3.1 Wavelet transform

Wavelet transform is the most used method for calculating multi-resolution signal representations. This is because the wavelet transform approach may localize information in both the temporal and frequency domains [27]. Equation illustrates the wavelet transform calculation formula (1). (1)

In Eq (1), j and i are the location parameter and the scaling factor, respectively. The primary goal of this transform is to approximate and interpret the signal using a collection of fundamental mathematical operations. An example of a function’s wavelet transform is shown in Eq (2): (2)

The function f(x) can be presented in the equation by its wavelet coefficients C_f(i, j) where i > 0, j ∈ R. The discrete wavelet transform is performed by i = 2^h, j = k2^h = ka for k, h ∈ Z². The method used to represent images is called a discrete curvelet transform. Image codes can edge more effectively using this method. This is due to the geometric component of the curvelet transform method. As a feature vector, the discrete curvelet transform approach’s coefficients are employed. A fast discrete curvelet transform (FDCT) technique was the focus of research by [28].

The variables used in this study include x as the spatial variable, w as the frequency domain variable, and r and θ as the polar coordinates in the frequency domain. Additionally, the study defines a pair of windows, namely W(r) and V(t). They serve, respectively, as radial and angular windows. While both W and V use real parameters supplied by t ∈ [−1, 1] and r ∈ (1/2, 2), respectively, all of the vectors are real, smooth, and positive, values. Non-negative real inputs are used by W(r) and V(t) respectively. Windows complies with the entrance requirements listed in [27]. Eqs (3) and (4) provide the following formulas for these windows: (3) (4)

For every h ≥ h₀, a window of frequency in the Fourier domain, U_h is defined by (5) Eq (5) uses the integer ⌊h/2⌋. The data from the radial window, W and angular window, V are also used to compute the polar wedge, U_h. The symmetrized form of Eq (5) can be derived by combining U_h(r, θ) + U_h(r, θ + π) to produce real-valued curvelets. Similarly, the Fourier transform can be used to define the waveform φ_h(x) as illustrated in Eq (6). U_j(w₁, w₂) can be found using Eq (5), where the windows specified in the polar coordinate system are w₁, w₂. The major curvelet φ_h is utilized in Eq (6) to represent all curvelets of scale 2^−h that can be obtained by rotating and translating φ_h. Now, we add rotation angles with l = 0, 1, … such that 0 ≤ θ ≤ 2π. These angles are θ_l = 2π.2^⌊−h/2⌋.l. k = (k1, k2) ∈ Z², reveals the order of the translational parameters. The curvelets are thereby defined as a function of scale 2^−h, orientation angle theta_l, and position (6)

The rotation by θ radians can be calculated by R_θ and its inverse by (7)

The inner product of an element f ∈ L²(R²) and a curvelet φ_h,l,k, is defined as curvelet coefficient. (8) where R denotes the real line. Length ≈ 2^−h/2, width = 2^−h, i.e., width ≈ length², known to be the anisotropy scaling relation or curve scaling law [28].

3.2 Wavelet and curvelet statistical characteristics

The proposed model utilizes ten descriptors or features to represent the coefficients in each sub-band of both wavelet and curvelet. These features are created based on the image’s overall distribution of grey levels. The ten features in our proposed strategy are statistical ones and are formally detailed below [29]. Mean is the mathematical average of a set of numeric data, z₁, z₂, …, z_m. [30]: (9)

Variance is a measure that illustrates the spread of the histogram, indicating the extent to which the grey levels (I[x, y]) differ from their mean (). Also, the variance provides insight into the range of a random variable’s values. The square of the deviations from the mean are averaged to create the variance. The square root of variance is known as the standard deviation [30]. (10) (11)

The third moment μ₃ of a standardized random variable is referred to as skewness, which determines the degree of asymmetry in the histogram around the mean. The clarity of the histogram is measured using kurtosis, denoted as μ₄. If significant skewness and kurtosis are present, the data may not be considered normal [29]. Eqs (12) and (13) define skewness and kurtosis as: (12) (13)

Energy is a statistical feature that represents the sum of the squared magnitude of coefficients in a sub-band. It is often used to measure the amount of signal content in a particular sub-band. The following formula [30] can be used to compute Energy: (14)

The measure of entropy is employed to assess the degree of randomness in an image, with smoother images resulting in lower entropy and rougher images resulting in higher entropy. The formula used to calculate entropy is presented below [30]: (15)

