Image-based Bilateral Beard Method for measuring weight-based short fiber contents in raw cotton and semi-finished slivers

Abstract

In this paper, an economical way for accurately determining weight-based short fiber contents in raw cotton and semi-finished slivers by utilizing special bilateral beard specimens and image processing was introduced. In the specimen preparation, cotton fibers were drawn by a manual device into a sliver, then the sliver was combed to form a bilateral beard specimen, and finally the bilateral beard was scanned to generate a grayscale image from which a relative fiber number curve was extracted. An algorithm for calculating the weight-based short fiber contents based on the curve was proposed. Five types of cottons were repetitively measured to investigate the robustness of the results of $S F C_{w} (12.7)$ and $S F C_{w} (16)$ , with the weight ratio of fibers shorter than 12.7 and 16 mm, respectively. The results showed that measuring two bilateral beards for each sample could keep the error rate lower than 15%, while four specimens kept the error rate lower than 10%. Compared with AFIS Pro 2, this Image-based Bilateral Beard Method provided results with lower standard deviations and variable coefficients, signifying its analogous or better robustness. In addition, 37 samples from some of the world’s major producing areas were measured by this method and AFIS Pro 2, and a Bland–Altman analysis confirmed a good agreement between the results from the two methods. As only a manual fiber drawing device and an office scanner are needed, this Image-based Bilateral Beard Method is clearly a cheap approach for accurately determining the short fiber contents in raw cotton and semi-finished slivers.

Keywords

Short fiber content bilateral beard cotton measurement

Naturally, cotton fibers possess wide length distributions, resulting from cotton breeds, growing conditions, harvesting methods, and ginning practices, and are usually characterized by a number of length parameters.^1–5 Short fiber content (SFC) is such a parameter that represents the weight-based or number-based proportion of fibers shorter than a threshold in a sample.^6–8 Because short fibers compromise yarn quality (e.g., strength, evenness, and hairiness), it is widely acknowledged that they play a negative role in yarn manufacturing.^9–13 Therefore, methods to measure SFCs in raw cotton continue to be focused upon in the textile testing and instrumentation research area.^14–18

A traditional approach is the array method, in which fibers are manually aligned at one end to form a baseline and then separated into a range of length groups to be weighed separately.⁷ A length–weight histogram of these length groups can be used to calculate several length parameters, including weight-based SFC. In this time-consuming manual operation, short fibers tend to bear a larger risk of being lost than long fibers.¹⁹

The High Volume Instrument (HVI) is an automatic system that is widely employed to evaluate the integrated quality of raw cotton because of its efficiency and completeness of measurement.^20–22 Fibers in raw cotton are randomly grabbed and hung on a row of needles and then combed to form a tapered beard. The beard is scanned by a light slit to generate a fibrogram showing the changes of fiber amount from the needle line to the tapered end. Many length parameters can be extracted from the fibrogram; however, it is not possible to measure SFC firsthand because the fibers near the needle line are likely to be tangled or folded.²³ Instead, the HVI provides a prediction parameter, the Short Fiber Index (SFI), based on other available HVI parameters.²⁴,²⁵ The low repeatability of the SFI, however, prevents it from being used in the United States Department of Agriculture’s (USDA's) raw cotton classification system.^26–29

Another widely employed automatic instrument is the Advanced Fiber Information System (AFIS), which has a built-in fiber opener for individualizing cotton fibers from a sliver, as well as a photoelectric sensor to test the fibers’ length when they are transported by a high-speed airflow.¹⁹ On the basis of at least 3000 single fibers and the assumption that the fibers have uniform linear density, AFIS can output length–number frequency distribution histograms, length–weight frequency distribution histograms, and a series of length parameters extracted from these histograms.²³,^30–33

In recent years, a new form of bilateral beard specimen had been introduced for simplifying the sample preparation in fiber length testing.^34–37 The bilateral beard specimen, which has two tapered ends in opposite directions, eliminates not only the fiber entanglements that severely alter the SFC but also the un-measurable areas of specimens. The Lengthcontrol system (LCT) is the first commercial system that can automatically produce bilateral beard specimens from a feeding sliver and provide SFC measurements.³⁸,³⁹ However, the fibrogram principle and validation results of the LCT have not been made publicly available.

