Introduction

As long as the significance of anthropogenic activities, variations in LULC are inconsistent. The abovementioned variations are like financial objectives, such as wood exploration, agriculture, and cattle ranching (Christovam et al. 2019). In particular, transformations in LULC will happen because of deforestation. Furthermore, certain expectations concerning deterioration as well as submergence circumstances will increase the LULC changes, and these problems raise an increase in greenhouse gas secretion and loss of biodiversity (Mas 1999; Mangan et al. 2022). Therefore, the data that the LULC has plays a vital role in climate and environmental change studies. Thus, the universal concern is to use all the requirements to get superior LULC maps. These maps will provide data for arranging and assessing natural resource management, developing sustainable practices, and modelling environmental variables (Adam et al. 2014). The images that are most prevalently used for LULC change classification and detection are Multispectral (MohanRajan and Loganathan 2021; 2022; 2023; MohanRajan et al. 2020), and Hyperspectral images (Navin and Agilandeeswari 2020a, b; Agilandeeswari et al. 2022). The main issues in the HSI over the past ten years have been spectral dimensionality and the requirement for particular spectral-spatial classifiers (Yadav et al. 2022).

Rapid, extensive changes in land cover are currently occurring in several areas of the world. Several of these nations, including Brazil, Columbia, Indonesia, Mexico, the Ivory Coast, Venezuela, and Zaire, are the focal points of most of this activity (Mas 1999). Due to the possible consequences of erosion, increased run-off and flooding, rising CO2 concentration, climatological shifts, and biodiversity loss, these changes in land cover, especially the destruction of tropical forests, have drawn attention (Fontan 1994). To map large areas and get multitemporal information from the large covered areas is a time-consuming and expensive task (Kavzoglu and Colkesen 2009). To overcome these issues, the satellite imagery technique is the most useful (Lv and Wang 2020). Since 1970, remote sensing images have contributed to persistent as well as legitimate knowledge of the land surface area (Petitjean et al. 2012). These satellite images have the proficiency to grab the entire land (Puletti et al. 2016). Some of the satellite hyperspectral sensors are AVIRIS, ROSIS, Hyperion, and MODIS (Haq et al. 2021a; Haq et al. 2020; Haq et al. 2021b). The captured images extract the LULC information using image processing techniques. An aforementioned satellite community has emerged the improvements in image classification approaches throughout the period of time to map the LULC (Haq 2022a). Through digital image processing, there is an extent for mapping the LULC. However, as a result of numerous factors corresponding to the procedure, building the LULC maps using image processing techniques is a challenging task (Manandhar et al. 2009).

Hyperspectral remote sensing imagery emerged a few decades ago and can gently enhance LULC mapping. A large number of uninterrupted spectral bands, along with diminutive bandwidths could be accumulated for hyperspectral imaging (Bioucas-Dias et al. 2013). These images will give more details about spectral bandwidths collected by multispectral sensors. And it will provide new opportunities for LULC applications (Chutia et al. 2016). And also, it will give a large quantity of information that leads to the Hughes phenomenon. To help the difference between classes, the amount of predictor features may add data. Also, it will raise the complexity. A small dataset sample size may not be sufficient to characterize this complexity. To decrease the complexity, we have to add more features rather than increase the dimensions of the data (Maxwell et al. 2018). As a consequence of the confined proportion of training data specimens, in addition to the nonessential details exhibited across all features, hyperspectral image classification (HSI) is a difficult task. And also, the uncertainties accompanying the atmospheric and geographic effects present in the spatial resolution. Some machine learning algorithms like supervised, unsupervised, and semi-supervised classification methods are available to get high LULC classification accuracy using hyperspectral images (Haq et al. 2021c). Supervised classification will be used when the user has labeled input samples. Some supervised classification algorithms are random forest (RF) and support vector machine (SVM) (Ghamisi et al. 2017; Haq et al. 2022). Unsupervised classification is used when the user has unlabelled samples; the training model will take the labels from them. Some unsupervised classification algorithms are k-nearest neighbors (Cariou et al. 2020). Semi-supervised classification will be used when the user has more labeled and fewer unlabelled samples. Some of the semi-supervised classification algorithms are self-organizing algorithms. Each algorithm will give the best results based on the datasets we have used (Ma et al. 2016).

In recent years, for hyperspectral image classification (HSIC) use of deep learning-based algorithms has been increasing, and they are achieving good results in LULC classification (Haq 2022b). To get high LULC classification accuracy, we must perform feature extraction and band selection before classification. This helps us remove the redundancy among the features, avoids the Hughes phenomenon, and decreases the computational cost. When we have high-dimensional input features in the dataset, we have to extract the features from that data. Here are some of the feature extraction methods, PCA (principal component analysis) (Hotelling 1933), autoencoders (AE), LDA (linear discriminant analysis), and t-SNE (stochastic neighboring embedding). PCA will be used to find the subspace of principal components from the input feature vectors (Fauvel et al. 2009). To distinguish the low-dimensional hyperspace based on miscellaneous classes LDA is used (Chang and Ren 2000). LDA and PCA are known as global linear algorithms, that could never perform efficiently in nonlinear scattered data circumstances (Kambhatla and Leen 1993). As a consequence of these arguments, a couple of researchers have suggested nonlinear feature extraction algorithms for hyperspectral images (Zhang et al. 2008). They are ISOMAP (isometric mapping), LLE (local linear embedding) (Bachmann et al. 2005), and spherical stochastic neighboring embedding (SSNE) (Lunga and Ersoy 2012). Particularly, the above-mentioned nonlinear feature extraction approaches to handle an individual feature as input, i.e., spectral features (Segl et al. 2003). A multiple-feature extraction technique to subordinate a probability-preserving projection structure to get more features from the data, i.e., t-SNE. A probability distribution is fabricated based on t-SNE for each feature (Devassy and George 2020).

To perform the non-destructive diagnosis to get significant information from the different bands in hyperspectral images, t-SNE is used. a t-SNE-based dimensionality reduction method to analyze the ink. Nonlinear equivalence features among the spectra are used to extract the features and measure them through a lower dimension t-SNE. To validate ink spectral information graphically and measurably, t-SNE is giving good results compared to other feature extraction methods (Devassy and George 2020). A modified stochastic neighbour embedding (MNSE) in favour of numerous feature dimensionality reduction. This will build a probability distribution function based on t-SNE for the respective feature. Compared to additional ongoing dimensionality reduction techniques, the intended approach has been utilizing a hyperspectral image’s spectral and spatial features. By using MNSE the hyperspectral image classification accuracy is also increased (Zhang et al. 2013).

