%, and 57.99 and 54.48%, respectively, for Fransis, Mamaei, and Rabei. Overall, the sphericity of almonds and their kernels (seeds) ranged from 52.53 to 68.34% and 45.02 to 60.21% making them difficult to roll on a surface because of the seed shape.[26,27] However, the flat shape of the seeds enables the seeds to slide and this property is important in the development of hopper and dehuller designs for almonds. This is because the flat seeds slide easier than the spherical seeds, which depend on structural surfaces.[28]

Table 1 Some statistical parameters of the measured physical parameters of three varieties of almonds

		Parameters
Varieties	Statistical parameter	Dix (mm)	Diy (mm)	Diz (mm)	Axy (mm^2)	Axz (mm^2)	Dp (mm)	Φ (%)
Fransis	Mean	33.35	21.00	14.28	447.41	244.64	21.53	64.76
	Min	27.10	16.55	12.70	221.11	116.50	17.85	61.42
	Max	39.25	25.60	15.80	779.98	441.22	24.89	68.34
	Std	4.40	2.75	1.04	222.34	133.93	2.36	2.20
Mamaei	Mean	36.27	22.01	15.19	457.44	292.18	22.94	63.19
	Min	31.80	18.30	12.50	200.99	108.99	19.87	57.93
	Max	43.35	27.75	21.15	891.20	649.33	28.81	66.47
	Std	3.87	2.90	2.48	272.58	196.82	2.84	2.43
Rabei	Mean	37.82	16.73	14.53	470.58	315.03	21.95	57.99
	Min	32.05	13.60	10.60	248.74	140.16	18.15	52.53
	Max	43.30	20.50	17.50	828.58	616.84	26.30	61.54
	Std	3.94	2.25	2.12	242.56	172.72	2.66	2.59

Table 2 Some statistical parameters of the measured physical parameters of kernels of three varieties of almonds

		Parameters
Varieties	Statistical parameter	Dix (mm)	Diy (mm)	Diz (mm)	Axy (mm^2)	Axz (mm^2)	Dp (mm)	Φ (%)
Fransis	Mean	25.52	13.13	7.87	383.69	242.92	13.78	54.28
	Min	19.65	10.95	7.45	173.77	105.45	11.83	50.98
	Max	30.05	19.20	9.05	605.11	386.40	16.19	60.21
	Std	3.49	2.22	0.41	152.42	98.50	1.43	2.83
Mamaei	Mean	25.89	12.67	7.80	433.12	263.45	13.64	52.71
	Min	21.45	10.85	5.75	191.25	104.65	11.92	45.02
	Max	30.60	15.65	10.90	604.98	383.28	17.14	59.43
	Std	3.13	1.70	1.64	166.71	107.67	1.93	3.94
Rabei	Mean	27.37	13.41	9.01	434.75	302.77	14.85	54.18
	Min	23.05	11.30	6.55	195.17	112.82	12.15	49.12
	Max	30.60	15.40	11.60	665.68	470.00	16.97	57.72
	Std	2.63	1.44	1.87	176.71	132.67	1.83	2.46

As shown in Tables 1 and 2, L, W, T, and D_p of seeds of Rabei variety were the highest values among the varieties, whereas Fransis showed the lowest values. These relations existed between three classified sizes: small, medium, and big, whereas the other parameters, A_xy, A_xz , and , didn't show closely related to classified sizes (Figs. 4 and 5). The following general expression can be used to describe the relationship among the average dimensions of Fransis, Mamaei, and Rabei seeds, respectively (relations (12–14)):

(12)

(13)

(14)

Figure 4 The dimensions and geometric mean diameter of almonds (a) and their kernels (b) of three varieties of almond. The modeled samples of each variety were divided into three classification (small, medium, big; 3 × 30) used to measure the dimensions.

Figure 5 The seeds areas, A_xy and A_xz , of three varieties of almonds. For the three varieties, A_xy is more than A_xz .