The above-mentioned commercial instruments have a high degree of automation, but they generally have the disadvantages of being expensive and complicated to maintain. In this paper, we developed a novel method for measuring weight-based SFCs in an economical and reliable way, called the Image-based Bilateral Beard Method (abbreviated as IBBM). This method utilized a transmission scanner to scan bilateral beard specimens, then extracted the fiber number change curve from one end of the bilateral beard to the other, and finally calculated the weight-based SFCs using a special algorithm. Beyond the measurement methods and algorithms, this paper also investigated the robustness and the agreement of the weight-based SFC results respectively from the IBBM and a reference method.

Method

Specimen preparation and scanning

Figure 1 presents the procedure of specimen preparation. Firstly, 0.65 ± 0.15 g cotton fibers from a cotton bale or semi-finished sliver are drawn to form a sample sliver of approximately 20 cm long and 5 cm wide, using the fiber drafting device of the array method, which can make tangled fibers straight and parallel without breaking them. Next, the sample sliver is randomly clamped by a clamp with an extra narrow jaw. The clamping line should be perpendicular to the axial direction of the sliver and at least 5 cm away from the sliver’s ends to avoid instability of the ends. Then, the floating fibers are gently combed away, and the sample sliver is converted to a bilateral beard specimen. Finally, the bilateral beard specimen is scanned by an office transmission scanner, which has ordinary parameters such as three-color light-emitting diode (LED) linear lighting, 16-bit gray depth, a linear-array charge-coupled device (CCD), 3200 dpi × 6400 dpi optical resolution, and 3.9 Dmax optical density. To balance the measurement accuracy, efficiency, and equipment cost, the images are set to 1000 ppi resolution and 8-bit gray depth.

Figure 1.
Procedure of specimen preparation.

Image processing

As the beard image is 8-bit grayscale, the gray values of image pixels are from 0 to 255, which precisely reflects the intensity of transmission light at different pixel points. As shown in Figure 2, the gray values are normalized by dividing by 255, converting into the transmissivity at each point. Then, the transmissivity is further transformed into the bilateral beard’s areal density at each point by using the optical algorithm presented in our previous publication for calculating the areal density of the laminar fiber assembly.⁴⁰

Figure 2.
Principle of image processing. RFN: relative fiber number; RLD: relative linear density.

Because each column of the image corresponds to an infinitesimal fragment of the bilateral beard, summing the areal densities of the points in one column gives the bilateral beard’s linear density at the corresponding infinitesimal fragment. It is expected that the clamping line has the highest linear density, so dividing the linear density of each infinitesimal fragment by the clamping line’s linear density can result in a normalized vertical axis value of the relative linear density (RLD) curve. Considering the image resolution is 1000 points per inch, which means the size of pixels is 0.0254 mm × 0.0254 mm, the location of each infinitesimal fragment on the horizontal axis can be established.

Eventually, the RLD curve of the bilateral beard specimen can be plotted as shown in Figure 2. Under the common assumption that fibers in the same sample have the same linear density, the linear density at each infinitesimal fragment has a linear relationship with the number of fibers there, so the RLD curve can be equivalent to the relative fiber number (RFN) curve from which the SFC values can be extracted.

Analysis of the symmetry of the bilateral beard

In this study, to reduce the random fluctuation caused by the nonuniformity of cotton fibers, the bilateral RFN curve in Figure 2 was transformed into a unilateral RFN curve, as shown in Figure 3, by averaging the vertical axis values of the symmetry points on the horizontal axis. Although the bilateral beards are intuitively symmetric, it still needs to be theoretically proved that there is no systematic difference between the two ends.

Figure 3.
Geometrical meaning of the SFC_w(α) calculation formula. RFN: relative fiber number. (Color online only.)

