LowRank and Sparse Matrix Recovery for Hyperspectral Image Reconstruction Using Bayesian Learning
Abstract
:1. Introduction
 (1)
 The proposed method gives the structure of the covariance matrix of the L&S signals, models HSI data with the L&S structure, and utilizes the CS and Bayesian learning methods to compress and reconstruct HSI data.
 (2)
 In the reconstruction part, the proposed method makes use of the relationship between multiple dimensions of highdimensional data and combines data reconstruction with HSI data acquisition. It can be used to realize the segmented acquisition, compression, and transmission of HSI data so as to reduce the amount of calculation and data transmission that must be performed in the satellite.
 (3)
 We demonstrate the superior performance of the proposed method in comparison with stateoftheart alternatives by conducting experiments on both synthetic signals and real signals.
2. TwoDimensional Reconstruction Methods for HSI Data
 (1)
 Estimation of $\mathbf{x}$.After the Bayesian posterior probability is obtained with the Bayesian rule, the maximum a posteriori (MAP) is used to obtain the estimation of $\mathbf{x}$:$$\begin{array}{cc}\hfill \widehat{\mathbf{x}}=\mathrm{vec}\left({\widehat{\mathbf{X}}}^{\top}\right)\triangleq {\mathbf{\mu}}_{\mathbf{x}}& ={(\lambda {\mathsf{\Sigma}}_{0}^{1}+{\mathbf{H}}^{\top}\mathbf{H})}^{1}\mathbf{H}{\mathsf{\Sigma}}_{0}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& ={\mathsf{\Sigma}}_{0}{\mathbf{H}}^{\top}{(\lambda \mathbf{I}+\mathbf{H}{\mathsf{\Sigma}}_{0}{\mathbf{H}}^{\top})}^{1}\mathbf{y},\hfill \end{array}$$
 (2)
 Estimation of $\lambda $.In order to obtain $\lambda $, we simplify the expression of ${\mathsf{\Sigma}}_{0}$ as$${\mathsf{\Sigma}}_{0}=\mathsf{\Gamma}\otimes \mathbf{B}.$$After maximizing the logarithm of the joint probability of $\mathbf{x}$ and $\mathbf{y}$, we take the derivative of it with respect to $\lambda $:$$\lambda \leftarrow \frac{\parallel \mathbf{y}\mathbf{H}{\mathbf{\mu}}_{\mathbf{x}}{\parallel}_{2}^{2}+{\lambda}^{\left(pre\right)}\left[mn\mathrm{tr}\left({\mathsf{\Sigma}}_{\mathbf{x}}{\mathsf{\Sigma}}_{0}^{1}\right)\right]}{pn}.$$
 (3)
 Estimation of $\mathbf{B}$.$$\begin{array}{cc}\hfill \mathbf{B}\phantom{\rule{1.em}{0ex}}& =arg\underset{\mathbf{X}}{min}\mathrm{tr}\left[{\mathbf{B}}_{0}^{1}\left({\mathbf{XX}}^{\top}+{\nabla}_{{\mathbf{B}}_{0}^{1}}\right)\right]+mlog\left{\mathbf{B}}_{0}\right\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\frac{1}{m}\left(\widehat{\mathbf{X}}{\widehat{\mathbf{X}}}^{\top}+{\nabla}_{{\mathbf{B}}_{0}^{1}}\right).\hfill \end{array}$$$${\mathbf{B}}_{0}\leftarrow \frac{1}{m}\sum _{i=1}^{m}\frac{\left({\mathsf{\Sigma}}_{\mathbf{x}}^{i}+{\mathbf{\mu}}_{\mathbf{x}}^{i}{\left({\mathbf{\mu}}_{\mathbf{x}}^{i}\right)}^{\top}\right)}{{\gamma}_{i}^{2}},$$$${\nabla}_{{\mathbf{B}}^{1}}=\sum _{i=1}^{m}\mathbf{B}\mathbf{B}{\mathbf{H}}_{i}^{\top}{\left(\mathbf{H}{\mathsf{\Sigma}}_{0}{\mathbf{H}}^{\top}+\lambda \mathbf{I}\right)}^{1}{\mathbf{H}}_{i}\mathbf{B}.$$
 (4)
