Nonlinear Unmixing via Deep Autoencoder Networks for Generalized Bilinear Model

Abstract

Hyperspectral unmixing decomposes the observed mixed spectra into a collection of constituent pure material signatures and the associated fractional abundances. Because of the universal modeling ability of neural networks, deep learning (DL) techniques are gaining prominence in solving hyperspectral analysis tasks. The autoencoder (AE) network has been extensively investigated in linear blind source unmixing. However, the linear mixing model (LMM) may fail to provide good unmixing performance when the nonlinear mixing effects are nonnegligible in complex scenarios. Considering the limitations of LMM, we propose an unsupervised nonlinear spectral unmixing method, based on autoencoder architecture. Firstly, a deep neural network is employed as the encoder to extract the low-dimension feature of the mixed pixel. Then, the generalized bilinear model (GBM) is used to design the decoder, which has a linear mixing part and a nonlinear mixing one. The coefficient of the bilinear mixing part can be adjusted by a set of learnable parameters, which makes the method perform well on both nonlinear and linear data. Finally, some regular terms are imposed on the loss function and an alternating update strategy is utilized to train the network. Experimental results on synthetic and real datasets verify the effectiveness of the proposed model and show very competitive performance compared with several existing algorithms.

1. Introduction

Hyperspectral images (HSIs), containing spatial and spectral information, are generated by imaging spectrometers mounted on various space platforms [1]. With the development of hyperspectral imaging technology, modern hyperspectral sensors can obtain hundreds of continuous spectral bands and reach nanometer-level spectral resolution [2,3]. The hyperspectral image is widely used in the areas of target detection and recognition, forest mapping, mineral exploitation, and other applications, due to its high spectral resolution. However, due to the limited spatial resolution of HSIs, several materials may appear simultaneously in the area covered by a single pixel, known as ‘‘mixed pixel’’ [1,4]. Mixed pixels have a significant impact on hyperspectral classification [5,6,7,8], target detection [9,10,11], and matching [12,13,14]. Therefore, hyperspectral unmixing (HU) is a hot research topic, which aims to address the decomposition problem of “mixed pixels” in hyperspectral images [15].

Hyperspectral data usually assumes that the spectrum of each pixel is a combination of a set of pure materials called endmembers and the corresponding proportion of each endmember, called abundances [16]. The purpose of HU is to estimate the endmember spectral signatures and the value of fractional abundances [3]. The mixture model associated with HU can be linear or nonlinear, depending on the hyperspectral image under study. Due to simplicity and physical interpretability, the linear mixing model (LMM) is the most widely used model in HU. LMM assumes that each observed pixel is a linear combination of endmember spectral signatures [17,18]. For the linear spectral unmixing, numerous classical algorithms have been proposed, such as minimum volume constrained nonnegative matrix factorization (MVCNMF) [19], vertex component analysis (VCA) [20], variable splitting and augmented Lagrangian sparse unmixing (SUnSAL) [21,22], and fully constrained least squares (FCLS) [23].

Although the linear mixing model (LMM) has shown excellent properties for macroscopic mixtures, LMM may lose its applicability in some natural scenes where there are small scale distributions of ground targets, large topographical fluctuations, and intimate mixing of specific materials. In response to the intractability of the nonlinear mixing effects, many nonlinear mixing models (NMM) have been developed. The kernel-based method is commonly used in nonlinear unmixing tasks. In [24,25], the radial basis kernel function is used to realize the unmixing of nonlinear mixed pixels. With the help of this technique, nonlinear data can be mapped onto a high-dimensional space where the relationship between the data can be stated linearly. In addition, some nonlinear unmixing methods are based on physical mixture models, such as the Hapke model [26,27] and geometric optics model [28]. The Hapke model, a mixing model based on specific ground materials, is the most prevalent and significant nonlinear model. However, the Hapke model needs lots of prior knowledge of ground materials, which limits its effective applications in practice. To simplify the nonlinear mixture effects, the bilinear mixing model (BMM) only considers the second-order scattering of the spectrum, and ignores the effect of high-order scattering [29]. The classic BMMs include the Nascimento model (NM) [30], the Fan model (FM) [31], and the generalized bilinear mixing model (GBM) [32]. The NM considers the second-order interactions between various endmembers, and the bilinear interaction terms can be regarded as extra endmembers. The FM assumes that the magnitude of the second-order products is related only to the abundances. In [32], the GBM, a generalization of the LMM and FM, was proposed, and different algorithms were studied to estimate the parameters of this bilinear model. Thereafter, different improved methods have been proposed for the GBM unmixing of hyperspectral images. Halimi et al. proposed a constrained projection optimal gradient method [33] and Yokoya et al. adopted semi-nonnegative matrix factorization as an optimization method for the GBM unmixing [34].

With the advent of deep learning, learning-based methods have many applications in hyperspectral unmixing. In [35], the nonlinear mixture model based on support vector machines and neural network-based techniques was used to estimate the fraction of abundances. In [36], the authors implemented a model-free unmixing method based on an auto-associative neural network to realize the mapping from the input pixel to the abundance percentages. In [37], the authors designed an end-to-end network for unmixing, and the convolutional neural network was used for extracting spatial information to improve the accuracy of unmixing. However, these methods require known ground truth abundance that is usually generated by other time-consuming labeling methods. The quality of the ground truth label drastically influences the performance of unmixing.