In the bilateral beard, for a fiber whose length is $l$ , it is divided by the clamping line into a left-hand segment and a right-hand segment whose lengths are denoted as $l_{l}$ and $l_{r}$ , respectively, so $l = l_{l} + l_{r}$ . Table 1 lists all possible relationships between $l$ , $l_{l}$ , $l_{r}$ , and $L$ , which is the independent variable of the RFN function and represents the distance along the beard axis from the clamping line. The probability of each case is also shown, and the bold red line represents the possible location range of the clamping point.

Table 1.
Possible relationships between the lengths of the left- and right-hand segments and a specific distance from the clamping line (color online only)

Fiber length relationship Clamping pointlocation range Probability Segment lengthrelationship

$l > 2 L$ Case 1: $\frac{l - 2 L}{l}$ $l_{l} \geq L$ and $l_{r} \geq L$

Case 2: $\frac{L}{l}$ $l_{l} \leq L$ and $l_{r} \geq L$

Case 3: $\frac{L}{l}$ $l_{l} \geq L$ and $l_{r} \leq L$

$2 L \geq l > L$ Case 4: $\frac{l - L}{l}$ $l_{l} \leq L$ and $l_{r} \geq L$

Case 5: $\frac{l - L}{l}$ $l_{l} \geq L$ and $l_{r} \leq L$

Case 6: $\frac{2 L - l}{l}$ $l_{l} \leq L$ and $l_{r} \leq L$

$l \leq L$ Case 7: 1 $l_{l} \leq L$ and $l_{r} \leq L$

If $N$ is the total number of fibers in a bilateral beard specimen, $f_{n} (l)$ is the number-based length probability density function of them, then $N f_{n} (l)$ should be the number of fibers with a length of $l$ . According to Table 1, among the fibers with a length of $l > 2 L$ , those whose left-hand segments are beyond $L$ fit into Cases 1 and 3, and their number is $\int_{2 L}^{l_{\max}} N \cdot f_{n} (l) \cdot (\frac{l - 2 L}{l} + \frac{L}{l}) d l$ , where $l_{\max}$ is the maximal length of the specimen fibers. Among the fibers with a length of $2 L \geq l > L$ , those whose left-hand segments are beyond $L$ fit into Case 5, and their number is $\int_{L}^{2 L} N \cdot f_{n} (l) \cdot \frac{l - L}{l} d l$ . Among the fibers with a length of $l \leq L$ , neither the left-hand segment nor the right-hand segment is beyond $L$ . Thus, the number of fibers extending to the clamping line’s left-hand side with a distance beyond $L$ can be computed by
$\begin{array}{l} F_{l} (L) = \int_{2 L}^{l_{\max}} N \cdot f_{n} (l) \cdot (\frac{l - 2 L}{l} + \frac{L}{l}) d l \\ + \int_{L}^{2 L} N \cdot f_{n} (l) \cdot \frac{l - L}{l} d l \\ = \int_{L}^{l_{\max}} N \cdot f_{n} (l) \cdot \frac{l - L}{l} d l \end{array}$
(1)

Similarly, the number of fibers extending to the right-hand side with a distance beyond $L$ is
$\begin{array}{l} F_{r} (L) = \int_{2 L}^{l_{\max}} N \cdot f_{n} (l) \cdot (\frac{l - 2 L}{l} + \frac{L}{l}) d l \\ + \int_{L}^{2 L} N \cdot f_{n} (l) \cdot \frac{l - L}{l} d l \\ = \int_{L}^{l_{\max}} N \cdot f_{n} (l) \cdot \frac{l - L}{l} d l \end{array}$
(2)

It is obvious that $F_{l} (L) = F_{r} (L)$ , which means there are the same number of fibers on the left- and right-hand sides of the clamping line at the same distance. Thus, the bilateral beard specimen is theoretically proved symmetric and the function of the RFN curve, $F (L)$ , is an even function that yields
$F (L) = F (- L)$
(3)

For experimental verification, 10 bilateral beard specimens from 10 types of cotton samples (marked as 1–10) were prepared and cut into two halves along the clamping line. The weights of the heavier half and the lighter half of each bilateral beard are shown in Table 2. The average weight ratio and the maximum weight ratio were 1.030 and 1.043, respectively, which confirmed that the bilateral beard specimens had good symmetry.

