Algorithm for predicting the length of each color fiber in mixed-wool fiber assemblies based on the transmission image

Abstract

A color-separation algorithm was proposed to predict the length of each color fiber in mixed-wool fiber assemblies based on a red, green, and blue transmission image. In this work, mixed-wool fiber assemblies consisted of different color wool fibers and a digital color image was obtained by a scanner. The relative thickness of the fiber assemblies was measured based on the Beer-Lambert theory. The color-separation formula was constructed to calculate the quantity of each color fiber at every point of the mixed-wool top to achieve the relative linear density curve and the average length. A series of systematic experiments demonstrated high consistency with the reference relative linear density curve and average length and confirmed the validity of the color-separation formula. This algorithm could be used for quality detection and control of mixed-wool tops. It could be also extended to uniformity detection of other mixed-color fiber assemblies.

Keywords

mixed wool fiber assemblies RGB Beer-Lambert color transmission image fiber relative linear density curve

Woolen products, especially worsted wool products, are mostly blended from different colors of wool fibers, which enrich the color of woolen products and meet the needs of the public. Semi-finished woolen products in the spinning process, such as slivers, rovings, and spun yarns, are generally blended from different color fibers. Fiber length and length distribution are very important indexes of fiber quality that can be used to configure the structure and properties of fiber assemblies in the textile industry.^1–3 Damage of fibers when processing the blended products implies not only the change of fiber length distribution but also quality issues of fiber products. Therefore, predicting the average length of each monochromatic fiber in the mixed-wool fiber assembly is important for the textile industry.

Beer-Lambert law (B-L) has been widely applied to many fields, such as textiles, chemistry, and physics. For example, in the 1970s, the High Volume Instrument system was developed by the Spinlab Corporation based on B-L to measure the cotton fiber relative linear density curve.^4–6 In this manuscript, B-L was also used to calculate the length of each fiber in mixed-fiber assemblies.

To predict the relative linear density curve of each monochromatic fiber in mixed-wool fiber assemblies based on B-L, first the problem of color separation of different fibers has to be solved. As we know, a material's absorption coefficient is different at varying wavelengths of light. For color-separation purposes, a digital color image is beneficial due to its various monochromatic lights.

Digital images include four color modes: red, green, blue (RGB); hue, saturation, brightness; Lab; and cyan, magenta, yellow, and black. The most common is RGB, which is the color standard of industries and is applied to projectors, scanners, and digital cameras. A vast number of research fields use digital color images to analyze a material's interior structure, such as food,⁷ agriculture,⁸ computer recognition,^9–12 textiles, etc. For example, in textiles, many studies use computer image processing and analysis to detect weave patterns and yarn color designs. Tae Jin Kang¹³ used a hue, saturation, value color model to differentiate or group similar yarn colors through computer image processing. Keiji Osaki¹⁴ tried to reproduce the desired colors in textiles based on the CIEL*a*b* color space. Fan Qinguo¹⁵ designed a special pattern to print on paper-supported pretreated fabric samples based on digital image analysis and optical microscopy.

Hence, in this manuscript, an optical algorithm based on an RGB color model and B-L theory was developed to predict the length of each color fiber in mixed-wool fiber assemblies according to digital color images. To solve the problem mentioned above, an optical algorithm was proposed to calculate the length of each fiber in the mixed-fiber assemblies. Moreover, the Almeter, the standard fiber length measurement of the International Wool Textile Organization (IWTO), is regarded as the reference method for evaluating the accuracy of the predicted length measured by the proposed optical algorithm. A series of experiments demonstrated the algorithm was an effective way to calculate the length of different fibers in a mixed-wool fiber assembly.

Materials and methods

Materials

The wool-fiber assemblies used in this work were supplied by Shandong Nanshan Textile Clothing Co., Ltd, including six kinds of monochrome wool top and six kinds of mixed-wool top, which consisted of two kinds of monochrome wool assemblies. The six kinds of monochrome wool top (1–6) and their average fiber lengths are listed in Table 1. The colors of different monochrome wool assemblies are showed in Figure 1. The mixed-wool top included two-color mixtures, 1/2, 1/4, 2/5, 3/4, 4/5, and 4/6.

Figure 1.

Photographs of different monochromatic wool-fiber assemblies and mixed-fiber assemblies.

Table 1.

