Color difference classification of solid color printing and dyeing products based on optimization of the extreme learning machine of the improved whale optimization algorithm

Abstract

To mitigate the problem of low classification accuracy in solid color printing and dyeing, a color difference classification model based on the differential evolution (DE) improved whale optimization algorithm (WOA) for extreme learning machine (ELM) optimization, named the DE–WOA–ELM, was developed in this study. Considering that the initial population of the WOA has a significant influence on the solution speed and quality, DE was used to generate a more suitable initial population for the WOA by avoiding local optima, thereby improving the performance. The method used an excellent global search ability to improve the WOA for optimization and obtained an optimal parameter combination for the ELM. Thus, the problem of randomly initializing the input weight and the hidden layer bias of the ELM, which leads to a nonuniform training model and unstable algorithm, was solved. Finally, by optimizing the input weight and hidden layer bias, the color difference classification model of the ELM with a strong generalization ability was constructed. The results of the color difference classification experiments on fabric images collected under standard light sources show that the average classification accuracy for the dataset is increased by 2.15%, 11.06%, 12.11%, and 0.47% compared with those of the ELM, support vector machine, back propagation neural network, and kernel ELM, respectively.

Keywords

color difference classification differential evolution extreme learning machine whale optimization algorithm

In the printing and dyeing industry, the detection of color differences in printed and dyed fabrics is an essential component of quality control. In the actual production process, companies use swatches provided by customers as a standard to produce textile products that meet the customer requirements. When customers accept the dyeing quality of the textile products, they will compare the color difference with the luster of the original samples and products to determine whether the color of the textile products meets their requirements. The detection of color difference in conventionally printed and dyed fabrics depends on the subjective judgment of human beings, which leads to large errors and does not meet the needs of automated production. Therefore, online detection of color differences in printed and dyed fabrics has become an urgent need.

In the printing and dyeing industry, the color feature-based RGB color difference detection technology is the most commonly used method. In this method, the image is preprocessed in the RGB space and then converted into the CIELAB color space for calculation using a color difference formula.¹ This method is only a numerical representation, which visually reflects the color difference between two images, and it does not classify the rank. Wong² proposed a Naive Bayesian (NB) method based on the genetic algorithm (GA), which showed a significant instability in the prediction of color difference in dyed products. Because the NB method depends on prior knowledge of training samples, the classification accuracy is greatly reduced when the number of training samples changes. Furthermore, the optimization process of the GA lasts for a long time, and the classification accuracy is not ideal. Zhang and Yang³ proposed a color difference detection algorithm based on the GA to optimize the support vector machine (GA–SVM). Although the optimization speed of the algorithm was faster, it increased the risk of falling into local optima. Zhou et al.⁴ presented a color difference classification method to optimize the SVM based on the improved gray wolf optimizer (GWO) by illumination correction of dyed fabrics.⁵ The excellent global search ability of the GWO was used to solve the problem of falling into local optima. A model based on the least squares support vector machine (LS–SVM),^6,7 which solved a large number of storage problems in quadratic programming required for the original SVM in large-scale tasks, was also proposed. Simultaneously, the parameters of the LS–SVM were selected by using the GWO algorithm in the LS–SVM–GWO⁸ model to improve the performance and overcome the shortcomings of the conventional parameter selection methods. There exist many studies on neural networks for color difference detection.^9–11 For example, based on the CIELAB color space, Li et al.¹⁰ proposed the Levenberg–Marquardt (LM) optimized back propagation (BP) algorithm for textile color difference detection. Li et al.¹¹ studied fabric color difference detection based on the Takagi–Sugeno (T–S) fuzzy neural network. The trained T–S fuzzy neural network was used to obtain the color characteristics of dyed fabrics. It avoided the large number of calculations in conventional color space conversion formulas and improved the detection efficiency to a certain extent. However, in general, the convergence speed of neural networks tends to be slow, they easily fall into local optima, and the real-time performance of the algorithms is poor.

To avoid the shortcomings of neural networks, in 2004, Huang et al.¹² proposed a single hidden layer feedforward neural network (SLFN) called the extreme learning machine (ELM). The ELM has a simple structure, few adjustment parameters, randomly generated hidden layer input weights and biases, and a high learning speed. Furthermore, it is suitable for different types of problems^13–16 and is applicable to different tasks. For example, Zhang et al.¹⁷ used the ELM for pathological brain detection. The random settings of the input layer weight and bias of the ELM may affect its performance.^18,19 To solve these shortcomings, global search algorithms can be used. Differential evolution (DE)^20,21 is used to optimize the input layer weights and biases of the ELM, but DE needs to control a larger number of parameters, and parameter selection is more dependent on empirical rules. The particle swarm optimization (PSO) algorithm is used to optimize the input layer weight and bias of the ELM,²² but PSO is unstable and can easily fall to the local optima solution, and the results can be easily affected by the parameter size.

The whale optimization algorithm (WOA), a new nature-based metaheuristic optimization algorithm, was proposed by Mirjalili and Lewis²³ in 2016. It has the advantages of simple operation, fewer adjustment parameters, and a strong ability to jump out of local optima. The WOA was tested using several metaheuristic optimization problems and structural design problems.²³ The optimization results show that the WOA algorithm has a larger optimization effect than the existing GA, DE, PSO, and other metaheuristic algorithms. The initial population of the WOA is randomly set; however, population initialization is key in the algorithm because it directly affects the convergence speed and the quality of the final solution. Rahnamayan et al.^24,25 proposed a new population initialization method that uses a reverse learning algorithm for population location initialization. Experiments on a group of various benchmark functions showed that using reverse learning instead of random initialization can speed up the convergence. In 2014, Hussein et al.²⁶ proposed a new initialization algorithm based on the patch concept and Lévy flight distribution to initialize the bee population in the bees algorithm (BA). When the enhanced BA variants are compared with the other variants and the most advanced variants of the artificial bee colony (ABC), the experimental results on several widely used and high-dimensional benchmark functions show that it has significant advantages in terms of population quality, convergence rate, and success rate. Gao et al.²⁷ proposed a new algorithm in 2016, named the DGABC combination, which combined DE²⁸ with the best-guide ABC (GABC) through an evaluation strategy, attempting to use more prior information from previous search experience to accelerate convergence. In addition, to improve global convergence, the chaotic reverse learning group initialization method was adopted in the initial population generation, which showed a superior performance.²⁷ Thus, the initial population with high diversity is helpful for improving the performance of the optimization algorithm.