The highest value in the provided matrix is referred to as the maximum value. The calculations are as follows. Where R and C represent rows and columns respectively: (16)

The consistency of the element distribution in the different shades of grey is measured using the term “homogeneity.” Its value ranges from 0 to 1. A smoother texture image results from a value that is near to 1, and so on. The definition of an image’s homogeneity in mathematics is [29]: (17)

Moment is a property of a picture that is utilized widely in image classification and pattern recognition [29]. (18)

3.3 Grey wolf optimizer (GWO)

Mirjalili et al. [31] proposed the grey wolf optimizer (GWO), which is a type of metaheuristic optimization approach. The algorithm’s leadership pyramid and hunting strategy are similar to those of grey wolves, therefore the name. A grey wolf pack typically has five to twelve members and is organized into the social classes alpha, beta, delta, and omega. Here, alpha is regarded as the alpha dog. All of the pack’s crucial decisions are made by the alpha. The alpha wolf is supported by the beta wolf in decision-making and other activities as it moves down the food chain. Delta wolves also referred to as sub-ordinate wolves, are dominant over omega wolves yet work closely with alpha and beta wolves.

The alpha and beta wolves provide guidance and safety to the pack, while the omega wolves are regarded as disposable members or scapegoats. Nonetheless, omega wolves play a critical role in scouting, warning the pack of potential threats, and defending against external threats. The symbols used to represent alpha, beta, delta, and omega are α, β, δ, and ω, respectively. To use the GWO for addressing the feature selection problem, the study in [32], proposed a competitive binary grey wolf optimizer. In our study, the GWO was modeled as given in Algorithm 1 which illustrates how GWO was used as a feature selection technique.

Algorithm 1 The pseudo-code for GWO for feature selection:

1: Initialize the population:

 • Generate random binary strings of length N (where N is the number of features) to form the initial population.

2: Define the fitness function: KNN (classification accuracy)

 • Evaluate the fitness of each individual in the population using a classification accuracy of KNN.

3: Initialize the positions of the alpha, beta, and delta wolves.

 • Select three individuals with the highest fitness scores and assign them as alpha, beta, and delta wolves.

4: Update the positions of the wolves.

 • Update the position of each individual in the population. using the following equations:

• For alpha wolf: x_alpha = x_alpha + A * D_alpha
• For beta wolf: x_beta = x_beta + A * D_beta
• For delta wolf: x_delta = x_delta + A * D_delta
• For the rest of the population: x_i = (x_alpha + x_beta + x_delta + x_i) / 4 + rand() * (x_alpha—x_beta)

 • where x_i is the position of the i th individual, A is the step size, and D_alpha, D_beta, and D_delta are the distance vectors of the alpha, beta, and delta wolves, respectively.

5: Evaluate the fitness of the new population.

 • Calculate the fitness of each individual in the population using the fitness function.

6: Repeat steps 4–5 for a certain number of iterations or until a satisfactory solution is found.

7: Select the best feature subset.

 • Select the individual with the highest fitness score as the best feature subset.

8: Return the best feature subset.

3.4 Particle swarm optimizer (PSO)

Kennedy and Eberhart [33] introduced PSO as a swarm-based algorithm that imitates animal social behavior such as bird flocking and fish schooling. Each potential solution is represented as a fast-moving particle, similar to a swarm of birds, traversing the problem space. The particle combines a portion of its best historical position and current location, determined by one or more agents of the swarm, with random disturbances to determine its future path through the search space. Once all particles have moved, the next iteration begins. PSO’s main advantage is that it requires fewer tuning parameters. However, the high-dimensional search space can slow down the convergence to the global optimum [34]. In our paper, the PSO was modeled as a feature selector as given in Algorithm (2).

Algorithm 2 The pseudo-code for a particle swarm optimization (PSO) algorithm for feature selection:

1: Initialization: Generate a population of particles, each representing a set of features.

2: Define the fitness function: KNN (classification accuracy)

 • Evaluate the fitness of each particle in the population using a classification accuracy of KNN.

3: Initialization: Set the best position (set of features) and fitness value found so far for each particle and the entire swarm.

4: Repeat steps 5–8 for a number of iterations or until maximum number of iterations.

5: Update particle velocity: For each particle, update its velocity based on its previous velocity, its distance to its best position and the swarm’s best position, and two tuning parameters (inertia weight and acceleration coefficients).