Several studies have been conducted on the band selection strategy algorithms; a Minimum Noise band selection technique is suggested in (Bajwa et al. 2004). Depending on the characteristics of each band, the minimum noise method will work, that bands are represented by the high signal-to-noise ratio (SNR) and low correlation. A progressive band selection method has been proposed in (Ettabaa and Salem 2018), which differs from all the traditional band selection methods. This is going to be approving the more than a few picked bands; out of that, distinct endmembers are exploited for spectral unmixing. Researchers are focusing on optimization-based band selection methods to improve classification accuracy. For band selection global optimization algorithms are used; they are the GSA (gravity search algorithm) (Zhang et al. 2019), GA (genetic algorithms) (Saqui et al. 2016), PSO (particle swarm optimization) (Zhang et al. 2017), and FA (firefly algorithm) (Su et al. 2015). Due to low convergence speed, the gravity search algorithm does not satisfy the global search band selection strategy. Genetic algorithms have a more significant number of parameters; owing to that, the execution is complex; there is a possibility to fall into local optimum readily. This is also not giving good results for global search band selection. The band discovery rate is low when firefly algorithms are used. The accuracy rate is low, and the convergence speed is also very low. To overwhelm these issues, a contemporary heuristic algorithm is suggested. That is Grey Wolf Optimization (GWO), which was introduced in 2014. GWO has fewer modifications in parameters, high execution, and rapid convergence compared to other optimization methods, but it still has a few flaws. For example, while solving multimodal functions, the convergence speed is slow, as well as that is uncomplicated to descend victim to the drawbacks of local extremes (Wang et al. 2022). Because of this action, we have used the global search-based grey wolf optimization (GWO) band selection method to select the bands because GWO gives very few computational results compared with other existing optimization methods.

Semi-supervised band selection using an optimal graph (BSOG) is proposed in (Teng et al. 2022). It will perform the band selection as well as it will learn the local structuring bands. This intellectual similarity matrix is accommodative in contradiction to demonstrating the input similarity matrix to find out greater local configuration. A particular superlative band subset might be picked by assessing the gained projection matrix W. The proposed method is giving better results than the other existing methods. A modified Grey wolf optimization (MGWO) to select the number of bands from a scene is proposed in (Wang et al. 2021). The operations are performed based on the index of grey wolves. The intended technique is evaluating the variation of a particular band and calculating the worst fitness function. The suggested technique accomplished higher results for hyperspectral image classification than the other one. A band selection approach based on a modified cuckoo search optimization surrounded by correlations following the initialization procedure is demonstrated in (Sawant et al. 2019). The metaheuristic cuckoo search algorithm might drop into the local optimum solution. At the same time, they have initiated a strategy under interconnections to avoid the decline toward the local optimum solution.

Hyperspectral imaging is used to capture images; these images have rich spectral and spatial information. Spectral and spatial information is observed from the earth, and the generated image will be stored in the hyperspectral data cube. This data cube is 3D, two sides of the cube have spectral information, and one side has spatial information. The procedure of hyperspectral image classification is to establish the land-cover class of individual hyperspectral pixels, which are presented in hyperspectral images. As a consequence of the absence of accessibility of hyperspectral datasets that are available in public and the large size of the land cover classes, hyperspectral image classification is difficult. Since 2012, researchers have been focusing more on deep learning methods for LULC classification, and the classification results are promising (Otter et al. 2020). The deep learning techniques are enumerated towards spectra-spatial-based and spectral-based approaches. Spectral-based approaches are familiar with the spectral signatures of a single pixel for hyperspectral images. The spectral-spatial-based methods are used to explore the adjacent hyperspectral pixels of hyperspectral image classification. The convolutional neural network (CNN) as well as the fortunate deep learning algorithm are the most prevalent (Haq et al. 2023; Haq 2022c). Because it will use the hidden layers to extract the deep features for LULC classification (Lee and Kwon 2017). First, for HSI classification CNN is turned out to be used through various hidden layers. Then, to classify HSIs directly from their spectral domain, a deep convolutional neural network (deep-CNN) is employed. To withdraw the spectral features from hyperspectral images, 1D-CNN is exploited. But it involves input in a one-dimensional vector. 2D-CNN is also introduced for both spectral and spatial hyperspectral image classification, this combines the spectral and spatial features to give better results for classification. However, it is missing some information from spectral features (Roy et al. 2019).

Land use/ land cover (LULC) data obtaining is an essential stage because the source of information is utilized to receive the environmental variables, to improve the high-quality LULC maps LULC classification is used. Hyperspectral images have several issues: holding abundant spectral data, high dimensionality of information, and a smaller amount of training instances. Because of these issues, LULC classification is difficult. To overcome these issues, (Christovam et al. 2019) proposed supervised classification methods. They are random forest (RF), support vector machines (SVM), and spectral angular mapping (SAM). RF and SVM are determined along with 176 surface reflectance bands. PCA is used for the dimensionality reduction, and for classification, SVM and RF were used. The hyperspectral image classification has achieved good results using the SAM, SVM, and RF. To minimize the spectral shift produced by the adjacency factor, a correlation coefficient-weighted spatial filtering operation technique is suggested (Yang et al. 2022). To introduce the operation into the kernel collaborative representation method with Tikhonov regularisation (KCRT), the weighted spatial, spectral KCRT method is used to construct the land cover classes. The main problem of this proposed method is to label a pixel in hyperspectral images due to small-sized labeled samples.

An attention mechanism was also introduced to get high classification accuracy. Since CNN struggled to represent long-distance dependencies to gather global context data, the bulk of attention techniques for HSI classification now in use is based on the convolution layer. As a result, classification accuracy might be improved. Adding the channel attention mechanism to a squeeze-sand-excitation network (SENet) enhanced the classification accuracy (Hu et al. 2018). To enhance the feature maps using a squeeze operator and an excitation operation, a spatial-spectral squeeze-and-excitation network (SSSE) has been proposed. Additionally, including the attention mechanism is a well-known model, that can significantly enhance categorization performance. To compute the attention map, the sigmoid function, as well as the tanh function, was used (Wang et al. 2019). An extreme learning machine (ELM) ensemble method was proposed in (Su et al. 2017), to achieve good accuracy. By using correlation analysis, they have divided the spectral bands into Multiview. They have used random rotation to view the multiple coordinate spaces. Then, ensembled strategy pruning is designed for low complementary and, consequently, classification results.

Very recently, a model called transformers was introduced for hyperspectral classification. Transformers are developed for natural language processing (Mishra et al. 2023). Transformers will be working based on the self-attention mechanism (He et al. 2021). Transformers will be used attention to design a global dependency across a concatenation by time of input. To transform the input data from one sequence to another, a self-attention encoder and decoder are used. These decoder and encoder sequences are known as “tokens”, in the model, the tokens are represented as feature vectors in the primary data. The transformers are also used to extract the features from sequence data. Transformers will give the best weights to the initial data by using a multi-layer perceptron (MP). The transformer’s mapping considers it when a piece of exhaustive information is implemented to an image data. And it leads to a sudden intensification in the model’s size and a significant computational and training overhead. Therefore, image feature extraction is limited in transformers. To classify an image, we have various transformer models: vision transformers (ViT) (Palani and Loganathan 2023a, b; Aberna et al. 2023), Swin transformers (SwinT) (Agilandeeswari and Meena 2023), DeiT, and so on. The tokens are fixed scale, vision transformer will take all the tokens in a fixed size scale; because of this ViT is unstable for the vision applications. Swin transformers are also used for LULC classification, but their computational complexity will depend on the image size. And also, it is used to enable the development of hierarchical feature maps (Zhang et al. 2022). A multiscale convolutional embedding module with transformers for hyperspectral images is proposed in (Jia and Wang et al. 2022). To make use of the unlabelled samples, a self-supervised pre-training task is also developed. For the mask autoencoder, the proposed pre-training technique addresses the masks only on equivalent tokens away from the central pixel in the encoder.

