Proceedings of the 7th Annual ISC Graduate Research Symposium

ISC-GRS 2013

April 24, 2013, Rolla, Missouri

Zengli Yang

Department of Electrical and Computer Engineering

Missouri University of Science and Technology, Rolla, MO 65409

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A Comparative Study of Sparse Methods for 3-D Synthetic aperture Radar Image Reconstruction

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Abstract

This paper compares four sparse methods for 3-D synthetic aperture radar (SAR) image reconstruction. The four different sparse methods are image denoising or compressed sensing (CS) approach combined with range migration algorithm (RMA) using Stolt transform or non-uniform fast Fourier transform (NUFFT). With undersampled measurements, the sparse methods decrease the load of data acquisition but still reconstruct the important features of 3-D SAR images. We adopt the structured similarity (SSIM) index to assess the quality of the 3-D reconstructed images. Numerical simulations demonstrate that the sparse methods based on the Stolt-RMA benefit little from the low-complexity Stolt transform, and the image denoising approach using the NUFFT-RMA achieves the best tradeoff between the reconstruction quality and computational costs.

1.  INTRODUCTION

Wideband 3-D synthetic aperture radar (SAR) imaging finds important applications in the area of nondestructive testing and evaluation (NDT&E) [1] for its feasibility to acquire high-resolution holographic images of specimen under test (SUT). Microwave and millimeter wave NDT&E techniques have been applied to diverse applications, such as detection and evaluation of corrosion under paint and composite laminates, fatigue crack detection and sizing in metal surfaces, dielectric material characterization, etc.

However, data collection for wideband 3-D SAR imaging imposes great challenges on application of microwave and millimeter wave. Conventional raster scanning takes about two hours to scan a 50 × 50 mm2 area at 2 mm spacing, hindering the practical applications of the 3-D SAR imaging system. Random undersampling can reduce the acquisition time, and advanced sparse methods using either image denoising or compressed sensing (CS) approach [2] can reconstruct image with good quality comparable to full sampling [3]–[7].

Conventional 3-D SAR imaging with planar aperture usually adopts the range migration algorithm (RMA) [8] with either Stolt transform [9] or non-uniform fast Fourier transform (NUFFT) [10]. For brevity, we call these two conventional RMAs as Stolt-RMA and NUFFT-RMA, respectively. The recovered image using the NUFFT-RMA usually has less artifacts and better resolution than that using the Stolt-RMA, however, the NUFFT-RMA has much higher computational costs than the Stolt-RMA. The procedure for converting the raw measurements to the 3-D image is called forward SAR imaging, and the reverse procedure is called reverse SAR imaging. Nonlinear transform exists when applying the RMAs for SAR imaging. With undersampled measurements, the image denoising or CS approach can be combined with the RMAs for image reconstruction, yielding four different sparse methods. For brevity, we denote the four sparse methods as Stolt-DN, NUFFT-DN, Stolt-CS, and NUFFT-CS, respectively.

Although both image denoising and CS approaches exploit the sparsity of the 3-D SAR image, differences between these two approaches exist when combined with RMA for 3-D SAR imaging. The image denoising approach is to denoise the SAR image that is obtained by applying RMA directly on the under-sampled measurements. However, the CS approach applies the forward and reverse RMAs during the reconstruction, and finds the solution to an underdetermined system. For each iteration, the CS approach has larger computational costs than the image denoising approach due to its more complicated transforms. Our prior works investigated the performance of the CS approach using the Stolt-RMA or NUFFT-RMA separately, and demonstrated their capability to reconstruct the important features of 3-D SAR images [4]–[7]. Therefore, it is interesting to compare the overall tradeoff between the reconstruction quality and computational costs for the four sparse methods.

In this paper, we compare the four different sparse methods using a synthesized SUT and its simulated measurements. To solve the l1-regularized image reconstruction problems, we adopt the split Bregman framework [2], [11] to achieve fast convergence and better numerical stability. To be more consistent with human eye perception, we adopt the structured similarity (SSIM) [12] index instead of the peak signal-to-noise ratio (PSNR) to assess the quality of the 3-D reconstructed images. Numerical simulations demonstrate that the sparse methods based on the Stolt-RMA benefit little from the low-complexity Stolt transform, and the image denoising approach using the NUFFT-RMA achieves the best tradeoff between the reconstruction quality and computational costs.

2.  cONVENTIONAL 3-d sar IMAGE RECONSTRUCTION WITH FULL SAMPLING

Consider a wideband monostatic stripmap 3-D SAR imaging system where data are collected by scanning the SUT over a 2-D planar aperture denoted as XY -plane. A point target is characterized by its reflectivity function. In 3-D Cartesian space, we define the dimension perpendicular to the 2-D aperture plane as the -dimension and its origin is located at the 2-D aperture plane. Without the far-field assumption, the wavefront curvature is no longer negligible. Without plane wave approximation for spherical wave, the received spherical waveform at position (x, y) with temporal angular frequency is given by

(1)

where with f being the temporal frequency, is the wavenumber with c being the propagation speed of microwave, and g(x′, y′, z′) denotes the 3-D reflectivity function of the SUT. With plane wave decomposition for spherical wave and Stolt transform [9], the 3-D reflectivity function of the SUT is give by [13]

(2)

which is known as the 3-D Stolt-RMA [13]. Heredenotes 2-D cross-range fast Fourier transform (FFT) along the XY - plane, denotes the 3-D inverse FFT (IFFT), and denotes the 1-D Stolt transform with the nearest neighbor interpolation. Note that the distinction between the primed and unprimed coordinate systems is now dropped because the coordinate systems coincide after the FFT and IFFT.

