A) Standard kernel

Suppose we have a set of HSI training samples $\varsigma = \{ ({\boldsymbol{x}_1},{y_1}),({\boldsymbol{x}_2},{y_2}), ..., ({\boldsymbol{x}_N},{y_N})\}$ with ${\boldsymbol{x}_i} \in {\Re ^d}$, where d denotes the dimensionality of a sample. The kernel measures the similarity between two samples. The most commonly used kernels are linear kernel, polynomial kernel, and RFB kernel. We choose the RBF kernel in our experiment due to its wide application for HSI classification, and the RBF kernel is calculated as

(1)\begin{equation} {\boldsymbol{K}_{ij}} = \boldsymbol{K}({\boldsymbol{x}_i},{\boldsymbol{x}_j}) = \exp \left( { - {{{{{\left \Vert {{\boldsymbol{x}_i} - {\boldsymbol{x}_j}} \right \Vert}^2}} \over {2{\delta^2}}}}} \right), \end{equation}

where $\delta$ is the band width of the RBF kernel.

B) Ideal kernel

The ideal kernel is defined as [Reference Pan, Lai and Shen20]

(2)\begin{equation} {\boldsymbol{T}_{ij}}{\rm =} \boldsymbol{T}({\boldsymbol{x}_i},{\boldsymbol{x}_j}) = \left\{ {\begin{array}{@{}c@{}} {1,{y_i} = {y_j}}\\ {0,{y_i} \ne {y_j}} \end{array}} \right.. \end{equation}

The ideal kernel is inspired by the idea that if two samples should be considered as “similar” if and only they belong to the same class. The ideal kernel has included the label information.

C) Ideal regularized kernel

The similarity measurement between samples in the standard kernel does not consider the labeled information, and a more desirable kernel can be learned by incorporating the labeled information. In [Reference Pan, Chen, Xu and Chen21], an ideal regularization (IR) learning framework is proposed, which can efficiently incorporate the label information into the standard kernel. The following equation learns a more desirable matrix K given the initial kernel ${\boldsymbol{K}_0}$ [Reference Pan, Chen, Xu and Chen21]

(3)\begin{equation} \mathop {\min} \limits_{\boldsymbol{K}\,{\succeq}\,0} (D(\boldsymbol{K},{\boldsymbol{K}_0}) + \gamma {\Omega} (\boldsymbol{K})), \end{equation}

where D represents the von Neumann divergence between $\boldsymbol{K}$ and ${\boldsymbol{K}_0}$, ${\Omega}$ is a regularization term, and $\gamma$ is the regularization parameter. $\boldsymbol{K}\,{\succeq}\,0$ indicates that $\boldsymbol{K}$ is a symmetric positive semidefinite matrix. The von Neumann divergence is expressed as

(4)\begin{equation} D(\boldsymbol{K},{\boldsymbol{K}_0}) = {\rm tr}(\boldsymbol{K}\log \boldsymbol{K} - \boldsymbol{K}\log {\boldsymbol{K}_0} - \boldsymbol{K} + {\boldsymbol{K}_0}), \end{equation}

where tr(•) represents the trace of a matrix and ${\Omega}$(K) is defined as ${\Omega} (\boldsymbol{K}) = - {\rm tr}(\boldsymbol{KT})$. The functionality of this term is to incorporate the label information into the standard kernel. Therefore, the final objective function is:

(5)\begin{equation} \mathop {\min} \limits_{\boldsymbol{K}\,{\succeq}\,0} {\rm tr}(\boldsymbol{K}\log \boldsymbol{K} - \boldsymbol{K}\log {\boldsymbol{K}_0} - \boldsymbol{K} + {\boldsymbol{K}_0} - \gamma {\rm tr}(\boldsymbol{KT})).\end{equation}

Taking the derivative regarding K and setting the equation to zero, the following equation for K is derived as

(6)\begin{equation} \boldsymbol{K} = \exp (\log {\boldsymbol{K}_0} + \gamma \boldsymbol{T}) = {\boldsymbol{K}_0} * \exp (\gamma \boldsymbol{T}), \end{equation}

where * represents the element-wise dot product between two matrices. Using the Taylor expansion of equation (6), it can be also expressed as:

(7)\begin{equation} \boldsymbol{K} = {\boldsymbol{K}_0} + \gamma {\boldsymbol{K}_0} * \boldsymbol{T} + {\textstyle{{{\gamma ^2}} \over {2!}}}{\boldsymbol{K}_0} * {\boldsymbol{T}^2} + .... \end{equation}

The above equation indicates that the IR kernel $\boldsymbol{K}$ can be considered as a linear combination of the standard kernel and the ideal kernels.