In this study, we developed a color difference classification model based on the DE improved WOA for ELM optimization to generate the initial population of the WOA and thus improve the quality of the WOA initial population. The main contributions are as follows.

For the first time, the ELM model was applied to the color difference classification of solid color printing and dyeing products to improve the classification accuracy.

Because the quality of the initial population affects the global convergence rate and final solution of the group optimization algorithm, the idea that the initial population of the WOA is generated by using the DE algorithm was introduced, and the DE–WOA method was established.

Because in the ELM, the input weight, and bias are randomly set, the classification accuracy and stability of the model will be poor. To overcome this problem, the improved WOA algorithm was used to optimize the input weight and hidden layer bias of the ELM in establishing the DE–WOA–ELM model.

The influence of the WOA parameters in the DE–WOA–ELM on the algorithm performance was analyzed. Furthermore, the influence of the number of hidden nodes in the ELM on the accuracy of the algorithm classification was investigated. The stability of the algorithm was analyzed by using an iterative analysis method and a box plot. A statistical significance analysis was performed on the proposed algorithm and its competitors using the T-test and the F-test.

The remainder of the paper is organized as follows: the preliminaries section introduces the theoretical background of the ELM classifier and related algorithms. In the extreme learning machine based on the optimized whale optimization algorithm by differential evolution section, the ELM based on the DEWOA is proposed. The experimental results section provides the experimental conditions and data sources, including a discussion and analysis of the experimental results. The Sensitivity, stability, and significance analyses of parameters section presents a discussion and analysis of the sensitivity, stability, and significance of the parameters, and finally, the study is concluded in the conclusion section.

Preliminaries

The ELM, DE, and WOA used in the process of establishing the color difference classification model, that is, the DE–WOA–ELM, for printed and dyed products, are introduced in the following sections.

Extreme learning machine

The ELM¹² is a SLFN. Suppose that there are N samples $(X_{j}, t_{j})$ , where $X_{j} = [x_{j 1}, x_{j 2}, \dots, x_{jn}] T \in R^{n}$ and $t_{j} = [o_{j 1}, o_{j 2}, \dots, o_{jm}] T \in R^{m}$ . A neural network that is a single hidden layer with L hidden layer nodes can be expressed as

\sum_{i = 1}^{L} β_{i} g (W_{i} • X_{j} + b_{i}) = o_{j}, j = 1, 2, \dots, N

(1)

where g(x) is the activation function,

W_{i} = [w_{i 1}, w_{i 2}, \dots, w_{in}]^{T}

is the input weight of the ith hidden layer unit,

b_{i}

is the bias of the ith hidden layer unit, and

β_{i} = [β_{i 1}, β_{i 2}, \dots, β_{im}]^{T}

is its output weight.

W_{i} \cdot X_{j}

represents the inner product of

W_{i}

and

X_{j}

The single hidden layer neural network is trained to obtain the optimal parameters¹² and possibly obtain ${\overset{\land}{w}}_{i}$ , ${\overset{\land}{b}}_{i}$ , and ${\overset{\land}{β}}_{i}$ to generate

‖ H ({\overset{\land}{w}}_{i}, {\overset{\land}{b}}_{i}) \cdot \overset{\land}{β} - T ‖ = mi n_{w, b, β} ‖ H (w, b) \cdot β - T ‖

(2)

where

i = 1, 2, \dots, L

. This is equivalent to minimizing the loss function

E = \sum_{j = 1}^{N} ‖ \sum_{i = 1}^{N} β_{i} \cdot g (w_{i} \cdot x_{j} + b_{i}) - t_{j} ‖ 2 2

(3)

The matrix of the output weight is calculated as

\overset{\land}{β} = H^{†} T

(4)

where

H^{†}

is the Moore–Penrose generalized inverse of matrix H, and the norm of the obtained solution for

\overset{\land}{β}

is minimal and unique.

Differential evolution

DE²⁸ is a heuristic global search algorithm used to simulate biological evolution mechanisms. It is primarily used to solve global optimization problems of continuous variables. The main steps include mutation, crossover, and selection. In the DE algorithm, the difference vector between two individuals randomly selected from the population is used as the random variation source of the third individual, and the difference vector is weighted and summed with the third individual according to a certain rule to generate a variant individual. The mutation operation to DE is defined as follows

D_{i} (T + 1) = X_{r 1} (T) + F \times (X_{r 2} (T) - X_{r 3} (T))

(5)

where T is the frequency of current iterations and F is the zoom factor, which has a range of [0, 2].

r_{1}

r_{2}

, and

r_{3}

are random integers that are not equal to i in [1, N], and are not equal to each other. N is the size of the population.

The basic principle of the cross-operation is that the parameters of the mutated individual and predetermined target individual are fused to generate the desired ones. For the ith individual in the jth dimension, the cross-operation is expressed as follows

U_{i, j} (T + 1) = {\begin{matrix} D_{i, j} (T + 1), & if rand < CRorj = D_{n} \\ X_{ij} (T), & otherwise \end{matrix}

(6)

where CR is the cross probability, rand is a random number in [0, 1], and

D_{n}

is a random dimension.

DE uses the greedy algorithm to select individuals of the next generation to ensure the evolution direction. That is, the fitness of the test individual is better than that of the target individual; therefore, the test individual replaces the target individual in the next generation. Otherwise, the target individual remains the same. The selection operation formula is as follows

X_{i} (T + 1) = {\begin{matrix} U_{i} (T + 1), & if f (U_{i} (T + 1)) \leq f (X_{i} (T)) \\ X_{i} (T), & otherwise \end{matrix}

(7)

Whale optimization algorithm

The WOA²³ is a newly proposed metaheuristic optimization algorithm that simulates humpback whales' hunting behavior. The algorithm includes three mechanisms: encircling prey, spiral update, and random search.