Table 2.
Weight contrast between the two halves of the specimens

Sample no. 1 2 3 4 5 6 7 8 9 10

Weight of the heavier half (mg) 31.4 31.5 37.1 25.5 32.3 30.7 38.1 27.3 33.3 29.7

Weight of the lighter half (mg) 30.1 30.7 35.8 24.5 31.5 29.8 37.4 26.8 32.1 29.0

Ratio 1.043 1.026 1.036 1.041 1.025 1.030 1.019 1.019 1.037 1.024

SFC algorithm

According to some previous reports on the IBBM,³⁴,³⁵,⁴¹ the weight-based length probability density function of raw cotton, $p_{w} (l)$ , has a relationship with the quadratic differential of the RFN curve function as
$p_{w} (l) = l F^{″} (l)$
(4)

Here, the variable $l$ represents the fiber length. Combining Equation (4) with the definition formula of the weight-based SFC of raw cotton
$S F C_{w} (α) = \int_{0}^{α} p_{w} (l) d l$
(5)
and using the integration by parts, we can get
$\begin{array}{l} S F C_{w} (α) = \int_{0}^{α} l F^{″} (l) d l \\ = {[l F^{'} (l)]}_{0}^{α} - \int_{0}^{α} F^{'} (l) d l \\ = α F^{'} (α) - {F (l) |}_{0}^{α} \\ = α F^{'} (α) - F (α) + 1 \end{array}$
(6)

Here, $α$ is the length threshold of short fibers.

As $F (L)$ is a decreasing function, the derivative $F^{'} (α)$ will be negative and Equation (6) can be diagrammatized as shown in Figure 3. The tangent of the RFN curve at $(α, F (α))$ intersects with the vertical axis, and the distance between the point of intersection and the peak point of the curve represents $S F C_{w} (α)$ , which is marked by a red bold line.

Considering that the length threshold of short fibers is different in different countries, for example, it is 0.5 inch (12.7 mm) in the USA and some other countries while it is 16 mm in China, both of these two standards were employed in this study, and Equation (6) became
$S F C_{w} (12.7) = 12.7 F^{'} (12.7) - F (12.7) + 1$
(7)
and
$S F C_{w} (16) = 16 F^{'} (16) - F (16) + 1$
(8)

Because $F (L)$ is not a continuous theoretical function, the values of $F^{'} (12.7)$ and $F^{'} (16)$ cannot be calculated by theoretical differential. As an alternative, the slope of the line between the points $(11.7, F (11.7))$ and $(13.7, F (13.7))$ , $\frac{F (11.7) - F (13.7)}{2}$ , was considered as $F^{'} (12.7)$ . Then, Equation (7) became
$\begin{matrix} S F C_{w} (12.7) = 12.7 \frac{F (11.7) - F (13.7)}{2} - F (12.7) + 1 \\ = 6.35 [F (11.7) - F (13.7)] - F (12.7) + 1 \end{matrix}$
(9)

Similarly, the slope of the line between $(15, F (15))$ and $(17, F (17))$ , $\frac{F (15) - F (17)}{2}$ , was considered as $F^{'} (16)$ , so Equation (8) became
$\begin{matrix} S F C_{w} (16) = 16 \frac{F (15) - F (17)}{2} - F (16) + 1 \\ = 8 [F (15) - F (17)] - F (16) + 1 \end{matrix}$
(10)

Analysis of the number of specimens needed

The current fiber length measurement methods generally require several specimens to obtain reliable results. It has been reported by many references that the SFC has lower robustness than most other length parameters, which may be primarily caused by the short fibers’ more serious unevenness from one cotton boll to another, and from one plant to another.⁷,¹⁰,¹³,¹⁴,²⁸ So, it is of great important to investigate the number of bilateral beard specimens needed for stable and reliable SFC results.