6: Update particle position: For each particle, update its position by adding its new velocity to its current position.

7: Evaluation: Evaluate the fitness of each particle’s new position.

8: Update best position: For each particle, update its best position and fitness value found so far, and update the swarm’s best position and fitness value if the particle’s fitness is better than the swarm’s best fitness.

9: Return the best position found, which represents the best subset of features for the classification task.

3.5 Genetic algorithms optimizer (GAO)

Genetic algorithms simulate biological evolution and model evolutionary processes to solve problems. Genetic algorithms can evolve sophisticated and intriguing structures in a simple computational framework. Population genetics is the basis for genetic algorithms. The population is randomly constructed, with each individual represented by a bit string that represents a potential answer to the problem at hand. Some individuals are fitter than others due to genetic variation within the population. Selection biases the next group of potential solutions based on these differences. By copying and eliminating successful individuals, selection creates a new population with some differences among its members. During this copying process, mutation, crossover, and other bit-string changes may occur. Mutation and crossover procedures create new samples with a better-than-average likelihood of being good by changing the existing group of good individuals. After many generations of appraisal, selection, and genetic operations, the population’s fitness improves, and its members become better “solutions” to the fitness function’s problem [35]. In our paper, the GAO was modeled as a feature selector as given in Algorithm (3).

Algorithm 3 The pseudo code for Genetic Algorithm (GA) for feature selection:

1: Initialize a population of chromosomes with random values within the search space. each representing a set of features.

2: Define the fitness function: KNN (classification accuracy)

3: Evaluation: Evaluate the fitness of each chromosome by measuring the classification accuracy rate of KNN on the subset of features it represents.

4: Selection: Select a subset of chromosomes to mate based on their fitness.

5: Crossover: Create new chromosomes by combining the features of the selected chromosomes.

6: Mutation: Randomly mutate some of the chromosomes by changing the value of some of their genes.

7: Evaluation: Evaluate the fitness of each particle’s new position.

8: Repeat steps 3–6 for a number of generations (maximum number of generations).

9: Return the best chromosome found, which represents the best subset of features for the classification task.

3.6 Classification: Random forests (RF)

Random forest method is a strategy for ensemble learning that combines many classifiers to improve a model’s performance. It is a decision tree-based supervised machine learning technique and is employed for both regression and classification issues [36]. The random forest’s ensemble notion states that merging a number of ineffective classifiers could result in a high classification rate. Multiple decision trees’ outputs were blended by random forest to increase prediction accuracy. The classification stage in this study was carried out using 10-fold cross-validation to evaluate the efficacy of the suggested methodology.

3.7 Classification: K-nearest neighbors (KNN)

The K-Nearest Neighbors classifier is a straightforward, yet powerful, classification algorithm that falls under the category of instance-based learning or lazy learning algorithms. During the training phase, KNN only stores the data without constructing any explicit models. To classify a new instance, KNN calculates its similarity with its K-nearest neighbors in the training set. KNN offers several benefits, such as being simple to implement, tolerant of noisy data, and capable of handling multi-class classification. However, KNN also has some drawbacks, such as its sensitivity to the choice of K and the distance metric employed.

3.8 Classification: Naive bayes (NB)

The Naive Bayes classifier is a well-known machine learning and data analysis algorithm that uses the Bayes theorem of conditional probability to make predictions. It is a probabilistic model that assumes the features are independent of each other, hence the name “naive”. Despite this simplifying assumption, the algorithm exhibits impressive performance in a broad range of applications, such as text classification, spam filtering, and sentiment analysis. One of the Naive Bayes’s significant advantages is its simplicity and speed, making it an efficient algorithm for handling large datasets. Additionally, it requires minimal training data and can work well even with high-dimensional data. While NB may not be the optimal algorithm for every problem, it is still a popular choice for many machine learning tasks.

5.1 Dataset

The public dataset, Terravic Facial infrared (IR) [41], was adopted in the evaluation of the proposed models. The dataset contains 4000 images for 20 different classes. In each of the classes, there is a total of 200 black-and-white images. The photos are JPEG files that are 8 bits grayscale and 320 pixels by 240 pixels in size. Each class is meant to represent a unique individual. Every individual possesses certain images that can be viewed in a variety of configurations (front, left, right; indoors/outdoors; glasses) to represent different life scenarios. As a result of the corruption that occurred in the fifth and sixth classes, this evaluation only utilised a total of 18 classes (i.e., 3600 images were used).