Encircling prey: humpback whales can identify their prey's location and surround it,²³ as indicated in the following formula

\vec{D} = | \vec{C} \cdot \vec{X^{*}} (t) - \vec{X} (t) |

(8)

\vec{X} (t) = \vec{X^{*}} (t) - \vec{A} \cdot \vec{D}

(9)

where t is the frequency of current iterations,

\vec{A}

and

\vec{C}

are the coefficient vectors,

\vec{X^{*}}

is the best position vector of the whale, and

\vec{X}

represents the position vector of the whale.

\vec{A}

and

\vec{C}

are calculated using the following formula

\vec{A} = 2 \vec{a} \cdot \vec{r} - \vec{a}

(10)

\vec{C} = 2 \cdot \vec{r}

(11)

where the value of

\vec{a}

decreases linearly from 2 to 0, and

\vec{r}

is a random number in [0, 1].

Spiral update: to model this whale behavior, assume that there is a 50% probability of choosing between the reduced shrinking circling mechanism and spiral model to update the position of whales during the optimization process.²³ The mathematical model is established as follows

\vec{X} (t + 1) = {\begin{matrix} \vec{X^{*}} - \vec{A} \cdot \vec{D} & if p < 0.5 \\ D' \cdot e^{bl} \cdot cos (2 π l) + \vec{X^{*}} (t) & if p \geq 0.5 \end{matrix}

(12)

where p is a random number in [0, 1].

Random search: to strengthen the exploration of the WOA, whales will search for prey randomly by using a position randomly selected from the current whale group to update their position accordingly.²³ The mathematical model is presented as follows

\vec{X} (t + 1) = \vec{X_{rand}} - \vec{A} \cdot \vec{D}

(13)

where

\vec{D} = | \vec{X_{rand}} - \vec{A} \cdot \vec{D} |

, and

X_{rand}

is a position selected randomly in the current whale group.

Extreme learning machine based on the optimized whale optimization algorithm by differential evolution

The properties of the ELM are decided by two pivotal parameters: the input weight w and hidden layer bias b. This study proposes an optimization of the ELM of DE based on the improved WOA (DE–WOA–ELM). The overall framework of the method is illustrated in Figure 1. The DE–WOA–ELM consists of two parts: parameter optimization and evaluation in classification. In parameter optimization, nine-tenths of the entire dataset is internally five-fold cross-validated. When the internal parameter optimization is terminated, the optimal parameter pair is used as the input of the ELM by an external 10-fold cross-validation, and the classification task for printed and dyed products is in the external cycle

Acc = \frac{\sum_{i = 1}^{M} T_{i}}{N} 100 %

(14)

where M is the number of sample classifications,

T_{i}

represents the number of samples correctly classified in each class, and N represents the number of samples in the sample set. Considering the accuracy rate of the classification when its fitness is calculated

F = avgACC = \frac{(\sum_{i = 1}^{K} testAC C_{i})}{K}

(15)

where

avgACC

in function F represents the test accuracy achieved by the ELM classifier through the five-fold crossover in the internal parameter optimization, and

ACC

is the accuracy rate of classification (Equation (14)). The pseudo code for the proposed ELM based on the optimized WOA by the DE algorithm is summarized in the Appendix.

Figure 1.

Flow chart of the proposed algorithm. WOA: whale optimization algorithm; ELM: extreme learning machine; DE: differential evolution.

Experimental results

The proposed algorithm was compared with the ELM,¹³ PSO–ELM,²² DE–ELM,²¹ WOA–ELM, GWO–kernel extreme learning machine (KELM),²⁹ Grid–KELM,³⁰ KELM,³⁰ GWO–LS–SVM,⁸ LS–SVM,⁶ GA–SVM,³ Grid–SVM,⁵ PSO–SVM,⁵ SVM,⁵ T–S fuzzy neural network,¹¹ LM–BP,⁹ and BP⁹ to evaluate its effectiveness. The training time and prediction accuracy were applied to estimate the performance of the algorithm. For the experiment, a PC with 2.5 GHz CPU, 8GB RAM, dual core processor, and the 2015a version of MATLAB were used.

Datasets

Device selection

As shown in Figure 2, A SONY SSC-DC398P color camera, which uses a BNC video connector to output a standard Phase Alternating Line (PAL) video signal, was used. The camera's imaging device is a 1/3-inch interline transfer HyperHAD charge-coupled device (CCD) sensor with 752 × 582 pixels. When working in the continuous acquisition state, the maximum frame rate can reach 60 fps, 1/50–1/100,000 s shutter range, and a high signal-to-noise ratio. The high-sensitivity feature allows it to be shot in high-speed and low-light environments, giving users the control flexibility.

The selection of illumination sources is especially important when performing color difference measurements in different industries. The research object of this study is a finished product fabric with cotton and polyester; hence, the material density of the cloth was not considered and the most commonly used illumination sources, that is, D65, A, and D50 were chosen.

Under the condition of standard light sources (such as D65, D50, or A), the image collection mechanism of the CCD camera cannot be adjusted according to the change in light source information. The illumination correction algorithm³¹ was introduced to eliminate the error in the color difference evaluation caused by the light source.

Selection of the color difference formula

In color difference detection systems, the selection of the color difference formula for different color spaces is important to better evaluate the color difference and meet the visual perception requirements of people. In this study, CIELAB, CMC, and CIEDE2000 formulas³² were used to calculate the color difference.

Figure 2.

Flow chart of image capture.

Under the same illumination, two frames ((a) and (b)) without a visual color difference and one frame (c) with a visual color difference are shown in Figure 3. Using one of the images without a visual color difference as the reference image, three color difference formulas were employed to calculate the color difference between the other image without a visual color difference and the reference image. Then, a color difference dataset, based on the pixel points whose mean value and variance can be calculated, was obtained (Table 1). Next, the color difference between the image with a visual color difference and the reference image was calculated, and another color difference dataset based on the pixel point was obtained (Table 2).

