A novel hybrid model using the rotation forest-based differential evolution online sequential extreme learning machine for illumination correction of dyed fabrics

Abstract

This study proposes an ensemble differential evolution online sequential extreme learning machine (DE-OSELM) for textile image illumination correction based on the rotation forest framework. The DE-OSELM solves the inaccuracy and long training time problems associated with traditional illumination correction algorithms. First, the Grey–Edge framework is used to extract the low-dimensional and efficient image features as online sequential extreme learning machine (OSELM) input vectors to improve the training and learning speed of the OSELM. Since the input weight and hidden-layer bias of OSELMs are randomly obtained, the OSELM algorithm has poor prediction accuracy and low robustness. To overcome this shortcoming, a differential evolution algorithm that has the advantages of good global search ability and robustness is used to optimize the input weight and hidden-layer bias of the DE-OSELM. To further improve the generalization ability and robustness of the illumination correction model, the rotation forest algorithm is used as the ensemble framework, and the DE-OSELM is used as the base learner to replace the regression tree algorithm in the original rotation forest algorithm. Then, the obtained multiple different DE-OSELM learners are aggregated to establish the prediction model. The experimental results show that compared with the textile color correction algorithm based on the support vector regression and extreme learning machine algorithms, the ensemble illumination correction method achieves high prediction accuracy, strong robustness, and good generalization ability.

Keywords

illumination correction rotation forest online sequential extreme learning machine differential evolution

With the development of artificial intelligence, computer vision, and image processing,^1,2 the color-difference detection of textiles based on computer vision has become an important research topic in the textile printing and dyeing industry. Therefore, in the color-difference detection system,³ to remove the influence of scene illumination, we must establish an accurate and effective illumination correction model as a pretreatment step of the color-difference detection system.

Different light sources result in different colors on the same object surface, and the illumination correction of a textile image is a very important part of the color-difference detection system. The illumination correction of the image is also called the color constant calculation or illumination estimation. The purpose of color constancy is to eliminate the influence of the illumination of the scene so that a computer can have the same color constancy as the human visual system.⁴ For an image taken under an unknown illumination, the first step is to estimate the illumination value of the image. Then, according to the estimated value of illumination, the image is corrected to the standard illumination; therefore, regardless of the image illumination, the same color will appear. Color constancy refers to the genre of techniques that aim to recognize the color of objects invariant of the color of the light source. Generally, it can be divided into a color constant calculation in single-illumination scenes or in multiple-illumination scenes.⁵ The study of color constant calculation in single-illumination scenes is to assume that the illumination is evenly distributed throughout the scene; that is, there is only one illumination in the entire scene. In past decades, there has been significant progress in color constancy algorithms that can be roughly divided into two major approaches: unsupervised and supervised.⁶

Literature review

The unsupervised color constancy algorithm is used to estimate the color of an image using the color feature of the image itself without relying on other prior knowledge. Unsupervised color constancy algorithms mainly include the max-RGB algorithm,⁷ the Grey–World algorithm,⁸ and the shades of grey (SoG) algorithm.⁹ Weijer et al.¹⁰ considered the color derivative distribution in the opposing color space (opponent color space) image and proposed a Grey–Edge hypothesis: the average of the reflectance differences in a scene is achromatic. Based on the Grey–Edge hypothesis, Weijer et al.¹⁰ proposed a unified framework for color constancy that included max-RGB, Grey–World, and SoG algorithms and extended the color constancy calculation to the higher-order derivative space of the image. In addition, Celik and Yetgin¹¹ proposed a color constancy algorithm called the Grey Wavelet that achieved good results by unifying the Grey–World and Grey–Edge color constancy algorithms. In general, the unsupervised color constancy algorithm has the advantage that the algorithm is simple and easy to calculate; however, its shortcoming is that the performance of the algorithm is heavily dependent on specific assumptions. For example, the max-RGB algorithm requires that the surface of the object have a white area in the scene. The Grey–World algorithm requires that the corrected image has a rich color. However, in the practical application of illumination correction, these conditions are often difficult to meet and are great limitations.

The supervised color constancy algorithm predicts image color under an unknown illumination by learning which colors will appear under various light conditions. From the current experimental effects of various color constancy algorithms, supervised color constancy algorithms are generally superior to unsupervised algorithms. Supervised methods establish the relationships between the image color distribution and the illumination color values through learning methods, including Bayesian color constancy,¹² back propagation (BP),¹³ and others.^14,15 Among them, the color constancy algorithm proposed by Xiong and Funt¹⁴ based on support vector regression (SVR) is superior to other methods. However, the SVR method also has shortcomings; first, the SVR-based color constancy method independently calculates each chrominance component of an illumination so that each component may be optimal, but the estimation of the optimal entire chrominance vector cannot be guaranteed. Second, the learning speed of SVR is relatively slow. In view of these shortcomings, Li et al.¹⁵ introduced a new fast-learning algorithm, the extreme learning machine (ELM) color constancy algorithm, based on a single hidden-layer feedforward neural network. Experiments on a variety of different image data sets show that the ELM-based color constancy algorithm performs better than the SVR algorithm.

However, the traditional ELM method is a batch method. First, the learning parameters need to be selected; then, when new data arrives, the used data and the new data must be retrained. In many industrial applications, it is very common that training data can only be obtained one-by-one or by chunk (a block of data). Thus, Liang et al.¹⁶ proposed an online sequential extreme learning machine (OSELM) learning algorithm. Compared to an ELM, an OSELM can learn data one-by-one or chunk-by-chunk with a fixed or varying chunk size. It is not only 10 times faster than the SVM method, but is also applied to multiple areas, such as data fitting and classification¹⁷ and prediction.¹⁸ However, compared with the original ELM algorithm,^19,20 the OSELM usually suffers from ill-posed problems because of the random determination of the input weights and hidden biases, which lead to unstable performance and affect the robustness of the illumination correction. To optimize the two parameters in the OSELM algorithm (input weight and hidden-layer bias), an optimization algorithm is introduced. There are many optimization algorithms, but some algorithms have long solution times, such as the particle swarm optimization (PSO) algorithm.²¹ Genetic algorithms have very complex programming and unstable parameters.²² The differential evolution (DE) algorithm, a heuristic global optimization algorithm widely used in recent years, has a simple structure, fast speed, and high efficiency.²³ Bazi et al.²⁴ proposed the use of the DE algorithm to select the input weight and hidden-layer bias of an ELM, and the effectiveness of the algorithm was verified by a large number of experiments. Therefore, this study uses the DE algorithm to optimize the input weights and hidden-layer bias of the OSELM to obtain a differential evolution online sequential extreme learning machine (DE-OSELM) model with good predictive performance.