To overcome the limitation of supervised methods, unsupervised blind unmixing methods based on an autoencoder network have developed rapidly. An autoencoder consists of the encoder and the decoder. The encoder can map the input to low-dimensional embedding, and the decoder tries to restore the original input from low-dimensional embedding. Generally, since abundances can be considered as a low-dimensional representation of hyperspectral data, autoencoder networks are ideal for spectral unmixing. In [38], a stacked autoencoder and a variational autoencoder were implemented for unmixing. The stacked autoencoder, which can mitigate the effects of outliers, extracted endmember spectral signatures to generate a good initialization for the variational autoencoder. In [39], Palsson et al. used a full convolutional autoencoder to extract spatial features of pixels in a hyperspectral image, to estimate endmembers and abundance. In [40], the authors combined a shallow autoencoder with generative adversarial network (GAN) for spectral unmixing, and GAN was employed to improve the quality of reconstructed pixels. However, these methods were all used to solve the linear unmixing problem. Few researchers have focused on the application of autoencoder networks for nonlinear unmixing. In [41], the authors regarded hyperspectral data as a combination of the linear mixing part and nonlinear fluctuations. A pre-training technique was utilized to realize the mapping of nonlinear fluctuations. However, it was not convincing to use the same pixels as input and output in the process of pretraining to achieve a mapping of nonlinear fluctuations. Moreover, most papers consider the nonlinear model as the addition of the linear part and the nonlinear part. In [42], two deep autoencoders were employed to model the linear mixture and the second-order scattering interactions, respectively. In addition, the multi-task learning strategy was used to optimize two autoencoders jointly. In [43], the authors imposed a specific component on both the encoder and decoder, and considered the mixing model as a nonlinear fluctuation over a linear mixture. The nonlinear part of these models is learned by neural networks. In our paper, we seek a reconstruction network that integrates a linear part and a nonlinear one. Considering the simplicity and effectiveness of the GBM, we propose a blind source unmixing model based on a specially designed autoencoder network (GBM_AE), which divides the decoder into the linear decoder and the nonlinear one based on the GBM, and utilizes a deep encoder to extract features. The amplitude of the nonlinear mixing part is obtained by network feedforward. The main parts of the work are as follows:

• Inspired by the widely used autoencoder network, we design a new deep autoencoder network structure based on the GBM, to achieve nonlinear unmixing. The deep encoder of the autoencoder is utilized to extract features. In the decoder part, the GBM is used to divide the decoder into a linear part and a nonlinear part. For the linear part of the decoder, we design a specific network layer to meet the constraints of ASC and ANC. For the nonlinear part of the decoder, the coefficients of nonlinear mixing terms are determined by a set of parameters, which can be learned during network training.
• To avoid overfitting, some regular constraint terms based on prior knowledge are added to the loss function. At the same time, we borrow some ideas from BCD’s (block-coordinate decent) method, and regard the optimization of nonlinear unmixing as two sub-problems. During the training process, the learnable parameters in the nonlinear decoder and the other part of the network are alternately trained. When training the linear decoder part, the parameters of the nonlinear decoder are fixed. The training for the nonlinear decoder is the same.
• Since the coefficients of the nonlinear parts are learned, the network can learn useful parameters adaptively for linear mixing data in HSI. To demonstrate the efficiency and superior performance of the proposed model, we conduct experiments on both linear and nonlinear synthetic data. In addition, we further verify the efficiency by using typical real HSIs.

The remaining sections of this paper are structured as follows: Section 2 introduces the generalized bilinear mixing model; Section 3 elaborates on the proposed network structure and implementation details; in Section 4, experiments on both simulated datasets and real HSIs are conducted, to demonstrate the performance of the proposed network; subsequently, a discussion is presented in Section 5; finally, the conclusion is presented in Section 6.

2. Generalized Bilinear Mixing Model

Notation: Scalars are represented by italic letters. Matrices and vectors are denoted by boldface uppercase and lowercase letters, respectively. Assume that HSI is a collection of $D$ mixed pixels. Let vector $\mathit{y}\in {\mathit{R}}^{L\times 1}$ represent an observed pixel with $L$ spectral bands, and $\mathit{E}=\left[{\mathit{e}}_1,{\mathit{e}}_2,\dots ,{\mathit{e}}_m\right]\in {\mathit{R}}^{L\times M}$ denote the endmember matrix with each column ${\mathit{e}}_i\in {\mathit{R}}^{L\times 1}$ being the $i^{\mathrm{th}}$ endmember spectral signature, where $M$ is the number of endmembers. $\mathit{a}={\left[a_1,a_2,\dots ,a_m\right]}^T\in {\mathit{R}}^{M\times 1}$ is the abundance vector associated with the corresponding pixel. This section provides a quick overview of their general forms, as well as the connection between the LMM and the GBM.

The LMM’s basic assumption is that the incident light only interacts with one material of the earth’s surface once before reaching the sensors [3]. In this case, each observed pixel $\mathit{y}\in {\mathit{R}}^{L\times 1}$ in HSI can be simply expressed as a linear combination of the endmembers, weighted by their associated abundances:

$$\mathit{y}=\mathit{E}\mathit{a}+\mathit{n}$$

(1)

where $\mathit{n}\in {\mathit{R}}^{L\times 1}$ denotes the additive noise in HSI. Since abundance has practical physical meaning, it has to meet two constraints, i.e., sum-to-one constraint (ASC) and nonnegativity constraint (ANC):

$${\displaystyle \sum }_{i=1}^M{a}_i=1,$$

(2)

$$a_i>0, i\in 1,\dots ,M$$

(3)

The LMM may not be adequate in some real-world scenes where the nonlinear mixture effects cannot be ignored. In these scenes, a nonlinear mixing model would be more reasonable than LMM [44]. Numerous nonlinear mixing models have been proposed for modeling the additional interactions among different materials. The bilinear mixing model (BMM) generalizes the linear mixing model by introducing second-order interaction terms, and successfully handles scattering effects occurring in the multilayered scene [15]. In this work, we consider the GBM as a foundation, which can be written as:

$$\mathit{y}=\mathit{E}\mathit{a}+{\displaystyle \sum }_{i=1}^{M-1}{\displaystyle \sum }_{j=i+1}^M{\gamma }_{ij}{a}_i{a}_j{\mathit{e}}_i\odot {\mathit{e}}_j+\mathit{n}$$

(4)

where $\odot$ is the Hadamard product operation, ${\mathit{e}}_i$ denotes the $i^{\mathrm{th}}$ endmember in matrix $\mathit{E}\in {\mathit{R}}^{L\times M}$ and $a_i$ denotes the fractional abundance of the $i^{\mathrm{th}}$ endmember. ${\gamma }_{ij}\in \left[0,1\right]$ is a nonlinear factor that quantifies the interactions between $i^{\mathrm{th}}$ endmember and $j^{\mathrm{th}}$ endmember, which makes the GBM more flexible than other BMMs. When ${\gamma }_{ij}=0$, the GBM degenerates into the linear mixing model, and when ${\gamma }_{ij}$ = 1, the GBM is the same as the form of the Fan model.