Figure 3.

Experiment for the selection of the color difference formula.

Table 1.

Color difference calculation values for no visual color difference

Color difference formula	CIEDE2000	CIELAB	CMC
Variance	0.449688235	1.031269872	0.012541917
Color difference	0.382608126	1.038748303	0.012614839

Table 2.

Color difference calculation values for existing visual color difference

Color difference formula	CIEDE2000	CIELAB	CMC
Variance	1.869743251	6.227070178	0.06284886
Color difference	4.810324829	16.06817076	0.221988966

In cloth images without a color difference, the smaller the color difference calculation value, the better. In cloth images with a color difference, the larger the calculated color difference value, the better. Tables 1 and 2 show that the mean value of the color difference calculated by CMC is a better representative when there is no color difference. When there is a color difference, the mean value calculated by CIEDE2000 is better.

To increase the reliability of the CIEDE2000 color difference formula, multiple sets of experiments were performed using the colors of the cloth pictures (Figures 4(a)–(g)). The color difference between the image before and after light denoising and the standard image was calculated. The first column is the standard image and the second column is the processed image; they were compared with the Datacolor 650 measurement results. The CIEDE2000 formula and the Datacolor 650 measurement results are in close agreement, as indicated in Table 3.

Figure 4.

Multicolor cloth images.

Table 3.

Measurement result of CIEDE2000 and Datacolor 650

Group	CIEDE2000	Datacolor 650
a	0.379	0.255
b	1.675	2.167
c	1.474	1.439
d	1.441	1.410
e	1.947	1.990
f	0.0186	0.0827
g	0.7619	0.7843

In this study, considering its applicability and accuracy³² and based on the analysis and integration of existing color difference models and visual evaluation data, the CIEDE2000 color difference formula was selected to calculate the color difference between two pure color images for online detection.

Datasets

To evaluate the performance of the proposed algorithm in classifying printed and dyed products, we verified the captured dataset. As shown in Figure 2, by using a high-precision color industrial camera (SSC-DC398BP) under a standard light source (such as D65, D50, or A), we obtained the same textile image as an experimental material and processed the acquired image to obtain the dataset. Taking Figure 5 as an example, the specific process can be described as follows.

In Figure 5, the acquired image has noise and thus was subjected to wavelet denoising preprocessing. After denoising, the image was converted from RGB to the HSV (hue, saturation, value) color space, and ΔH, ΔS, and ΔV were calculated. Then, the image was then converted into the CIELAB color space, and ΔL, Δa, and Δb eigenvalues were calculated.

The color difference between the two solid color images was calculated using the following CIEDE2000 formula

\begin{matrix} Δ E = [(Δ L / K_{L} \cdot S_{L}) 2 + (Δ C / K_{C} \cdot S_{C}) 2 + (Δ H / K_{H} \cdot S_{H})^{2} \\ + rot (Δ C / K_{C} \cdot S_{C}) (Δ H / K_{H} \cdot S_{H})]^{\frac{1}{2}} \end{matrix}

(16)

where

Δ L

is the luminance difference,

Δ C

is the saturation difference, and

Δ H

is the tone difference.

S_{L}

S_{C}

, and

S_{H}

are the weights of

Δ L

Δ C

, and

Δ H

, respectively. “rot” represents the angle of rotation. The unit of color difference is NBS³ (National Bureau of Standards) (Table 4), and the calculated color difference is classified into five levels (Table 4).

Repeating steps (1) and (2), the calculated level and ΔH, ΔS, ΔV, ΔL, Δa, and Δb eigenvalues of color difference were grouped into different datasets. In the trial, 1400 datasets were selected; 1400 × seven-dimensional color difference data. Some of the data are presented in Table 5, Figure 6 gives the three-dimensional representation (ΔL, Δa, and Δb) of the dataset, and Figure 7 shows the color distribution of the fabric sample.

Experimental conditions

The classification models used in this study are the following: the DE–WOA–ELM classification model, ELM model, WOA–ELM model, DE–ELM model, PSO–ELM model, conventional SVM classification model, LS–SVM model, GA–SVM model, Grid–SVM model, PSO–SVM model, BP model, and novel classification models applied in images, such as the GWO–KELM model, Grid–KELM model, KELM model, GWO–LS–SVM model, T–S fuzzy neural network model, and LM–BP model. The parameters of these models and the algorithm were set as follows.

Figure 5.

Images of some of the textiles.

Figure 6.

Three-dimensional scatter plot (ΔL, Δa, and Δb).

Figure 7.

The color distribution of the fabric sample.

Table 4.

Color difference criterion

$Δ E$ (NBS)	Human eye perception	level
0–0.5	Infinitesimal	1
0.5–1.5	Slight	2
1.5–3.0	Obvious	3
3.0–6.0	Very obvious	4
>6.0	Strong	5

NBS: National Bureau of Standards.

Table 5.

Partial dataset

Level	$Δ E$	$Δ H$	$Δ S$	$Δ V$	$Δ L$	$Δ a$	$Δ b$
2	0.90335	0.02963	0.06356	0.00872	0.06994	0.00486	0.01640
2	1.36349	0.03271	0.07930	0.02135	0.18126	0.00457	0.02011
3	2.02213	0.50555	0.06797	0.00872	0.22105	0.01508	0.05059
3	2.18328	0.00308	0.01573	0.01264	0.11132	0.00030	0.00370
5	9.8687	0.53827	0.14727	0.01264	0.03980	0.01052	0.03048
3	2.18328	0.00308	0.01573	0.01264	0.11132	0.00030	0.00370
4	3.88778	0.05537	0.02869	0.06885	0.54620	0.01095	0.00594
2	1.06322	0.29612	0.11610	0.00915	0.14566	0.00584	0.02900
1	0.24103	0.63192	0.06709	0.00349	0.03178	0.00277	0.04070
4	4.79961	0.24074	0.08740	0.07800	0.69186	0.00510	0.02306
4	4.08695	0.57654	0.03839	0.07233	0.57798	0.01372	0.03477
2	0.84242	0.33580	0.04901	0.00566	0.11388	0.00861	0.01171
5	6.12711	0.01599	0.01353	0.13508	0.84800	0.10506	0.07734
3	2.56836	0.14373	0.025427	0.02266	0.48042	0.08387	0.05069
4	4.02102	4.25510	4.25511	0.01324	0.09281	0.10892	0.02160