The contribution of this paper is to classify the Land Use/Land Cover classification using hyperspectral images with a deep kernel attention transformer algorithm to get high classified accuracy.

The remaining paper is arranged like this, The proposed method is elaborated in Section II, the dataset description is given in Section III, evaluation metrics are explained in Section IV, and Section V demonstrates the experimental results followed by limitations of the proposed method and future scope. Finally, the conclusions are drawn in Section VI.

Proposed method

There has been a relatively new attempt to apply deep learning approaches to classify the land use/land cover (LU/LC) using hyperspectral images. However, there are still some difficulties in this area due to limited labels. To overwhelm above mentioned difficulties, a deep kernel attention transformer (DKAT) is suggested to classify the land use/ land cover (LULC) classification using hyperspectral images. First, we’re extracting the features from the dataset using t-distributed stochastic neighboring embedding (t-SNE); this will apply the probability function to get the features. To pick the informative bands from the extracted features, a grey wolf optimizer (GWO) is applied. Then we apply the classification method on the selected bands, i.e., deep kernel attention transformer (DKAT), to get the accurate classified results. Figure 1 shows the proposed framework of the DKAT-based land use/land cover (LULC) classification. In satellite image classification approaches, the single-feature extractors will recognize the majority of selective features. Without taking into consideration, the spatial features of an adjacent pixel, single-feature extractors want to investigate respective pixels one at a time through their spectral features. To obtain efficient data across all the pixels, extracting complicated characteristics based on each spectral and spatial region is crucial in LULC classification using hyperspectral images. Here are the reasons to use the t-SNE in this work; the uppermost significant point is that t-SNE holds the ability to maintain both global as well as local forms of the data. An additional purpose of t-SNE is that it will be used to designate the probability equivalence in connection with the high- to low-dimensional feature interval. Eventually, t-SNE presents good results for both nonlinear as well as linear data. The reason for selecting the GWO, it is a global optimization algorithm that can find the best possible solution for the band selection problem, even if it’s non-convex. GWO is indeed an efficient algorithm. It can find the global optimal solution in a reasonable amount of time, which makes it a great tool for solving complex problems. GWO’s robustness is one of its greatest strengths, and it makes it suitable for a wide range of applications, including band selection problems where the initial conditions can be tricky to determine. With GWO, we can have confidence that we will get accurate results regardless of the initial conditions. The reason for selecting the Deep Kernel Attention Transformer is that it is a recently proposed classifier model for hyperspectral images. Learning the long-range dependencies between pixels is critical for hyperspectral images since the spectral bands contain information about different parts of the image. To overcome these flaws, the DKAT model is proposed, which can take the features from a kernel attention mechanism to capture the dependencies. DKAT is an excessive model for hyperspectral image classification because it is robust to noise. Even though hyperspectral images can be noisy, DKAT can ignore the noise and focus on the important features of the image to achieve high accuracy. Its kernel attention mechanism is particularly useful for capturing spectral band information, making it a promising classifier for improving hyperspectral image classification.

Fig. 1
figure 1

The proposed architecture of DKAT-based classification

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t-SNE (t- distributed stochastic neighbouring embedding)

Vander Maaten and Hinton developed the t-SNE algorithm in 2008. This algorithm can convert the higher dimensional values into lower dimensions. The elaborated divergence of traditional SNE (stochastic neighbouring embedding) is t-SNE, it is designed to meet the needs of a single-feature nonlinear dimension diminution. This is going to work in familiar with the standardization about the halved expanses as long as a combined probability dissemination beyond the contribution of representative pairs. High-dimensional intervals among data points towards the conditional probabilities will be transformed initially by SNE. In such a way that it will portray the common features. The resemblance of \({X}_{k}\) to measurement point \({X}_{l}\) is announced through the conditional probability \({P}_{k/l}\). And also, the probability of \({X}_{l}\) will be identified as employing \({X}_{k}\) as its neighbour, if the neighbours are incorporated in the same way to their probability density subject to the Gaussian centered at \({X}_{k}\). It is represented by

$${P}_{k/l }=\frac{exp(-|\left|{X}_{k}-{X}_{l}\right|{|}^{2}/2{\sigma }^{2})}{{\sum }_{k\ne l }\mathrm{exp}(-|\left|-{X}_{k}-{X}_{l}\right|{|}^{2 }/2{\sigma }_{i}^{2})}$$
(1)

Here, the variance of the Gaussian is represented as \({\sigma }_{i}\), it is concentrated explicitly on datapoint\({X}_{k}\). Additionally, the t-SNE algorithm acknowledges the independent variables as the “perplexity”. It could be ascertained as an advance concerning more than a few efficient neighbours. Perplexity is determined scientifically as in the following equation

$$perp\left({P}_{i}\right)={2}^{{H(p}_{i})}$$
(2)

Here, \({H(p}_{i})\) is represented as the Shannon entropy, \({P}_{i}\) is represented to calculate in bits.

$${H(p}_{i})= -\sum_{k}{P}_{k/l }{log}_{2}{P}_{k/l}$$
(3)

This approach relies on the pairwise intervals among the points; it will spontaneously identify the variance of \({\sigma }_{i}\), it ensures that the adequate number of neighbours matches the perplexity input by the user. To prevent overcrowding, the student t-distribution is used by t-SNE. It utilizes just one degree of freedom. The allocation of the probability at a lower dimension is \({q}_{l/k}\), it can be mathematically defined in the below equation

$${q}_{l/k}= \frac{(1+||{y}_{l}-{y}_{k}|{|}^{2}{)}^{-1}}{\sum_{k\ne l}(1+||{y}_{k}-{y}_{l}|{|}^{2}{)}^{-1}}$$
(4)

The lower dimensional equivalents \({y}_{k}\) and \({y}_{l}\) of the high-dimensional data points \({X}_{k}\) and\({X}_{l}\). To calculate the conditional probability Eq. (1) is used. If the lower-dimensional equivalents of \({y}_{k}\) and \({y}_{l}\) are used to model the high-dimensional data points \({X}_{k}\) and\({X}_{l}\), then the conditional probability of \({P}_{k/l}\) and \({q}_{k/l}\) is equal. If a mismatch has occurred between the \({P}_{k/l}\) and \({q}_{k/l}\), SNE will find the lower-dimension representations to avoid the mismatch. To evaluate the correspondence among two distributions, Kullback-Leiber (KL) divergence is used. To underestimate the measurement of comprehensive data points of KL divergence, SNE will be using the gradient descent method. As long as the KL divergence is not symmetric, an immense volume of mistakes has been eventuated. To underestimate the KL divergence among the conditional distributions of \({P}_{k/l}\) and\({q}_{k/l}\), a joint probability distribution P is a single KL divergence in the lower-dimensional space. A joint probability distribution of Q in the lower dimension is diminished. The cost function has been provided as

$$C=KL (P|\left|Q\right)=\sum_{l}\sum_{k}{p}_{lk}\mathrm{log}\frac{{p}_{lk}}{{q}_{lk}}$$
(5)

where, \({p}_{lk}\) is given by in Eq. (6) and \({q}_{lk}\) can also be used the Eq. (4). Minimizing the cost function in Eq. (5) is now mentioned as a symmetric SNE.