For simplicity of expression, the GBM for HSI with $D$ pixels can be mathematically expressed as follows:

$$\mathit{Y}=\mathit{E}\mathit{A}+F\mathit{B}+\mathit{N}$$

(5)

where $\mathit{Y}\in {\mathit{R}}^{L\times D}$ is the HSI data matrix; $\mathit{A}\in {\mathit{R}}^{M\times D}$ is the abundance matrix, which can be denoted as $\left[{\mathit{a}}^{\left(1\right)},{\mathit{a}}^{\left(2\right)},\cdots ,{\mathit{a}}^{\left(D\right)}\right]$ or $\left[{\mathit{a}}_{\mathbf{1}}^{\mathit{T}},{\mathit{a}}_{\mathbf{2}}^{\mathit{T}},\cdots ,{\mathit{a}}_{\mathit{M}}^{\mathit{T}}\right]$, and ${\mathit{a}}^{\left(\mathit{i}\right)}\in {\mathit{R}}^{M\times 1}$, ${\mathit{a}}_{\mathrm{i}}\in {\mathit{R}}^{D\times 1}$ are the abundance vector of the $i^{th}$ pixel and the fractional abundances of all pixels regarding the $i^{th}$ endmember; $\mathit{F}=\left[{\mathit{e}}_1\odot {\mathit{e}}_2,\cdots ,{\mathit{e}}_{M-1}\odot {\mathit{e}}_M\right]\in {\mathit{R}}^{L\times \frac{M\left(M-1\right)}{2}}$ denotes the bilinear endmember matrix, $\mathit{B}=\left[{\gamma }_{1,2}{\mathit{a}}_1\odot {\mathit{a}}_2,\cdots ,{\gamma }_{M-1,M}{\mathit{a}}_{M-1}\odot {\mathit{a}}_M\right]\in {\mathit{R}}^{\frac{M\left(M-1\right)}{2}\times D}$ denotes the bilinear coefficient matrix, and $N\in R^{L\times D}$ is the noise matrix. The constraints imposed on the GBM can be written as follows:

$$\mathit{A}\ge \mathbf{0},$$

(6)

$$\mathbf{0}\le {\mathit{B}}_{\left(i,j\right),k}\le {\mathit{A}}_{i,k}{\mathit{A}}_{j,k},$$

(7)

$${\displaystyle \sum }_{j=1}^M{\mathit{A}}_{\mathit{i}\mathit{j}}=1,i\in \left[1,D\right]$$

(8)

3. Proposed Model

In this section, we put forth a GBM-based autoencoder network, which introduces a nonlinear branch in the decoder by unfolding the nonlinear part of the GBM. The framework of the AE consists of two parts, namely the encoder and the decoder. The encoder $f_E$ is utilized for learning low-dimensional representation $\mathit{z}$ of the mixed pixel $\mathit{y}$. Then, the decoder $f_D$ uses the low-dimensional representation $\mathit{z}$ to reconstruct the original mixed pixel. We carefully designed the decoder, which has a linear part and a nonlinear part like the GBM. More details will be provided in the following subsection. Figure 1 shows the structure of the proposed autoencoder network based on the GBM.

3.1. Encoder

The deep encoder is utilized for compressing the high-dimensional input $\mathit{y}$ into low-dimensional feature $\mathit{z}$:

$$\mathit{z}=f_E\left(\mathit{y}\right)$$

(9)

where $f_E:{\mathit{R}}^{L\times 1}\to {\mathit{R}}^{M\times 1}$ represents a transformation.

Table 1. Structure of the Proposed Autoencoder Network for GBM.

	Layers		Activation	Units	Bias
Encoder	Input layer		-	L	No
	Hidden layer 1		Leaky ReLU	9∗M	No
	Hidden layer 2		Leaky ReLU	6∗M	No
	Hidden layer 3		Leaky ReLU	3∗M	No
	Hidden layer 4		Leaky ReLU	M	No
	Batch Normalization		-	M	-
Decoder	Linear part	ANC + ASC	-	M	-
		Linear output layer	-	L	No
	Nonlinear part	Hidden layer	$Ø$	M(M − 1)/2	No
		Custom layer	-	M(M − 1)/2	No
		Nonlinear output layer	-	L	No

contains the deep encoder’s structural information. The encoder has four hidden layers, each of which has fewer neurons than the one before it. The dimension of the last hidden layer is the number of endmembers, M. In the hidden layer, either Sigmoid, ReLU, or Leaky ReLU can be chosen as the activation function $\mathit{g}$. Currently, Leaky ReLU is employed in the model. The majority of autoencoders favor using no bias in spectral unmixing problem. We follow the no bias setting, since the experiment demonstrates that the bias has no influence on the outcome. The transformation in the hidden layer can be expressed as:

$${\mathit{h}}^{\left(\mathit{l}\right)}=\mathit{g}\left({\mathit{W}}^{\left(\mathit{l}\right)}{\mathit{h}}^{\left(\mathit{l}-1\right)}\right)$$

(10)

where ${\mathit{h}}^{\left(\mathit{l}\right)}$ denotes the output of the ${\mathit{l}}^{th}$ layer, ${\mathit{W}}^{\left(\mathit{l}\right)}$ is the learnable parameter in the ${\mathit{l}}^{th}$ layer.

At the end of the encoder, we perform Batch Normalization (BN) on the low-dimension representation. BN is a special neural network layer. In addition to accelerating the model’s rate of convergence, it can also alleviate “gradient dispersion”, making the training of neural network more stable [45]. Assume that $\mathcal{B}={\left\{{\mathit{h}}_i\right\}}_{\mathit{i}=\mathbf{1}}^{\left|\mathcal{B}\right|}$ is the batch data input for BN and $\left|\mathcal{B}\right|$ is the batch size. The output ${\mathit{z}}_i$ of BN can be represented as follows:

$${\mathit{z}}_i=B{N}_{\gamma ,\beta }\left({\mathit{h}}_i\right)=\alpha {\widehat{\mathit{h}}}_i+\beta$$

(11)

$${\mathit{\mu }}_{\mathcal{B}}=\frac{1}{\left|\mathcal{B}\right|}{\displaystyle \sum }_{i=1}^{\left|\mathcal{B}\right|}{\mathit{h}}_i,$$

(12)

$${\mathit{\sigma }}_{\mathcal{B}}^2=\frac{1}{\left|\mathcal{B}\right|}{\displaystyle \sum }_{i=1}^{\left|\mathcal{B}\right|}{({\mathit{h}}_i-{\mathit{\mu }}_{\mathcal{B}})}^2,$$

(13)

$${\widehat{\mathit{h}}}_i=\frac{{\mathit{h}}_i-{\mathit{\mu }}_{\mathcal{B}}}{\sqrt{{\mathit{\sigma }}_{\mathcal{B}}^2+ϵ}}$$

(14)

where ${\mathit{\mu }}_{\mathcal{B}}$ and ${\mathit{\sigma }}_{\mathcal{B}}^2$ are the mean and variance of the batch data $\mathcal{B}$, respectively. ${\widehat{\mathit{h}}}_i$ is the normalized data. $ϵ$ is a small positive value, used to avoid division error. To avoid simply whitening the data, BN introduces the learnable parameters $\alpha$ and β to scale and shift the normalized data. To sum up, through the forward computation of the encoder, the mixed pixel $\mathit{y}$ is compressed into a feature $\mathit{z}$, with the same dimension as the abundance vector.