In the ELM classification model, the number of hidden layer nodes is L = {5,10,15,20,25,30,35,40, 45,50,100,200,300,400,500}, and there are different activation functions, including sigmoid, Sine, Hardlim, Sign, and Gaussian. Because the random setting of the input weight and bias may have some influence on the performance of the ELM, the DE–WOA, WOA, DE, and PSO were used to optimize the ELM; the parameters of each algorithm are presented in Table 6.

Table 6.

Parameter setting for the extreme learning machine optimization algorithm

Algorithm	Parameter	Value
WOA²⁴	Number of populations	{10, 20, 30, 40, 50}
	Maximum iteration times	[20, 100]
	A	[1, 0] [2, 0] [4, 0]
	Size of population	[−1, 1]
DE^21,22	Number of populations	20
	Maximum iteration times	100
	F	1
	CR	0.8
PSO²³	Number of populations	20
	Maximum iteration times	100
	Magnitude of speed	[−1, 1]
	Number of populations	[−5, 5]
	C1	2
	C2	2

WOA: whale optimization algorithm; DE: differential evolution; PSO: particle swarm optimization.

In the SVM classification model, the radial basis function (RBF) was selected as the kernel function.³ The RBF function was used because it can better manage the nonlinear relationship between the evaluation index and the evaluation level; c and γ were set randomly. To determine the parameters of the penalty factors c and γ in the kernel function of the standard SVM, the GA, Grid, and PSO (GA–SVM model, Grid–SVM model, and PSO–SVM model) were used to optimize the SVM. The parameters of each algorithm are presented in Table 7.

Table 7.

Parameter setting for the support vector machine optimization algorithm

Algorithm	Parameter	Value
GA³	Number of populations	20
	Maximum iteration times	100
	Crossover probability	0.4
	Mutation probability	0.01
Grid⁵	G	[−8, 8]
	C	[−8, 8]
PSO⁵	c1	1.5
	c2	1.7
	Number of populations	20
	Maximum iteration times	100
	W	[0.2, 0.9]

GA: genetic algorithm; PSO: particle swarm optimization.

In the BP classification model, the number of hidden layer nodes is L = {5,10,20 30,40,50}, and the other parameters are acquiescent in the BP toolbox. In the LM–BP model, the number of hidden layer nodes is L = {5,10,20,30,40,50}. In the T–S fuzzy neural network, the number of membership functions is 12, which include the Gaussian membership functions. In the LS–SVM, the RBF was selected as the kernel function.⁶ To improve the LS–SVM performance, the GWO was used to optimize the parameter in the LS–SVM (LS–SVM–GWO). The parameter setting of the GWO is presented in Table 8. In the KELM, the RBF was selected as the kernel function;³¹ c and γ were randomly set. To ascertain the parameters of the penalty factors c and γ in the kernel function of the model, the GWO and Grid were employed for optimization. The parameters of each algorithm are listed in Table 8.

Table 8.

Parameter setting for the kernel extreme learning machine optimization algorithm

Algorithm	Parameter	Value
GWO³⁰	Number of populations	20
	Maximum iteration times	100
Grid³¹	c	$[2^{- 8}, 2^{8}]$
	γ	$[2^{- 8}, 2^{8}]$

GWO: gray wolf optimizer.

All the input values (variables) were normalized to [−1, 1] using the min–max normalization so that they can be in the same range. Each feature x is linearly transformed from [min, max] toward y in the new interval [NewMin, NewMax] according to Equation (17)

\begin{matrix} y = (\frac{x - min}{max - min}) \times (NewMax - NewMin) \\ + NewMin \end{matrix}

(17)

Experimental results for datasets

The proposed DE–WOA–ELM algorithm was compared with the other 16 algorithms to evaluate its performance in the classification of printed and dyed products. To obtain statistically significant results, each algorithm was run 10 times independently. The average classification accuracy (Avg), standard deviation (Stdv), best classification accuracy (Best), and worst classification accuracy (Worst) were used as indicators to evaluate the performance of the algorithms. The results are presented in Tables 9–12 and indicate the operation of each algorithm in the best state. Tables 9–11 show that the results of the proposed algorithm (DE–WOA–ELM) are optimal in all indicators (i.e., 0.9969, 0.0011, 0.9995, and 0.9954, respectively) compared with those of the other algorithms in the respective tables. However, in Table 12, the standard deviation and best classification accuracy of the proposed algorithm are not optimal although the best average classification accuracy and worst classification accuracy are optimal compared with those of the GWO–KELM, Grid–KELM, and KELM.

Table 9.

Performance comparison of the DE–WOA–ELM, WOA–ELM, DE–ELM, PSO–ELM, and ELM (the optimal results are shown in bold)

Measure	DE–WOA–ELM	WOA–ELM	DE–ELM	PSO–ELM	ELM
Avg	0.9969	0.9912	0.9874	0.9849	0.9754
Stdv	0.0011	0.0025	0.0024	0.0017	0.0031
Best	0.9995	0.9947	0.9904	0.9875	0.9799
Worst	0.9954	0.9881	0.9838	0.9831	0.9709

DE: differential evolution; WOA: whale optimization algorithm; ELM: extreme learning machine; PSO: particle swarm optimization.

Table 10.

