A RPCABased ISAR Imaging Method for Micromotion Targets
Abstract
:1. Introduction
2. Signal Model and Problem Formulation
 1.
 Pointscattering model can be satisfied, i.e., the radar echo is assumed to be a sum of dominant scatterers.
 2.
 The radar echo satisfies the stop–go assumption, i.e., the target is assumed to be static during one pulse duration.
 3.
 The 2D imaging plane is unchanged in CPI.
 4.
 The translational motion is compensated completely, thus, the target is equivalent to rotate around the image center, which indicates that the target can be stated as a turntable model.
 5.
 The change of aspect angle of the target is so small that the instantaneous range can be approximated by its firstorder Taylor expansion.
 6.
 The range migration among the scatterers is so small that it can be ignored in CPI.
2.1. Signal Model
2.2. Preliminary
2.3. Proposed Optimization Problem
3. Proposed Algorithms
3.1. Algorithm 1
 1.
 solution for (13)To solve the optimization problem (13), the ADMM method is employed, and the main procedures are derived in the following. For the sake of simplicity, the superscripts $\left(\mathrm{k}\right)$ and $\left(\mathrm{k}+1\right)$ are omitted.To apply the ADMM method, introducing an auxiliary variable $\tilde{\mathbf{X}}\in {\mathbb{C}}^{K\times N}$ and the Lagrange multiplier $\mathbf{dX}\in {\mathbb{C}}^{K\times N}$ is required. Then we split the variable $\mathbf{X}$ as $\mathbf{X}=\tilde{\mathbf{X}}$, having the augmented Lagrangian function as$$\begin{array}{c}J=\underset{\phantom{\tilde{\mathbf{X}}}\mathbf{dX}}{\mathrm{max}}\underset{\mathbf{X},\tilde{\mathbf{X}}}{\mathrm{min}}{\lambda}_{x}\Vert \tilde{\mathbf{X}}{\Vert}_{1}+0.5{\Vert {\mathbf{S}}_{\mathbf{X}}\mathbf{FX}\Vert}_{\mathrm{F}}^{2}\\ +0.5\beta \Vert \mathbf{X}\tilde{\mathbf{X}}+\mathbf{dX}/\beta {\Vert}_{\mathrm{F}}^{2}\end{array}$$For the problem (15), we alternately solve the following subproblems as$$\begin{array}{c}\mathbf{X}:=\underset{\mathbf{X}}{\mathrm{argmin}}\frac{1}{2}{\Vert {\mathbf{S}}_{\mathbf{X}}\mathbf{FX}\Vert}_{\mathrm{F}}^{2}\end{array}$$$$\begin{array}{c}+\frac{\beta}{2}{\Vert \mathbf{X}\tilde{\mathbf{X}}+\mathbf{dX}/\beta \Vert}_{\mathrm{F}}^{2}\end{array}$$$$\begin{array}{c}\tilde{\mathbf{X}}:=\underset{\tilde{\mathbf{X}}}{\mathrm{argmin}}{\lambda}_{x}\Vert \tilde{\mathbf{X}}{\Vert}_{1}+\frac{\beta}{2}{\Vert \mathbf{X}\tilde{\mathbf{X}}+\mathbf{dX}/\beta \Vert}_{\mathrm{F}}^{2}\end{array}$$$$\begin{array}{c}\mathbf{dX}:=\mathbf{dX}+\beta (\mathbf{X}\tilde{\mathbf{X}})\end{array}$$Problem (16) involves a quadratic cost and leads to a closedform solution, which can be obtained by setting the firstorder derivative of its objective function with respect to $\mathbf{X}$ as zero. We obtain$$\begin{array}{c}\hfill \mathbf{X}:={\left({\mathbf{F}}^{\mathrm{H}}\mathbf{F}+\beta \mathbf{I}\right)}^{1}\times \left({\mathbf{F}}^{\mathrm{H}}{\mathbf{S}}_{\mathbf{X}}+\beta \tilde{\mathbf{X}}+\mathbf{dX}\right)\end{array}$$Problem (17) has a closed solution involving ${\ell}_{1}$ norm shrink operator [31]:$$\begin{array}{c}\hfill \tilde{\mathbf{X}}:=shrink\left(\mathbf{X}+\mathbf{dX}/\beta ,{\lambda}_{x}/\beta \right)\end{array}$$
 2.
