A) Kernel extreme learning machine

ELM [Reference Huang, Ding and Zhou9] is a random generation single-hidden layer feed-forward neural network. KELM is the evolution of ELM, which utilizes the mapping kernel function to replace the hidden layer of ELM. It shows a better performance compared with other methods.

Given N different training samples$\{\textbf{x}_i, {\boldsymbol{z}} _i\} _{i = 1}^N$, where ${\textbf{x}_i} = \left[ {{x_{i1}},{x_{i2}}, \ldots ,{x_{iD}}} \right] \in {{\mathbb R}^D}$, D stands for the number of the spectral bands or dimensionality of the HSI, L is the number of the classes, N represents the number of training samples. The row vector z_i = [z _i1, …z _ik…, z _iL] ∈ ℝ^L determines which class the sample belongs to, we define z _ik ∈ {0, 1} and 1 ≤ k ≤ L. If z _ik = 1 and any other elements of z_i are zero, the sample belongs to the kth class. The output function of the ELM having P hidden neurons defined as follows:

(1)\begin{equation} f(\textbf{x}_i) = \sum_{j = 1}^P {\boldsymbol{\upbeta} _j\cdot G(\boldsymbol{\upomega} _j\cdot \textbf{x}_i + b_j)} = \boldsymbol{z} _i,\quad i = 1,2,\ldots, N \end{equation}

where G(·) represents a non-linear activation function (e.g. RBF), $\boldsymbol{\omega} _j\in {\mathbb R}^D$ is the input weight vector connecting the jth hidden neuron and input neurons, $\boldsymbol{\upbeta} _j\in {\mathbb R}^L$ is the output weight vector connecting thejth hidden neuron and the output neurons, and b _j is the bias of the jth hidden neuron. $\boldsymbol{\upomega} _j\cdot \textbf{x}_i$ denotes the inner product of $\boldsymbol{\upomega} _j$ and $\textbf{x}_i$. With N samples, equation (1) can be briefly written as

(2)\begin{align} &\textbf{H}\cdot \boldsymbol{\upbeta} = \textbf{Z}\end{align}

\begin{align} &\textbf{H}(\boldsymbol{\omega} _1, \ldots, \boldsymbol{\omega} _P,b_1, \ldots, b_P, \ldots, \textbf{x}_1, \ldots, \textbf{x}_N)\notag\\ &\quad = \left[ {\begin{matrix} {G({\textbf{x}_1})}\\ \vdots \\ {G({\textbf{x}_N})} \end{matrix}} \right]\notag\\ &\quad = {\left[ {\begin{matrix} {G({\boldsymbol{\omega }_1} \cdot {\textbf{x}_1} + {b_1})}& \cdots &{G({\boldsymbol{\omega }_P} \cdot {\textbf{x}_1} + {b_P})}\\ \vdots & \ddots & \vdots \\ {G({\boldsymbol{\omega }_1} \cdot {\textbf{x}_N} + {b_1})}& \cdots &{G({\boldsymbol{\omega }_P} \cdot {\textbf{x}_N} + {b_P})} \end{matrix}} \right]_{N \times P}}\notag\end{align}

where

\begin{equation*} \boldsymbol{\beta } = {\left[ {\begin{matrix} {{\boldsymbol{\beta }_1}}\\ {{\boldsymbol{\beta }_2}}\\ \vdots \\ {{\boldsymbol{\beta }_P}} \end{matrix}} \right]_{P \times L}},\textbf{Z} = {\left[ {\begin{matrix} {{\textbf{z} _1}}\\ {{\textbf{z} _2}}\\ \vdots \\ {{\textbf{z} _L}} \end{matrix}} \right]_{N \times L}}.\end{equation*}

Equation (2) can be written as

(3)\begin{equation} \boldsymbol{\beta} = \textbf{H}^\dagger \textbf{Z}, \end{equation}

where $\textbf{H}^\dagger$ is the Moore–Penrose generalized inverse of $\textbf{H}$. It can be transformed into the form like $\textbf{H}^\dagger = \textbf{H}^T(\textbf{H}\textbf{H}^T)^{-1}$. In order to achieve more stability and generalized inverse classification result, a positive value ρ ⁻¹ is added to the diagonal elements of $\textbf{H}\textbf{H}^T$. Therefore, the output function of ELM classifier is expressed as follows