The probabilities across the innovative distance are demonstrated in the following equation

$${P}_{l/k }= \frac{({P}_{l/k }+{P}_{k/l })}{2n}$$
(6)

Here, n represents the size of the dataset. Figure 2 illustrates the extracted features from the Pavia University and Salinas datasets.

Fig. 2
figure 2

a Pavia University, b Salinas datasets extracted features

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Algorithm
figure a

for Hyperspectral image classification using DKAT

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Band selection using Grey Wolf Optimisation

Band selection is used to choose the informative bands from the hyperspectral data cube. By using the combinational optimization problem, band selection is formulated. Depending on the class separability evaluators and the classification accuracy in this paper, we have proposed a new fitness function. We have selected a new meta-heuristic method designated as a grey wolf optimization (GWO) to optimize that fitness function. Meta-heuristic means, a high-level independent algorithmic framework (developed for optimization algorithms). Mirjaliali Mohammed and Lewis developed grey wolf optimization in 2014 from Griffith University, Australia. This meta-heuristic algorithm has been used to solve many different fields. It is developed by taking the inspiration of nature and is also related to physical phenomena, evolutionary concepts, or animal behaviour. GWO is stimulated through the social hierarchy and hunting procedure of grey wolves. These grey wolfs are from the candidate family. Grey wolves are established in an immensely structured pack, and the volume of the pack is 5 to 12. Various ranks have been given to the pack of wolves, they are Alpha, Beta, Delta, and Omega wolves. Figure 3 represents (a) the Hierarchy of grey wolf organization, and (b) the grey wolf hunting process.

Fig. 3
figure 3

a Hierarchy of Grey Wolfs, b Grey wolf hunting process

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Alpha wolf

The grey wolf family’s captain is called the alpha wolf. A male alpha wolf and a female alpha wolf will represent the pack’s leader. Other members will follow the instructions given by the Alpha. It is accountable for decision-making regarding sleeping places, hunting, time to wake up, and many others.

Beta wolf

A beta wolf is represented as a secondary stage of the grey wolf hierarchy. The superior contender to lead the wolves is this wolf. The alpha wolf will benefit from it for both decision-making and other purposes. The alpha wolf will receive input from the beta wolf.

Delta wolf

If the wolf is neither Alpha, Beta, nor Omega, it is Delta wolf. It dominates the omega wolf; to avoid the hazards, it will work for the pack. Delta wolf will deliver the food to the pack.

Omega wolf

The lowest order of grey wolf is the Omega wolf. This will play the challenge of a scapegoat (a victim who is blamed for the mistakes or faults of others). Scouts, hunters, elders, and caretakers will come under this category. They are allowed to eat last, to avoid the internal conflicts and difficulties in the pack it will work.

Working principle of grey wolf optimization

GWO algorithm impersonates grey wolves’ leadership and hunting mechanism.

The primary stages for grey wolf hunting are:

  1. 1.

    Piercing for the prey

  2. 2.

    Chasing, tracking, and approaching the prey

  3. 3.

    Encircling, pursuing, and harassing the prey until it stops moving

  4. 4.

    Attacking the prey

  • Step 1: piercing for the prey: grey wolf targets the prey

  • Step 2: if the selected prey runs away, the pack will chase the prey to kill. Example: once the target is entered into the territory, i.e., wolf territory. A group of animals defines Wolf territory against another.

Chasing animal (i.e., the prey/ target) to other waiting wolves.

The following are the specifications for the mathematical model of the grey wolf optimization (GWO) social hierarchy:

The method of hunting is regarded as an optimization process,

  • The optimum solution is determining the prey,

  • Fitness solution as an alpha wolf (α)

  • The second-greatest resolution is Beta wolf (β)

  • The third finest interpretation is Delta wolf (δ), and

  • Omega wolf (ω) follows these three wolves.

Mathematical model of encircling the prey

Throughout the hunting, grey wolves encircle the prey. Encircling behaviour is modeled as:

$$\overrightarrow{D}=|\overrightarrow{C}. \overrightarrow{{X}_{p}}-\overrightarrow{X}(t)|$$
(7)
$$\overrightarrow{X}\left(t+1\right)=|\overrightarrow{{X}_{p}}\left(t\right)-\overrightarrow{A}.\overrightarrow{D}|$$
(8)

Here, the current iteration is represented as t, and the location of the prey is determined as \(\overrightarrow{{X}_{p}}\), the position of the grey wolf is represented as \(\overrightarrow{X}\), coefficient vectors are given as \(\overrightarrow{A}, \overrightarrow{C}\), and \(\overrightarrow{D}\) is the distance of the vector.

\(\overrightarrow{A}, \overrightarrow{C}\) is calculated as:

$$\overrightarrow{A}=2\overrightarrow{a}. \overrightarrow{{r}_{1}}-\overrightarrow{a}$$
(9)
$$\overrightarrow{C}=2.\overrightarrow{{r}_{2}}$$
(10)

Here, the random vectors are represented as \(\overrightarrow{{r}_{1}}, \overrightarrow{{r}_{2}}\), ranges from [0,1], component \(\overrightarrow{a}\) is linearly decreased from 2 to 0 around reputations.

To improve the grey wolf location, the above equations are used. In accordance with the location of the grey wolf Eqs. (7) and (8) are various places approximately the best example of investigative agents might be able to concern about the contemporary location by regulating the standards of A & C. Equations (9), and (10) are used for calculating the vectors.

  • Step 3: grey wolf hunting: The grey wolf starts hunting by locating the prey. They will target the weak or elderly ones. Large animals like moose may stand on their ground and fight. Wolf may choose to try other prey rather than risk attacking large animals willing to fight.

Mathematical model of hunting the prey

The hunting process is directed by Alpha; it is assumed that \(\alpha ,\beta , \delta\) have enhanced understanding of the position of the prey (i.e., the optimum solution). Additional wolves will upgrade their positions according to the position of \(\alpha ,\beta , \delta\).

$$\overrightarrow{{D}_{\alpha }}=|{C}_{1}.\overline{{X }_{\alpha }}-\overrightarrow{X}(t)|$$
(11)
$$\overrightarrow{{D}_{\beta }}=|{C}_{2}.\overline{{X }_{\beta }}-\overrightarrow{X}(t)|$$
(12)
$$\overrightarrow{{D}_{\delta }}=|{C}_{3}.\overline{{X }_{\delta }}-\overrightarrow{X}(t)|$$
(13)
$$\overrightarrow{{X}_{1}}=|\overrightarrow{{X}_{\alpha }}-{A}_{1}.\overrightarrow{{D}_{\alpha }})|$$
(14)
$$\overrightarrow{{X}_{2}}=|\overrightarrow{{X}_{\beta }}-{A}_{2}.\overrightarrow{{D}_{\beta }})|$$
(15)
$$\overrightarrow{{X}_{3}}=|\overrightarrow{{X}_{\delta }}-{A}_{3}.\overrightarrow{{D}_{\delta }})|$$
(16)

Here, \({A}_{1}, {A}_{2}, {A}_{3}\) and \({C}_{1}, {C}_{2}, {C}_{3}\) are the coefficient vectors. \(\overrightarrow{{X}_{\alpha }}, \overrightarrow{{X}_{\beta }}, \overrightarrow{{X}_{\delta }}\) are the positions of the vectors of \(\alpha ,\beta ,\delta\).