Since the single OSELM model still suffers from poor robustness, researchers have integrated ELMs to achieve better robustness and generalization. The idea of ensemble learning was initially used to solve the classification problem, and as research on ensemble learning deepened, it was gradually used to solve the problem of regression. In 2009, Lan et al.²⁵ proposed an ensemble of online sequential extreme learning machine (EOS-ELM). The ensemble system used the OSELM as the member classifier and learned the data set one-by-one or group-by-group, where the number of samples can be fixed or not fixed. The experimental results showed that the EOS-ELM was faster than other sequential learning algorithms and had a good generalization performance on many standard regression, classification, and timing prediction problems. Although the experimental results of the EOS-ELM for three different types of problems were good, this method did not deal with the original sample set during construction of the ensemble classifier system; therefore, the difference between the base classifiers did not improve. The rotation forest (RF) algorithm²⁶ is an ensemble algorithm developed on the basis of random forests. Its main characteristic is the disjoint segmentation and data set transformation of the feature set, and a decision tree is used as the base classifier to establish the ensemble classification model. The algorithm randomly separates the attribute set of the sample and transforms the attribute subset by means of linear transformation to increase the difference between subsets. It then uses the transformed subset of attributes to select the sample to train different classifiers. The RF algorithm not only provides an integrated classification algorithm but also provides an ensemble learning framework that can be used for classification and regression.^27,28

In this study, we use the RF algorithm as the ensemble framework and adopt the optimized OSELM to replace the regression tree algorithm in the original RF algorithm as the base learner. A new textile illumination correction ensemble DE-OSELM model based on the RF framework was established. The main contributions of this work are as follows.

The low-dimensional and efficient image features extracted by the Grey–Edge framework as the input vector of the OSELM improve the training and learning speed of the OSELM.

Since the input weight and hidden bias of the OSELM are randomly selected, this leads to poor prediction accuracy and poor robustness. To overcome the shortcomings of the OSELM, this study uses the DE algorithm to optimize the OSELM input weight and hidden bias and to improve the prediction accuracy of the OSELM.

In this study, the RF algorithm is used as the ensemble framework, and we adopt the optimized OSELM to replace the regression tree algorithm in the original RF algorithm as the base learner. In addition, the RF algorithm is used to integrate the different base learners (DE-OSELMs) to build a robust ensemble RF-DE-OSELM illumination correction model that enhances the generalization ability and robustness of the model.

The remainder of this paper is organized as follows. The third section provides a brief overview of the Grey–Edge algorithm, OSELM, DE algorithm, and RF algorithm. In the fourth section, the developmental steps of the hybrid prediction model and the model's application on the RF-DE-OSELM are described. A detailed description of the experiment is provided in the firth section, and the sixth section draws conclusions.

Preliminaries

In this section, in the process of establishing the textile illumination correction RF-DE-OSELM model, the Grey–Edge algorithm, OSELM, DE algorithm, and RF algorithm are utilized. The following is a brief introduction to the mathematical theory needed in the modeling process.

Grey–Edge algorithm

Most of the existing supervised color constancy algorithms use two-dimensional (2D) or three-dimensional (3D) binarized image chromaticity histograms as input vectors for learning algorithms. Although this binarized chromaticity histogram feature is widely used and achieves good results, its greatest drawback is that the feature dimension is too high. This high input vector will inevitably lead to a significant reduction in learning speed. To improve the efficiency of the algorithm, we abandon this inefficient and high-dimensional feature according to Li et al.¹⁵ and use an unsupervised color constant algorithm to propose an efficient low-dimensional feature as the input vector.

Recently, Weijer et al.¹⁰ proposed a new color constancy algorithm based on the Grey–Edge assumption that extended the low-level-image feature-based method to incorporate derivative information. Based on the introduction of a Minkowski norm and Gaussian filter, this algorithm presented a unified framework that generated various unsupervised color constancy algorithms. The framework is defined as

(\int | \frac{\partial^{n} f^{σ} (X)}{\partial X^{n}} |^{p} d X)^{1 / p} = k e^{n, p, σ}

(1)

where n is the order of the derivative, x is the spatial coordinate, and p is the Minkowski norm.²⁹

In this study, the Grey–Edge algorithm was used as the framework to generate the low-dimensional color feature used as the input data for the OSELM. Thus, its advantages are fully utilized while avoiding the problem of parameter selection.

Online sequential extreme learning machine

Compared with the ELM, the OSELM uses a sequential learning strategy instead of a batch learning strategy. The learning process of the OSELM output weight for the single hidden-layer neural network is divided into two parts: the initial stage is to obtain the output weight β through a small number of samples, and the second stage is the online learning process where the output weight β of the single hidden-layer feedforward neural network learned at the initial stage is updated by using a single sample or sample data block. In the initial stage, when there are $N_{0}$ arbitrary training samples $(X_{i}, t_{i})$ , where $X_{i} = [x_{i 1}, x_{i 2}, \dots, x_{in}]^{T} \in R^{n} and t_{i} = [t_{i 1}, t_{i 2}, \dots, t_{im}]^{T} \in R^{m}$ , it is desirable to obtain a value of $β_{0}$ that satisfies the minimum of $‖ H_{0} β - T_{0} ‖$ using the basic ELM algorithm, where

\begin{matrix} H_{0} = [\begin{matrix} g (a_{1}, b_{1}, X_{1}) & \dots g (a_{\tilde{N}}, b_{\tilde{N}}, X_{1}) \\ : & : \\ g (a_{1}, b_{1}, X_{N_{0}}) & \dots g (a_{\tilde{N}}, b_{\tilde{N}}, X_{N_{0}}) \end{matrix}]_{N_{0} \times \tilde{N}} \end{matrix}

(2)

and

\begin{matrix} T_{0} = [\begin{matrix} t_{1}^{T} \\ : \\ t_{{\tilde{N}}_{0}}^{T} \end{matrix}]_{N_{0} \times m} \end{matrix}

(3)

According to the generalized inverse method, we can calculate $β_{0}$

β_{0} = K_{0}^{- 1} H_{0}^{T} T_{0}

(4)

where

K_{0} = H_{0}^{T} H_{0}

. When there are

N_{1}

new samples in the model, we use the basic extreme learning algorithm to solve

β^{(1)}

satisfying the following equation

\begin{matrix} ‖ [\begin{matrix} H_{0} \\ H_{1} \end{matrix}] β - [\begin{matrix} T_{0} \\ T_{1} \end{matrix}] ‖ \end{matrix}

(5)

According to the generalized inverse method, the value of $β^{(1)}$ is

\begin{matrix} β^{(1)} = K_{1}^{- 1} [\begin{matrix} H_{0} \\ H_{1} \end{matrix}]^{T} [\begin{matrix} T_{0} \\ T_{1} \end{matrix}] \end{matrix}

(6)

where

\begin{matrix} K_{1} = [\begin{matrix} H_{0} \\ H_{1} \end{matrix}]^{T} [\begin{matrix} H_{0} \\ H_{1} \end{matrix}] \end{matrix}

(7)

For online learning, we need to denote $β^{(1)}$ as a function of $β^{(0)}$ , $K_{1}$ , $H_{1},$ and $T_{1}$ , and $K_{1}$ can be expressed as

\begin{matrix} K_{1} = [\begin{matrix} H_{0}^{T} & H_{1}^{T} \end{matrix}] [\begin{matrix} H_{0} \\ H_{1} \end{matrix}] = K_{0} + H_{1}^{T} H_{1} \end{matrix}

(8)

and

\begin{matrix} [\begin{matrix} H_{0} \\ H_{1} \end{matrix}]^{T} [\begin{matrix} T_{0} \\ T_{1} \end{matrix}] = H_{0}^{T} T_{0} + H_{1}^{T} T_{1} \\ = K_{0} K_{0}^{- 1} H_{0}^{T} T_{0} + H_{1}^{T} T_{1} \\ = K_{0} β^{(0)} + H_{1}^{T} T_{1} \\ = (K_{1} - H_{1}^{T} H_{1}) β^{(0)} + H_{1}^{T} T_{1} \\ = K_{1} β^{(0)} - H_{1}^{T} H_{1} β^{(0)} + H_{1}^{T} T_{1} \end{matrix}