3.2. Decoder

The work of data reconstruction in blind unmixing is handled by the decoder. Since what we are considering is the generalized linear model (GBM), the reconstruction process can be divided into a linear part and a nonlinear part.

3.2.1. Linear Decoder

The linear part of the decoder, which aims to simulate linear mixture, has two layers. Abundances have to meet the ANC and ASC constraints, so we utilize an ANC-ASC layer to implement the two constraints imposed on abundances. Assuming that the input vector of this layer is $\mathit{z}\in {\mathit{R}}^{M\times 1}$ and the output is $\mathit{a}\in {\mathit{R}}^{M\times 1}$, the operations implemented in this layer are as follows:

$${\widehat{\mathit{z}}}_i=max\left(0,{\mathit{z}}_i\right)$$

(15)

$${\mathit{a}}_i=\frac{{\widehat{\mathit{z}}}_i}{{\sum }_{j=1}^M{\widehat{\mathit{z}}}_j}$$

(16)

The ANC-ASC layer makes the output vector meet the two constraints of ANC and ASC, so the output $\mathit{a}$ can be treated as the abundance vector. We define the operation of this layer as $\mathscr{H}$, and so this layer can be expressed as:

$$\mathit{a}=\mathscr{H}\left(\mathit{z}\right)$$

(17)

The second layer of the linear decoder is a fully connected layer without bias, and the weight $W_D$ is equivalent to the endmember matrix $\mathit{E}$ in the GBM. The output of this layer can be expressed as:

$${\mathit{y}}^{lin}={\mathit{W}}_D\cdot \mathit{a}$$

(18)

where ${\mathit{y}}^{lin}$ is the output of the linear decoder, which can be regarded as the linear mixing component in the GBM. Therefore, the reconstruction of the linear part can be written as:

$${\mathit{y}}^{lin}={\mathit{W}}_D\cdot \mathscr{H}\left(\mathit{z}\right)$$

(19)

where $\mathit{z}$ denotes the output of the encoder.

3.2.2. Nonlinear Decoder

The nonlinear decoder consists of three layers: a hidden layer, a custom layer, and a nonlinear output layer. The role of the hidden layer is to learn the nonlinear factor $\mathit{\gamma }$ in the GBM. Therefore, the output of this layer has $M\left(M-1\right)/2$ neurons. The activation function ϕ can be expressed as:

$$\varphi \left(\mathit{x}\right)=min\left(1,max\left(0,\mathit{x}\right)\right)$$

(20)

Supposing that the weight of the hidden layer is ${\mathit{W}}_h$ and bias is ${\mathit{b}}_h$, then the output of the hidden layer can be written as:

$$\mathit{\gamma }=\varphi \left({\mathit{W}}_h\mathit{z}+{\mathit{b}}_h\right)$$

(21)

The custom layer in the nonlinear part requires the help of the output $\mathit{a}$ of the ANC-ASC layer in the linear part. The operations we implement in this layer are as follows:

$$\mathit{B}=\mathit{\gamma }.\star \widehat{\mathit{{\rm B}}}$$

(22)

$$\widehat{\mathit{{\rm B}}}=\left[{\mathit{a}}_1\odot {\mathit{a}}_2,{\mathit{a}}_1\odot {\mathit{a}}_3,\dots ,{\mathit{a}}_1\odot {\mathit{a}}_M,{\mathit{a}}_2\odot {\mathit{a}}_3,\dots ,{\mathit{a}}_{M-1}\odot {\mathit{a}}_M\right]$$

(23)

where $\widehat{\mathit{{\rm B}}}\in {\mathit{R}}^{\frac{M\left(M-1\right)}{2}\times 1}$ is the bilinear abundance matrix, calculated by the output $\mathit{a}$ of the ANC-ASC layer in the linear decoder. The output of the nonlinear hidden layer $\mathit{\gamma }$ performs the dot product operation with $\widehat{\mathit{B}}$. The output $\mathit{B}$ can be considered as the bilinear coefficient matrix in GBM. Let $\mathcal{g}$ denote the operation of the nonlinear custom layer, then this layer can be expressed as:

$$\mathit{B}=\mathcal{g}\left(\mathit{\gamma }\right)$$

(24)

The weight of the nonlinear output layer is not trainable, and is determined by the weight of the linear output layer in the linear part. This layer uses no bias, and the weight matrix ${\mathit{W}}_l$ is given by:

$${\mathit{W}}_l=\left[{\mathit{W}}_{D,1}\odot {\mathit{W}}_{D,2},{\mathit{W}}_{D,1}\odot {\mathit{W}}_{D,3},\dots ,{\mathit{W}}_{D,1}\odot {\mathit{W}}_{D,M},\dots ,{\mathit{W}}_{D,M-1}\odot {\mathit{W}}_{D,M}\right]$$

(25)

where ${\mathit{W}}_{D,i}$ denotes the $i^{\mathbf{th}}$ endmember’s spectra, and ${\mathit{W}}_l$ can be treated as the bilinear endmember matrix in GBM. The output ${\mathit{y}}^{nonlin}$ of this layer can be expressed as:

$${\mathit{y}}^{nonlin}={\mathit{W}}_l\mathit{B}$$

(26)

In general, the nonlinear decoder can now be written as:

$${\mathit{y}}^{nonlin}={\mathit{W}}_l\mathcal{g}\left( \varphi \left({\mathit{W}}_h\mathit{z}+{\mathit{b}}_h\right)\right)$$

(27)