The location of the grey wolf is upgraded by

$$\overrightarrow{X}\left(t+1\right)= {}^{\overrightarrow{{X}_{1}}+\overrightarrow{{X}_{2}}+\overrightarrow{{X}_{3}}}\!\left/ \!{}_{3}\right.$$
(17)

  • Step 4: killing the prey: The alpha wolf will terminate the hunt by attacking the prey. Once the prey has been hunted, the alpha wolf will eat first.

A mathematical model for attacking the prey

If the prey interrupts stimulating, the wolves should attack to complete the hunting. During the iteration, the model finishes by decreasing \(\overrightarrow{a}\) from 2 to 0. As \(\overrightarrow{a}\) decrease, \(\overrightarrow{A}\) also decreases. \(A<1\) forces the wolf to attack the prey. |A|> 1 diverse from prey and locate greater prey. C vectors are random values; interval ranges from [0,2]. C prevents placing a few additional weights on the prey to make it complicated for the wolves to find it.

Fitness function

The degree to which a particular design solution adheres to the stated objectives is summarised using a fitness function. It is a specific kind of objective function. In genetic programming and optimization problems, fitness functions are used to direct simulations toward the best possible design solutions. To select the bands using the grey wolf optimization, we are using the fitness function in three different ways; they are

  1. i.

    Correlation coefficient Measure: It is used to find the correlation between the bands. The correlation coefficient measure between two neighboring bands is utilized as the decision criterion to determine whether the two bands are substantially correlated. The two bands must be merged if this correlation exceeds a particular threshold, at which point they are deemed redundant. The correlation is calculated by using the below formula

    $$CorM({b}_{i},{b}_{j})=\frac{{\sum }_{{b}_{i},{b}_{j}}}{\sqrt{\sigma ({b}_{i})\sigma ({b}_{j})}}$$
    (18)

    Here, CorM \(({b}_{i},{b}_{j})\) is the correlation coefficient between the \({b}_{i}\; and\; {b}_{j}\) bands, \({\sum }_{{b}_{i},{b}_{j}}\) is the covariance between the \({b}_{i}\; and\; {b}_{j}\) bands, σ is the variance.

    The correlation coefficient between the band varies from band \({b}_{i}=-1\; and\; band\; {b}_{j}=+1\). If the correlation is close to + 1 and -1, it demonstrates the presence of a strong linear dependency between the two bands. And the two bands are supposed to be incorporated. Whereas 0 indicates no linear dependence.

  2. ii.

    Distance Calculation: Distance calculation is used to calculate the distance between bands. Here, we consider the minimum distance between the band and to band. To calculate the distance between the bands and measure the class separability for selected bands, the Jeffries-Matusita (JM) distance is used. If we are considering the two classes i and j, then the JM distance between the i class and j class is provided in the following equation

    $${J}_{ij}=\sqrt{2(1-{e}^{{B}_{ij}})}$$
    (19)

    Here, \({B}_{ij}\) is the Bhattacharyya distance. It is defined as

    $${B}_{ij}= \frac{1}{8}({m}_{i}-{m}_{j}{)}^{T}(\frac{{\sum }_{i}+{\sum }_{j}}{2})({m}_{i}-{m}_{j})+\frac{1}{2}In[\frac{|({\sum }_{i}+{\sum }_{j})/2|}{|{\sum }_{i}{|}^{1/2}|{\sum }_{j}{|}^{1/2}} ]$$
    (20)

    Here, \({m}_{i}, {m}_{j}\) are representing the mean vector of the classes and \({\sum }_{i},{\sum }_{j}\) are representing the class covariance.

To decide which bands to use in a binary classification task, the \({J}_{ij}\) distance is used. But for band assertions in a multi-class classification problem, we must discover the bands that give the average JM distance. The average distance is given in Eq. (21)

$$D({b}_{i},{b}_{j})=\sum\nolimits_{i=1}^{c}\sum\nolimits_{j=1}^{c}p\left({\omega }_{i}\right)p({\omega }_{j}){J}_{ij}$$
(21)

Here, \({J}_{ij}\) is JM distance between the α, β, δ, and ω wolves, i and j are bands. p \(\left({\omega }_{i}\right)\; and\; p({\omega }_{j})\) are representing the class prior probabilities. C represents the total number of classes.

  1. iii.

    Maximum information Entropy (MIE): To select the bands with maximum information to improve the classification accuracy, maximum information entropy (MIE) is used. The optimum probability distribution is the one with the maximum information entropy. Shannon entropy is the fundamental unit of information in information theory. Let R be a vector; its information quality can be quantified discretely

    $$E\left(R\right)=-{\sum }_{i}p({y}_{i}){log}_{2}p({y}_{i})$$
    (22)

    Here, E(R) is the entropy of R, \(p({y}_{i})\) is the probability and \({y}_{i}\) is the component of R.

    A straightforward way is directly collecting those features with high entropy to determine the most distinct feature subset, in which raw data information is taken notice of the maximum extent. If the feature subset has s features, the issue can be mathematically described as,

    $$MIE ({b}_{i},{b}_{j})= \frac{1}{s}\sum\nolimits_{i,j=1}^{s}E({R}_{i,j})$$
    (23)

    where \(E({R}_{i,j})\) is the correlation between the \({i}^{th}\) and \({j}^{th}\) bands.

Imagine, N grey wolves are inputted across the early population; this is going to be indispensable to establish an entropy of the particular grey wolf at the beginning of the population using the Eq. (22). correspondingly, to designate the grey wolves, the first three maximum entropies should be interchangeable, and are allocated as α, β, and δ wolf one after another. The leftover grey wolves are contemplated as ω wolves. Deficiency of the location of the prey might be useful to interchange the adequate location vector of the prey with the position of the α wolf.

Every grey wolf is accomplished to track down the prey and confine it. Surrounding the prey throughout the hunt can perhaps be the Eqs. (7) to (10) and setting a=2.

Consequently, every wolf is surrounded by the prey is essential to enhance at the right time in the context of the locations of α, β, and δ wolves to enclose the prey. The hunting process is fulfilled by calculating from Eqs. (11) to (17). Repeating the encircling and searching process frequently will lead to the finest result.

The fitness function for the grey wolf optimizer (GWO) is defined as

$${F}_{n}=CorM({b}_{i},{b}_{j})+{D}_{b}({b}_{i},{b}_{j})+MIE({b}_{i},{b}_{j})$$
(24)

Here, \({F}_{n}\) is the fitness function of α, β, λ and ω. \(CorM({b}_{i},{b}_{j})\) is representing the correlation coefficient between the bands \({b}_{i}\; and\; {b}_{j}\). \({D}_{b}({b}_{i},{b}_{j})\) is the distance calculation of the bands \({b}_{i}\; and\; {b}_{j}\). \(MIE({b}_{i},{b}_{j})\) is representing the maximum information entropy of the bands \({b}_{i}\; and\; {b}_{j}\). The selected band subset of the Salinas dataset is depicted in Fig. 4.