 solution for (14)For the problem (14), the splitting variable $\mathbf{D}=\tilde{\mathbf{D}}\in {\mathbb{C}}^{K\times N}$ and the Lagrange multiplier $\mathbf{dD}\in {\mathbb{C}}^{K\times N}$ are required. Then we have the augmented Lagrangian function as:$$\begin{array}{cc}\hfill J& =\underset{\phantom{\tilde{\mathbf{D}}}\mathbf{dD}}{\mathrm{max}}\underset{\mathbf{D},\tilde{\mathbf{D}}}{\mathrm{min}}{\lambda}_{d}\Vert \tilde{\mathbf{D}}{\Vert}_{*}+0.5{\Vert {\mathbf{S}}_{\mathbf{D}}\mathbf{FD}\Vert}_{\mathrm{F}}^{2}\hfill \\ \hfill \hspace{1em}& +0.5\tau \Vert \mathbf{D}\tilde{\mathbf{D}}+\mathbf{dD}/\tau {\Vert}_{\mathrm{F}}^{2}\hfill \end{array}$$For problem (21), we alternately solve the following subproblems:$$\begin{array}{c}\mathbf{D}:=\underset{\mathbf{D}}{\mathrm{argmin}}\frac{1}{2}{\Vert {\mathbf{S}}_{\mathbf{D}}\mathbf{FD}\Vert}_{\mathrm{F}}^{2}\end{array}$$$$\begin{array}{c}+\frac{\tau}{2}{\Vert \mathbf{D}\tilde{\mathbf{D}}+\mathbf{dD}/\tau \Vert}_{\mathrm{F}}^{2}\end{array}$$$$\begin{array}{c}\tilde{\mathbf{D}}:=\underset{\tilde{\mathbf{D}}}{\mathrm{argmin}}{\lambda}_{d}\Vert \tilde{\mathbf{D}}{\Vert}_{*}+\frac{\tau}{2}{\Vert \mathbf{D}\tilde{\mathbf{D}}+\mathbf{dD}/\tau \Vert}_{\mathrm{F}}^{2}\end{array}$$$$\begin{array}{c}\mathbf{dD}:=\mathbf{dD}+\tau (\mathbf{D}\tilde{\mathbf{D}})\end{array}$$Problem (22) has a closed solution, which is represented as$$\begin{array}{c}\hfill \mathbf{D}:={\left({\mathbf{F}}^{\mathrm{H}}\mathbf{F}+\tau \mathbf{I}\right)}^{1}\times \left({\mathbf{F}}^{\mathrm{H}}{\mathbf{S}}_{\mathbf{D}}+\tau \tilde{\mathbf{D}}+\mathbf{dD}\right)\end{array}$$The problem (23) involves a nuclear norm minimization problem, which can be solved by SVT computation in [30]:$$\begin{array}{c}\hfill \tilde{\mathbf{D}}:=svt\left(\mathbf{D}+\mathbf{dD}/\tau ,{\lambda}_{d}/\tau \right)\end{array}$$The whole algorithm is summarized in Algorithm 1 (Microdoppler Extraction based on RPCA).
Algorithm 1 MERPCA. 

3.2. Algorithm 2
 3.