Monochrome wool materials

Samples	1	2	3	4	5	6
Color	Light grey	Light blue	Dark grey	White	Grey	Dark blue
Average length/mm	64.03	69.6	63.58	79.16	69.83	66.97

Experimental procedure

In this work, wool dual beard made of wool top was prepared based on our team's previous work, using the random dual-beard preparation method.^16, ¹⁷ This method used a holder to clamp the wool top tightly. The wool top was then combed to remove floating fibers, without pulling out any clamped fibers. Therefore, the relative linear density curve in this dual beard was the same as in the wool top. A scanner was used to obtain the dual beard's color digital transmission image, which included the dual beard's digital information, such as RGB values. Figure 2 shows the wool dual beard.

Figure 2.

Color digital transmission image of wool dual beard.

The RGB values on each pixel were extracted from the digital image and represented the transmitted light intensity under different monochromatic light. The resolution of the scanner was 600 dpi and the width of each pixel was 0.00423 cm.

Measurements

As we know, based on a material's transmitted light intensity, B-L¹⁸ has been widely used to calculate the material's thickness, such as fiber assembly¹⁹ etc. For example, ultraviolet spectrophotometry was used to calculate the thickness of transparent media based on B-L, which is also the basis of colorimetry and photoelectric colorimetry.

B-L showed the relationships between material thickness, incident light intensity, and transmitted light intensity, using Equation 1.

kx = ln (I_{0} / I) = ln (1 / T)

(1)

where k is the absorption coefficient of material to monochromatic light, x is the thickness of material at one point,

I_{0}

is the incident light intensity, I is the transmitted light intensity, and T is the transmittance of material under a certain monochromatic light, which is the ratio of I to

I_{0}

In this work, H was defined as the optical depth that represented the degree of light attenuation caused by the material thickness and absorption coefficient at the same time, calculated by Equation 2.

H = kx = ln (I_{0} / I) = ln (1 / T)

(2)

Under different monochromatic lights the absorption coefficient of the material changes, resulting in different optical depths.^20,21 According to this, the optical depth of material is calculated with different equations under each monochromatic light, using Equations 3–5. In the scanner, the incident light ,which was white, is split into three monochromatic lights, red, green and blue.

H_{R} = ln (R_{0} / R) = ln (1 / T_{R})

(3)

H_{G} = ln (G_{0} / G) = ln (1 / T_{G})

(4)

H_{B} = ln (B_{0} / B) = ln (1 / T_{B})

(5)

where

H_{R}

H_{G},

and

H_{B}

are the optical depth of material under red, green, and blue light, respectively.

R_{0}, G_{0},

and

B_{0}

are incident light intensity of red, green and blue, respectively.

R, G,

and B values were transmitted light intensity of red, green, and blue light, respectively.

T_{R}

T_{G,}

and

T_{B}

are the transmittance of material under red, green, and blue light, respectively.

Results and discussion

Light absorption of wool fibers under different monochromatic lights

This work Photoshop was used to obtain the RGB values of wool fibers with a different quantity on each pixel, ranging from 0 to 4. When the number of fibers was 0, RGB values represented the incident light intensity of red, green, and blue, which were $R_{0}, G_{0}$ , and $B_{0}$ respectively. The transmittance was the ratio of transmitted light intensity to incident light intensity.

The transmittance curves of six kinds of wool fiber at different quantities on each pixel under varying monochromatic lights are presented in Figures 3(a), 4(a), and 5(a). Each curve was plotted as the average of 20 points. As shown in these three figures, as the number of fibers increased, the transmittance decreased monotonically and the transmittance was 100% when the number of fibers was 0. With the number of wool fibers increasing, the transmittance of wool fiber 1 (light grey) decreased by 8.35% and 3 (dark grey) decreased by 15.67% under red light, as shown in Figure 3(a). This demonstrated that the quantity of light absorption of different wool fibers under the same light was different. Comparing Figure 3(a) with 4(a), the transmittance of wool fiber 5 (grey) decreased by 15.33% under green light and 10.90% under blue light when there was one fiber. This demonstrated the quantity of light absorption of the same wool fibers differed significantly under different monochromatic lights.

Figure 3.

The relationship between quantity of fiber on each pixel and (a) transmittance and (b) optical depth under red light.

Figure 4.

The relationship between quantity of fiber on each pixel and (a) transmittance and (b) optical depth under green light.

Figure 5.