According to the relations (12–14), the particular parameter, , of Rabei variety was the highest in three varieties and the Mamaei variety had the lowest value of in the varieties. As shown in the relations, the particular parameters , , , and , of Fransis variety were similar to Mamaei variety.

According to the result, the correlation coefficients of the seed dimensions and area were significant at the 1% level. Furthermore the correlation coefficient (Table 3) shows that the , , , and , , and ratios were highly significant.

Table 3 Correlation coefficients (R) between particulars of dimensions, area, and mass for the three varieties of almonds

Particulars	Lk/La	Wk/Wa	Tk/Ta	MK/Ma	Dpk/Dpa	ϕ k /ϕ a	Axyk/Axya	Axzk/Axza
Lk/La	1
Wk/Wa	0.435**	1
Tk/Ta	0.364**	0.458**	1
MK/Ma	0.473	0.332**	0.557**	1
Dpk/Dpa	0.860**	0.950**	0.753**	0.668**	1
φ k /φ a	−0.633**	0.375	0.594**	0.314	0.323	1
Axyk/Axya	0.575**	0.812**	0.547**	0.355	0.761**	0.023	1
Axzk/Axza	0.440**	0.563**	0.628**	0.291	0.792**	0.123	0.458**	1
**Significant level at 1%.

Modeling the Mass and Size Distributions

About 78% of the almonds have a length ranging from 30 to 35 mm; about 90%, a width ranging from 19.5 to 23 mm, about 85%, a thickness ranging from 12.5 to 16.0 mm, and about 92%, a mass ranging from 2.5 to 4.5 gr. About 69% of the kernels have a length ranging from 24 to 27 mm; about 93%, a width ranging from 11.5 to 15.5 mm; about 87%, a thickness ranging from 6.5 to 10 mm, about 93%, a mass ranging from 0.9 to 1.5 gr.

The results showed that Eq. (2) provides a good description of the size and mass distributions of almonds and their kernels. In comparing our density functions to the histogram of the class aggregate data, the correspondence was quite good. The frequency distribution histogram for the mean values of the size and mass showed a trend towards a Weibull distribution. As shown in Table 4, only a slight deviation of the average coefficient of determination from unity was shown. The Weibull PDF performed a good prediction of the length, width, thickness, and mass classes of seeds. The RMSE ranges of the Weibull WDF were generally smaller than 0.07 (Table 4). The η has the same units as X, Eq. (3), such as mass units, size units, etc. It is proposed that η may be used as a criterion to determine the level of small seeds in different varieties of products. Thus, a low value of η indicates that the product consists of a high level of small almonds or kernels. For three varieties, the parameters β and η varied from 2 to 23 and 1.5 to 37.9, respectively. Corresponding values of thickness distribution were the most values in the dimensions. The corresponding values of mass distribution were less than the dimension distributions of three varieties. For the parameters, such as thickness, mass, length, and width, the β varied from 5 to 23, 2 to 11, 3 to 21, and 5 to 15 and the η varied from 26.4 to 37.9, 1.5 to 13.5, 8.2 to 17, and 12.5 to 23.1, respectively.

Table 4 The estimated values of Weibull constants and the statistical parameters derived between measured value and estimated value

Variety	Product	Parameter	η	β	RMSR	R ^2
Fransis	Almond	Length	14.6	21	0.055	0.97
		Width	21.8	13	0.045	0.97
		Thickness	34	23	0.041	0.90
		Mass	4	3	0.038	0.80
	Kernel	Length	10.5	7	0.055	0.90
		Width	13.5	5	0.019	0.70
		Thickness	29	5	0.014	0.90
		Mass	13.5	2	0.050	0.80
Mamaee	Almond	Length	17	11	0.045	0.93
		Width	23.1	15	0.042	0.98
		Thickness	37.9	19	0.036	0.97
		Mass	4.4	3	0.044	0.99
	Kernel	Length	8.2	5	0.049	0.92
		Width	13.3	15	0.042	0.95
		Thickness	27.5	15	0.043	0.97
		Mass	1.9	7	0.060	0.88
Rabei	Almond	Length	16.5	11	0.055	0.90
		Width	21.1	15	0.045	0.94
		Thickness	36.5	19	0.046	0.97
		Mass	3.8	3	0.060	0.95
	Kernel	Length	8.4	5	0.050	0.80
		Width	12.5	11	0.045	0.98
		Thickness	26.4	17	0.040	0.98
		Mass	1.5	11	0.070	0.80