For this purpose, five laboratory samples, including upland cottons (samples 11–13) and island cottons (samples 14 and 15), were used, and 10 specimens from each sample were made and tested. These SFC results were analyzed by using Equation (11), where $n$ represents the number of specimens needed for each sample, $t_{α}$ is the critical value of the t-distribution, $C V$ is the coefficient of variation, and $d$ is the permissible error rate. For a parameter with unknown $C V$ , firstly, prepare and test $n_{0}$ specimens. Next, calculate the $C V$ of these $n_{0}$ results. Then, choose a special error rate $d$ and confidence level, and calculate $n$ by using Equation (11). Finally, contrast $n$ with $n_{0}$ . If $n \leq n_{0}$ , $n$ is confirmed to be the number of specimens for credible results. If $n > n_{0}$ , $n_{0}$ should be enlarged and the whole process needs to be done again
$n = {(\frac{t_{α} \times C V}{d})}^{2}$
(11)

Table 3 lists the SFC results of the multiple tests and some analysis data when $n_{0} = 10$ , $d = 10 %$ and 15%, and the confidence level is 95%. It is indicated that under the confidence level of 95%, the number of specimens needed to keep the error rate at less than 15% was two, while if the error rate was expected to be less than 10%, four specimens were needed. In addition, it seemed that the measurement results of $S F C_{w} (12.7)$ had less stability than those of $S F C_{w} (16)$ , which may result from the more unequal distribution of the shorter fibers.

Table 3.
Analysis of the number of specimens needed for error rate control

Test no. $S F C_{w} (12.7)$ (%)
$S F C_{w} (16)$ (%)

11 12 13 14 15 11 12 13 14 15

1 9.08 9.27 9.36 6.22 4.97 18.72 14.61 18.87 11.57 10.49

2 10.16 8.60 9.71 5.72 5.07 17.90 15.33 18.67 9.93 8.96

3 9.59 8.76 8.41 5.78 5.86 20.57 16.24 16.48 11.56 9.53

4 8.50 7.58 9.05 6.10 6.20 17.86 15.66 16.75 10.53 10.53

5 9.31 8.25 7.81 5.49 5.28 20.24 14.90 19.32 12.26 9.38

6 9.60 9.17 9.07 4.97 5.29 18.72 14.48 18.03 10.55 10.53

7 10.50 9.01 7.72 5.19 6.07 20.03 15.11 16.83 10.53 9.24

8 8.70 8.72 8.47 4.87 5.24 19.56 16.63 16.71 12.03 10.31

9 8.51 7.61 9.11 5.95 6.16 18.44 15.32 18.40 10.77 10.63

10 9.37 9.05 8.11 6.17 5.61 20.08 14.33 17.86 11.73 9.82

Average (%) 9.33 8.60 8.68 5.65 5.58 19.21 15.26 17.79 11.15 9.94

SD (%) 0.64 0.61 0.68 0.50 0.47 1.00 0.75 1.03 0.78 0.63

CV (%) 6.81 7.04 7.79 8.79 8.37 5.23 4.91 5.81 6.98 6.32

Error rate d = 10%

n 2.25 2.40 2.94 3.75 3.39 1.32 1.17 1.63 2.36 1.94

Round up 3 3 3 4 4 2 2 2 3 2

Error rate d = 15%

n 0.99 1.07 1.31 1.67 1.51 0.59 0.52 0.73 1.05 0.86

Round up 1 2 2 2 2 1 1 1 2 1

Results and Discussion

Repeatability comparison with AFIS

To examine the repeatability level of the SFC results from the IBBM, the AFIS Pro 2 was employed as a reference method to measure the above five samples (samples 11–15) again. The results are based on plenty of direct data of individual fibers, making AFIS a good reference for fiber property measurement studies.