(9)

It can be seen that $β^{(1)}$ is

\begin{matrix} β^{(1)} = K_{1}^{- 1} [\begin{matrix} H_{0} \\ H_{1} \end{matrix}]^{T} [\begin{matrix} T_{0} \\ T_{1} \end{matrix}] = K_{1}^{- 1} (K_{1} β^{(0)} - H_{1}^{T} H_{1} β^{(0)} + H_{1}^{T} T_{1}) \\ = K_{1}^{- 1} (K_{1} β^{(0)} - H_{1}^{T} H_{1} β^{(0)} + H_{1}^{T} T_{1}) \\ = β^{(0)} + K_{1}^{- 1} H_{1}^{T} (T_{1} - H_{1} β^{(0)}) \end{matrix}

(10)

Thus, a recursive formula for online learning can be obtained

K_{k + 1} = K_{k} + H_{k + 1}^{T} H_{k + 1}

(11)

Calculate the output weight $β^{(k + 1)}$

β^{(k + 1)} = β^{(k)} + K_{k + 1}^{- 1} H_{k + 1}^{T} (T_{k + 1} - H_{k + 1} β^{(k)})

(12)

and

K_{k + 1}^{- 1}

can be obtained from the Woodbury formula

\begin{matrix} K_{k + 1}^{- 1} = (K_{k} + H_{k + 1}^{T} H_{k + 1})^{- 1} \\ = K_{k}^{- 1} - K_{k}^{- 1} H_{k + 1}^{T} (I + H_{k + 1} K_{k}^{- 1} H_{k + 1}^{T})^{- 1} \times H_{k + 1} K_{k}^{- 1} \end{matrix}

(13)

Let $P_{k + 1} = K_{k + 1}^{- 1}$ , then (12) and (13) can be expressed as

P_{k + 1} = P_{k} - P_{k} H_{k + 1}^{T} (I + H_{k + 1} P_{k} H_{k + 1}^{T})^{- 1} \times H_{k + 1} P_{k}

(14)

β^{(k + 1)} = β^{(k)} + P_{k + 1} H_{k + 1}^{T} (T_{k + 1} - H_{k + 1} β^{(k)})

(15)

The learning process can be continuous by letting k = k + 1.

The above process is the principle of the OSELM algorithm. First, the output weight $β^{(0)}$ of the single hidden-layer feedforward neural network is initialized by fewer samples. After the initialization phase, the online sequential learning stage begins and the output weight β of the single hidden-layer feedforward neural network is adjusted by the sample data block composed of each sample or a plurality of samples.

The working principle of the OSELM can be divided into two stages¹⁶: initialization and sequential learning. In the initialization phase, $H_{0}$ is established, and the number of samples to build $H_{0}$ should be at least equal to the number of hidden-layer nodes. In most instances, the number of training data in the initialization phase should be equal to or close to the number of hidden-layer nodes. After the initialization stage, data enter the sequence learning phase and can be one or a group of a group to reach the network. One or a set of data is discarded once used and is no longer used in the next learning process.

From the derivation of the OSELM, we can see that when the rank of $H_{0}$ is equal to the number of hidden nodes, and the OSELM and ELM can achieve the same learning performance (training error and accuracy). To make the rank of $H_{0}$ equal to the number of hidden-layer nodes, the number of data initialized should not be less than the number of hidden-layer nodes.

Differential evolution

DE, like other evolutionary algorithms, is a stochastic model for simulating biological evolution. It is achieved by repeated iterations whereby individuals adapting to the environment are preserved. However, compared with the evolutionary algorithm, DE preserves the population-based global search strategy that reduces the complexity of genetic operation by using real-number coding and simple mutation based on difference and a one-to-one competitive survival strategy. At the same time, it can dynamically track the current search situation to adjust its search strategy because of its unique memory ability. It has a strong global convergence ability and robustness and does not need to use the characteristic information of the problem, which is suitable for solving optimization problems in complex environments that cannot be solved by the conventional mathematical programming method.

The DE algorithm²⁴ is mainly used to solve the global optimization ability of continuous variables. The main working steps are basically the same as other evolutionary algorithms, including mutation, crossover, and selection. The basic idea of the algorithm is to start with a randomly generated initial population and use the difference vector of two individuals randomly selected from the population as the random variation source of the third individual and then divide the difference vector according to certain rules, and the third individuals are summed to produce mutated individuals, called mutations. Then, the mutated individuals are mixed with a predetermined target individual to generate an experimental individual called a cross. If the fitness value of the experimental individual is owing to the fitness value of the target individual, the experimental individual replaces the target individual in the next generation; otherwise, the individual is saved and is called the selection. In each evolutionary process, each individual vector is taken as the target individual once. During the iterative calculation, the algorithm retains the optimal individual and eliminates the poor individual, thus guiding the search process approximate to the global optimal solution.

Rotation forest

The RF algorithm²⁶ is an ensemble learning algorithm based on principal component analysis (PCA). The algorithm uses PCA to transform the original training set to construct the base classifier and generate the training subset for each classifier. The RF strategy mainly deals with the original sample characteristics of the ensemble classifier and increases the difference between the individuals of the ensemble classifier by adopting the feature extraction transformation to obtain the new samples needed for the ensemble and to guarantee the accuracy of classification. The strategy operates as follows.

The initial sample set S (N × D), where N and D are the number of samples and the number of sample features, respectively, is given. First, the feature of the sample is randomly divided into K feature subsets. The characteristic number of each feature subset is M (M = D/K), and K samples are obtained based on the feature subset ${S_{i}} (i = 1, \dots, K)$ . If the number of features is not divisible, the remaining features are added to the K group. Then, the PCA transform method is used to transform the sample subset $S_{i}$ to obtain M eigenvectors. If the eigenvectors corresponding to the M' nonzero eigenvalues are selected to form an eigenvector matrix $a_{i} = [a_{i 1}, \dots, a_{i M^{'}}]$ , the K feature vector matrices are combined to obtain a matrix R , where $a_{i}$ is located at the ith M and M columns of R . Finally, the value and the initial position (dimension) of the eigenvector in the initial sample are determined, and each eigenvector according to the corresponding feature initial position is rearranged to form the new eigenvector matrix R*. Therefore, a new sample $S_{new} = S \cdot R^{*}$ is generated by R*.

The RF strategy is proposed for two aspects: the diversity between ensemble classifiers and the accuracy of ensemble classifiers. Researches have shown that if the number of integrated classifiers chosen is 15–25, better classification performance is obtained but the computation time increases. Therefore, the training samples of each ensemble classifier were constructed using the RF strategy, and the ensemble running time was effectively shortened by reducing the number of integrated individuals, while the accuracy of integrated classification was ensured.

