Leveraging joint sparsity in 3D synthetic aperture radar imaging

Dylan Green; JR Jamora; Anne Gelb

doi:10.3934/ammc.2023005

Article Contents

2023, Volume 1, Issue 1: 61-86. Doi: 10.3934/ammc.2023005

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Leveraging joint sparsity in 3D synthetic aperture radar imaging

1.
Department of Mathematics, Dartmouth College, Hanover, NH, USA
2.
The Air Force Research Laboratory, Wright-Patterson AFB, OH, USA

^*Corresponding author: Dylan Green
^*Corresponding author: Dylan Green

Received: July 2023

Revised: August 2023

Early access: September 2023

Published: September 2023

This work is partially supported through the Autonomy Technology Research Center by AFRL contracts #FA8650-18-2-1645 (DG & JJ) and # FA8650-22-C-1017 (DG & JJ), and by the NSF grant DMS #1912685 (AG), AFOSR grant #F9550-22-1-0411 (DG & AG), DOE ASCR #DE-AC05-0 0OR22725 (AG), and ONR MURI grant #N00014-20-1-2595 (AG)

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Abstract

Three-dimensional (3D) synthetic aperture radar (SAR) imaging is an active and growing field of research with various applications in both military and civilian domains. Sparsity promoting computational inverse methods have proven to be effective in providing point estimates for the volumetric image. Such techniques have been enhanced by leveraging sequential joint sparsity information from nearby aperture windows. This investigation extends these ideas by introducing a Bayesian volumetric approach that leverages the assumption of sequential joint sparsity. In addition to obtaining a point estimate, our new approach also enables uncertainty quantification. As demonstrated in simulated experiments, our approach compares favorably to currently used methodology for point estimate approximations, and has the additional advantage of providing uncertainty quantification for two-dimensional projections of the volumetric image.

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Mathematics Subject Classification: 15A29, 62F15, 65F22, 94A12.

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Figure 1. A graphical depiction of Theorem 2.3

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Figure 2. A graphical depiction of SAR PHD $ \hat{{\mathit{\boldsymbol{g}}}} $ (3) in $ k $-space as well as the partitioning of the data into $ N_\theta $ partitions according to the azimuthal angle sets $ \Theta_n $ given by (9)

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Figure 3. Different views at various dB thresholds of the 3D reconstruction of the synthetic cube data set in the ideal case; ground truth point cloud is displayed in black. The threshold values chosen to best demonstrate reconstruction quality

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Figure 4. Cross-sections of the (left) 2D IRB and (middle) 3D SRCI reconstructions of the cube data set with no additional noise. (right) MHD values at various dB threshold values when either technique is used on the cube data set with no additional noise; the minimum MHD value calculated for the 2D IRB method is 0.7017cm, and for the 3D SRCI, the minimum MHD is 0.60140.6014cm

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Figure 5. Different views at various dB thresholds of the 3D reconstruction of the B747 data set with no added noise; ground truth CAD model is displayed in black

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Figure 6. Cross-sections of the (left) 2D IRB and (middle) 3D SRCI reconstructions of the B747 data set with no additional noise. (right) MHD values at various dB threshold values when either technique is used on the B747 data set with no additional noise; the minimum MHD value calculated for the 2D IRB method is 1.832cm, and for the 3D SRCI, the minimum MHD is 1.440cm

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Figure 7. Threshold value vs. MHD for the cube (left) and B747 (right) data sets comparing the 2D IRB and 3D SRCI for both the JHBL and MLE approximations. (top) SNR $ \approx 0 $ dB; (bottom) SNR $ \approx -24 $ dB. In all plots, the dashed blue lines are the MLE MHD values, while the solid red lines are the JHBL MHD values. In all cases, it is straightforward to infer the rest of the characterization of the MHD values by continuing the trends in (A)-(H)

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Figure 8. Slices of the (left) MLE and (right) JHBL reconstructions of the B747 with SNR of -30dB-30dB using the 3D SRCI approach

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Figure 9. Slices of the (left) 2D IRB and (right) 3D SRCI reconstructions of the B747 with SNR of approximately -34dB. Note that for interpretability, the threshold dB scale is different for each figure

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Figure 10. Different views at various dB thresholds of the 3D reconstruction of the B747 sub-sampled data set using our reconstruction techniques with no additional noise added; ground truth CAD model is displayed in black

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Figure 11. Cross-sections of the (left) 2D IRB and (middle) 3D SRCI reconstructions of the sub-sampled B747 data set using the parameters in Tables 4 and 5. (right) MHD values at various dB threshold values

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Table 1. Parameters of data sets used for experimentation

Parameter	Dataset Value
Elevation Range	$ [-3^\text{o},3^\text{o}] $
Elevation Sampling	$ 0.5^\text{o} $
Frequency Range	[27, 39]GHz
Frequency Sampling	50MHz
Bandwidth	12GHz
Center Frequency	33GHz
Azimuth Range	$ [0^\text{o},359.9^\text{o}] $
Azimuth Sampling	$ 0.1^\text{o} $

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Table 2. Sizes of the inputs and outputs of Algorithm 3 for our numerical experiments

Parameter	2D IRB (Algorithm 4)	3D SRCI (Algorithm 5)
Image Size	201×201	201×201×201
Data Size	241×13×2	241×13×100
Data Partitions	1800	36

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Table 3. Minimum MHD (cm) achieved across tested dB thresholds

	2D IRB (Algorithm 4)		3D SRCI (Algorithm 5)
	MLE	JHBL	MLE	JHBL
Cube, High SNR	0.7281	0.6774	0.5925	0.5222
Cube, Low SNR	0.6820	0.6408	0.5542	0.5570
B747, High SNR	1.868	2.150	1.354	1.401
B747, Low SNR	2.728	2.882	1.357	1.369

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Table 4. Parameters of sub-sampled data set used for experimentation

Parameter	Sub-sampled Value
Elevation Range	$ [-3^\text{o},3^\text{o}] $
Elevation Sampling	$ 0.5^\text{o} $
Frequency Range	[31, 35]GHz
Frequency Sampling	150MHz
Bandwidth	4GHz
Center Frequency	33GHz
Azimuth Range	$ [0^\text{o},359.9^\text{o}] $
Azimuth Sampling	$ 0.3^\text{o} $

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Table 5. Sizes of the parameter inputs and outputs for Algorithm 3 for the sub-sampled data experiments

Parameter	2D IRB (Algorithm 4)	3D SRCI (Algorithm 5)
Image Size	201×201	201×201×201
Data Size	27×13×2	27×13×30
Data Partitions	600	40

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