The out-of-sample extension is investigated for samples that are not encountered before. Denote $\boldsymbol{S} = {{{\rm K}}_0}^{ - 1}(\boldsymbol{K} + {\boldsymbol{K}_0}){\boldsymbol{K}_0}^{ - 1}$, then the kernel between the two new points $\boldsymbol{s}$ and $\boldsymbol{t}$ can be calculated as

(8)\begin{equation} \boldsymbol{K}(\boldsymbol{s},\boldsymbol{t}) = - {\boldsymbol{K}_0}(\boldsymbol{s},\boldsymbol{t}) + \sum\limits_{i,j = 1}^N {\boldsymbol{S}(i,j)} {\boldsymbol{K}_0}(\boldsymbol{s},{\boldsymbol{x}_i}){\boldsymbol{K}_0}({\boldsymbol{x}_j},\boldsymbol{t}). \end{equation}

D) Majority voting-based ensemble approach

We propose to use a majority voting-based ensemble approach to combine the output of SVM using the IR kernel with different types of features. Suppose we are using m features and ${f_i}(\boldsymbol{x})$ represents the output using the i-th feature, the final output is calculated as

(9)\begin{equation} \tilde{y} = {\rm mode}\{ {f_1}(\boldsymbol{x}),{f_2}(\boldsymbol{x}), ..., {f_m}(\boldsymbol{x})\}. \end{equation}

A) Experimental dataset

Three popular HSI datasets are used in our experiments.

1. (1) Indian Pines: This dataset is collected by the Airborne Visible and Infrared Imaging Spectrometer (AVIRIS) sensor over the Indian Pines site in June 1992. The spatial size is 145 × 145 with the spatial resolution of 20 m/ pixel. The 200 spectral bands are ranging from 400 to 2500 nm. After bad bands removal, 200 spectral bands remain. There are a total of 16 classes in the scene. The false color infrared of bands 50, 27, and 17 is shown in Fig. 2(a), and the groundtruth is shown in Fig. 2(b).

2. (2) University of Pavia: The second dataset is acquired by the Reflective Optics System Imaging Spectrometer (ROSIS) sensor, and the scene covers the University of Pavia with 115 spectral bands ranging from 0.43 to 0.86 µm. The spatial size is 610 × 340 with the spatial resolution of 1.3 m/pixel including nine classes. After removing the noisy band, 103 bands remain. Figures 3(a) and 3(b) show the color infrared composite of bands 60, 30, 2 and the groundtruth, respectively.

3. (3) Salinas: The third dataset is collected by the AVISIS sensor over Salinas Valley, California. The spatial size is 512 × 217 with the spatial resolution of 3.7 m/pixel. After removing the 20 water absorption bands, 200 bands remain. The color infrared composite of bands 47, 27, 13 and the groundtruth is shown in Fig. 4(a), and the groundtruth of this dataset is shown in Fig. 4(b). There are a total of 16 classes.

Fig. 2.

Indian pines dataset. (a) Color-infrared-composite of bands 50, 27, and 17. (b) Groundtruth.

Fig. 3.

University of Pavia dataset. (a) Color-infrared-composite of bands 60, 30, and 2. (b) Groundtruth.

Fig. 4.

Salinas dataset. (a) Color-infrared-composite of bands 47, 27, and 13. (b) Groundtruth.

B) Experimental setup

The number of training samples per class is denoted as N. In the experiment, N is chosen to be $\{3, 5, 7,\ldots,13\}$, and for those classes less than N samples, half of the total samples are chosen for training. The rest of the labeled samples are chosen as the testing set. Gaussian kernel is investigated, and the LIBSVM [Reference Chang and Lin33] is used to implement SVM. The number of PCs for the Gabor features, EMAP features, and LBP features are set to 10, 4, and 4, respectively [Reference Li and Du26–Reference Dalla Mura, Atli Benediktsson, Waske and Bruzzone28]. The EMAP features are generated according to [Reference Dalla Mura, Atli Benediktsson, Waske and Bruzzone28], and the parameters for LBP are set according to [Reference Li, Chen, Su and Du27]. Each experiment is repeated 50 times to avoid bias.

D) Parameters analysis

The parameters in this paper are analyzed in this section. For the SVM, we will first analyze the RBF kernel σ and the regularization term C first. For σ, it is chosen from the range {2⁻⁴, 2⁻³, …, 2⁴}. C is chosen in the range {1, 10, 100, 10⁵}. Figures 7(a)–7(c) show the OA with different σ and C for EMAP features on Indian Pines, University of Pavia, and Salinas, respectively, using 13 training samples per class. A holdout validation sample set is used for testing. It can be concluded from Fig. 7 that C does not affect the OA much especially when it is larger than 100, and any number above 100 will guarantee good performance. For σ, a relatively small number will generate good classification performance. In order to analyze the effect of the $\gamma$ parameter in IR, in the following experiment, C is set as 1000, and σ is chosen as 2⁻¹ for the EMAP features of different datasets to ensure good performance.

Fig. 7.

Parameters tuning using EMAP feature. (a) σ and C for Indian Pines. (b) σ and C for the University of Pavia. (c) σ and C for Salinas. (d) $\gamma$ for all datasets.

The parameter $\gamma$ plays an important role in the classification performance. We investigate the effect of $\gamma$ on the EMAP feature for Indian Pines, University of Pavia, and Salinas, respectively. Figure 7(d) shows the OAs of the three datasets with respect to different $\gamma$ in the range {1e⁻³, 5e⁻³, …, 5e⁻¹}. It can be seen that the OA stays stable when $\gamma$ is very small, and when $\gamma$ increases, the OAs of all three datasets will decrease. This may be due to the reason that the label information is dominant in the generated kernel while ignoring the spectral similarity between the training samples. In our experiment, we set C as 1000, σ as 2⁻¹, and $\gamma$ as 5e⁻³ for the EMAP features for three datasets. For other types of features, a similar parameter tuning process is adopted.