Proposed method

Data representation

Since illumination correction is more concerned with the chrominance information of an image (the direction information of the RGB vector) rather than the image luminance information (the mode of the RGB vector), the chrominance information of the image is used. Converting RGB color space to r-g chromaticity space is an important role in eliminating the effect of light intensity on color. The RGB space of an image can be converted to r-g space by

\begin{matrix} {\begin{matrix} r = \frac{R}{R + G + B} \\ g = \frac{G}{R + G + B} \\ b = \frac{B}{R + G + B} \end{matrix} \end{matrix}

(16)

where the b component can be calculated by r-g. Therefore, in general, only the r-g component is used to represent the chromaticity space. In the r-g chromaticity space, the color vectors R , G , and B have the same chrominance values as kR, kG, and kB (k ≠ 0), thus eliminating the effect of light intensity on color. Second, there is an advantage to using r-g chromaticity space: the computational complexity is reduced from the 3D RGB color space to the 2D r-g chromaticity space, thereby saving time for data preprocessing during model training.

In this study, we set $n \in {0, 1, 2}$ , $p \in {1, 2, \dots, 10}$ , and $σ \in {1, 3, 5, 7, 9}$ . The different combinations can produce a total of 150 kinds of statistical results by equation (1), which can be marked as $e_{i} = e^{n, p, σ} = (R_{i}, G_{i}, B_{i})^{T}$ and $i = 1, 2, \dots, 150$ . Converting $e_{i}$ to chroma space, the chromaticity value is

c e_{i} = (r_{i}, g_{i}) = (\frac{R_{i}}{R_{i} + G_{i} + B_{i}}, \frac{G_{i}}{R_{i} + G_{i} + B_{i}})

(17)

Therefore, the input vector of the base learner OSELM is made up of $c e_{i}$ as

I_{i} = [c e_{1}, c e_{2}, \dots, c e_{150}]^{T}

(18)

In addition, the output vector is composed of the illumination chromaticity information $o e_{i}$ , which can be denoted as

o e_{i} = (r_{o}, g_{o})^{T} = (\frac{R_{o}}{R_{o} + G_{o} + B_{o}}, \frac{G_{o}}{R_{o} + G_{o} + B_{o}})^{T}

(19)

Parameter optimization for the OSELM based on the DE algorithm

Compared with the traditional ELM, the OSELM has a greater advantage in adding data by doing so one-by-one or chunk-by-chunk. It avoids the problem of retraining historical data; thus, it can save training time. However, the accuracy of the algorithm is not ideal and needs to be improved, because the parameters, such as input weight and bias, are randomly selected. To optimize these two parameters in the OSELM algorithm, an optimization algorithm the OSELM based on a DE algorithm is introduced in this study. The DE algorithm has the advantages of simple structure, fast speed, and high efficiency.

The general flowchart of the DE-OSELM algorithm is shown in Figure 1. The specific steps of the DE-OSELM illumination correction model are as follows.

Figure 1.

Online sequential extreme learning machine (OSELM) process optimized by the differential evolution (DE) algorithm.

Assume a training sample set $N_{t} = {(x_{i}, t_{i}) | x_{i} \in R^{n}, i = 1, 2, \dots, N}$ and L hidden-layer nodes.

Step 1: initialize the population. Set the number of hidden nodes L, the population number NP, the value of the cross-probability CR, the optimal range of the scaling factor F, and the maximum number of iterations $G_{max}$ .

Step 2: set the population. Randomly generate a set of initial population $G_{NP \times D}$ , where D is the dimension of population G. Select the input weight and hidden-layer bias as a member of the population to form each population.

Step 3: calculate the fitness function value. Using an individual in the initial population, the regression model is established and the training data is used to predict the regression and the fitness function is calculated. Using the fitness function to measure the merits of an individual, the fitness function is defined as the mean absolute error, $f = MAPE$ , as

{min}_{f} = \sum_{i = 1}^{N} \frac{| a_{i} - p_{i} |}{a_{i}} / N * 100 %

(20)

where N is the number of training samples,

a_{i}

is the true value in the training data, and

p_{i}

is the predicted value. It can be seen from the formula that if the fitness function value is smaller, an individual in the population has larger competitiveness and is more able to survive into the next generation. Therefore, according to the value of f, the individual having the smallest fitness value is selected and the global optimal

x_{best}

is preserved.

Step 4: test the termination condition. If the fitness function value $f (x_{best})$ is within 0–0.001 or the evolutionary algebra G reaches the maximum value $G_{max}$ , go to Step 8, otherwise, proceed to the next step.

Step 5: mutation. For each parameter, four different random integers $r_{1}$ , $r_{2}$ , $r_{3}$ , and $r_{4}$ are generated. Perform the following mutations to generate new individuals

v_{i, G + 1} = x_{best} + F [(x_{r_{1}, G} - x_{r_{2}, G}) + (x_{r_{3}, G} - x_{r_{4}, G})]

(21)

If the parameters in the generated new individual exceed their specified range, make the following adjustments

x_{new} = x_{min} + (x_{max} - x_{min}) * rand (1)

(22)

where

x_{new}

is the adjusted mutation parameter and

x_{max}

and

x_{min}

are the maximum and minimum values, respectively, of the parameters.

Step 6: crossover. According to the crossover principle of DE, the individuals after mutation are crossed to improve the population diversity; that is, the parameters are more diverse.

Step 7: selection. The regression model is established again according to each new individual in the new generation population; that is, each group of new parameters. According to the fitness function, the new optimal population $u_{best, G + 1}$ in the new population is obtained. The local optimal individual $f (u_{best, G + 1})$ is compared with the fitness function value $f (x_{best})$ of the global optimal individual. If it is smaller, the global optimal is substituted as

\begin{matrix} x_{best} = {\begin{matrix} u_{best, G + 1} & f (u_{best, G + 1}) \leq f (x_{best}) \\ x_{best} & f (u_{best, G + 1}) > f (x_{best}) \end{matrix} \end{matrix}

(23)

Step 8: obtain the optimal input weight and hidden-layer bias using the optimization algorithm for the entire data. Then input the data into equation (11), and the hidden-layer output weight matrix is obtained.

Hybrid forecasting model based on the RF-DE-OSELM

The idea of ensemble learning was used to solve the problem of classification, and recently it has gradually been used to solve the problem of regression. The RF algorithm is an algorithm for training ensemble classifiers based on the PCA, proposed by Rodriguez et al.²⁶ Zhang et al.²⁷ extended the RF ensemble classification algorithm to solve the regression problem and adopted the regression tree as the base learning algorithm. However, as for the learning process of the regression tree, the complexity of the learner results led to over-fitting problems. In addition, the OSELM has better self-adaptability; therefore, it has greater advantages in solving over-fitting problems. However, the generalization ability of a single OSELM algorithm in textile image illumination estimation is poor. In this study, we propose a new ensemble algorithm, an ensemble DE-OSELM algorithm based on the RF algorithm.

The RF-DE-OSELM algorithm first uses the original RF algorithm to deal with the training set of each learner to improve the diversity of each. Then, the DE-optimized OSELM model is used instead of the regression tree as the base learner to solve the over-fitting problem of the original algorithm and improve the prediction accuracy. Finally, the outputs of the base learners are combined with the average method to obtain the final illumination estimation model. The use of the average method also prevents the learning process from over-fitting so as to improve the prediction effect of the algorithm and enhance its generalization ability. The main flow of the RF-DE-OSELM algorithm is shown in Figure 2. The implementation process of the illumination correction model based on the RF-DE-OSELM can be described as follows.