Fig. 4
figure 4

Selected band subset from Salinas dataset

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Deep kernel attention transformers

In this article, we have suggested a novel transformer-based model named Deep Kernel Attention Transformer (DKAT). Deep kernel attention transformer working flow is designed based on Visual Transformer (ViT). ViT is a model developed for image classification; it is used to apply transformers like architecture over image patches. Vision Transformers require partitioning the input image into patches of the same shape and vectorization of the patches. Vectorization means reshaping a tensor into a vector. If the patches are defined as \({d}_{1}\times {d}_{2}\times {d}_{3}\) tensors, and then the vectors \({d}_{1}{d}_{2}{d}_{3}\times 1\) dimensional vectors. After the image is split into n layers from \(\mathrm{1,2},3\dots \dots ..,n\), then we have to apply the dense layer. The output of the dense layer will be \({Z}_{1}= {Wx}_{1}+b\), here linear activation function is not applied. The dense layer is only the activation function. Here, W and b are the parameters; likewise, all the vectors will get the dense layer output. So, that dense layer has the same parameters. The Deep Kernal attention transformer (DKAT) and working procedure of DKAT are depicted in Fig. 5.

Fig. 5
figure 5

a Deep Kernel Attention Transformer (DKAT), and b Working procedure of DKAT

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In addition, we need to add the position encoding to the vectors \({Z}_{1}, {Z}_{2}, {Z}_{3},\dots \dots \dots ,{Z}_{n}.\) An input image is broken down into n patches; each patch has a position that is an integer between 1 to n. Positional encoding maps an integer to a vector, and the shape of the vector is the same as Z. Add the positional encoding vectors to the Z vectors. In this way, the ‘Z’ vector captures a patch’s content and position. \({X}_{1}\) to \({X}_{n}\) are the vectorization of n patches. Let vectors \({Z}_{1}\) to \({Z}_{n}\) be the best results of linear transformation and positional encoding. They are the representation of the n patches. They capture both the content and positions of the patches. Aside from the m patches, we use the cls token for classification. We are using the cls token because the output of the transformer is an embedding layer, it takes input as a cls token, and it will provide the output as \({Z}_{0}\). \({Z}_{0}\) has the shape of the other Z vectors. The output of the kernel Attention layers is a sequence of n + 1 vectors. And then adding another layer of kernel attention layer and dense layers. The kernel attention layer and dense layers constitute the transformer encoder network. The outputs are the sequence of n + 1 vectors denoting the kernel attention layers, feed-forward layers, and multi-layer perceptron (MLP). Network vector \({c}_{0,\dots \dots ,n}\) are the output of the transformer. To perform the classification task, we need only \({c}_{0}\). It is the feature vector extracted from the image. Feed \({c}_{0}\) into a SoftMax classifier. The classification results will be based on \({c}_{0}\). The classifier output vector is P. P’s shape, depending on the amount of classes in the input. For example, the Indian Pines dataset has 16 classes. So, vector P has 16 classes. These vectors also indicate the classification results. During training, we compute the cross entropy of vector P and the ground truth of the dataset. Then, we will add the gradient of cross-entropy loss concerning the model parameters and the performance of gradient descent to update the parameters. This allows DKAT to learn hierarchical context information and adjust hyperparameters like `N` to control the scope of self-attention calculation. Additionally, DKAT uses a hierarchical training strategy to improve its generalization capabilities, allowing it to learn more generalizable features. In the first stage of training, DKAT is trained on a small dataset of patches, and in the second stage, DKAT is fine-tuned on a larger dataset of patches. Table 1 shows the hyperparameter tuning of DKAT.

Table 1 Hyperparameter tuning of DKAT
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Dataset description

Dataset characteristics

When endeavouring to create a model for hyperspectral data cubes, one must take into account the various dataset characteristics that play a pivotal role in determining its performance and generalization. This is because of the factors such as the size of the hyperspectral data cubes, spatial and spectral resolutions, noise levels, and potential class imbalance can have a significant impact on the model’s ability to accurately and reliably perform in real-world scenarios. Through the careful optimization of these key factors, we can ensure that our model is capable of delivering accurate, reliable, and effective results in a variety of real-world applications. The dataset characteristics table is given in Table 2.

Table 2 Dataset characteristics
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Indian Pines (IP)

Using an AVIRIS sensor at a place with mixed vegetation in northwest Indiana, this dataset was collected. It includes 145 × 145 pixels, here individual pixel includes 224 spectral bands wavelengths extending from 0.4–2.5 µm. This IP dataset has 16 number classes. As an example, ‘Oats’ contain only 20 labeled samples, although ‘Soyabean-mintill’ has 2455 labeled pixels. In Fig. 6, (a) false-color composite (FCC) image, (b) ground truth reference map is shown, and (c) detailed classes, with their training and testing samples are described in Table 3. We can download the Salinas dataset by using the below link

Fig. 6
figure 6

a IP RGB image, b ground truth, c classes, d Salinas RGB image, e ground truth, f classes, g PU RGB image, h ground truth, i classes, j Botswana RGB image, k ground truth, l classes, and m KSC RGB image, n ground truth, and o classes

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Table 3 Explanation of the Indian Pines (IP) and Salinas dataset
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https://www.ehu.eus/ccwintco/index.php/Hyperspectral_Remote_Sensing_Scenes#Indian_Pines.

Salinas

Using AVIRIS (Airborne Visible/Infrared Imaging Spectrometer), the Salinas dataset was collected. The dimensions of every image are 512 × 217 pixels. It has a high spatial resolution of 3.7 m per pixel. This dataset encompasses 204 spectral bands, but some low signal-to-noise-ratio bands were eliminated. It also has 16 classes. In Fig. 6, (d) false-color composite (FCC) image, (e) ground truth reference map is specified as well as in (f) detailed classes with their training and testing samples are described. In Table 3, we can download the Salinas dataset by using the below link

https://www.ehu.eus/ccwintco/index.php/Hyperspectral_Remote_Sensing_Scenes#Salinas.

Pavia University (PU)

In the location of northern Italy in 2001, using a Reflective Optics System Imaging Spectrometer sensor this dataset was acquired. The aforementioned dataset capability is 610 × 340 pixels. This dataset has nine (9) urban land-cover classes, the spatial resolution of each image is 1.3 m per pixel, as well as the wavelength range is from 430 to 860 nm. Pavia university has 103 spectral bands. In Fig. 6, (g) false-color composite (FCC) image, (h) ground truth reference map are given as well as in (i) detailed classes with their training and testing samples are described in Table 4, and we can download Pavia University dataset from below link

Table 4 Description of Botswana, Kennedy Space Centre, and Pavia University dataset
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https://www.ehu.eus/ccwintco/index.php/Hyperspectral_Remote_Sensing_Scenes#Pavia_University_scene

Botswana

This dataset is collected using NASA EO-1 sensors in Okavango Delta, Botswana. This has 145 bands with a pixel resolution of 1476*256. In Fig. 6, (j) false-color composite (FCC) image, (k) ground truth reference map is given as well as in (l) detailed classes with their training and testing samples are described in Table 4, This dataset is accessible in the lower link, so download it freely.

http://www.ehu.eus/ccwintco/index.php/Hyperspectral_Remote_Sensing_Scenes#Botswana.