Finally, the output of the decoder is the sum of the linear and nonlinear components. The reconstructed pixel $\widehat{\mathit{y}}$ can be written as:

$$\widehat{\mathit{y}}={\mathit{y}}^{lin}+{\mathit{y}}^{nonlin}$$

(28)

3.3. Loss Function

The traditional autoencoder in DL aims to restore the input data as much as possible, of which the loss function only considers the reconstruction error measured by the mean square error (MSE) of the input and output. However, this does not apply to the hyperspectral unmixing issue. It is well known that a complex nonlinear mixing model is very likely to yield regression overfitting, and obtain the wrong estimated endmembers and abundance. To avoid this, some regular constraint terms based on prior knowledge are imposed in the work. Therefore, we combine MSE with spectral angular distance (SAD) and $L_{1/2}$ sparse regularization terms as the loss function. The MSE and SAD function is given by:

$$J_{\mathrm{MSE}}\left(\mathit{Y},\widehat{\mathit{Y}}\right)=\frac{1}{2}{\Vert \mathit{Y}-\widehat{\mathit{Y}}\Vert }_2^2$$

(29)

$$J_{\mathrm{SAD}}\left(\mathit{Y},\widehat{\mathit{Y}}\right)=\frac{1}{D}{\displaystyle \sum }_{i=1}^D\arccos \left(\frac{⟨{\mathit{y}}_i,{\widehat{\mathit{y}}}_i⟩}{{\Vert {\mathit{y}}_i\Vert }_2{\Vert {\widehat{\mathit{y}}}_i\Vert }_2}\right)$$

(30)

where $\mathit{Y}\in R^{L\times D}$ denotes original mixed pixels, and $\widehat{\mathit{Y}}\in R^{L\times D}$ denotes reconstructed pixels. Prior knowledge indicates that the abundance of HSI is sparse, since the majority of pixels only contain one or two materials. The use of $L_1$ sparse regularization has become so widespread that it could arguably be considered the “modern least squares”. However, for many applications, the solutions of the $L_1$ regularization are often less sparse than expected. Therefore, we use a more powerful alternative approach $L_{1/2}$ regularization [4]. $L_{1/2}$ sparse regularization is used in the loss function, to constrain the output of the ANC-ASC layer in the linear decoder and guide the sparsity of abundance vectors. $L_{1/2}$ sparse regularization is given by:

$$J_{L_{\frac{1}{2}}}\left(\mathit{A}\right)=\frac{1}{M}{\displaystyle \sum }_{i=1}^M{\left|{\mathit{a}}_i\right|}^{\frac{1}{2}}$$

(31)

Now, we can provide the loss function formula as follows:

$$Loss\left({\theta }_e,{\theta }_{ld},{\theta }_{nld}\right)=J_{\mathrm{MSE}}\left(\mathit{Y}, \widehat{\mathit{Y}}\right)+\alpha ·J_{\mathrm{SAD}}\left(\mathit{Y}, \widehat{\mathit{Y}}\right)+\beta ·J_{L_{\frac{1}{2}}}\left(\mathit{A}\right)$$

(32)

where $\alpha$, $\beta$ are the hyperparameters, which can balance the relationship between reconstruction error and the regular constraint. ${\theta }_e$, ${\theta }_{ld}$, ${\theta }_{nld}$ are the learnable parameters in the encoder part, linear decoder part, and nonlinear decoder part, respectively.

As for the training of the network, the parameter ${\mathit{W}}_D$ in the linear output layer is initialized with the estimates for VCA methods instead of a random initialization. During training, we utilize BCD’s strategy, which solves the optimization problem by updating just one or a few blocks of variables at a time, rather than updating all the blocks together. Firstly, we fix the learnable weight ${\mathit{W}}_D$ in the first few epochs. With this technique, we can quickly obtain a somewhat decent abundance $\mathit{a}$ from the network, and begin the optimization process from a more trustworthy initialization. The parameters in the encoder and the nonlinear decoder are then trained alternately. We adopt an alternate update strategy, which can be express as follows:

$${\theta }_{nld}^{k+1}=\arg Loss\left({\theta }_e^k,{\theta }_{ld}^k,{\theta }_{nld}\right)$$

(33)

$${\theta }_e^{k+1},{\theta }_{ld}^{k+1}=\arg Loss\left({\theta }_e,{\theta }_{ld},{\theta }_{nld}^{k+1}\right)$$

(34)

When updating the parameters of the hidden layer in nonlinear decoder, the parameters of the other part are fixed. When optimizing the parameters of the linear decoder and the encoder, the parameters of the hidden layer in the nonlinear decoder part are fixed. The update applied to each block can be one or a few batch gradient descent steps. With the help of parameter freezing technology, this strategy can be easily realized.

Batch size is a crucial hyperparameter for neural network optimization. In practice, a large batch size can produce a training error equal to that of a small batch size, but the generalization error on the test set is worse [46]. This means that a smaller batch size can lead to a relatively better model. Meanwhile, the smaller the batch size, the fewer the epochs when the model converges to the minima [47]. From [48], Palsson et al. verified that too large a batch size usually resulted in a poor solution for the least represented endmember. Therefore, a relatively small batch size is chosen and tuned in different datasets for better performance. At the same time, the network is trained using the Adam optimizer.

4. Experiments

In this section, to verify the efficacy of the proposed approach, experiments are conducted on both synthetic data and real HSIs. The proposed approach, named GBM_AE, is compared with five typical and state-of-the-art unmixing algorithms. The comparison methods include: (1) a typical blind source unmixing algorithm with minimum volume constraint (MVCNMF) [19]; (2) endmembers extracted by VCA, and abundances estimated by the SUnSAL (VCA + SUnSAL); (3) a deep autoencoder network based on a linear mixing model (LinearAE) [48]; (4) initialization with VCA, a nonlinear mixing method based on a multilinear mixing model (VCA + MLM) [49]; (5) a deep autoencoder network based on a free model for nonlinear unmixing (FMAE) [41].