Step 1: suppose that the given n-dimensional sample set S : ${(x_{i} y_{i})} i = 1 N$ , where F represents the set of attributes of the sample. T is the number of base learner DE-OSELMs in the ensemble algorithm, K is the number of subsets, and the maximum number of iterations is L.

Step 2: $D_{1}, D_{2}, \dots, D_{L}$ are the base learners for prediction. $H_{ij}$ represents the subset of the jth feature corresponding to the training set used by the base learner $D_{i}$ , where $1 \leq i \leq L and 1 \leq j \leq K$ . The sample set is randomly sampled m times to form the sample subset $Z_{i}$ ; $m = n / K$ ; $Z_{ij}$ denotes the subset of samples corresponding to $H_{ij}$ . Select a function for $Z_{ij}$ to perform the PCA and arrange its feature vector to produce a new coefficient matrix $C_{ij}$ .

Step 3: repeat the above steps K times. For each $Z_{i}$ , a coefficient matrix is produced by PCA.

Step 4: combine the resulting coefficient matrices into a large sparse matrix to generate the rotation matrix $R_{i}$ for the base learner $D_{i}$

\begin{matrix} R_{i} = [\begin{matrix} \begin{matrix} \begin{matrix} C_{i 1} & 0 \end{matrix} \\ \begin{matrix} 0 & C_{i 2} \end{matrix} \end{matrix} & \begin{matrix} \dots & 0 \\ \dots & 0 \end{matrix} \\ \begin{matrix} \begin{matrix} : & : \end{matrix} \\ \begin{matrix} 0 & 0 \end{matrix} \end{matrix} & \begin{matrix} : & : \\ \dots & C_{ik} \end{matrix} \end{matrix}] \end{matrix}

(24)

The learning set used by learner

D_{i}

Z R_{i}

Step 5: train the DE-OSELM-based learner on the transformed data set $Z R_{i}$ and repeat the above steps to obtain the base learner DE-OSELMs.

Step 6: use the average method to combine the output results of the ensemble learning machine to obtain the final illumination estimation model.

Figure 2.

Ensemble rotation forest differential evolution online sequential extreme learning machine (RF-DE-OSELM) framework. PCA: principal component analysis.

Experiments

Experimental conditions

In this study, a SONY SSC-DC398BP high-resolution industrial camera with an SSV0358GNB industrial lens was used to capture fabric images; the distance between the lens and the cloth surface was 0.7 m. It is important for different industries to choose the correct light source for color-difference measurement because of the different characteristics of various light sources. The material of this study was cotton and polyester finished fabric, so we selected the most commonly used lighting sources in the textile industry—D65, A, D50, and D55. We collected 500 polyester and cotton solid-color finished-fabric dyeing images with the industrial camera. All image sets were randomly assigned; 450 were used as a training image set, and the remaining 50 images as a test set. All tests were processed on a computer with the Windows 7 operating system (OS); the hardware configuration included an Intel core i3 processor clocked at 2.3 GHz with 4-Gb memory. In this study, the corresponding algorithms were developed on the PC using MATLAB R2012b simulation optimization and then transplanted to the digital signal processing system to achieve miniaturization and the convenience of a color-difference detection system. A schematic diagram of the acquisition process is given in Figure 3. The libsvm toolbox provided by Chang and Lin³⁰ was used for the SVR algorithm.

Figure 3.

Block diagram of the acquisition process.

Measurement criteria

In this section, we evaluate and analyze the illumination estimation algorithm mainly using the chromaticity error and angle error to judge the performance of each algorithm. In this case, the chromaticity error and the angle error¹⁵ were used to measure the difference between the real illumination and the predicted illumination. We calculated the root mean square error (RMSE), the maximum (MAX) error, and the median (MED) error of each algorithm as the evaluation index for each model.

Chromaticity error Chromaticity error is an important performance evaluation index of the lighting correction algorithm. The color $(R, G, B)$ space is projected onto the plane $R + G + B = 1$ to eliminate the influence of the color intensity to obtain a chromaticity normalized color space $(r, g, b)$

\begin{matrix} {\begin{matrix} r = \frac{R}{R + G + B} \\ g = \frac{R}{R + G + B} \\ b = 1 - r - g \end{matrix} \end{matrix}

(25)

It can be seen from equation (25) that chromaticity b = 1-r-g is redundant information; therefore, we generally used the (r, g) 2D chromaticity value. Assuming that the true colorimetric value of the image is

c_{i} = (r_{i}, g_{i})

, the estimated chromaticity of the image is

c_{j} = (r_{j}, g_{j})

, then the Euclidean distance of chromaticity will be the chromaticity error

E_{l}

E_{l} = \sqrt{(r_{i} - r_{j})^{2} + (g_{i} - g_{j})^{2}}

(26)

Angle error The illumination chromaticity is essentially a 3D vector, and for the illumination estimation algorithm, we are concerned with the accuracy of the illumination estimation and not the light intensity. Therefore, for an image, only the error measurement of illumination estimation needs to be calculated by the algorithm. The estimated illumination chromaticity is $e_{i} = (R_{i}, G_{i}, B_{i})$ , the true illumination chromaticity is $e_{j} = (R_{j}, G_{j}, B_{j})$ , and the angle error is $E_{a}$

E_{a} = {cos}^{- 1} (\frac{e_{i} \cdot e_{j}}{\sqrt{R_{i}^{2} + G_{i}^{2} + B_{i}^{2}} \times \sqrt{R_{j}^{2} + G_{j}^{2} + B_{j}^{2}}}) \frac{\times^{360} \circ}{2 π}

(27)

where

{cos}^{- 1}

denotes the inverse cosine function and

e_{i} \cdot e_{j}

denotes the inner product of two vectors.

The closer the two illumination chromaticities, the smaller the angle difference, which means that the estimated illumination chromaticity is closer to the illumination chromaticity of the true scene; when $E_{a} = 0$ , the two are exactly the same. When the value of $E_{a}$ is larger, the difference between the estimated illumination chromaticity and the real scene illumination chromaticity is larger. If there are M images in the image test set, where the angular error of the kth image is $E_{i} (k)$ , the RMSE is defined as

RMSE = \sqrt{\frac{1}{M} \sum_{i = 1}^{M} E_{i}^{2} (k)}

(28)

Parameter setting

To verify the performance of the RF-DE-OSELM illumination estimation model, comparative tests of prediction error and stability were performed with the single OSELM, DE-OSELM, and RF-OSELM models and the SVR and ELM traditional illumination estimation models. The parameters of these models were set as follows.

ELM illumination estimation model: the number of hidden-layer nodes (L) was 100 by cross-validation, and the Gaussian radial basis function (RBF) $G (a, b, x) = exp (‖ - x - a ‖^{2} / b)$ and the sigmoid additive $G (a, b, x) = 1 / (1 + exp (- (a \cdot x + b)))$ were selected as the activation functions of the hidden-layer node.

SVR illumination estimation model: RBF was selected as the kernel function, as used by Xiong and Funt.¹⁴

OSELM illumination estimation model: the number of hidden-layer nodes was 5, 10, 15, 20, and 25; the number of training data in the initialization phase $N_{0} = L + 25$ , the size of learning data for each step in the learning phase block was 40, and the Gaussian radial basis function (RBF) $G (a, b, x) = exp (- ‖ x - a ‖^{2} / b)$ and the sigmoid additive $G (a, b, x) = 1 / (1 + exp (- (a \cdot x + b)))$ were selected as the activation functions of the hidden-layer node.