Performance comparison of the DE–WOA–ELM, GWO–LS–SVM, LS–SVM, GA–SVM, Grid–SVM, PSO–SVM, and SVM (the optimal results are shown in bold)

Measure	DE–WOA–ELM	GWO–LS–SVM	LS–SVM	GA–SVM	Grid–SVM	PSO–SVM	SVM
Avg	0.9969	0.9705	0.9262	0.9017	0.9352	0.9050	0.8851
Stdv	0.0011	0.0045	0.0091	0.0071	0.0058	0.0086	1.0268
Best	0.9995	0.9804	0.9388	0.9132	0.9447	0.9202	0.8969
Worst	0.9954	0.9645	0.9113	0.8896	0.9235	0.8928	0.8663

DE: differential evolution; WOA: whale optimization algorithm; ELM: extreme learning machine; GWO: gray wolf optimizer; LS–SVM: least squares support vector machine; GA: genetic algorithm; PSO: particle swarm optimization.

Table 11.

Performance comparison of the DE–WOA–ELM, T–S fuzzy neural network, LM–BP, and BP (the optimal results are shown in bold)

Measure	DE–WOA–ELM	T–S fuzzy neural network	LM–BP	BP
Avg	0.9969	0.9788	0.9699	0.8758
Stdv	0.0011	0.0213	0.0078	0.0158
Best	0.9995	0.9910	0.9833	0.8986
Worst	0.9954	0.9211	0.9588	0.8489

DE: differential evolution; WOA: whale optimization algorithm; ELM: extreme learning machine; T–S: Takagi–Sugeno; LM–BP: Levenberg–Marquardt back propagation.

Table 12.

Performance comparison of the DE–WOA–ELM, GWO–KELM, Grid–KELM, and KELM (the optimal results are shown in bold)

Measure	DE–WOA–ELM	GWO–KELM	Grid–KELM	KELM
Avg	0.9969	0.99563	0.99561	0.9922
Stdv	0.0011	0.0043	0.0008	0.0023
Best	0.9995	1	0.9975	0.9969
Worst	0.9954	0.9869	0.9947	0.9877

DE: differential evolution; WOA: whale optimization algorithm; ELM: extreme learning machine; GWO: gray wolf optimizer; KELM: kernel extreme learning machine.

Sensitivity, stability, and significance analyses of parameters

To determine the optimal parameters of the proposed algorithm and examine its performance, the influence of the number of hidden layer nodes on the algorithm classification accuracy was investigated and the stability and significance of the parameters were analyzed.

Evaluation of the WOA parameters

To determine the influence of the standard parameters of the WOA in the DE–WOA–ELM, different values of the main parameters in the WOA algorithm (population number (m) and maximum number of iterations (t) were used under the criterion that the number of ELM hidden layer nodes is 100) and the vector in Equation (10) were extensively tested on the data. To test different combinations of these parameters, m and t were allowed to take five different values (10, 20, 30, 40, or 50) and seven different values (20, 35, 50, 65, 80, 90, and 100), respectively. The last parameter (vector a) varies linearly in three different intervals: [1, 0], [2, 0], and [4, 0].

The dataset was tested by simultaneously changing the values of m, t, and a to illustrate the effect of these parameters on the performance of the WOA, and 105 parameter combinations were calculated for the dataset. For each set of parameters in the dataset, the algorithm used a 10-fold cross-validation to calculate the average accuracy and reliably compare the results. Figures 8–10 show the corresponding performance indicators when the parameters (m, t, and vector a) in the dataset were changed. In Figures 8–10, the bubble size represents the level of average accuracy. The evaluation of the WOA parameters on the dataset shows that 100 iterations, in most cases, are sufficient to obtain the optimal result. When a is in [2, 0], the average accuracy is more stable than when it is in [1, 0] and [4, 0]. Moreover, when the population is 20, the average classification accuracy is the highest.

Figure 8.

Change in average accuracy with respect to the changes in the main controlling parameters of the proposed whale optimization algorithm-based method in the case of $\vec{a}$ in [2, 0].

Figure 9.

Change in average accuracy with respect to the changes in the main controlling parameters of the proposed whale optimization algorithm-based method in the case of $\vec{a}$ in [1, 0].

Figure 10.

Change in average accuracy with respect to the changes in the main controlling parameters of the proposed whale optimization algorithm-based method in the case of $\vec{a}$ in [4, 0].

The choice of activation function in the ELM

To minimize the effect of the activation function on the classification accuracy in the ELM, the average classification accuracy of different activation functions was compared in 10 groups of experiments. The results are presented in Table 13. The sigmoid function produced the most accurate results and thus was used as the activation function in the ELM.

Table 13.

The average classification accuracy of different activation functions

Number	Activation function
Number	Sigmoid	Sine	Hardlim	Sign	Gaussian
1	0.96403	0.95238	0.90062	0.89241	0.89362
2	0.98171	0.94611	0.91026	0.87755	0.91391
3	0.98160	0.95205	0.84416	0.88158	0.92969
4	0.96250	0.95480	0.95238	0.88571	0.90728
5	0.96250	0.92105	0.89474	0.88439	0.89024
6	0.97368	0.95152	0.89726	0.92258	0.91975
7	0.95906	0.95954	0.89032	0.87429	0.90798
8	0.93571	0.94340	0.91667	0.89313	0.89542
9	0.98726	0.96178	0.94737	0.94366	0.90341
10	0.97590	0.96000	0.93491	0.87179	0.94375

Effect of the number of hidden nodes in the ELM on the accuracy of algorithm classification

There is no established rule on setting the number of hidden neurons in SLFNs. To investigate the effect of the number of hidden layer nodes in the DE–WOA–ELM on its performance, a simulation was performed for 5–50 neurons (5 neurons were added each time). After searching for the optimal weight in [5, 100], the range was extended by increasing the number of neurons starting from 100 to 500, adding 100 neurons each time. The main reason for this increase was to use larger networks to test the performance of the algorithms.