The relationship between the quantity of fiber on each pixel and (a) transmittance and (b) optical depth under blue light.

The optical depth was then calculated by solving Equations 3–5. The optical depth curves of fibers with a different quantity on each pixel under each different monochromatic light are presented in Figures 3(b), 4(b), and 5(b). It was observed that the optical depth increased linearly with the wool fiber quantity on each pixel. For sample 1, increasing the quantity of wool fiber on each pixel to 1 led to the optical depth increasing by 0.089 under red light, 0.125 under green light, and 0.142 under blue light. Apparently, the wavelength of light may lead to a different light absorption for the same quantity of fiber.

It was observed that the quantity of wool fibers on each pixel was proportional to the optical depth. So the linear-fitting method was carried out to obtain the linear formula between the relative quantity of fiber and the optical depth under each monochromatic light; the fitting degree was above 0.99, as shown in Equation 6. For wool-fiber assemblies that include different diameters, the following assumptions are made: (1) wool fibers used in the experiment have the same diameter; (2) the diameter of fiber would not change during combing; (3) the air inside the fiber assembly and the assembly's fluffy performance would not influence the optical depth.

H_{ih} = C_{ih} + C_{ih}' x_{i}

(6)

where

H_{ih}

was the optical depth of wool fiber i (i represented the kind of fiber, from 1–6) under the monochromatic light h (h = R, G, B) at a certain point.

x_{i}

was the relative quantity of fiber.

C_{ih}

and

C_{ih}'

were constants that represented the material's light absorption property.

The linear formula between the relative quantity of fiber and optical depth under each monochromatic light can be calculated by Equation 6. The constants and coefficients (

C_{ih}

and

C_{ih}'

) of the linear formula are listed in Table 2.

Table 2.

Constants and coefficients of the linear formula between the relative quantity of fiber on each pixel and optical depth under each monochromatic light

Sample	1	2	3	4	5	6
Monochromatic light	Red
C _iR	0.02071	0.02067	0.02259	0.01975	0.02473	0.01780
C_iR′	0.08905	0.15038	0.16991	0.11244	0.11133	0.15586
Monochromatic light	Green
C _iG	0.02810	0.03004	0.03532	0.02775	0.03257	0.02814
C_iG′	0.12513	0.12533	0.21247	0.10529	0.16460	0.19109
Monochromatic light	Blue
C _iB	0.04507	0.03832	0.04811	0.03939	0.03600	0.03339
C_iB′	0.13775	0.15504	0.21873	0.10803	0.13016	0.22740

Note: C_iR, C_iG and C_iB were equation's constants which represented the material's light absorption property under Red, Green, and Blue light, respectively. C_iR′, C_iG′ and C_iB′ were equation’s coefficients which represented the material's light absorption property under Red, Green, and Blue light, respectively.

Algorithm for predicting the quantity of each fiber on each pixel of the mixed-fiber assembly images

To predict the quantity of each fiber in the mixed-fiber assembly images, the authors' previous work on the color-separation formula of mixed polyethylene terephthalate (PET) films was applied.²² In the previous work, the color-separation formula of mixed PET films was the sum of each kind of PET film's linear formula. A series of systematic experiments has demonstrated the extremely high accuracy of this method.

Therefore, the color-separation formula of mixed-fiber assemblies was the sum of each kind of fiber's linear formula. The linear formula between the relative quantity of fiber and optical depth under each monochromatic light was described above. When the mixed-fiber assemblies consisted of two different wool fibers, the quantity of each fiber could be calculated using the color-separation formula presented by Equation 7.

H_{ab h} = H_{ah} + H_{bh} = C_{ah} + C_{bh} + C_{ah}' x_{a} + C_{bh}' x_{b}

(7)

where a and b (a, b represented any two kinds of fibers among 1–6) were different color fibers that made up the mixed-fiber assemblies, h was the monochromatic light (h = R, G, B),

H_{abh}

was the optical depth of the mixed-fiber assemblies under the monochromatic light h, which could be calculated by Equations 3–5.

H_{ah}

and

H_{bh}

were the optical depth of the mixed fiber a and fiber b assemblies under the monochromatic light h, respectively.

x_{a}

and

x_{b}

were the relative quantities of fiber a and fiber b.

C_{ah}

and

C_{ah}'

were constants and coefficients of fiber a’s linear formula under the monochromatic light h, listed in Table 2.