DE-OSELM illumination estimation model: the number of hidden-layer nodes was 5, 10, 15, 20, and 25; the range of input weight of the hidden layer was [−1,1], the range of bias of the hidden layer was [0,1], the number of training data in the initialization phase $N_{0} = L + 25$ , the size of learning data for each step in the learning phase block was 40, and the Gaussian radial basis function (RBF) $G (a, b, x) = exp (- ‖ x - a ‖^{2} / b)$ and the sigmoid additive $G (a, b, x) = 1 / (1 + exp (- (a \cdot x + b)))$ were selected as the activation function of the hidden-layer node. The parameters in the DE algorithm were NP = 30, F = 0.5, and CR = 0.75.

RF-OSELM illumination estimation model: the number of hidden-layer nodes was 5, 10, 15, 20, and 25; the number of training data in the initialization phase $N_{0} = L + 25$ , the size of learning data for each step in the learning phase block was 40, and the Gaussian radial basis function (RBF) $G (a, b, x) = exp (- ‖ x - a ‖^{2} / b)$ and the sigmoid additive $G (a, b, x) = 1 / (1 + exp (- (a \cdot x + b)))$ were selected as the activation function of the hidden-layer node. The number of sub-features selected for constructing the rotation matrix was fixed at three.

RF-DE-OSELM illumination estimation model: the number of hidden-layer nodes was 5, 10, 15, 20, and 25; the number of training data in the initialization phase $N_{0} = L + 25$ , the size of learning data for each step in the learning phase block was 40, and the Gaussian radial basis function (RBF) $G (a, b, x) = exp (- ‖ x - a ‖^{2} / b)$ and the sigmoid additive $G (a, b, x) = 1 / (1 + exp (- (a \cdot x + b)))$ were selected as the activation function of the hidden-layer node. The number of sub-features selected for constructing the rotation matrix was fixed at three. The parameters in the DE algorithm were NP = 30, F = 0.5, and CR = 0.75.

Simulation results and discussion

In this section, the performance of the proposed method is verified by comparing it to other illumination correction models. To verify the performance of the RF-DE-OSELM, 50 experiments were repeated on the textile image data set. The error of each experiment was derived from equations (26) and (27). All experiments were carried out on 500 polyester dyeing fabric images. The average values of RMSE, MAX error, and MED error of the 50 experiments were taken as the evaluation indexes of the performance of the various illumination correction algorithms.

For the problem of optimizing OSELM parameters, when grid research is used to select the parameters, the parameters can only grow in an exponential form of the grid width, which has poor regulation and is greatly influenced by human factors. At present, the most commonly used intelligent optimization algorithms are the DE algorithm and the PSO algorithm. Therefore, we first compare the DE algorithm and the PSO algorithm for OSELM parameter optimization. Figures 4 and 5 show the process of optimizing the parameters of the training set using the PSO and DE algorithms.

Figure 4.

Parameter optimization curve of OSELM for particle swarm optimization (PSO). MSE: Mean Square Error.

Figure 5.

Parameter optimization curve of OSELM for differential evolution (DE). MSE: Mean Square Error.

Figures 4 and 5 describe the precision transformation process of the prediction accuracy of the PSO-OSELM and DE-OSELM on the image data set with the number of iterations. As can be seen from Figure 4, the convergence rate of the PSO algorithm is very slow during the iteration, and the algorithm converges to the optimal solution at the 95th iteration. From Figure 5, we can see that the DE algorithm converges rapidly from the start and converges to the optimal solution at the 60th iteration, and the prediction accuracy remains unchanged until the algorithm stops.

Comparing Figures 5 and 4, we can see that when optimizing the OSELM parameters, the performance of the DE model is better than that of the PSO model. DE not only allows the OSELM to obtain a higher solution on the data set but also converges faster than PSO. This is mainly because the realization mechanism of the PSO algorithm lacks variation ability; it only uses the global optimal particle and individual optimal particle to change the velocity of each particle. The particles gradually approach the optimal solution; therefore, it is easy to converge to the local optimal solution, resulting in poor prediction accuracy. The DE model preserves the global search strategy of the population. It can effectively avoid the population falling into the local optimal solution, which allows the OSELM to obtain better parameters and improves the prediction accuracy of the model. Moreover, the DE adopted differential mutation operation, and one-to-one competitive survival strategy reduced the complexity of the genetic operation and greatly reduced the algorithm running time. Therefore, the DE algorithm is used to optimize the OSELM parameters and make full use of its optimization process, which is simple and easy to control; the iterative speed of convergence is fast, and the efficiency of searching the global optimal solution is high. The optimal parameters of the OSELM are obtained through optimization.

Compared with the single model, the ensemble model can effectively improve the prediction performance, but the choice of the number of base learners may have a greater impact on the final prediction results. Therefore, this study discusses the impact of the integration scale on the algorithm. For the RF-OSELM, another parameter that needs to be determined is the number of OSELM networks in the ensemble. We set this parameter to be 3, 6, 9, and 12 in our simulations. For different sizes of hidden-layer nodes, we performed the experiment 100 times and computed the average. The SD is also given and the best results for different numbers of hidden nodes are reported in bold. An example of selecting the number of OSELM networks in the ensemble for the RF-OSELM network with the sigmoid activation function for the fabric image set is shown in Table 1.

Table 1.

Rotation forest online sequential extreme learning machine (RF-OSELM) model selection for the fabric image set

Number of OSELM networks	Testing (RMSE)	Number of hidden-layer nodes L					Training time (s)
Number of OSELM networks	Testing (RMSE)	5	10	15	20	25	Training time (s)
3	Mean	0.0326	0.0324	0.0321	0.0323	0.0225	5.5158
3	SD	0.0040	0.0041	0.0043	0.0047	0.0049	5.5158
6	Mean	0.0324	0.0321	0.0318	0.0320	0.0323	12.0290
6	SD	0.0036	0.0038	0.0039	0.0041	0.0042	12.0290
9	Mean	0.0325	0.0323	0.0318	0.0315	0.0323	27.1407
9	SD	0.0036	0.0038	0.0039	0.0040	0.0042	27.1407
12	Mean	0.0321	0.0319	0.0321	0.0312	0.0319	42.1031
12	SD	0.0029	0.0031	0.0032	0.0035	0.0036	42.1031

RMSE: root mean square error.

It can be observed from Table 1 that as the size of ensemble increases, the estimation error of each algorithm is reduced, implying that the overall trend of the prediction accuracy increased. However, it is not true that the larger the integration scale, the better the integration scale, because a large scale of integration leads to a significant increase in training time and the accuracy will reach a certain threshold. Therefore, the integration scale of the RF-DE-OSELM illumination estimation model should not be too large; otherwise, it will increase the number of calculations and reduce the training speed. Therefore, to balance the model accuracy and training time, we used six OSELMs in the RF-OSELMs and RF-DE-OSELMs in our experiments.

In addition, Tables 2 and 3 show the mean and SD of the 100 experimental results of the algorithm with different numbers of hidden-layer nodes under different activation functions (Sigmoid and RBF).