5.2 Performance metrics

The effectiveness of the proposed method was assessed utilizing a series of commonly-used benchmark metrics that were derived from the confusion matrix obtained during the classification stage. These metrics include:

Accuracy is a metric that quantifies the percentage of accurate predictions made during the classification process. (19) where TP is the number of true positive instances, TN is the number of true negative instances, FP is the number of false positive instances, and FN is the number of false negative instances.

Precision is a metric that determines the ratio of true positive outcomes to all the predicted positive outcomes. (20)

Recall, also known as Sensitivity or True Positive Rate, is a performance metric that quantifies the ability of a model to correctly identify the positive instances. (21)

The F-Measure is a performance metric which combines the values of precision and recall into a single score. It is computed as the harmonic mean of precision and recall and can be expressed mathematically as follows: (22)

The Receiver Operating Characteristic (ROC) curve’s Area Under the Curve (AUC) is a widely-used metric that assesses the effectiveness of classification models. The ROC curve illustrates the true positive rate (sensitivity) versus the false positive rate (1-specificity) at various classification thresholds. An AUC score of 1.0 indicates that the model has a perfect performance, whereas an AUC score of 0.5 suggests a random guess. The ROC curve provides a comprehensive summary of the classifier’s overall performance.

The Matthews Correlation Coefficient (MCC) is a performance metric utilized to assess the binary classifier’s effectiveness. It depends on the entries in the confusion matrix and measures the correlation between the observed and predicted binary classifications. The MCC is mathematically expressed as: (23)

The MCC can take values from -1 to 1, where -1 indicates a completely incorrect classifier, 1 indicates a perfect classifier, and 0 indicates a random classifier. The MCC is a robust metric that provides a balanced evaluation of the performance of a classifier, taking into account both false positive and false negative errors. These metrics provide a comprehensive evaluation of the performance of the proposed method.

5.3 Experiments environment setting

All the experiments in this paper were executed under the following settings. The specification of the used PC was: Intel(R) Core(TM) i7–10700 CPU @ 2.90GHz with 8.00 GB RAM and the experiments were implemented using MATLAB R2019a under Windows 10. Also, all the experiments in the scenarios below were conducted in 10 independent runs because the optimization process starts with random solution(s) in Meta-Heuristic. Hence, it is basically affected by the parameter’s initialization. Therefore, ten independent runs were conducted. The average of the ten runs was then calculated and reported.

A good training approach for a model with a dataset must be found. The model should have enough instances to train on without over-fitting it, but if there aren’t enough, the model won’t be properly trained and will perform poorly when tested [42, 43]. Using the k-fold cross-validation, with a large value of k (i.e., k approaching the number of instances in the dataset) or leave-one-out cross-validation can help ensure that the model is evaluated on as much of the data as possible. This can provide a more accurate estimate of the model’s performance than a simple train-test split, as each data point is used for both training and validation [42]. It was justified in [42] that the k-fold cross-validation is better to be used over hold-out validation. Therefore, in our case, k-fold cross-validation has been used to evaluate the performance of the proposed models.

Unlike, splitting the dataset into 70% for training and 30% for testing a model, in a k-fold cross-validation the dataset is split into k subsets, or folds, of approximately equal size. The model is then trained on k-1 folds, and the remaining fold is used for validation/testing. This process is repeated k times, with each fold used as the validation set exactly once. The results of the k-folds are then averaged to obtain an overall estimate of the model’s performance. This approach is useful because it allows for a more accurate estimation of the model’s performance, as each data point is used both for training and validation at some point in the process [42, 43].

5.4 Parameter settings

This section presents an explanation of the values of the parameters used in each classifier and feature selection method. An experiment was conducted to determine the best value of KNN parameter, k. The results, given in Table 2, show that the best value of k is 5 which was then used in all experiments of KNN below.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 2. Impact of K value on the KNN performance.

https://doi.org/10.1371/journal.pone.0287349.t002

Random Forest parameters are given in Table 3. The Max_Depth parameter can affect the performance of random forest, so we run an experiment aimed at finding the best value and giving the best results. It was observed that the best value of Max_Depth is to make it as long as possible, meaning that Nodes are expanded until all leaves are pure, as shown in Table 4. Consequently, this value was adopted for all subsequent experiments involving random forest. The parameters values of the feature selection algorithms used in the proposed method are given in Tables 5–7.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 3. Parameter settings of random forest.