C_{bh}

and

C_{bh}'

were constants and coefficients of fiber b’s linear formula.

Similarly, when the mixed-fiber assemblies were composed of two different wool fibers, the quantity of each fiber could be calculated using the color-separation formula presented by Equation 8.

\begin{matrix} H_{abch} = H_{ah} + H_{bh} + H_{ch} = C_{ah} + C_{bh} \\ + C_{ch} + C_{ah}' x_{h} + C_{bh}' x_{i} + C_{ch}' x_{j} \end{matrix}

(8)

where

H_{abch}

was the optical depth of the mixed fiber assemblies under the monochromatic light h, which could be calculated by Equations 3–5.

H_{ch}

was the optical depth of the fiber c assemblies under the monochromatic light h.

In this work, the scanner has three monochromatic lights. First, the digital color image was obtained by scanning the mixed-fiber assemblies. Next, the color-separation formula was written into Matlab program codes. Third, for mixed-fiber assemblies with two different wool fibers, the quantity of each fiber was obtained from two simultaneous equations using Matlab, which are listed in Table 3. Other indexes, such as length distribution, linear density curve and weighted average length, are calculated based on the quantity of each fiber.

Table 3.

Color-separation formula of different mixed fiber assemblies under each monochromatic light

Mixed-fiber assemblies	Color separation formula
1/2	Y_R=0.04138 + 0.08905×1 + 0.15038×2 Y_G=0.05814 + 0.12513×1 + 0.14593×2
1/4	Y_R=0.04046 + 0.08905×1 + 0.11244×4 Y_G=0.05585 + 0.12513×1 + 0.10529×4
3/4	Y_R=0.04234 + 0.16991×3 + 0.11244×4 Y_G=0.06307 + 0.21247×3 + 0.10529×4
4/5	Y_R=0.04448 + 0.11244×4 + 0.11133×5 Y_G=0.06032 + 0.10529×4 + 0.16460×5
2/5	Y_R=0.04540 + 0.15038×2 + 0.11133×5 Y_G=0.06261 + 0.14593×2 + 0.16460×5
4/6	Y_R=0.03755 + 0.11244×4 + 0.15586×6 Y_G=0.05589 + 0.10529×4 + 0.19109×6

Note: Y_R was the optical depth of the mixed fiber assemblies under the monochromatic light R.

Table 3 shows the color-separation formula for different mixed-fiber assemblies under each monochromatic light. The color-separation formula of mixed-fiber assemblies was the sum of each kind of fiber's linear formula between quantity of each fiber on each pixel and optical depth of the mixed fiber assemblies. In Table 3, 1/2 represents the mixed-fiber assemblies, which consisted of wool fibers 1 and 2.

A comparison between the predicted relative linear density curve of each kind of fiber in the mixed-fiber assemblies and the Almeter curve of monochromatic fiber assemblies

To verify these color-separation formulas, the predicted relative linear density curve of each kind of fiber assembly in the mixed-fiber assemblies was compared with the actual curve obtained by an Almeter, a standard method for measuring wool-fiber lengths that is certified by the IWTO.

As shown in Figure 2, the relative quantity of fibers on each pixel of the mixed-fiber assembly transmission color images was obtained using Matlab software. The relative linear density curve of the dual beard was calculated according to the sum of the fiber quantity in each column of the image, as listed in Figures 6–11. In these figures, the red line represents the predicted relative linear density curve and the black line represents the Almeter result.

Figure 6.

Comparing the predicting relative linear density curve of each color fibers in mixed-fiber assemblies 1/2 and its Almeter result, (a) fiber 1 and (b) fiber 2.

Figure 7.

Comparing the predicting relative linear density curve of each color fibers in mixed-fiber assemblies 1/4 and its Almeter result, (a) fiber 1 and (b) fiber 4.

Figure 8.

Comparing the predicting relative linear density curve of each color fiber in mixed-fiber assemblies 3/4 and its Almeter result, (a) fiber 3 and (b) fiber 4.

Figure 9.

Comparing the predicting relative linear density curve of each color fiber in mixed-fiber assemblies 4/5 and its Almeter result, (a) fiber 4 and (b) fiber 5.

Figure 10.

Comparing the predicting relative linear density curve of each color fiber in mixed-fiber assemblies 2/5 and its Almeter result, (a) fiber 2 and (b) fiber 5.