 Estimation of $\mathsf{\Gamma}$.The algorithm calculates the singular values of $\mathbf{X}$ through singular value decomposition (SVD) for each slice of the HSI data and sorts the singular values in descending order. Depending on the singular value of the sequence, we can obtain the value of the corresponding ${\gamma}_{i}$$(i=1,\cdots $, $m)$. The ${\gamma}_{i}$ corresponding to the larger singular value is 1, or is otherwise 0. For HSI data, a large singular value distribution can be obtained by analyzing only one segment of its 2D slice signal in advance. The expression of $\mathsf{\Gamma}$ is$$\mathsf{\Gamma}=\left[\begin{array}{cccc}{\gamma}_{1}{\gamma}_{1}& {\gamma}_{1}{\gamma}_{2}& \cdots & {\gamma}_{1}{\gamma}_{m}\\ {\gamma}_{2}{\gamma}_{1}& {\gamma}_{2}{\gamma}_{2}& \cdots & {\gamma}_{2}{\gamma}_{m}\\ \vdots & \vdots & \ddots & \vdots \\ {\gamma}_{m}{\gamma}_{1}& {\gamma}_{m}{\gamma}_{2}& \cdots & {\gamma}_{m}{\gamma}_{m}\end{array}\right].$$
 (5)
 Estimation of ${\mathbf{S}}_{opt}$.
2.1. TwoDimensional HSI Reconstruction Algorithm—L&SbSBL (1)
 (1)
 According to the different bands M, we use the tensor to expand with ${\mathbf{F}}_{1}1,\cdots $, ${\mathbf{F}}_{m}1,\cdots $, ${\mathbf{F}}_{M}1$;
 (2)
 let ${\mathbf{F}}_{m}1,m\in \{1,2,\cdots $, $M\}$ turn into ${\mathbf{F}}_{m},m\in \{1,2,\cdots $, $M\}$ by using the $\mathrm{vec}$ operator;
 (3)
 let $\mathbf{F}=[{\mathbf{F}}_{1}^{\top},{\mathbf{F}}_{2}^{\top},\cdots $, ${\mathbf{F}}_{M}^{\top}]$;
 (4)
 obtain the corresponding value of $\mathbf{F}$ in the DCT domain $\mathbf{X}\in {\mathbb{R}}^{M\times NQ}$.
2.2. TwoDimensional HSI Reconstruction Algorithm—L&SbSBL (2)
 (1)
 Let slices ${\mathbf{F}}_{q}\in {\mathbb{R}}^{M\times N\times 1}$ and $q\in \{1,2,\cdots $, $Q\}$ as ${\mathbf{F}}_{q}\in {\mathbb{R}}^{M\times N}$;
 (2)
 obtain the corresponding value of ${\mathbf{F}}_{q}\in {\mathbb{R}}^{M\times N}$ in the DCT domain ${\mathbf{X}}_{q}$, $q\in \{1,2,\cdots $, $Q\}$;
 (3)
 let ${\mathbf{X}}_{q}$ turn into ${\mathbf{x}}_{q}$, $q\in \{1,2,\cdots $, $Q\}$ by using the vec operator.
3. L&S Reconstruction Algorithm Combined with Acquisition Methods
3.1. Problem Formulation and Signal Model
3.2. Proposed Method
Algorithm 1 Proposed method 

3.3. Simulation Experiments
4. Discussion
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
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Zhang, Y.; Huang, L.T.; Li, Y.; Zhang, K.; Yin, C. LowRank and Sparse Matrix Recovery for Hyperspectral Image Reconstruction Using Bayesian Learning. Sensors 2022, 22, 343. https://doi.org/10.3390/s22010343
Zhang Y, Huang LT, Li Y, Zhang K, Yin C. LowRank and Sparse Matrix Recovery for Hyperspectral Image Reconstruction Using Bayesian Learning. Sensors. 2022; 22(1):343. https://doi.org/10.3390/s22010343
Chicago/Turabian StyleZhang, Yanbin, LongTing Huang, Yangqing Li, Kai Zhang, and Changchuan Yin. 2022. "LowRank and Sparse Matrix Recovery for Hyperspectral Image Reconstruction Using Bayesian Learning" Sensors 22, no. 1: 343. https://doi.org/10.3390/s22010343