Four well known metrics are employed to evaluate the unmixing performance. For abundance, MSE (the mean square error of abundance) and AAD (the abundance angle distance) are selected as the evaluation metrics. MSE aims to evaluate the errors between all the element of the reference abundance vector ${\mathit{a}}_i$ and the estimated abundance vector ${\widehat{\mathit{a}}}_i$, and AAD is used to measure the structural similarity between ${\mathit{a}}_i$ and ${\widehat{\mathit{a}}}_i$. For endmembers, SAD (spectral angle distance) is utilized to evaluate the performance of extracted endmembers. In addition, the RE (reconstruction error) aims to perform reconstruction, which calculates the errors between observed and reconstructed spectra. These metrics are defined as:

$$\mathrm{MSE}=\frac{1}{DM}{\displaystyle \sum }_{i=1}^D{\Vert {\mathit{a}}_i-{\widehat{\mathit{a}}}_i\Vert }^2$$

(35)

$$\mathrm{AAD}=\frac{1}{D}{\displaystyle \sum }_{i=1}^D{\cos }^{-1}\left(\frac{{\mathit{a}}_i^T{\widehat{\mathit{a}}}_i}{\Vert {\mathit{a}}_i\Vert \Vert {\widehat{\mathit{a}}}_i\Vert }\right)$$

(36)

$$\mathrm{SAD}=\frac{1}{M}{\displaystyle \sum }_{i=1}^M{\cos }^{-1}\left(\frac{{\mathit{e}}_i^T{\widehat{\mathit{e}}}_i}{\Vert {\mathit{e}}_i\Vert \Vert {\widehat{\mathit{e}}}_i\Vert }\right)$$

(37)

$$\mathrm{RE}=\frac{1}{DM}{\displaystyle \sum }_{i=1}^D{\Vert {\mathit{y}}_i-{\widehat{\mathit{y}}}_i\Vert }^2$$

(38)

4.1. Experiments on Synthetic Data

In this subsection, two experiments are conducted on the synthetic data, to verify the effectiveness of the proposed GBM_AE. In order to make the synthetic data closer to the real HSIs, the synthetic data is generated by the following steps:

Firstly, a certain number of endmembers from the United States Geological Survey (USGS) digital spectrum library (Available here: http://speclab.cr.usgs.gov/spectral-lib.html (accessed on 1 February 2022)) are utilized to construct the endmember matrix $\mathit{E}$ of the synthetic data. Each endmember contains 224 spectral bands, which are distributed between 0.38 and 2.5 μm. Since we already know the endmember matrix $\mathit{E}$, the bilinear endmember matrix $\mathit{F}$ can be obtained easily. To generate the abundance matrix $\mathit{A}$, the method provided in the paper [50] is used, instead of using the Dirichlet distribution. From Figure 2, it can be found that the abundance produced by the method we use (Code is Available at: https://github.com/Growingg/Generate_data (accessed on 20 March 2022)) is sparser, and more similar to the abundance distribution of real HSI than that produced by the Dirichlet distribution. More specifically, we partition a known hyperspectral image into block regions, using super-pixel segmentation, and then randomly generate an abundance matrix that satisfies the ANC and ASC requirements for each block. The abundance matrix rank ranges from 1 to 4. Next, by setting the nonlinear coefficient $\mathit{\gamma }$ between 0 and 1, the bilinear coefficient matrix $\mathit{B}$ can be created. Finally, two kinds of noise are utilized to simulate the gaussian noise and impulse noise of real HSI:

• Additive white Gaussian noise: Add Gaussian noise with a signal-to-noise ratio (SNR) of 30 dB for all bands;
• Impulse noise: Add 20% Impulse noise to a 10% band, randomly selected.

4.1.1. Experiment on Synthetic Nonlinear Data and Linear Data

In this experiment, we synthesize both linear and nonlinear data with 6 endmembers using LMM and GBM under identical conditions. Figure 3 displays the six selected endmember spectral curves. For each synthetic image, a total of 10,000 pixels are generated. Both the rows and columns are 100. The parameters of the experiment are as follows: the input unit $L$ is 224 and the encoder output unit $M$ is 6. The learning rate is set to ${10}^{-4}$ and the batch size is 16.

Table 2. Comparison of different methods on Nonlinear Mixed Data. Best results are reported in bold.

	MVCNMF	VCA + SUnSAL	LinearAE	VCA + MLM	FMAE	Proposed
MSE	0.0115	0.0147	0.0142	0.0110	0.0111	0.0030
AAD	0.2764	0.2959	0.3511	0.2676	0.3211	0.1486
SAD	0.0721	0.0801	0.0731	0.0672	0.0918	0.0377
RE	0.0042	0.0031	0.0041	0.0068	0.0023	0.0019

reports the performance of different methods on nonlinear mixed data. It can be seen that the nonlinear unmixing methods, i.e., MLM, FMAE, and GBM_AE, are superior to other linear unmixing methods. The proposed GBM_AE produces better results on all metrics than the other five approaches. Figure 4 shows the comparison of the six endmember spectra extracted by GBM_AE with the ground-truth endmembers spectra. The endmember spectra extracted by GMB_AE are very similar to the ground truth, indicating that GBM_AE performs well in endmember extraction. For the abundance estimation, our method obtains the lowest MSE and AAD. Figure 5 provides the visual comparison of the abundance maps obtained by all considered methods and ground-truth abundance maps. It can be noted that GBM_AE’s abundance maps are sparser than those obtained by other unmixing methods. In addition, the smallest RE demonstrates that GBM_AE can accurately reconstruct pixels in this nonlinear dataset.

The proposed GBM_AE also performs well on linear mixed data. From

Table 3. Comparison of different methods on Linear Mixed Data. Best results are reported in bold.

	MVCNMF	VCA + SUnSAL	LinearAE	VCA + MLM	FMAE	Proposed
MSE	0.0112	0.0111	0.0107	0.0158	0.0120	0.0092
AAD	0.2736	0.2731	0.2956	0.3035	0.3137	0.2921
SAD	0.0587	0.0801	0.0643	0.0817	0.0911	0.0322
RE	0.0003	0.0012	0.0054	0.0035	0.0059	0.0018

, we observe that, except for GBM_AE, the linear unmixing methods, i.e., MVCNMF, SUnSAL, and LinearAE, outperform other nonlinear unmixing methods. The proposed GBM_AE achieves the smallest MSE and SAD. In addition, the RE of GBM_AE is also relatively small, compared with other nonlinear methods. The nonlinear part of GBM_AE can be adjusted by the coefficient $\gamma$, and a small $\gamma$ would be learned after training by linear data. This makes our method more flexible than other nonlinear unmixing methods and still achieves competitive performance on linear data. MVCNMF obtain the second smallest SAD, illustrating that the minimum volume constraint is beneficial for endmember extraction in the linear mixture scene.