Figure 11.

Comparing the predicting relative linear density curve of each color fiber in mixed-fiber assemblies 4/6 and its Almeter result, (a) fiber 4 and (b) fiber 6.

As shown in Figures 6–11, the relative linear density curve of each color fiber in the mixed-fiber assemblies was calculated by the color-separation formula and was roughly the same as the actual curve obtained by the Almeter. This indicated the color-separation formula was applicable to fiber assemblies. However, there were differences between two curves. The predicting curve was slightly higher than the actual curve obtained by the Almeter due to B-L, which ignored light scattering and reflection.

The wool beard was a mixture of fiber and air. The incident light reflected and scattered on the fiber-air interfaces, especially where the wool dual beard was thicker. As shown in Figure 2, the dual beard was made through combing the wool top to remove floating fibers without any clamped fibers being pulled out. Due to the larger numbers of fibers in the middle of the dual beard, the light scattered and reflected more at that location, which led to a deformation at the head of the relative linear density curve, as shown in Figure 12(a). These three curves were not normalized. The black line represents the Almeter result, which was certified by the IWTO. The blue line represents the linear density curve of monochromatic wool fiber 2, with the dual beard calculated by B-L. The red line represents the linear density curve of fiber 2 in the mixed-fiber assemblies calculated by B-L and the color-separation formula. Apparently, the more light reflects and scatters, the less the light is absorbed, which makes the calculated fiber quantity smaller than other locations. Therefore, the heads of the red and blue lines were lower than that of the black line. The three curves in Figure 12(a) were normalized and are shown in Figure 12(b). The two figures show the difference between the predicting relative linear density curves of each color fiber in mixed-fiber assemblies and the Almeter curves.

Figure 12.

Comparing the predicting relative linear density curve of fiber 2 in mixed-fiber assemblies 2/5 (red line), the actual curve of monochromatic wool fiber 2 calculated by Almeter (black line), and the actual curve of monochromatic wool fiber 2 calculated by Beer-Lambert law (blue line), (a) not normalized and (b) normalized.

A comparison of the predicted average length of each kind of fiber in the mixed-fiber assemblies and the Almeter result of monochromatic-fiber assemblies

The color-separation formula was verified in the last section to predict the relative linear density curve of each kind of fiber in the mixed-fiber assemblies and the actual curve of monochromatic-fiber assemblies. This was a qualitative comparison. In this section, the average length of each kind of fiber in the mixed-fiber assemblies is calculated to compare with the actual average length according to the relative linear density curve. Therefore, these two results can be quantitatively compared.

As we know, double integration was applied to a relative linear density curve to obtain the weight frequency density function of the fiber length, which could be used to calculate the weight of the average length B. Therefore, the weight of the average length (B) of all samples in last section was calculated, as listed in Table 4. The predicted B was compared with the Almeter result, which was certified by the IWTO. For example, the predicted B of 1 in mixed-fiber assemblies 1/2 was 70.29 mm, and the Almeter result of monochromatic fiber 1 was 64.07 mm.

Table 4.

A comparison between the predicted average length B of each kind of fiber in the mixed-fiber assemblies and the actual length B (Almeter result) of monochromatic-fiber assemblies

Mixed-fiber assemblies	1/2		1/4		3/4
Each kind of fiber	1	2	1	4	3	4
Almeter B/mm	64.07	72.89	64.07	79.16	63.58	79.16
Predicting B/mm	70.29	80.45	70.59	86.79	64.86	88.39

Mixed-fiber assemblies	4/5		2/5		4/6
Each kind of fiber	5#	4#	2#	5#	4#	6#
Almeter B/mm	69.83	79.16	72.89	69.83	79.16	67.48
Predicting B/mm	73.68	87.41	77.09	73.28	88.33	70.13

Almeter B was the monochromatic-fiber length measured by an Almeter.

Predicted B was the length of each color-fiber length in mixed-color fiber assemblies predicted by the color-separation formula.

These results showed a trend that agreed with the actual length (Almeter B). However, the predicted B was a little higher than actual B measured by Almeter due to B-L, which ignored light scattering and reflection. The reason is explained in the last section. The largest difference rate (difference rate = (actual value−predicting value)/actual value* 100%) was 11.66% and the average difference rate was 8.31%.