Kennedy Space Center (KSC)

This dataset is collected from the Kennedy space center in Florida using NASA AVIRIS sensors. This has 176 bands and 13 classes. 512*614 pixels in the wavelength 400-2500 nm of the electromagnetic spectrum. In Fig. 6, (m) false-color composite (FCC) image, (n) ground truth reference map are given as well as in (o) detailed classes, with their training and testing samples are described in Table 4. By using below link we can freely access the dataset

http://www.ehu.eus/ccwintco/index.php/Hyperspectral_Remote_Sensing_Scenes#Kennedy_Space_Center_.28KSC.29.

Evaluation metrics

To understand the efficiency of the suggested algorithm for LULC classification, there are some evaluation metrics. They are Kappa coefficient (KC), overall accuracy (OA), and average accuracy (AA). The techniques mentioned above are exploited to test the pixels and confusion metrics [\({T}_{{cc}{\prime}}\)] are used to precise the given classifier.

  1. 1.

    Overall accuracy: Overall accuracy refers to the percentage of pixels correctly categorized into all the pixels. It is defined as

    $$OA= \frac{1}{T}\sum\nolimits_{c=1}^{C}{T}_{cc}$$
    (25)

    Here, T represents the confusion matrix of the chosen classifier, and \({T}_{cc}\) is represents the number of testing pixels.

  2. 2.

    Average accuracy: Average accuracy calculates the average per-class classification accuracy, whereas the proportion of pixels in a specific class is correctly classified to all of the pixels in that class is known as per-class accuracy.

    $$AA=\boldsymbol{ }\frac{1}{C}\sum\nolimits_{c=1}^{C}\frac{{T}_{cc}}{\sum\nolimits_{{c}^{\prime}}^{C}{T}_{{cc}^{\prime}}}$$
    (26)

    Here, T represents the number of difficult pixels, and Tcc’ signifies the confusion matrix of a given classifier.

  3. 3.

    Kappa coefficient: The Kappa coefficient tries to fix OA by lowering its worth in the presence of an agreement that could be reached by chance.

    $$KC= \frac{\frac{1}{T}\sum_{c}{T}_{cc}-\frac{1}{{T}^{2}}({\sum }_{{c}{\prime}}{T}_{{cc}{\prime}})({\sum }_{{c}{\prime}}{T}_{{c}{\prime}c})}{1-\frac{1}{{T}^{2}}({\sum }_{{c}{\prime}}{T}_{{cc}{\prime}})({\sum }_{{c}{\prime}}{T}_{{c}{\prime}c})}$$
    (27)

Experimental results

The purpose is to assess the anticipated method’s accuracy, the outcomes are compared in this section to those of various land use/land change (LU/LC) classification algorithms that are currently in use. The existing methods used to compare the Pavia University, Indian Pines, and Salinas datasets are PCA-CNN (Chen et al. 2021), SVM (Chen et al. 2014), CNN-PPF (Li et al. 2016), GCN-CNN (Hong et al. 2020), DHC-Net (Zhu et al. 2018). Principal component analysis (PCA) is the dimensionality reduction technique for feature extraction and classification of convolutional neural networks (CNN). And the combination of the PCA-CNN gives better results for hyperspectral image classification. It’s a classic machine learning approach used for training on small datasets. For SVM, the kernel function is used. Cost value and gamma values are used for the kernel and combinedly selected to get the average performance of the classified output. A CNN framework is developed based on PPF. PPFs are employed to increase the number of training samples. This can compensate for the lack of training sample data. It stands for graph convolutional network-convolutional neural network. It is a new concatenated fusion framework. In this method, the extracted features from CNN have been given to the GCN classifier. The combined use of CNNs and GCNs gives more accurate results for Hyperspectral image classifications. A deformable convolutional neural network is used to get the convolutional sampling locations. The size and shape of the hyperspectral images are composite because of spatial circumstances. Classifying land use/ land cover (LULC) classes using hyperspectral images is difficult by a reason of a lower amount of training samples. To overcome this issue, we have proposed a deep kernel attention transformer (DKAT) for classifying the land use/ land cover (LULC) classes (Tables 5 and 6).

Table 5 Computational complexity of DKAT for different datasets
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Table 6 Execution time in seconds for Proposed vs. state-of-the-art models
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Interpretability

By understanding Deep kernel attention transformers (DKAT), we can unlock the potential for greater interpretability in our transformers. With kernels, we can measure the similarity between tokens in a sequence, providing us with valuable insights. DKAT takes this one step further, allowing us to easily visualize attention weights and comprehend why certain tokens are being emphasized. The traditional matrix approach can be clumsy, but with DKAT’s kernel set, we can simplify the visualization process. Imagine the impact this can have on real-world applications, where accuracy and reliability are paramount. Embrace the power of DKAT and unlock the full potential of our transformers. With the use of DKAT, not only we can enhance interpretability, but we can also simplify the process of explaining the model’s predictions. Traditional transformers can be challenging to comprehend as predictions are based on attention weights alone. However, DKAT goes a step further by taking into account kernels when making predictions. This allows us to identify the significant tokens for the prediction, leading to a clearer understanding of the model’s prediction. This aspect of DKAT is particularly helpful in real-world scenarios where predictions need to be explained.

Training data

The Deep Kernel Attention Transformer (DKAT) is a novel approach to hyperspectral image classification that overcomes the challenge of insufficient training data. DKAT leverages a kernel attention mechanism to capture extended relationships between pixels, resulting in enhanced accuracy in categorizing images even with a limited number of training examples. We have learned that the kernel attention mechanism is an effective method for capturing extended relationships between pixels in hyperspectral image classification. It works by first learning a set of kernels that represent different spatial relationships between pixels, which are then used to attend to different parts of the image. This helps to capture long-range dependencies that are often important for classification, and enhances accuracy even when there is a limited amount of training data available. DKAT involves learning a set of kernels that represent different spatial relationships between pixels, which can help capture long-range dependencies and improve accuracy, even with limited training data.

Comparison with other algorithms

To verify the classification performance of the proposed method, we have compared it with some of the state-of-the-art algorithms. The algorithms are PCA-CNN, SVM, CNN-PPF, GCN-CNN, are DHC-Net. Specifically, randomly we have taken 20% of training samples from each class to construct the training set, and the remaining samples are used to validate the effectiveness of the proposed method. By using the OA, AA, and KC we have obtained the per-class classification accuracy for the proposed method including state-of-the-art methods. For the Pavia University dataset, the proposed algorithm (DKAT) has been obtaining good classification accuracy. Compared with the SVM-based algorithms, the proposed method has achieved more OA, AA, and KC accuracy. When compared to the conventional CNN, the proposed algorithm yields higher accuracy. Figure 7 represents the Indian Pines dataset’s original image, ground truth image, and existing methods results, along with the suggested method. Table 7 depicts the classification performance of the IP dataset with existing methods, and also the proposed method. Figure 8 depicts the (a) overall accuracy (OA), (b) average accuracy (AA), and (c) kappa coefficient (KC) of the existing methods and the proposed method for the IP dataset. Figure 9 represents the Salinas dataset’s original image, ground truth image, and existing methods results, along with the suggested method. Table 8, classification performance of Salinas dataset with existing methods, and also proposed method. Figure 10 depicts the (a) overall accuracy (OA), (b) average accuracy (AA), and (c) kappa coefficient (KC) of the existing methods and the proposed method for the Salinas dataset. Figure 11 represents the PU dataset’s original image, ground truth image, and existing methods results, along with the suggested method. Table 9, classification performance of the Salinas dataset with existing methods, and also the proposed method. Figure 12 depicts the (a) overall accuracy (OA), (b) average accuracy (AA), and (c) kappa coefficient (KC) of the existing methods and the proposed method for the PU dataset. Figure 13 represents the KCS dataset’s original image, ground truth image, and existing methods results, along with the suggested method. Table 10 depicts the classification performance of the KSC dataset with existing methods, and also the proposed method. Figure 14 depicts the (a) overall accuracy (OA), (b) average accuracy (AA), and (c) kappa coefficient (KC) of the existing methods and the proposed method for the KSC dataset. Figure 15 depicts the (a) overall accuracy (OA), (b) average accuracy (AA), and (c) kappa coefficient (KC) of the existing methods and the proposed method for the Botswana dataset. Figure 16 represents the PU dataset’s original image, ground truth image, and existing methods results, along with the suggested method. Table 11 depicts the classification performance of the Salinas dataset with existing methods, and also the proposed method.