(4)\begin{equation} f(\textbf{x}_i) = G(\textbf{x}_i)\boldsymbol{\upbeta} = G(\textbf{x}_i)\textbf{H}^T\left( {\displaystyle{\textbf{I} \over \rho} + \textbf{H}\textbf{H}^T} \right)^{-1}\textbf{Z}, \end{equation}

where Z ∈ ℝ^N×L is the training samples label set, similar to SVM, ELM can be generalized to kernel ELM using a kernel trick. The activation function can be replaced by a kernel function; the output function of KELM is expressed as below

(5)\begin{equation} \begin{aligned} f({\textbf{x}_i}) &= G({\textbf{x}_i}){\textbf{H}^T}{\left( {\frac{\textbf{I}}{\rho } + \textbf{H}{\textbf{H}^T}} \right)^{ - 1}}\textbf{Z} \\ &= {\left[ {\begin{matrix} {{K_{MF}}({\textbf{x}_i},{\textbf{x}_1})}\\ \vdots \\ {{K_{MF}}({\textbf{x}_i},{\textbf{x}_N})} \end{matrix}} \right]^T}{\left( {\dfrac{\textbf{I}}{\rho } + {\textbf{K}_{MF}}} \right)^{ - 1}}\textbf{Z} \\ &= {\textbf{K}_{(\textbf{x}_{i})MF}}{\left( {\frac{\textbf{I}}{\rho } + {\textbf{K}_{MF}}} \right)^{ - 1}}\textbf{Z} \end{aligned} \end{equation}

where $\textbf{K}_{MF} = [K _{MF}(\textbf{x}_q,\textbf{x}_t)]_{q,t = 1}^N$, $\textbf{K}_{(\textbf{x}_i)MF}{\rm =} [K _{MF}(\textbf{x}_i,\textbf{x}_1), \ldots,\,K _{MF}(\textbf{x}_i,\textbf{x}_N)]$.

Note that the label of the test sample is determined by the index of the output node with the largest value. The traditional RBF kernel function is replaced by MF kernel in KELM, and we observe an excellent generation result by doing this.

B) Hyperspectral image classification based on mean filtering kernel extreme learning machine

Although the RBF kernel used in KELM has achieved promising classification performance, it does not consider the underlying data structure of HSI. In order to reflect data relations in a kernel, we adopt the MF kernel [Reference Huang, Ding and Zhou9] and incorporate it into the KELM, and the MF kernel computes the mean value of the spatial neighboring pixels in the kernel space to estimate the central pixel.

Given $\textbf{x}^m\in \Omega ^*$, m = 1, 2, …, ω ², and Ω* represents the spatial window, ω is the size of the window, and $\textbf{x}$ is the center pixel. Let us denote $\phi (\textbf{x}^m)$ as the image of $\textbf{x}^m$ under the map ϕ. Then, the MF can be represented as follows

(6)\begin{equation} MF(\phi (\textbf{x})) = \displaystyle{1 \over {\omega ^2}}\sum_{m = 1}^{\omega ^2} {\phi (\textbf{x}^m)}. \end{equation}

In order to describe the similarity between different pixels, we define the function like

(7)\begin{equation} \begin{aligned} {{\mathop{\rm K}} _{MF}}({\textbf{x}_i},{\textbf{x}_j}) &= \left\langle {MF(\phi ({\textbf{x}_i})),MF(\phi ({\textbf{x}_j}))} \right\rangle \\ &= \left\langle {\frac{1}{{{\omega ^2}}}\sum_{m = 1}^{{\omega ^2}} {\phi (\textbf{x}_i^m)} ,\frac{1}{{{\omega ^2}}}\sum_{n = 1}^{{\omega ^2}} {\phi (\textbf{x}_j^n)} } \right\rangle \\ &= \frac{1}{{{\omega ^4}}}\sum_{m = 1}^{{\omega ^2}} {\sum_{n = 1}^{{\omega ^2}} {{\mathop{\rm K}} (\textbf{x}_i^m,\textbf{x}_j^n)} } . \end{aligned} \end{equation}

Where $\textbf{x}_i$ and $\textbf{x}_j$ represent the two different pixels, respectively.

Algorithm 1 describes the combination of the MF and KELM methods, and we named it MF-KELM.

Image denoising is an image preprocessing work, which has been widely used in the field of HSI processing and has shown a good performance. The following part C is the introduction of bilateral filtering for HSI denoising based on MF-KELM.