4.1.2. Effect of the Endmember Number

In this subsection, we conduct a study to assess the effect of the endmember number on the unmixing performance. In the experiment, the number of endmembers in nonlinear synthetic data varies from 5 to 20. Figure 6 shows the performance of GBM_AE as a function of the number of endmembers. In terms of RE, as the number of endmembers ($M$) increases, the reconstruction error does not change much. In terms of MSE and SAD, it can be observed that when the number of endmembers is relatively large, the value of MSE and SAD just decreases slightly. This means that the unmixing performance of the proposed GBM_AE is not fatally affected. Regarding running time, it exhibits an approximate quadratic relationship with $M$. To sum up, the number of endmembers alone has little effect on the performance of GBM_AE, but the computational cost of GBM_AE is greatly affected by the number of endmembers.

4.2. Experiments on Real Data

4.2.1. Jasper Ridge

Jasper Ridge, a common hyperspectral data with 512 × 614 pixels, is widely used for hyperspectral processing. Each pixel is recorded at 224 channels with a spectral resolution of up to 10 nm, ranging from 400 nm to 2500 nm. In this experiment, a sub-image of 100 × 100 pixels with ground truth labeling is considered. The false color image of the chosen Jasper Ridge can be seen in Figure 7a. Removing the noisy bands that are contaminated with water vapor density and atmosphere, 198 channels are kept. There are four endmembers in this dataset: road, soil, water, and tree. The parameters of the experiment are as follows: the input unit $L$ is 198 and the encoder output unit $M$ is 4. The learning rate is set to ${10}^{-4}$ and the batch size is 20.

In the real HSIs, there is no doubt that the linear mixing and nonlinear mixing pixels exist at the same time. From

Table 4. Comparison of different methods on Jasper Ridge. Best results are reported in bold.

	MVCNMF	VCA + SUnSAL	LinearAE	VCA + MLM	FMAE	Proposed
MSE	0.0264	0.0241	0.0216	0.0234	0.0260	0.0185
AAD	0.4617	0.4300	0.4274	0.3246	0.4363	0.2134
SAD	0.1773	0.1726	0.3161	0.2413	0.1275	0.0869
RE	0.0029	0.0028	0.0030	0.0028	0.0022	0.0005

, it can be observed that the performances of other linear or nonlinear methods are very similar, while the proposed GBM_AE shows obvious improvement. For SAD, GBM_AE is significantly better than other methods, indicating the superior performance of GBM_AE in endmember extraction. Figure 8 shows the comparison of endmember spectra extracted by GBM_AE with the reference one. As can be seen, the road has a somewhat worse spectral signature than the tree, water, and dirt. The reason may be that the road material is underrepresented in the Jasper Ridge dataset. For abundance estimation, GBM_AE achieves the lowest MSE and AAD, reflecting the fact that abundance maps estimated by GBM_AE are closest to the references. To make a visual assessment, Figure 9 depicts the abundance maps estimated by different methods and the reference abundance maps. We observe that the abundance maps estimated by MVCNMF and SUnSAL are not sparse enough. In addition, the soil and tree maps of GBM_AE contain more edge and detailed information, and are closer to the references than other compared methods. For further observation, we also show the difference maps between the abundance maps estimated by different methods and the reference abundance maps in Figure 10. It is evident that the difference is mainly concentrated at the boundary, and GBM_AE has a small value in difference maps of tree and soil. For the road map, it can be seen that our method has sharper lines and textures than other methods. Additionally, GBM_AE and other methods work equally on water, and the road may be misidentified as water. Figure 11 depicts the maps of the nonlinear factor $\gamma$; the value of $\gamma$ differs in different areas because it is obtained according to the forward propagation of the pixel itself. Nonlinear interactions are more likely to occur in the zone of contact between distinct components [33], and in maps of $\gamma$, i.e., a road-tree map, road-soil map and water-soil map, the value is relatively larger in the boundary areas between distinct materials. This shows that adaptive gamma values can be learned from GBM_AE. In summary, GBM_AE has significant potential for the unmixing of real HSI where both linear and nonlinear mixtures exist simultaneously.

4.2.2. Urban

The Urban dataset has been extensively studied in hyperspectral unmixing. The original dataset consists of 307 × 307 pixels, with each pixel corresponding to a 2 × 2 m² ground area. There are 210 bands, ranging from 400 nm to 2500 nm, in each pixel. Because of the steam and air pollution, 162 bands are kept after removing the corrupted band. The false color image of Urban can be seen in Figure 7b. There are five materials in the observed scene: asphalt road, grass, tree, roof, and dirt. The experimental parameters are as follows: the input unit $L$ is 162 and the encoder output unit $M$ is 5. The learning rate is set to ${10}^{-4}$ and the batch size is 6.

Table 5. Comparison of different methods on Urban. Best results are reported in bold.

	MVCNMF	VCA + SUnSAL	LinearAE	VCA + MLM	FMAE	Proposed
MSE	0.0739	0.0573	0.0452	0.0864	0.0511	0.0336
AAD	0.8452	0.6702	0.6104	0.7525	0.5751	0.4951
SAD	0.3032	0.3641	0.2031	0.2667	0.1958	0.1908
RE	0.0061	0.0058	0.0050	0.0047	0.0015	0.0015

shows the performance of all the considered methods on Urban. It can be seen that learning-based methods, i.e., LinearAE, FAE, and GBM_AE, are significantly better than traditional methods in general. For the quality of extracted endmember, GBM_AE has the lowest SAD, and Figure 12 shows that the endmember spectra extracted by GBM_AE are very similar to the references. In the terms of MSE and AAD, the proposed GBM_AE outperforms other compared approaches, which reflects the fact that GBM_AE has great advantages in estimating abundance. Figure 13 presents the abundance maps estimated by all methods. For reconstruction, both FMAE and GBM_AE have the smallest RE, and the MLM also obtain smaller RE than the other linear methods. These results indicate that it may be more competitive to consider nonlinear mixing effects on Urban dataset.