As shown in Table 4, the mass distribution of almonds of three varieties approached the normal PDF because the β was between 2.6 and 3.7 and the dimension distributions of seeds of three varieties had negatively skewed because the β was more than 3.7. The distribution of kernel mass of Fransis variety was positive skewed of the Weibull PDF because the β was less than 2.6. As illustrated in Fig. 6, the width distribution of Fransis, length distribution of Mamaei, and thickness distribution of Rabei variety were the negatively skewed Weibull PDF. According to Fig. 6, increasing the value of η while holding β constant had the effect on stretching out the Weibull PDF. Since the area under a Weibull PDF curve is a constant value of one, the “peak” of the PDF curve also decreases with the increase of η. As indicated in Fig. 6 (parts (c, f) and (b, d)), when the η is increased while the β is kept the same, the distribution gets stretched out to the right and its height decreases. The mean value of parameter η of Fransis was more than Mamaei and Mamaei was more than Rabei; consequently, the distribution height of Fransis variety was less than Mamaei and Mamaei was less than Rabei. The peak height of the mass distribution of Rabei variety was more than the length distribution of Mamaei variety and for the mass distribution of Fransis variety was more than the mass distribution of Mamaei variety. Finally the Weibull WDF of dimensions was better predicted than mass and the width distribution was better predicted than the other sizes for three varieties.

Figure 6 The measured values and estimated densities for some size and mass of almond varieties (color figure available online).

The experimental data were compared with the predicted model for three varieties of almond. Comparison between examined data and compared model of the class aggregate data, showed a nearly good correspondence. Only slight deviation of the average coefficients of determination from unity was seen between data. The kernel–thickness of Fransis variety had the least RMSE and the kernel–mass of Rabei had the most RMSE in the varieties. The R ² for the almond length of Fransis variety was the most and for the kernel width of Fransis variety it was the least.

CONCLUSIONS

The L, W, and T of almonds of Rabei variety were more than Mamaei and these parameters of Mamaei were more than Fransis. The volume has the most correlation with geometric mean diameter and area. Hence, if the difference of volume of almond and its kernel was used to estimate the gap between shell and kernel, the best parameters to estimate the gap are the areas and geometric mean diameter. For three varieties, the Weibull distribution of width was better predicted than the other sizes and the distribution of dimensions mainly were better predicted than the mass. The mass distribution of almonds of three varieties was approached to the normal PDF and the dimension distributions of seeds of three varieties had negatively skewed. The η may be used as a criterion to determine the level of small seeds in different varieties of product. Thus, a low value of η indicates that the product is consists of a high level of small almonds or kernels.

NOMENCLATURE

L	=	Length, mm
W	=	Width, mm
T	=	Thickness, mm
Axy	=	Horizontal area, mm2
Axz	=	Vertical area, mm2
Dp	=	Geometric mean diameter, mm
	=	Sphericity, %
M	=	Mass, g
β	=	Shape parameter
η	=	Scale parameter
γ	=	Location parameter
Subscripts	=
a	=	Almond
k	=	Kernel
F	=	Fransis
M	=	Mamaei
R	=	Rabei
Abbreviations	=
WDF	=	Weibull distribution function
PDF	=	probability density function
CDF	=	cumulative density function
RMSE	=	root mean square error

ACKNOWLEDGMENT

The authors would like to express a special thanks to Mrs. Maryam Haghparast and Mr. Mojtaba Taheri at the University of Tehran for their contributions at the early stage of the project.