C) Spectral band-subsets-wise Bilateral MF-KELM method

Considering the highly similar structural characteristics of neighborhood spectral bands in HSI, which is urgent to exploit the redundant structural information in terms of a better structure characteristics preservation. In this case, based on the structural similarity index metric [Reference Wang, Bovik, Sheikh and Simoncelli23], the spectral-adaptive band-subsets partition [Reference Lu and Li17] method is introduced as follow.

Let I = [I₁, I₂, …, I_D]^T represents an HSI cube, where I_i ∈ ℝ^P×Q is i th band of HSI. Then, we measure the structural similarity between spectral bands I_i and I_i+1 as shown in the following:

(8)\begin{equation} SSIM(\textbf{I}_i,\textbf{I} _{i + 1}) = \displaystyle{{(2\mu _{\textbf{I}_i}\mu _{\textbf{I}_{i + 1}} + C_1)(2\sigma _{\textbf{I}_i,\textbf{I}_{i + 1}} + C_2)} \over {(\mu _{\textbf{I} _i}^2 + \mu _{\textbf{I}_{i + 1}}^2 + C_1)(\sigma _{\textbf{I}_i}^2 + \sigma _{\textbf{I}_{i + 1}}^2 + C_2)}},\end{equation}

where $\mu _{\textbf{I}_i}(\mu _{\textbf{I}_{i + 1}})$ and $\sigma _{\textbf{I} _i}(\sigma _{\textbf{I} _{i + 1}})$ represent the mean value and the standard deviation of I_i(I_i+1), respectively, here, C ₁ and C ₂ are constants. When applying equation (8) on adjacent spectral bands, a correlation curve C would generate, where C(i) = SSIM(I_i, I_i+1). Shen et al. [Reference Shen, Xu and Li15] suggest the continuous spectral bands with high structural similarity correspond to relatively stable trend, while obvious drops existing in curve C demonstrate the adjacent spectral bands have much lower structural similarity. Thus, we according to the detection of sharp drops in C to achieve the band-subset partition. Let us define the band-subset as S _q, where S _q contains B _q spectral bands with the size of P × Q.

The bilateral filtering [Reference Tomasi and Manduchi22] defined as the combination of the range and domain filtering, which combines the nearby image values in a non-linear way, and preserves image features with a smooth image while preserving edges. Since the homogeneous regions are commonly contained by HSI, [Reference Shen, Xu and Li15] argues that for each pixel in a band-subset S _q, its neighboring pixels will likely share similar spectral characteristics or have the same class membership. Furthermore, the bilateral filtering, which has the ability to influence both the spatial distance and spectral dissimilarity relative to the center pixel, need to extend the general form to the vector form [Reference Peng and Rao21]. We denote S _q(a, b) as a pixel of S _q, where a and b are the spatial positions of S _q, we adopt the same bilateral filtering definition as provided in [Reference Tomasi and Manduchi22]:

(9)\begin{equation} \tilde{S}_q(a,b) = \displaystyle{{\sum_{(x,y)\in \Omega (a,b)} {S_q(x,y)\cdot w(x,y;a,b)}} \over {\sum_{(x,y)\in \Omega (a,b)} {w(x,y;a,b)}}}, \end{equation}

where S _q(x, y) stands for the neighboring pixel of S _q(a, b) within a spatial search window. A a × b matrix neighborhood Ω(a, b) is centered on the target pixel. The weight w(x,y; a, b) can be calculated as

(10)\begin{align} w(x,y;a,b) &= \exp\left( {\frac{{\left\| {{S_q}(x,y) - {S_q}(a,b)} \right\|_2^2}}{{2\sigma _r^2}}} \right)\notag\\ & \quad \times \exp\left( {\frac{{\left\| {x - a} \right\|_2^2 + \left\| {y - b} \right\|_2^2}}{{2\sigma _d^2}}} \right),\end{align}

σ _r and σ _d are the filter parameters and adaptive corresponding to different S _q.

After exploiting the spectral-spatial average information of each band-subset S _q via bilateral filtering, we merge all the band-subsets into a new three-dimensional data cube, and the cube will act as robust spectral signatures for the input of MF-KELM classifier to predict the final classification results. Figure 1 shows the flowchart of the proposed Bilateral MF-KELM method. Algorithm 2 is the complete description of Bilateral MF-KELM.

Fig. 1. The flowchart of the proposed Bilateral MF-KELM method.