According to AFIS operating regulations, 10 sliver specimens of 0.5 g were prepared from each of the samples, then put into the AFIS test tube and tested one by one. The repeatability statistics of SFC results from the IBBM and AFIS are presented in Table 4, where the difference ratio of the 10 measurements was the difference between the maximum and minimum measurements divided by the average.

Table 4.
The repeatability statistics of short fiber content (SFC) results from the Image-based Bilateral Beard Method (IBBM) and Advanced Fiber Information System (AFIS)

$S F C_{w} (12.7)$ (%)
$S F C_{w} (16)$ (%)

Sample no. 11 12 13 14 15 11 12 13 14 15

IBBM

Maximum 10.50 9.27 9.71 6.22 6.20 20.57 16.63 19.32 12.26 10.63

Minimum 8.50 7.58 7.72 4.87 4.97 17.86 14.33 16.48 9.93 8.96

Average 9.33 8.60 8.68 5.65 5.58 19.21 15.26 17.79 11.15 9.94

Difference ratio 21.44 19.56 22.90 23.88 22.03 14.11 15.07 15.96 20.90 16.80

SD 0.64 0.61 0.68 0.50 0.47 1.00 0.75 1.03 0.78 0.63

CV 6.81 7.04 7.79 8.79 8.37 5.23 4.91 5.81 6.98 6.32

AFIS

Maximum 11.37 10.17 10.63 8.58 7.59 22.41 18.22 21.11 15.25 11.93

Minimum 9.01 8.19 8.59 6.73 5.81 18.88 15.79 17.77 12.27 9.48

Average 9.89 9.52 9.96 7.14 6.77 20.48 16.96 19.07 13.50 10.80

Difference ratio 23.86 20.80 20.48 25.91 26.29 17.24 14.33 17.51 22.07 22.69

SD 0.71 0.71 0.75 0.64 0.55 1.31 1.15 1.24 0.96 0.82

CV 7.18 7.46 7.53 8.96 8.12 6.40 6.78 6.50 7.11 7.59

Comparing the repeatability evaluation index of the two methods, it can be seen that the difference ratios, the standard deviations (SD), and the coefficients of variation (CV) of SFCs from the IBBM are similar or sometimes lower than those from AFIS, signifying that the former has analogous or better robustness compared with the latter. Besides, the SFCs from AFIS appear a little higher than those from the IBBM, one possible reason being that fiber breakages in the opening process have brought a bias into the measurement, as many studies have indicated.¹³,¹⁴,¹⁷,²⁴,²⁵ For both methods, in addition, the results of $S F C_{w} (12.7)$ have larger difference ratios and variable coefficients than those of $S F C_{w} (16)$ , indicating that the shorter fibers distribute more unevenly in raw cotton.

Analysis of the substitutability of the SFC results from the IBBM and AFIS

In order to investigate the correlation and the consistency of the SFC results from the IBBM and AFIS, a total of 37 types of cotton samples (including the 15 samples above) from some of the world’s major cotton producing areas were collected and tested, which included 25 upland cottons from the USA, China, India, Australia, and Mexico, and 12 island cottons from China, Uzbekistan, and Egypt. In the former method, the final result of each sample was the average of four bilateral beard specimens. And in the latter method, the final result was based on four sliver specimens.

Figure 4 shows the correlation analysis results between the SFC results from the IBBM and AFIS. The Pearson’s correlation coefficient is r = 0.9043 for $S F C_{w} (12.7)$ and r = 0.9469 for $S F C_{w} (16)$ . The determination coefficient of the linear correlation equation is $R^{2} = 0.8125$ for $S F C_{w} (12.7)$ and $R^{2} = 0.8937$ for $S F C_{w} (16)$ , which means up to 81.25% and 89.37%, respectively, of independent variable changes are caused by dependent variables. Besides, the significance index is sig = 0.00 for both $S F C_{w} (12.7)$ and $S F C_{w} (16)$ . In other words, the correlation analysis illustrated that the SFC results from the two methods had a significant correlation.