Fig. 7
figure 7

Classification Results of Indian Pines dataset a Benchmark dataset, b Ground Truth, c PCA-CNN (Chen et al. 2021), d SVM (Chen et al. 2014), e CNN-PFF (Li et al. 2016), f GCN-CNN (Hong et al. 2020), g DHC-Net (Zhu et al. 2018), and h Proposed DKAT

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Table 7 Classification performance of different algorithms for the Indian Pines (IP) dataset
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Fig. 8
figure 8

Classification performance of existing methods along with the proposed method of the Indian Pines dataset, a Overall accuracy, b Average accuracy, and c Kappa coefficient

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Fig. 9
figure 9

Classification results of Salinas dataset a Benchmark Data, b Ground Truth, c PCA-CNN, d SVM, e CNN-PPF, f GCN-CNN, g DHC-Net, and h proposed method

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Table 8 Classification performance of different algorithms for the Salinas dataset
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Fig. 10
figure 10

Classification performance of existing methods along with the proposed method of the Salinas, a Overall accuracy, b Average accuracy, and c Kappa coefficient

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Fig. 11
figure 11

Classification results of PU dataset a Benchmark Data, b Ground Truth, c PCA-CNN, d SVM, e CNN-PPF, f GCN-CNN, g DHC-Net, and h proposed method

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Table 9 Classification performance of different algorithms for the Pavia University dataset
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Fig. 12
figure 12

Classification performance of existing methods along with the proposed method of Pavia University, a Overall accuracy, b Average accuracy, and c Kappa coefficient

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Fig. 13
figure 13

Classification results of KSC dataset, a Benchmark Data, b Ground Truth, c PCA-CNN, d SVM, e CNN-PPF, f GCN-CNN, g DHC-Net, and h proposed method

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Table 10 Classification performance of different algorithms for the Kennedy Space Centre dataset
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Fig. 14
figure 14

Classification performance of existing methods along with the proposed method of KSC dataset a Overall accuracy, b Average accuracy, and c Kappa coefficient

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Fig. 15
figure 15

Classification performance of existing methods along with the proposed method of Botswana dataset a Overall accuracy, b Average accuracy, and c Kappa coefficient

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Fig. 16
figure 16

Classification results of Botswana dataset, a Benchmark Data, b Ground Truth, c PCA-CNN, d SVM, e CNN-PPF, f GCN-CNN, g DHC-Net, and h proposed method

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Table 11 Classification performance of different algorithms for the Botswana dataset
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Computational complexity

We have achieved low computational complexity for all the algorithms used in our work because we have done each part individually. First, we extracted the features separately for each dataset using the t-SNE algorithm. In this, the high-dimensional intervals among data points towards the conditional probabilities will be transformed initially by SNE. Kullback-Leiber (KL) divergence is used, to evaluate the correspondence among two distributions. So that, we will get the useful features from the data. Then, we selected the bands using GWO from the extracted features. In GWO, we have defined a fitness function to select the appropriate bands, which are useful for classification. Here, the use of the fitness function is to select the relevant bands. After selecting a number of bands, which are having more information, the bands will be sent to the classifier model i.e., DKAT. Then the classifier model will classify the LU/LC cover classes. It will give the classification map as well as OA, AA, and KC. We can also use this model for real-time applications and also for large-scale applications. But we have to select the number of features, and bands according to the problem with which we are going to work.

The time complexity of the DKAT model is calculated using the O(n*s*k). Here, n is the number of samples, s is the size of the image i.e., height, width, bands, and k is the attention heads. The Indian pines dataset has 1000 training samples, the image size is 100*100*10, and the number of attention heads in our model is 8, so the computational complexity of the Indian pines dataset is O(1000*100*100*10*8) = 10^8 The computational complexity for the deep learning models increases with the number of samples, the number of dimensions, and the number of hidden layers. The computational complexity of the DKAT model is shown in Table 5, and the computation time in seconds for different models is depicted in Table 6.

Limitations and future scope

The number of kernels used in DKAT can have a significant impact on the accuracy of its classifications. If the number of kernels is too small, it may not be able to capture the long-range spatial dependencies in the data, while if the number of kernels is too large, it may become too computationally expensive to train and run. It’s important to find the right balance to ensure optimal performance. The size of the kernels used in DKAT can also affect the classification accuracy. If the kernels are too small, then the model may not be able to capture the important features in the data. If the kernels are too large, then the model may be less sensitive to noise. It’s important to find the right balance to ensure optimal performance. The accuracy of DKAT’s classification can be affected by the quality of its training data. If the training data does not accurately represent the test data, DKAT may not perform well in generalizing.

In the future, classifying land use land cover using hyperspectral images with Multiple attentions in Deep kernel attention transformer (DKAT) is looking very promising. Even though it is a new method, DKAT has already shown impressive results. As research in this area progresses, DKAT may become even more capable and adaptable. One possible improvement is making the kernels more flexible to capture even more intricate spatial relationships. Another way to optimize the kernel attention mechanism is to reduce its computational cost and make it more efficient.

Conclusion

This article proposes a Deep Kernel Attention Transformer (DKAT) for land use/land cover (LU/LC) classification using hyperspectral images. To classify LULC, we have used two methods. One is feature extraction; the other is band selection. To extract the features from the land cover, t-distributed stochastic neighbouring embedding is used. Searching-based band selection using grey wolf optimizers (GWO) is used for selecting the informative bands from the dataset. After selecting the bands from the land cover, we have classified the land cover by using the DKAT. In DKAT, we have used multiple dense networks and kernel attention layers to enhance classification accuracy. We have calculated the accuracy using overall accuracy (OA), average accuracy (AA), and kappa coefficient (KC). The proposed method performance has been checked with five publicly available datasets, namely Indian Pines, Salinas, Pavia University, Botswana, and Kennedy Space Centre. For real-world scenarios, our proposed method gives robust results. In the real world, by using our proposed method can work with any dataset for classification. Coming to LU/LC change classification, we can know how much land cover classes have been spread or decreased around the land, and how much the land use classes have been used. As a result, we got the best classification accuracy compared with the state-of-the-art methods.