https://doi.org/10.1371/journal.pone.0287349.t003

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 4. Impact of changing the Max_Depth value on the performance.

https://doi.org/10.1371/journal.pone.0287349.t004

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 5. Parameter settings of grey wolf optimizer.

https://doi.org/10.1371/journal.pone.0287349.t005

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 6. Parameter settings of particle swarm optimizer.

https://doi.org/10.1371/journal.pone.0287349.t006

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 7. Parameter settings of genetic algorithms.

https://doi.org/10.1371/journal.pone.0287349.t007

6.1 Scenario 1: Identification of the minimum set of features

The aim of this scenario was to investigate which type of meta-heuristic algorithms (evolutionary or swarm) would be the best feature selection technique. For the evolutionary type, we used GA while for the swarm type, we used PSO and GWO. The main goal was to identify the best set of features which improve face recognition accuracy while minimizing the computational time. To achieve this goal, GA, PSO, and GWO were used as feature selection techniques on the features extracted by the wavelet and curvelet transformation.

The experiment was conducted in ten independent steps because the optimization process in metaheuristic starts with random solutions. Hence, it is basically affected by the parameter’s initialization. Therefore, ten independent experiments were conducted, and the average of the ten runs was calculated. The average accuracy of each optimizer was recorded and compared as well as the time CPU taken was also calculated and compared. In addition, the percentage of the selected features for each method was reported. This was to show the amount of feature reduction each algorithm (GA, PSO, and GWO) can make. The summary of the results of this scenario is given in Table 8.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 8. Feature selection by GA, GWO and PSO.

https://doi.org/10.1371/journal.pone.0287349.t008

From these results, presented in this table, the following remarks can be made. Firstly, the results of all algorithms (GA, PSO, and GWO) using curvelet features were better than using wavelet features for thermal face images. The curevelet-based results were better than the wavelet-based results with about %12 on average (98% for curvelet and 86% for wavelet). This superiority of curvelet transformation is attributed to its ability to divide the image into 16 angles, as opposed to wavelet transformation which only divides the image into a few horizontal, vertical, and diagonal directions. Thus, the curvelet can extract more discriminating features than wavelet can.

Secondly, the average reduction rate of the wavelet features was much better higher than that of the cuverlet features. For the wavelet features, the average reduction rate was 89.69%, 91.31%, and 61.87% for GA, GWO, and PSO, respectively. For the curevelt features, the average reduction rate was 67.54%, 83.41%, and 55.94% for GA, GWO, and PSO, respectively. It was also noticed the GWO has achieved the highest reduction rate for both type of features while giving the highest accuracy too.

Thirdly, the results in Table 8 indicate that GWO was the best in terms of accuracy, time consumption, and percentage of the selected features (i.e., selected the minimum set of features giving the highest accuracy). The average accuracy achieved by the GWO was 98.83%, while the average time taken was 62.05 seconds. Furthermore, the average number of features selected was 134.4, which constituted only 16.59% of the total features. On the other hand, the GA gave an average accuracy of 98.61%, an average time consumption of 167.87 and an average number of 263.70 for selected features. These results demonstrate that the GWO method outperforms the GA in terms of both accuracy and time consumption, i.e., the swarm methods is better than the evolutionary method.

6.1.1 Convergence analysis.

Convergence of metaheuristics algorithms is a very important indicator of their performance. Thus, an experiment was conducted to investigate the relation between fitness value against the number of iterations of each of the used algorithms (GA, GWO, and PSO). Note that in this study, the fitness function was to minimize this error rate which is calculated for the classification accuracy of the model on the dataset, i.e., maximizing the accuracy. The best way to illustrate this relation is by plotting it. Figs 2–4 show the convergence of each method in finding the best solution. The x-axis represents the number of iterations, and the y-axis represents the best fitness value achieved by the algorithm at that iteration.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 2. The convergence of GWO to find the best set of features.

https://doi.org/10.1371/journal.pone.0287349.g002

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 3. The convergence of PSO to find the best set of features.

https://doi.org/10.1371/journal.pone.0287349.g003

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 4. The convergence of GA to find the best set of features.

https://doi.org/10.1371/journal.pone.0287349.g004

From these figures, it can be seen that as the number of iterations increases, there is a decreasing trend in the fitness value, indicating that the algorithm is making progress towards finding a good solution. When comparing the three convergence curves, it can been noticed that the GWO is the best solution with a fitness value of 0.002, while PSO and GA found solutions with fitness values of 0.0074 and 0.0043, respectively. These results suggest that GWO outperforms both PSO and GA as feature selection methods.