Conclusions

A color-separation algorithm was proposed to predict the length of each color fiber in the mixed-wool fiber assemblies based on the RGB transmission image in this manuscript. In addition, taking an Almeter result as the reference method, a series of experiments were carried out to verify the accuracy of the color-separation algorithm. The conclusions are as follows.

Monochromatic wool fiber assembly tests demonstrated a linear relationship between the optical depth and the quantity of wool fiber on each pixel of the digital image. The degree of fit was above 0.99. Each kind of fiber's linear formula was obtained and used to find the color-separation formula.

The color-separation formula of mixed-fiber assemblies under each monochromatic light was the sum of each kind of fiber's linear formula. The relative density curve and length of each color fiber can be calculated by solving two color-separation equations using Matlab. The results verified the algorithm by comparing the predicted length with the Almeter length.

In conclusion, the accuracy of the color-separation algorithm is good. However, it can be improved if light scattering is taken into consideration in the future.

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 authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors of this article gratefully acknowledge the support of the Chinese National Science Foundation (NSFC 51673036), and the Fundamental Research Funds for the Central Universities and Graduate Student Innovation Fund of Donghua University (CUSF-DH-D-2019031).

References

Ikiz

Rust

Jasper

, et al. Fiber length measurement by image processing. Text Res J 2001; 71(10): 905–910.

Zeidman

Sawhney

. Influence of fiber length distribution on strength efficiency of fibers in yarn. Text Res J 2002; 72(3): 216–220.

, et al. Snippet counting for cotton length distribution measurement using image analysis. Text Res J 2008; 78(4): 336–341.

Naylor

GRS

Delhom

Cui

, et al. Understanding the influence of fiber length on the High Volume Instrument (TM) measurement of cotton fiber strength. Text Res J 2014; 84(9): 979–988.

Hertel

Zervigon

. An optical method for the length analysis of cotton fibres. Textile Res 1936; 6(7): 331–339.

Hertel

. A method of fibre-length analysis using the fibrograph. Textile Res 1940; 10(12): 510–520.

Godinho

MdS

Pereira

Ribeiro

KdO

, et al. Carbonated soft drink classification based on image analysis and PCA. Quimica Nova 2008; 31(6): 1485–1489.

Milanez

Pontes

MJC

. Classification of edible vegetable oil using digital image and pattern recognition techniques. Microchemical J 2014; 113: 10–16.

Mery

Pedreschi

. Segmentation of colour food images using a robust algorithm. J Food Engineering 2005; 66(3): 35–360.

10.

Subr

Soler

Durand

. Edge-preserving Multiscale Image Decomposition based on Local Extrema. Acm Transactions Graphics 2009; 28(5): 147.

11.

Tan

Lien

Gingold

. Decomposing Images into Layers via RGB-Space Geometry. Acm Transactions Graphics 2017; 36(1): 7.

12.

McCann

Pollard

. Local Layering. Acm Transactions on Graphics 2009; 28(3): 84.

13.

Kang

Kim

. Automatic recognition of fabric weave patterns by digital image analysis. Text Res J 1999; 69(2): 77–83.

14.

Osaki

. High quality color reproduction on jacquard silk textile from digital color images. AUTEX Res J 2003; 3(4): 173–179.

15.

Fan

Kim

Perruzzi

, et al. Fabric pretreatment and digital textile print quality. J Imaging Sci Tech 2003; 47(5): 400–407.

16.

Jingye

Bugao

Fumei

. Measurement of short fiber contents in raw cotton using dual-beard images. Text Res J 2016; 88(1): 14–26.

17.

Wang

. Dual-beard sampling method for fibre length measurements. Indian J Fibre Text Tes 2014; 39(1): 72–78.

18.

Strong

. Theoretical basis of Bouguer-Beer law of radiation absorption. Anal Chem 1952; 24(2): 338–342.

19.

H-Y

Wang

F-M

. A new method for fiber length measurements - Dual-beard method. Industria Textila 2015; 66(3): 117–122.

20.

Mitschele

. Beer-Lambert Law. J Chemical Education 1996; 73(11): A260–A1.

21.

Calloway

. Beer-Lambert law. J Chemical Education 1997; 74(7): 744.

22.

Chen

Wang

. Algorithm of the thickness and mass proportion of color materials in the multilayer high-transparency specimen based on its scanned image. J Donghua University, natural sciences 2019; 45(1): .