Table 2.

Comparison of online sequential extreme learning machine (OSELM), differential evolution online sequential extreme learning machine (DE-OSELM), rotation forest online sequential extreme learning machine (RF-OSELM), and rotation forest differential evolution online sequential extreme learning machine (RF-DE-OSELM) for Sigmoid hidden nodes

Algorithm	Testing (RMSE)	Number of hidden-layer nodes L
Algorithm	Testing (RMSE)	5	10	15	20	25
OSELM	Mean	0.0543	0.0524	0.0538	0.0530	0.0566
OSELM	SD	0.0066	0.0061	0.0071	0.0066	0.0076
DE-OSELM	Mean	0.0528	0.0517	0.0532	0.0525	0.0553
DE-OSELM	SD	0.0059	0.0060	0.0072	0.0065	0.0073
RF-OSELM (six OSELMs)	Mean	0.0324	0.0321	0.0318	0.0320	0.0323
RF-OSELM (six OSELMs)	SD	0.0036	0.0038	0.0039	0.0041	0.0042
RF-DE-OSELM (six OSELMs)	Mean	0.0227	0.0225	0.0223	0.0226	0.0228
RF-DE-OSELM (six OSELMs)	SD	0.0034	0.0036	0.0038	0.0040	0.0041

RMSE: root mean square error.

Table 3.

Algorithm	Testing (RMSE)	Number of hidden-layer nodes L
Algorithm	Testing (RMSE)	5	10	15	20	25
OSELM	Mean	0.0547	0.0528	0.0541	0.0532	0.0565
OSELM	SD	0.0062	0.0063	0.0070	0.0075	0.0077
DE-OSELM	Mean	0.0528	0.0519	0.0532	0.0525	0.0553
DE-OSELM	SD	0.0058	0.0062	0.0066	0.0071	0.0072
RF-OSELM (six OSELMs)	Mean	0.03331	0.0329	0.0322	0.0325	0.0328
RF-OSELM (six OSELMs)	SD	0.0037	0.0038	0.0040	0.0041	0.0043
RF-DE-OSELM (six OSELMs)	Mean	0.0248	0.0229	0.0225	0.0227	0.0231
RF-DE-OSELM (six OSELMs)	SD	0.0035	0.0035	0.0037	0.0039	0.0042

RMSE: root mean square error.

From the experimental results, the illumination estimation error and SD of the OSELM algorithm were greater than those of other algorithms, regardless of the use of the Sigmoid or RBF activation functions. The DE-OSELM algorithm had fewer errors than the OSELM algorithm, but its SD was still relatively large, indicating that the algorithm performance was still not stable enough. Compared with the OSELM and DE-OSELM algorithms, the mean error and SD of the RF-OSELM and RF-DE-OSELM algorithms were significantly decreased, and the performance of the algorithm was greatly improved. Among them, the RF-DE-OSELM achieved the optimal illumination estimation results on different hidden nodes and different activation functions.

From the experimental results of Tables 2 and 3, we can see that the OSELM algorithm achieved the optimal illumination estimation result when the number of hidden nodes was 10. Since the OSELM input layer weight and hidden-layer bias are randomly accessed, the accuracy of the OSELM algorithm is not high and not stable enough, and the difference between the results of the illumination estimation are relatively large. It can be observed from Tables 2 and 3 that the SD of the single OSELM algorithm was greater than that of the RF-OSELM and the RF-DE-OSELM, showing that the single OSELM model is unstable. In the OSELM, we only need to adjust the hidden-layer node number parameter; therefore, the number of hidden-layer nodes has a great influence on the performance of the OSELM algorithm. Increasing the number of hidden nodes can improve the performance of the OSELM algorithm, but the number of hidden nodes cannot be too large.

Compared with the OSELM algorithm, the illumination estimation error of the DE-OSELM algorithm was reduced. As can be seen from Tables 2 and 3, the DE-OSELM algorithm achieved the optimal illumination estimation result at L = 10. However, the SD of the algorithm was very large and the stability was not high, and the DE optimization algorithm did not improve the stability of the OSELM algorithm. This is mainly because the DE algorithm randomly initializes the population at each iteration of the input layer weight and the hidden-layer bias is not the same, resulting in instability.

In the RF-OSELM algorithm, a total of six different OSELMs were integrated. As can be seen from Tables 2 and 3, the OSELM algorithm with the RF algorithm not only reduced the error of illumination estimation but also improved the stability of the algorithm. Taking the Sigmoid activation function as an example, compared with the minimum mean error (0.0524) of the OSELM algorithm and the minimum mean error (0.0519) of the DE-OSELM algorithm, the minimum mean error (0.0318) of the RF-OSELM algorithm was reduced by 39.3% and 38.7%, respectively. This is mainly because the RF algorithm effectively improved the performance of the unstable learning algorithm. It differentiates the basic learner OSELM by randomly dividing the sample attribute set and then using the feature transformation strategy. Therefore, it obtained a good integration effect and improved the generalization ability of the learning algorithm.

In the RF-DE-OSELM algorithm, the RF algorithm was used to integrate six DE-OSELMs. From Tables 2 and 3, we can see that the RF-DE-OSELM algorithm proposed in this study achieved the best illumination estimation results in both the Sigmoid and the RBF activation functions. Taking the activation function Sigmoid as an example, compared with the minimum mean error (0.0524) of the OSELM algorithm and the minimum mean error (0.0519) of the DE-OSELM algorithm, the minimum mean (0.0223) of the proposed algorithm was reduced by about 57.4% and 57.1%, respectively. In addition, the SD of the RF-DE-OSELM algorithm was smaller than that of the OSELM and DE-OSELM for different numbers of hidden nodes. The performance of this algorithm was slightly better than that of the RF-OSELM algorithm. The precision and diversity of the base learner are two factors that affect the integrated performance of RF algorithms. In this study, we mainly optimized the OSELM by DE to improve the prediction accuracy of the base learner. However, since the optimized DE-OSELM was unstable, we used the RF algorithm to integrate the optimized DE-OSELM to improve the accuracy of the illumination prediction by increasing the diversity of the base learner.

In addition, to better assess the performance of each model, the angular errors and training time of SVR, the ELM, and the RF algorithm in the textile sample test set are listed in Table 4. In the input vector of the SVR illumination estimation model, except for the same binarization chrominance vector (referred to as the SVR (2D) neural network method), the color intensity information

I = R + B

was also added to the 2D r-g chromaticity space to form the 3D color space, constructing a new binarized image feature representation vector called SVR (3D) in the 3D color space.

Table 4.