Overall, the results of this scenario support the conclusion that the GWO-based method is a superior feature selection method in terms of accuracy, time consumption, and the number of selected features. Also, the best accuracy, 99.28% was achieved by the set of features, GWO_3. Furthermore, the results showed that our proposed method could achieve 89% (using GWO_7 and GWO_10) and 94% (using GWO_4 and and GWO_10) reduction rate by GWO-based method for curvelet features and wavelet features respectively. Such high reduction rate of the features will not require high computation cost to recognise users. This means that our proposed GWO-based features selection method for thermal face recognition could supports limited resources, e.g., IoT applications.

6.2 Scenario 2: Best learning strategy of classifiers

Given the best-selected features (i.e., curvelet features from Scenario 1), in Scenario 2 it was aimed to investigate the best-supervised learning strategy. Those strategies are instance-based learning, ensemble-based learning and probabilistic-based learning, and the classifiers are KNN, RF and NB, respectively. From the first experiment, it was proved that the GWO was the best feature selector for both wavelet and curvelet features in terms of the minimum number of features and the highest accuracy rate. So, in Scenario 2, we designed two sub-experiments (wavelet-based features and curvelet-based features) in which the three classifiers were applied to the sets of features produced by GWO. In both sub-experiments, the classifiers, KNN, RF, and NB were applied to wavelet and curvelet features selected by GWO and the results were reported as given in Figs 5 and 6. The presented figures illustrated the average of ten runs for each classifier with best selected features from GWO method.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 5. Results of KNN, RF and NB classifiers using wavelet features.

https://doi.org/10.1371/journal.pone.0287349.g005

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 6. Results of KNN, RF and NB classifiers using curvelet features.

https://doi.org/10.1371/journal.pone.0287349.g006

From Figs 5 and 6, two main remarks can be noticed. Firstly, the RF classifier outperformed the other classifiers in all performance metrics. RF achieved average results of 99.4% with some set of features achieving above 99.6% by all evaluation metrics. In other words, the ensemble-based learning strategy was better than the instance-based learning and probabilistic-based learning in identifying users through their thermal faces. Secondly, like in Scenario 1, the curvelet features showed to give classification results better than the wavelet features in all evaluation metrics although the wavelet features can identify the users using a number of features less than that are needed by curvelet features. In conclusion, the results of all classifiers can be sorted as follows, RF, KNN and NB. The results of this scenario motivated us to further apply the random forest classifier over the features selected by three optimization algorithms as presented in Scenario 3.

6.3 Scenario 3: Best classifier performance

The aim here is to investigate which metaheuristic algorithm among GA, GWO and PSO would select the set of features (from wavelet and curvelet) which can give the highest classification performance using the best classifier identified from Scenario 2 (i.e., RF). In other words, this scenario does not consider the computational time nor the minimum number of features as in Scenario 1, the only concerned with the accuracy and other performance metrics. To achieve this, in Scenario 3, we designed two sub-experiments (wavelet-based features and curvelet-based features) in which, the features selected by three metaheuristic algorithms (GA, GWO and PSO) were given to the RF classifier (best classifier from Scenario 2) and the results of these two experiments were reported as given in Figs 7 and 8.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 7. Comparing GA, GWO and PSO as feature selectors using the random forest classifier on wavelet features.

https://doi.org/10.1371/journal.pone.0287349.g007

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 8. Comparing GA, GWO and PSO as feature selectors using the random forest classifier on curvelet features.

https://doi.org/10.1371/journal.pone.0287349.g008

The obtained results, from the curvelet features, showed that the average of the performance metrics of PSO features is outperforming the other two feature selectors, i.e., GWO and GA. The GWO and GA came second and third, respectively. Although the GWO was not the best in accuracy and other measures, it gave comparable results with the smallest average of the number of features needed to accurately identify the users. On the other hand, the area under curve in ROC showed an equal performance for the three FS methods reaching 100%. This means that the three-feature selection (FS) methods being compared in the study performed equally well in terms of their ability to discriminate between the different classes. An AUC of 100% indicates perfect discrimination between all users, which means that the classifier is able to correctly authenticate all users without any errors.