Figure 4.
Correlation analysis between the short fiber content (SFC) results from the Image-based Bilateral Beard Method (IBBM) and Advanced Fiber Information System (AFIS).

As shown in Figures 5 and 6, a consistency check was conducted by the Bland–Altman method, which is usually used to test the consistency of two subjects at the 95% confidence level. The ordinate data are the differences between the two SFC results of the two methods, and the abscissa data are their means. The lines of “Mean+1.96SD” and “Mean–1.96SD” respectively represent the upper 95% limit and the lower 95% limit. For $S F C_{w} (12.7)$ and $S F C_{w} (16)$ , the data points all locate between the upper and lower 95% limit lines and approximately distribute on both sides of the “mean” lines, signifying that the differences between the two methods are not significant. However, the means of the result differences are –0.58 and –0.62, showing that the SFC results from the IBBM tend to be slightly lower than those from AFIS. Considering that many measures, including the windless environment, black work surface, black velvet boards, and sufficient clamping force had been taken into account in the IBBM to ensure the integrity of the samples as much as possible, this slight difference, just like the differences between many current methods, was more likely to be a systematic error caused by the difference in detection principles. In summary, the SFC results from the IBBM have a good agreement with those from AFIS.

Figure 5.
Bland–Altman analysis for SFC_w(12.7) results from the two methods.

Figure 6.
Bland–Altman analysis for SFC_w(16) results from the two methods.

Conclusions

This paper presents the IBBM for accurately measuring the SFCs in raw cotton or semi-finished slivers. A new type of specimen, called the bilateral beard, was used, which should be scanned into grayscale images for extracting RFN curves. In this paper, general algorithms for extracting the weight-based SFCs from the RFN curves was proposed. Five types of cotton samples were repetitively measured to investigate the robustness of the results of $S F C_{w} (12.7)$ and $S F C_{w} (16)$ , which represented the weight ratio of fibers shorter than 12.7 and 16 mm, respectively. The results showed that measuring two bilateral beard specimens for each cotton sample could keep the error rate lower than 15%, while if the error rate was expected to be less than 10%, four specimens were needed. The lower SDs and CVs of the results from the IBBM signified its analogous or better robustness compared with AFIS Pro 2. In addition, 37 types of cotton samples from some of the world’s major cotton producing areas were measured by the IBBM and AFIS. For $S F C_{w} (12.7)$ and $S F C_{w} (16)$ , the correlation coefficients between the results from the two methods were 0.90 and 0.95, respectively, meaning a significant correlation. Besides, the Bland–Altman statistical analysis also confirmed a good agreement between the results from the two methods. Because only a manual fiber drawing device and an office transmission scanner are needed, this IBBM has the potential to be a cheap approach for accurately determining the SFCs of raw cotton and semi-finished slivers, especially for small industries and laboratories in academic institutions.

Fiber length relationship	Clamping pointlocation range	Probability	Segment lengthrelationship
$l > 2 L$	Case 1:	$\frac{l - 2 L}{l}$	$l_{l} \geq L$ and $l_{r} \geq L$
Case 2:	$\frac{L}{l}$	$l_{l} \leq L$ and $l_{r} \geq L$
Case 3:	$\frac{L}{l}$	$l_{l} \geq L$ and $l_{r} \leq L$
$2 L \geq l > L$	Case 4:	$\frac{l - L}{l}$	$l_{l} \leq L$ and $l_{r} \geq L$
Case 5:	$\frac{l - L}{l}$	$l_{l} \geq L$ and $l_{r} \leq L$
Case 6:	$\frac{2 L - l}{l}$	$l_{l} \leq L$ and $l_{r} \leq L$
$l \leq L$	Case 7:	1	$l_{l} \leq L$ and $l_{r} \leq L$

Sample no.	1	2	3	4	5	6	7	8	9	10
Weight of the heavier half (mg)	31.4	31.5	37.1	25.5	32.3	30.7	38.1	27.3	33.3	29.7
Weight of the lighter half (mg)	30.1	30.7	35.8	24.5	31.5	29.8	37.4	26.8	32.1	29.0
Ratio	1.043	1.026	1.036	1.041	1.025	1.030	1.019	1.019	1.037	1.024