Performance comparison of several algorithms on the fabric image set

Algorithm	Function	Angular error			Training Time (s)	Number of nodes
Algorithm	Function	MAX	RMSE	MED	Training Time (s)	Number of nodes
ELM	RBF	13.5312	4.8024	2.7153	0.4651	100
ELM	Sigmoid	13.0195	4.7231	2.6917	0.4580	100
SVR (2D)	RBF	15.7124	6.2533	4.2851	264.8400
SVR (3D)	RBF	13.0195	4.8572	2.9432	318.9284
OSELM	RBF	19.4773	8.0123	6.5146	0.6084	10
OSELM	Sigmoid	13.6901	4.8067	3.2576	0.3588	10
DE-OSELM	RBF	16.1722	5.6564	4.2033	8.9545	10
DE-OSELM	Sigmoid	13.0832	4.7959	3.1220	8.7985	10
RF-OSELM (six OSELMs)	RBF	13.3215	4.9023	3.1245	12.8124	15
RF-OSELM (six OSELMs)	Sigmoid	11.8167	4.5670	2.9356	12.0290	15
RF-DE-OSELM (six OSELMs)	RBF	13.2326	4.4691	2.9061	36.9591	15
RF-DE-OSELM (six OSELMs)	Sigmoid	11.3351	4.3567	2.4642	35.3104	15
RF		13.3754	4.8125	2.9246	4.2452

ELM: extreme learning machine; SVR: support vector regression; 2D: two-dimensional; 3D: three-dimensional; OSELM: online sequential extreme learning machine; DE-OSELM: differential evolution online sequential extreme learning machine; RF-OSELM: rotation forest online sequential extreme learning machine; RF-DE-OSELM: rotation forest differential evolution online sequential extreme learning machine; RF: rotation forest; RBF: radial basis function; RMSE: root mean square error.

As shown in Table 4, the training time (35.3104 s) of the proposed RF-DE-OSELM method was between that of the SVR (2D) (264.84 s), SVR (3D) (318.9284 s), and ELM (0.458 s); however, the prediction error was the smallest, but the best light estimation result was obtained. Compared to the traditional SVR and ELM algorithms, the RF-DE-OSELM method reduced the mean square error of illumination angle by 10.3% and 7.8%, respectively. The RF algorithm trains each base learner with different sub-sample data, so the differences between the base learners greatly increase, contributing to the improvement of the accuracy of integration. The prediction accuracy of the proposed algorithm is still better than that of the original RF algorithm, which is predicted by the regression tree, the learner of which is too complicated, resulting in over-fitting problems. However, the OSELM has better adaptability, so it has a great advantage in solving the problem of over-fitting. Experimental results show that the DE-optimized OSELM used in this study achieved very good results.

To show that the application of the proposed method is not limited to texture and non-texture cotton fabrics, polyester, or other fabric materials, a standard data set of colorful objects was tested. The 321-SFU image set³¹ consisted of 30 different scenes recorded under 11 different light sources. For comparison reasons, the leave-one-out cross-validation procedure was used to evaluate the proposed method in which the number of folds equaled the number of images in the data set. Thus, in the leave-one-out procedure, one image was selected for testing, and the remaining 320 images were used for training. The proposed method was compared to several state-of-the-art methods on the 321 images of the data set. The results of multiple methods are summarized in Table 5. From Table 5, it can be observed that the proposed RF-DE-OSELM method attained the best color constancy performance. The maximum angular error (MAX) and the RMSE of the proposed method were considerably lower than that of the max-RGB, Grey–World, SoG, Grey–Edge, BP neural networks, SVR (2D), and SVR (3D) methods and are comparable to the ELM (Sigmoid). The MED angular error of the proposed method was only 1.01, which is a reduction of 9.8% compared to the ELM methods.

Table 5.

Performance of color constancy algorithms on SFU data set containing 321 images

Method	Parameters	RMSE	MAX	MED
max-RGB		12.28	36.24	6.44
Grey–World		13.58	37.31	7.04
SoG	$ρ = 6$	9.03	28.70	3.97
Grey–Edge	$e^{1, 7, 4}$	8.30	31.57	3.20
BP		10.59	60.96	2.90
SVR (2D)	2D	10.06	22.99	4.65
SVR (3D)	3D	8.07	24.66	2.17
ELM (sigmoid)	L = 200	3.04	14.27	1.12
RF-DE-OSELM	L = 15, 6 OSELMs	2.85	13.92	1.01

Note: results of max-RGB, Grey–World, SoG, Grey–Edge, BP, SVR (2D), SVR (3D), and ELM algorithms are taken from Xiong and Funt¹⁴ and Li et al.¹⁵

RMSE: root mean square error; SoG: shades of grey; BP: back propagation; SVR: support vector regression; 2D: two-dimensional; 3D: three-dimensional; ELM: extreme learning machine; RF-DE-OSELM: rotation forest differential evolution online sequential extreme learning machine; OSELM: online sequential extreme learning machine.

In the experiment of image illumination correction, the color constant algorithm was used to estimate the illumination value of the scene. Then, based on the estimated illumination value, the original image was corrected to the standard illumination using the Von Kries model. To test the results more intuitively, some examples of textile image illumination correction results are shown in Figure 6 along with examples of unsupervised color constancy algorithms. Figure 6(a) shows corrected textile images based on the max-RGB method. Figure 6(b) shows corrected textile images based on the Grey–World method. Figure 6(c) shows corrected textile images based on the SoG method. Figure 6(d) shows corrected textile images based on the Grey–Edge method. Figure 6(e) shows corrected textile images based on the proposed method. Figure 6(f) shows the results of the SVR method. Figure 6(g) shows some example results of the ELM-based color constancy algorithm. Figure 6(h) shows images that are to be corrected. Figure 6(i) shows some target textile images. The first to fourth lines are the images corrected by the unsupervised illumination correction algorithm. The fifth to seventh lines are the images corrected by the supervised illumination correction algorithm. The image in the eighth column is the image to be corrected collected under a D55 light source, and the image in the ninth column is the standard image collected under the standard D65 light source. As can be seen from the figure, the results of the correction of the image by the supervised illumination correction algorithm were generally better than those of the unsupervised illumination correction. In the supervised illumination correction algorithm, the correction result of the RF-DE-OSELM was very close to that of standard illumination. According to the above analysis and the color correction examples shown in Figure 6, it can be concluded that the proposed illumination correction method is the most robust and accurate method.

Figure 6.

Color constancy example results of textile image set: (a) max-RGB; (b) Grey–World; (c) shades of grey; (d) Grey–Edge; (e) our proposed method; (f) support vector regression; (g) extreme learning machine; (h) input image; and (i) target image.

Conclusions

An improved DE-OSELM illumination correction model based on the RF algorithm was proposed because of the low accuracy and poor stability of the traditional OSELM-based illumination correction algorithm. Furthermore, a new, efficient, and low-dimensional color-feature extraction method based on the Grey–Edge framework was adopted to replace the traditional high-dimensional binary chromaticity histogram used to represent the input data of the RF-DE-OSELM to improve the training and learning speed. In our method, the DE algorithm was adopted to optimize the input weights and hidden biases of the single hidden-layer feedforward neural network to increase the prediction accuracy of the OSELM base learner. The RF algorithm greatly increases the difference between the base learners and improves the ensemble accuracy. Therefore, we used the RF algorithm as the ensemble framework and adopted the optimized OSELM as the base learner to replace the regression tree algorithm in the original RF. The average value of outputs of each DE-OSELM in the ensemble system was used as the final illumination chromaticity estimation results. The experimental results show that the proposed model has high prediction accuracy, strong generalization ability, and good robustness compared with traditional unsupervised and supervised algorithms (ELM and SVR algorithms). It effectively solves the problem of textile illumination estimation and achieves the purpose of the preprocessing step in the follow-up color-difference detection system.

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), Zhejiang Provincial Natural Science Foundation of China (No. LY18F030018) and Zhejiang Top Priority Discipline of Textile Science and Engineering.

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