6.4 Scenario 4: Most efficient model

This scenario conducted to measure the relation between the number of iteration of each algorithm (GA, GWO, and PSO) and the CPU time needed to reach the highest accuracy (i.e., to find out the most efficient model). As the random forest classifier with the curvelet features gave the highest results, see Table 8, we used them in measure the computational time of using GA, GWO, and PSO as feature selection. Fig 9 illustrates that the GWO outperformed both GA and PSO in the computational time consumed. This confirms the results presented and discussed in Scenario 1. This scenario provides additional insights into the computational efficiency of the algorithms, which is an important factor to consider when selecting an optimization algorithm for a particular application. The findings suggest that GWO may be a good choice for applications where computational efficiency is a critical factor.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 9. The computational time between three different feature selection methods GA, GWO and PSO.

https://doi.org/10.1371/journal.pone.0287349.g009

6.5 Statistical analysis

In general, the significance analysis provides a rigorous evaluation of the proposed methods and helps to ensure the reliability of the results obtained. By using appropriate statistical tests, the authors would be able to draw robust conclusions about the performance of the different methods and their relative strengths and weaknesses. In this study, due to the diversity of the obtained results (see results of Scenario 1–3), the authors conducted a significance analysis to evaluate the differences between the proposed methods. This analysis was conducted in two phases. In the first phase, the authors investigated whether the data follow a normal distribution or not. This was done by using the Shapiro-Wilk test. The results of this test showed that the data did not follow a normal distribution. Therefore, in the second phase, the authors proceeded to conduct a non-parametric test, namely the Wilcoxon signed-rank test, to determine if the differences between the employed algorithms were statistically significant. The results of the Wilcoxon test are presented in Tables 9–14, and a level of confidence of 95% was used for the statistical analysis. Note that the significance analysis was not done for the accuracy of the algorithms but also for all other performance metrics (precision, recall, F-Score, and MCC and ROC). This would further increase the confidence of selecting an algorithm of the other.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 9. Statistical analysis Scenario 1 using wavelet features.

https://doi.org/10.1371/journal.pone.0287349.t009

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 10. Statistical analysis Scenario 1 using curvelet features.

https://doi.org/10.1371/journal.pone.0287349.t010

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 11. Statistical analysis of Scenario 2 using wavelet features.

https://doi.org/10.1371/journal.pone.0287349.t011

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 12. Statistical analysis of Scenario 2 using curvelet features.

https://doi.org/10.1371/journal.pone.0287349.t012

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 13. Statistical analysis of Scenario 3 using wavelet features.

https://doi.org/10.1371/journal.pone.0287349.t013

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 14. Statistical analysis of Scenario 3 using Curvelet features.

https://doi.org/10.1371/journal.pone.0287349.t014

The statistical analysis was done for Scenarios 1–3 (presented above). For these scenarios, both wavelet and curvelet features were employed to study their classification performance. The first scenario is aiming to compare the three feature selection methods (i.e., GA, GWO and PSO-based methods). A comparison of the significance of the three methods was conducted in terms of accuracy, time consumed to determine the set of features and the number of selected features. The aim of this step was to study whether the average performance of the three feature selection methods is equal or not using Wilcoxon test. Table 9 shows that there is no statistically significant difference in accuracy between GAO, GWO, and PSO. Howver, in terms of time consumed, the results showed that there is statistical significance between GWO versus PSO or GAO. Hence, it generally seems that GWO was able to outperform both PSO and GAO when employing wavelet features. Moreover, in the case of the curvelet feature, GWO was better than PSO and GAO in terms of time consumed and the number of features as shown in Table 10.

Scenario 2 aims to study the performance of three different learning strategies (see above). An statistical analysis of their significance, as given in Table 11, shows that there is a statistical difference between RF and KNN or NB in the case of using wavelet features. While in the case of using curvelet features, as illustrated in Table 12, the difference is significant between RF against both KNN and NB.

In Scenario 3, the objective was to check the performance of using RF classifier with the different feature selection methods (i.e., GAO, PSO, and GWO-based methods), see above. The statistical analysis of the these results are summarized in Tables 13 and 14. The obtained results show that in the case of using wavelet features, the difference between GWO and PSO is statistically significant and also between the PSO and GAO methods. While in the case of using curvelet features, there is no statistical significance between the three methods since all P-values are greater than 0.05.