4.2.3. AVIRIS Cuprite

The hyperspectral image used in this subsection is the Cuprite dataset of Nevada, which is obtained by the well-known Airborne Visible Infrared Imaging Spectrometer (AVIRIS). Due to high similarity in materials spectra, Cuprite is the most widely used and challenging benchmark for HU research [51]. It has 400 $\times$ 350 pixels and 224 wavelength bands, ranging from 0.36 to 2.48 μm. Because of the effects of water absorption and low SNR, a total of 188 spectral bands are left after removing bands 1–2, 105–115, 150–170, and 223–224. In the experiment, a subset of 250 $\times$ 191 $\times$ 188 is considered, as shown in Figure 14. Regarding the endmembers in this subset, the researchers have their own thoughts, resulting in different versions of ground truths. Here, the version of 12 endmembers are chosen as the reference endmember, as in some previous studies [1,52]. The materials in cuprite include alunite, andradite, buddingtonite, dumortierite, kaolinite1, kaolinite2, muscovite, montmorillonite, nontronite, pyrope, sphene, and chalcedony. The experimental parameters are as follows: the input unit $L$ is 188 and the encoder output unit $M$ is 12. The learning rate is set to ${10}^{-4}$ and the batch size is 30.

Since the reference abundance of cuprite is unavailable,

Table 6. Comparison of different methods on cuprite. Best results are reported in bold.

	MVCNMF	VCA + SUnSAL	LinearAE	VCA + MLM	FMAE	Proposed
SAD	0.1172	0.1212	0.1393	0.1052	0.1038	0.0937
RE	3.41 $\times {10}^{-3}$	4.01 $\times {10}^{-3}$	1.54 $\times {10}^{-3}$	1.07 $\times {10}^{-3}$	7.56 $\times {10}^{-4}$	3.05 $\times {10}^{-4}$

only shows the obtained SAD and RE results from different methods. It can be seen that GBM_AE outperforms other methods on both SAD and RE. Figure 15 shows the 12 endmember spectral signatures extracted by GBM_AE compared with the references. In general, the endmember spectral signatures obtained by GBM_AE match well with the references. The number of abundance maps obtained by algorithms is 12, but some materials have very little distribution on the HSI, and the corresponding abundance map is difficult to observe with the naked eye. For observation and analysis, we show only the abundance maps of four materials with relatively clear images in Figure 16. The corresponding materials from left to right are sphere, dumortierite, buddingtonite, and nontronite. The proposed GBM_AE provides clearer maps of several specific emphasis positions. In summary, these results show the effectiveness of GBM_AE in complex scenarios.

4.3. Computational Complexity

In this subsection, we discuss the running time of all the considered methods. The experiments are performed on the same computer configuration having an eight-core CPU (Intel Core i5 8300H) and 16 GB of memory.

Table 7. The average running time in second (s) for all methods on each dataset.

	Syn Linear	Syn Nonlinear	Jasper Ridge	Urban	Cuprite
MVCNMF	10	34	396	31	559
VCA + SUnSAL	1	1	6	1	5
LinearAE	68	66	230	28	181
VCA + MLM	128	98	487	37	54
FMAE	203	211	1175	176	774
GBM_AE	142	154	558	50	689

reports the average running time over 5 runs carried out on each dataset. It can be noted that deep-learning-based methods generally take more time than traditional methods. The FMAE has to consider the time of pretraining for the nonlinear part, so it takes the most time. The LinearAE and GBM_AE perform with the same epoch and batch sizes, and it can be found that the time cost of GBM_AE relative to LinearAE increases with the number of endmembers.

5. Discussion

It is generally known that the decoder of the autoencoder represents the spectral mixing process in HU tasks. Most of the autoencoder networks for blind unmixing are proposed for LMM without considering the nonlinearity of HSI, which will affect the unmixing accuracy in the real scene. In our work, we carefully designed a decoder based on the GBM, and promote the performance of unmixing.

Through experiments on synthetic nonlinear data, the proposed model outperforms comparison methods, which verifies the effectiveness of the proposed model. On synthetic linear data, even if not optimal on all metrics, the proposed method still shows competitive performance, because relatively small coefficient of the nonlinear part can be learned adaptively on linear mixing pixels to reduce the impact of nonlinear mixing part on spectral unmixing. This mean that it is an efficient extension of the linear model. In fact, not all materials have approximately linear reflective properties in real HSIs. Therefore, nonlinear and linear mixtures exist at the same time. In the experiments on real datasets, compared with traditional methods, GBM_AE borrows the powerful fitting ability and feature extraction ability of deep learning, and this helps to achieve a better performance. Compared with other nonlinear AEs, the proposed model considers only the second-order interaction, thus it is a relatively inexpensive nonlinear unmixing model. In addition, to avoid overfitting and obtaining the wrong estimated abundance and endmember, some regular constraint terms based on prior knowledge are constructed and added to the loss function of the network, e.g., SAD, and $L_{1/2}$. In addition, the decoder of the proposed model is completely constructed according to the GBM, and the nonlinear factors γ of second-order interaction terms are obtained by forward propagation of pixels, which means that interpretable γ can be learned according to the feature of the input pixel after training, which greatly improves the adaptation of the proposed model.

Additionally, the method still has the following problems to be solved: (1) the unmixing accuracy and computational efficiency can still be further optimized and improved; (2) the interpretability of the network needs to be explored further; (3) it can be extended by introducing effective constraints such as noise reduction and spatial information.

All in all, we have effectively combined a traditional nonlinear mixing model GBM with deep learning techniques for nonlinear unmixing. This type of unmixing method may be a trend towards solving the mixed pixel issue in the future.

6. Conclusions

In this paper, an autoencoder network based on GBM is proposed for unmixing problems. This method is applied to a general mixing model composed of a linear mixed component and a nonlinear mixed component, and the endmembers and abundance are obtained simultaneously. We implement two custom layers to meet the requirements of abundance constraints and bilinear interactions, respectively. In order to improve the performance of unmixing, SAD and $L_{1/2}$ sparse regularization terms are introduced into the loss function. In particular, compared with other nonlinear approaches, the nonlinear mixing component in the proposed model can be adjusted by a set of learnable parameters, which enables the method to perform well on linear data, too. The comparative study on synthetic data and real hyperspectral images verifies the superiority of the proposed method.

This proposed method still has space for improvement. For instance, the autoencoder’s feature extraction only takes a single pixel into account, and ignores the spatial structure of the hyperspectral image. Spatial information may positively promote the learning of nonlinear parts. To further improve performance, we may combine spectral-spatial information with unmixing techniques in the future.