Test no.	$S F C_{w} (12.7)$ (%)	$S F C_{w} (16)$ (%)
1	9.08	9.27	9.36	6.22	4.97	18.72	14.61	18.87	11.57	10.49
2	10.16	8.60	9.71	5.72	5.07	17.90	15.33	18.67	9.93	8.96
3	9.59	8.76	8.41	5.78	5.86	20.57	16.24	16.48	11.56	9.53
4	8.50	7.58	9.05	6.10	6.20	17.86	15.66	16.75	10.53	10.53
5	9.31	8.25	7.81	5.49	5.28	20.24	14.90	19.32	12.26	9.38
6	9.60	9.17	9.07	4.97	5.29	18.72	14.48	18.03	10.55	10.53
7	10.50	9.01	7.72	5.19	6.07	20.03	15.11	16.83	10.53	9.24
8	8.70	8.72	8.47	4.87	5.24	19.56	16.63	16.71	12.03	10.31
9	8.51	7.61	9.11	5.95	6.16	18.44	15.32	18.40	10.77	10.63
10	9.37	9.05	8.11	6.17	5.61	20.08	14.33	17.86	11.73	9.82
Average (%)	9.33	8.60	8.68	5.65	5.58	19.21	15.26	17.79	11.15	9.94
SD (%)	0.64	0.61	0.68	0.50	0.47	1.00	0.75	1.03	0.78	0.63
CV (%)	6.81	7.04	7.79	8.79	8.37	5.23	4.91	5.81	6.98	6.32
Error rate d = 10%
n	2.25	2.40	2.94	3.75	3.39	1.32	1.17	1.63	2.36	1.94
Round up	3	3	3	4	4	2	2	2	3	2
Error rate d = 15%
n	0.99	1.07	1.31	1.67	1.51	0.59	0.52	0.73	1.05	0.86
Round up	1	2	2	2	2	1	1	1	2	1

	$S F C_{w} (12.7)$ (%)	$S F C_{w} (16)$ (%)
IBBM
Maximum	10.50	9.27	9.71	6.22	6.20	20.57	16.63	19.32	12.26	10.63
Minimum	8.50	7.58	7.72	4.87	4.97	17.86	14.33	16.48	9.93	8.96
Average	9.33	8.60	8.68	5.65	5.58	19.21	15.26	17.79	11.15	9.94
Difference ratio	21.44	19.56	22.90	23.88	22.03	14.11	15.07	15.96	20.90	16.80
SD	0.64	0.61	0.68	0.50	0.47	1.00	0.75	1.03	0.78	0.63
CV	6.81	7.04	7.79	8.79	8.37	5.23	4.91	5.81	6.98	6.32
AFIS
Maximum	11.37	10.17	10.63	8.58	7.59	22.41	18.22	21.11	15.25	11.93
Minimum	9.01	8.19	8.59	6.73	5.81	18.88	15.79	17.77	12.27	9.48
Average	9.89	9.52	9.96	7.14	6.77	20.48	16.96	19.07	13.50	10.80
Difference ratio	23.86	20.80	20.48	25.91	26.29	17.24	14.33	17.51	22.07	22.69
SD	0.71	0.71	0.75	0.64	0.55	1.31	1.15	1.24	0.96	0.82
CV	7.18	7.46	7.53	8.96	8.12	6.40	6.78	6.50	7.11	7.59

Footnotes

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by National Natural Science Foundation of China (Grant No: 51673036, 51803108), Fujian Provincial Natural Science Foundation of China (Grant No: 2019J05106), Quanzhou City Science & Technology Program of China (Grant Nos: 2019G028, 2018G011), and Quanzhou Home-bay Recruitment Program of Global Talents (Grant No: 2017ZT002).

ORCID iDs

Chenhong Lang

Jingye Jin

Bugao Xu

Yiping Qiu

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