 solution for (28).The augmented Lagrangian form of (28), after simple mathematic manipulation, is$$\begin{array}{cc}\hfill J& =\underset{\mathbf{D},\mathbf{U},\mathbf{V}}{\mathrm{argmin}}{\lambda}_{d}\left({\Vert \mathbf{U}\Vert}_{\mathrm{F}}^{2}+{\Vert \mathbf{V}\Vert}_{\mathrm{F}}^{2}\right)+{\Vert {\mathbf{S}}_{\mathbf{D}}\mathbf{FD}\Vert}_{\mathrm{F}}^{2}\hfill \\ \hfill \hspace{1em}& \phantom{J}+\gamma \Vert \mathbf{D}\mathbf{U}{\mathbf{V}}^{\mathrm{H}}+\tilde{\mathbf{D}}/\gamma {\Vert}_{\mathrm{F}}^{2}\hfill \end{array}$$According to (29), the resulting ADMM steps are expressed as follows:$$\begin{array}{c}\mathbf{D}:=\underset{\mathbf{D}}{\mathrm{argmin}}{\Vert {\mathbf{S}}_{\mathbf{D}}\mathbf{FD}\Vert}_{\mathrm{F}}^{2}\end{array}$$$$\begin{array}{c}+\gamma \Vert \mathbf{D}\mathbf{U}{\mathbf{V}}^{\mathrm{H}}+\tilde{\mathbf{D}}/\gamma {\Vert}_{\mathrm{F}}^{2}\end{array}$$$$\begin{array}{c}\mathbf{U}:=\underset{\mathbf{U}}{\mathrm{argmin}}{\lambda}_{d}{\Vert \mathbf{U}\Vert}_{\mathrm{F}}^{2}+\gamma {\Vert \mathbf{D}\mathbf{U}{\mathbf{V}}^{\mathrm{H}}+\tilde{\mathbf{D}}/\gamma \Vert}_{\mathrm{F}}^{2}\end{array}$$$$\begin{array}{c}\mathbf{V}:=\underset{\mathbf{V}}{\mathrm{argmin}}{\lambda}_{d}{\Vert \mathbf{V}\Vert}_{\mathrm{F}}^{2}+\gamma {\Vert \mathbf{D}\mathbf{U}{\mathbf{V}}^{\mathrm{H}}+\tilde{\mathbf{D}}/\gamma \Vert}_{\mathrm{F}}^{2}\end{array}$$$$\begin{array}{c}\tilde{\mathbf{D}}:=\tilde{\mathbf{D}}+{\tau}_{1}(\mathbf{D}\mathbf{U}{\mathbf{V}}^{\mathrm{H}})\end{array}$$Obviously, all of the problems associated with (30), (31) and (32) are least squares problems, so that their optimal solutions can be obtained by setting the firstorder derivative of corresponding objective functions with respect to the target variables. After some manipulations we have$$\begin{array}{c}\mathbf{D}:={\left({\mathbf{F}}^{\mathrm{H}}\mathbf{F}+\gamma \mathbf{I}\right)}^{1}\left({\mathbf{F}}^{\mathrm{H}}{\mathbf{S}}_{\mathbf{D}}+\gamma \mathbf{U}{\mathbf{V}}^{\mathrm{H}}\tilde{\mathbf{D}}\right)\end{array}$$$$\begin{array}{c}\mathbf{U}:=(\tilde{\mathbf{D}}+\gamma \mathbf{D})\mathbf{V}{\left({\lambda}_{d}\mathbf{I}+\gamma \mathbf{V}{\mathbf{V}}^{\mathrm{H}}\right)}^{1}\end{array}$$$$\begin{array}{c}\mathbf{V}:={(\tilde{\mathbf{D}}+\gamma \mathbf{D})}^{\mathrm{H}}\mathbf{U}{\left({\lambda}_{d}\mathbf{I}+\gamma {\mathbf{U}}^{\mathrm{H}}\mathbf{U}\right)}^{1}\end{array}$$The whole algorithm is summarized in Algorithm 2 (Microdoppler Extraction based on Low Complexity RPCA).
Algorithm 2 MELCRPCA. 

3.3. Convergence Analysis
4. Experiments
5. Conclusions
Author Contributions
Acknowledgments
Conflicts of Interest
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Algorithm 1  Algorithm 2  

Continuously  Entropy  1.61  1.61 
sampling  CPU time  7.1 s  4.6 s 
Randomly  Entropy  1.61  1.61 
sampling  CPU time  7.3 s  4.7 s 
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Lu, L.; Chen, P.; Wu, L. A RPCABased ISAR Imaging Method for Micromotion Targets. Sensors 2020, 20, 2989. https://doi.org/10.3390/s20102989
Lu L, Chen P, Wu L. A RPCABased ISAR Imaging Method for Micromotion Targets. Sensors. 2020; 20(10):2989. https://doi.org/10.3390/s20102989
Chicago/Turabian StyleLu, Liangyou, Peng Chen, and Lenan Wu. 2020. "A RPCABased ISAR Imaging Method for Micromotion Targets" Sensors 20, no. 10: 2989. https://doi.org/10.3390/s20102989