Each algorithm was tested in the experiment using different numbers of hidden neurons, and then the best performance and the number of neurons in that performance were reported. Figure 11 and Table 14 show the relationship between the average accuracy of five algorithms and the number of hidden neurons after performing 10-fold cross-validations of the data. The average accuracy of five algorithms improved as the number of neurons increased. In addition, by increasing the number of neurons to several hundreds, the average accuracy of the other algorithms was reduced. The WOA-optimized training models, including the WOA–ELM and DE–WOA–ELM, showed a better performance; in particular, the DE–WOA–ELM more consistently had high accuracy (up to 99.57%).

Figure 11.

Relationship between average accuracy and the number of hidden neurons: (a) number of hidden nodes in [5, 50]; (b) number of hidden nodes in [100, 500]. DE: differential evolution; WOA: whale optimization algorithm; ELM: extreme learning machine; PSO: particle swarm optimization.

Figure 12.

Classification accuracy distribution of the DE–WOA–ELM, WOA–ELM, DE–ELM, PSO–ELM, and ELM. DE: differential evolution; WOA: whale optimization algorithm; ELM: extreme learning machine; PSO: particle swarm optimization.

Figure 13.

Classification accuracy distribution of DE–WOA–ELM, GA–SVM, Grid–SVM, PSO–SVM, SVM, LS–SVM, and GWO–LS–SVM. DE: differential evolution; WOA: whale optimization algorithm; ELM: extreme learning machine; GA: genetic algorithm; SVM: support vector machine; PSO: particle swarm optimization; GWO: gray wolf optimizer; LS–SVM: least squares support vector machine.

Figure 14.

Classification accuracy distribution of the DE–WOA–ELM, T–S, LM–BP, and BP. DE: differential evolution; WOA: whale optimization algorithm; ELM: extreme learning machine; T–S: Takagi–Sugeno; LM–BP: Levenberg–Marquardt back propagation.

Table 14.

Average accuracy of different algorithms with different numbers of neurons

Number of neurons	Average accuracy
Number of neurons	DE–WOA–ELM	WOA–ELM	DE–ELM	PSO–ELM	ELM
5	0.9585	0.9418	0.8974	0.5418	0.7902
10	0.9730	0.9561	0.9416	0.7462	0.8683
15	0.9798	0.9684	0.9588	0.7462	0.9163
20	0.9832	0.9752	0.9673	0.9537	0.9346
25	0.9846	0.9764	0.9706	0.9659	0.9465
30	0.9872	0.9785	0.9713	0.9672	0.9545
35	0.9910	0.9815	0.9719	0.9695	0.9556
40	0.9906	0.9817	0.9743	0.9709	0.9556
45	0.9902	0.9847	0.9763	0.9699	0.9585
50	0.9914	0.9865	0.9741	0.9733	0.9628
100	0.9958	0.9907	0.9860	0.9849	0.9746
200	0.9967	0.9908	0.9874	0.9850	0.9754
300	0.99656	0.9912	0.9764	0.9696	0.9584
400	0.9964	0.9905	0.9554	0.9176	0.9346
500	0.9969	0.9904	0.9452	0.8882	0.9183

DE: differential evolution; WOA: whale optimization algorithm; ELM: extreme learning machine; PSO: particle swarm optimization.

The fact that the DE–WOA–ELM has a higher average accuracy than the other algorithms with the same number of neurons (Table 17) not only improves the accuracy of the network, but also reduces its size. The DE–WOA–ELM adopts a more compact network; that is, a higher classification accuracy is obtained with a smaller number of hidden layers and therefore the proposed algorithm is superior to the other algorithms. The DE–WOA–ELM uses the ELM to train the SLFN in the dataset with only five hidden nodes, and the average accuracy is approximately 94.19%, whereas the conventional ELM has an accuracy of 93.46% with 20 hidden nodes in the SLFN.

Table 15.

Comparison of p-values in the F-test ( $p \geq 0.05$ are underlined)

Comparison algorithm p-values
WOA–ELM 0.000887102	DE–ELM 0.023287378	PSO–ELM 0.191667732	ELM 0.003325246
GA–SVM 3.69776E-06	PSO–SVM 7.4011E-07	Grid–SVM 2.07701E-05	SVM 1.52499E-07
LS–SVM 4.43341E-07	T–S fuzzy neural network 2.31601E-10	LM–BP 1.63301E-06	BP 3.29095E-09
GWO–LS–SVM 0.000188777	GWO–KELM 0.000255526	Grid–KELM 0.480040601	KELM 0.029984518

DE: differential evolution; WOA: whale optimization algorithm; ELM: extreme learning machine; PSO: particle swarm optimization; GA: genetic algorithm; SVM: support vector machine; LS–SVM: least squares support vector machine; T–S: Takagi–Sugeno; LM–BP: Levenberg–Marquardt back propagation; GWO: gray wolf optimizer; KELM: kernel extreme learning machine.

Table 16.

Comparison of p-values in the T-test with equal variance hypothesis ( $p \leq 0.05$ is underlined)

Algorithm	Comparison algorithm
DE–WOA–ELM	p-values
	PSO–ELM 0.015643123	Grid–KELM 0.98780759

DE: differential evolution; WOA: whale optimization algorithm; ELM: extreme learning machine; PSO: particle swarm optimization; KELM: kernel extreme learning machine.

Effect of fabric color on the classification of color difference

To verify the effect of fabric color on the proposed algorithm, four fabrics of different colors were randomly selected and numbered as (a)–(d), as shown in Figure 18. The classification accuracy of these four colors was tested separately, and the results are presented in Table 18. When the colors of the fabrics are different, the proposed DE–WOA–ELM algorithm has the same average classification accuracy under the same conditions, which demonstrates that the algorithm is independent of the fabric color.

Figure 18.

Different colors of fabrics. (Color online only.)

Table 17.

Comparison of p-values in the T-test with heteroscedastic hypothesis ( $p \leq 0.05$ are underlined)

Algorithm DE–WOA–ELM	Comparison algorithm
Algorithm DE–WOA–ELM	p-values
	WOA–ELM 2.51453E-05	DE–ELM 0.00286074	ELM 8.16047E-07
	GA–SVM 6.87345E-13	PSO–SVM 9.27069E-08	Grid–SVM 1.52608E-09	SVM 0.000185134
	LS–SVM 3.92561E-07	T–S fuzzy neural network 1.22543E-08	LM–BP 0.242964847	BP 2.58596E-10
	GWO–LS–SVM 3.34206E-09	GWO–KELM 1.93903E-10	KELM 0.000958066

Table 18.