7.1 Research questions discussion

In the introduction, we identified two research questions (RQ1- which is better (wavelet or curvelet transform) in extracting thermal face features to accurately and accurately recognise a user? and RQ2- which metaheuristic algorithms, GWO, GA and PSO, can select the minimum set of features that efficiently and accurately recognise users from their thermal face images?).

For RQ1, the results, reported above section, showed all curvelet-based experiments were better than the results of wavelet-based experiments. In other words, the curvelet transformation was found more effective at producing discriminative features that can accurately describe the curves in thermal face images, which in turn led to higher classification performance in all evaluation metrics. One reason for this improved performance is that curvelet analyzes images from more angles than wavelet, which only analyze images in three directions (horizontal, vertical, and diagonal). Curvelets also have scale, location, and orientation parameters that allow them to tune to various orientations and scales, while wavelets only have scale and location parameters. Additionally, the curvelet covers the entire spectrum in the frequency domain. This means that there is no loss of information when capturing frequency information from the face images. Furthermore, the curvelet is particularly useful for representing singularities over geometric structures in images, whereas wavelet is better suited for point singularities [37]. Overall, the results of the scenarios above suggest that curvelet transforms offer advantages over wavelet transforms when it comes to accurately describing curves in thermal face images and extracting discriminative features for the classification of thermal face images (i.e., authenticating users through their thermal face).

For RQ2, the results obtained from Scenario 3 show two remarks. First, PSO could select a set of features which can achieve slightly higher classification performance than the cases of using GWO and GA. However, this higher performance required more number of features than the case of GWO and GA. Thus, the model with PSO-based features could be adopted by applications where high computational cost is not an issue (e.g., cloud-based face recognition applications). Second, with the minimum set of features, GWO could achieve the second-best classification performance, 99.5% accuracy. This accuracy is less than the PSO result by only 0.02% but GWO results were achieved by features less PSO results with 30%, see Table 8. This means that the model with GWO-based features could be adopted by limited resources applications such as IoT applications.

7.2 Possible applications

Thermal face recognition has a wide range of applications in different industries, particularly in situations where traditional face recognition technology may not be effective, such as in low-light or high-contrast environments. In security applications, thermal face recognition with high performance, like the proposed models, can detect and identify individuals wearing masks or other face coverings, which can be difficult for traditional visible light face recognition systems. This makes thermal face recognition an important tool for law enforcement. The wavelet-based model (see Table 8) could be a good candidate for this application as it is lightweight enough to be installed on a portable camera. In addition, thermal face recognition can be used to identify individuals at a distance, which can be useful for security applications in large areas such as airports, train stations, or sports stadiums. The proposed curvelet-based model (see Table 8) would be an ideal solution in such applications where it can be deployed on a cloud or fog computing model.

However, there are some limitations to thermal face recognition technology. For example, it may not be as effective in identifying individuals with similar thermal signatures, such as twins, and it may not be able to accurately identify individuals who have undergone plastic surgery or other facial alterations.

7.3 Comparison with related work

We compared the results of the proposed approach with the results of the related works that used the same public dataset (Terravic Facial I) and a summary of this comparison is given in Table 15.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 15. Comparison with related work used Terravic Facial IR dataset.

https://doi.org/10.1371/journal.pone.0287349.t015

From the table below, two main remarks can be noticed. Firstly, the impact of the proposed GWO-thermal-face-feature selection (GWO-TFFS) on the computation time and the efficiency of the thermal face-based authentication method. Our proposed model is the only model which gave above 99.5% for all evaluation metrics with only 16% of the features space. This would lead to less computational cost which makes our proposed authentication method energy efficient and suitable for IoT applications. Secondly, the suggested model has been thoroughly evaluated using the benchmark evaluation metrics (accuracy, precision, recall, F1-Score, MCC, and ROC curve) but all other related work has only been evaluated using accuracy which is not enough to assess the rigour of the face-based biometric authentication.

A recent work [17] has reported very good results (100% accuracy) but the proposed method not clearly described. It was not clear how the ROI (Face area) was extracted, the purpose of wavelet was not clear, the parameter of SVM and ANN were not specified, finally the reduction percentage of feature selection was not reported. Thus, the results seem not reliable.