Average classification accuracy of fabrics with different colors

	Different color numbers
Algorithm	a	b	c	d
DE–WOA–ELM	0.996618584	0.996563947	0.992883214	0.995401868

DE: differential evolution; WOA: whale optimization algorithm; ELM: extreme learning machine.

Stability analysis of the proposed algorithm

The stability of the proposed algorithm was validated by analyzing the box plot and the impact of the number of iterations.

Analysis of the box plot

Figures 12–15 show the distribution of the classification accuracy of each algorithm. Each box in the block diagram represents 10 runs of each algorithm used in the experiment. The median of each box plot and the spacing of the upper and lower four digits show that the precision of the DE–WOA–ELM is very symmetrical and compact, which indicates that it is more stable than the other algorithms.

Impact of the number of iterations on the algorithm

Figure 15.

Classification accuracy distribution of the DE–WOA–ELM, KELM, Grid–KELM, and GWO–KELM. DE: differential evolution; WOA: whale optimization algorithm; ELM: extreme learning machine; KELM: kernel extreme learning machine; GWO: gray wolf optimizer.

The convergence curves of the DE–WOA–ELM and other algorithms are shown in Figures 16 and 17. Note that the y-axis represents the average values of the best solutions obtained in each iteration of 30 runs. Figure 17 shows that the model based on the WOA optimization converges to the optimal value after it exceeds 30 iterations because the WOA does not find a suitable solution to use in the initial iteration phase when it avoids the local optimum. Therefore, this algorithm searches for a suitable solution to converge in the search space. However, for the initial population optimized by DE, the DE–WOA–ELM converges in less than 10 iterations to the optimal result. The other algorithms converge slowly or very poorly.

Significance analysis

Student's t-test (T-test) was used to detect whether there is a significant difference between the proposed algorithm and the other algorithms (Equation (18))

t = \frac{\bar{x_{1}} - \bar{x_{2}}}{\sqrt{\frac{δ_{x_{1}}^{2} + δ_{x_{2}}^{2} - 2 γ δ_{x_{1}} δ_{x_{2}}}{n - 1}}}

(18)

The T-test analyzes the significance of the mean difference between two samples. To use the T-test, the variances of two samples should be equal. Therefore, the joint hypotheses test (F-test) was performed before the T-test, as shown in Equations (19) and (20)

S^{2} = \frac{\sum (\bar{X_{1}} - \bar{X_{2}}) 2}{n - 1}

(19)

F = \frac{S_{1}^{2}}{S_{2}^{2}}

(20)

Figure 16.

Convergence curves of the DE–WOA–ELM, PSO–SVM, GA–SVM, and GWO–LS–SVM. DE: differential evolution; WOA: whale optimization algorithm; ELM: extreme learning machine; GA: genetic algorithm; SVM: support vector machine; PSO: particle swarm optimization; GWO: gray wolf optimizer; LS–SVM: least squares support vector machine.

Figure 17.

Convergence curves of the DE–WOA–ELM, PSO–ELM, WOA–ELM, and DE–ELM. DE: differential evolution; WOA: whale optimization algorithm; ELM: extreme learning machine; PSO: particle swarm optimization.

Tables 15–17 present the results of the F-test and T-test for the equal variance hypothesis and the heteroscedasticity hypothesis, respectively. Table 15 shows that $p > 0.05$ between the DE–WOA–ELM and both the PSO–ELM and Grid–KELM, which is in accordance with variance homogeneity. In contrast, the p-values between the DE–WOA–ELM and WOA–ELM, DE–ELM, ELM, GA–SVM, PSO–SVM, Grid–SVM, SVM, LS–SVM, T–S fuzzy neural network, LM–BP, BP, GWO–LS–SVM, GWO–KELM, Grid–KELM, and KELM do not meet the variance homogeneity condition. The T-test was used with the equal variance hypothesis, and the Grid–KELM showed a significant difference, as indicated in Table 16. In the case that the variance homogeneity was not consistent, the heteroscedastic T-test was used. The WOA–ELM, DE–ELM, ELM, GA–SVM, PSO–SVM, Grid–SVM, SVM, LS–SVM, T–S fuzzy neural network, BP, GWO–LS–SVM, GWO–KELM, and KELM showed significant differences, as indicated in Table 17.

Conclusion

A single hidden layer feedforward network training method based on the optimization of the ELM of the DE improved WOA was proposed. The main conclusions are as follows.

DE was used to produce quality populations for the WOA. The proposed DE-WOA method can adjust its parameters adaptively to prevent premature convergence in the iteration stage. In addition, it can prevent the problem of slow convergence at the end of the iteration stage.

The proposed DE–WOA–ELM model used the WOA to optimize the input weight and hidden layer bias. The Moore–Penrose generalized inverse model was used to determine the output weight. The proposed model was used to solve the problem that the original ELM randomly generates input weights and hidden layer bias.

The performance of the DE–WOA–ELM was evaluated using a dataset obtained from printed and dyed products as the experimental material. Statistical significance analysis using the T-test and F-test was performed on the proposed algorithm and its competitors.

Compared with the SVM technology and optimized SVM, neural networks, the conventional ELM, PSO–ELM, DE–ELM, KELM, and optimized KELM algorithms, the proposed DE–WOA–ELM is superior and its accuracy is extremely high. Furthermore, the proposed DE–WOA–ELM reduces the size of the network and is more stable.

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: This work is supported by NSFC-Zhejiang Joint Fund for the Integration of Industrialization and Informatization (No.U1609205), the Zhejiang Provincial Natural Science Foundation of China (No. LY18F030018), the Science Foundation of Zhejiang Sci-Tech University (No. 18032232-Y), and the Zhejiang Top Priority Discipline of Textile Science and Engineering.

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