Let represent the 2-D cross-range Fourier transform of , and represent the 3-D Fourier transform of the reflectivity function , where and are the wavenumbers in the x, y, and z dimensions, respectively, then due to curvature in radar wave propagation. It is necessary to obtain equispaced in for 3-D IFFT operation under uniform measurement grid, and direct interpolation is known as Stolt transform. Alternatively, nonuniform spaced in can be transformed to the reflectivity image by applying NUFFT,

(3)

which is known as the 3-D NUFFT-RMA [10]. Here denotes 2-D cross-range IFFT, and is the 1-D inverse NUFFT along. With more complicated interpolation, the

NUFFT-RMA has larger computational costs than the Stolt-RMA. The complete procedures for the conventional 3-D SAR image reconstruction methods are summarized in Fig. 1.

3. SPARSE METHODS FOR 3-D SAR IMAGE RECONSTRUCTION

To take the advantage of sparse methods for 3-D SAR imaging, the radar probe measures only at a small percentage of randomly-selected positions on the uniform XY grid. The

Fig. 1. Conventional 3-D SAR image reconstruction methods: Stolt-RMA and NUFFT-RMA.

backscatter data at these positions are collected and saved as the raw data. Image can be reconstructed from the undersampled measurements by two iterative sparse approaches: one is image denoising based on the full image data and another is CS based on the undersampled raw measurement. Both approaches exploit the sparsity of the 3-D SAR image and rely on the l1-norm and total variation (TV) minimizations [2]. For each approach, the Stolt-RMA or NUFFT-RMA can be applied yielding four different sparse methods: Stolt-DN, NUFFT-DN, Stolt-CS, and NUFFT-CS.

3.1. Image Denoising Approach

For the image denoising approach, we first obtain the full image data by applying the RMA on the under-sampled measurements as preprocessing. Then the following denoising procedure is performed [2],

(4)

where is the -norm, is the vectorized 3-D SAR image of the interrogated scene obtained directly from the undersampled measurements, is the vectorized estimated 3-D image, and is the tolerance. The cost function represents some regularization term with respect to and is selected as

(5)

where denotes the -norm, is the linear operator that transforms from voxel representation into a sparse representation, and D is the discrete 3-D isotropic TV operator. Depending on the type of the RMA in the preprocessing, we denote the two sparse methods using image denoising technique as Stolt-DN and NUFFT-DN.

3.2. Compressed Sensing Approach

The CS approach always values the consistency between the estimated measurements and actual received data during the reconstruction. For the 3-D SAR imaging, the CS approach can be interpreted as [7]

(6)

where has the same definition as (5), and (m < n) is the measurement matrix, which reflects the acquisition of the vectorized received measurements for the scene’s reflectivity. For the Stolt-CS and NUFFT-CS, we adopt the reverse Stolt-RMA and NUFFT-RMA for, respectively. According to (2) and (3), we can write the measurement operator as

(7)

(8)

if omitting the phase compensation term for brevity. Here † represents the pseudoinverse 1-D Stolt transform with nearest neighbor interpolation, denotes 3-D FFT, and denotes the binary matrix, which is to select estimated received waveform at these random positions.

Although (4) and (6) have the same objective function, (6) is actually an underdetermined system because of its constraint. Also, nonlinear forward and reverse RMAs are involved when finding the solution to (6). Apparently, tradeoff between the reconstruction quality and computational complexity exists for the image denoising approach and the CS approach.

4. Simulation and results

Our previous publications reported that undersampled measurements have been successfully performed with the 3-D SAR imaging system, and experimental results have demonstrated good image qualities [4]–[7]. However, in this paper, we use simulations to compare the difference of the four sparse methods because simulation provides ground truth image.

Suppose a uniform measurement grid in the XY-plane being 2 mm in both X and Y directions, and the stepped-frequencies ranging from 35.04 GHz to 44.64 GHz in the Q-band with a step-size of 0.64 GHz. We simulated the backscatter data over the square area with 128×128 mm2 in the noiseless and electromagnetic interference (EMI)-free environment. The SUT contains three objects square-pad, cross-profile, and circle-profile distributed at different depths of −28 mm, −58 mm, and −88 mm, respectively. Since the maximum range with being the frequency interval, mm for our benchmark. We set the sampling interval along the Z-dimension as 2 mm, so the data cube of the 3-D image had the dimension of 64×64×59. The raw measurements were generated from the simulated SUT according to (1), and the undersampled measurements were randomly selected from the raw measurements at the XY positions for the scanning area.

When solving (4) or (6) for the four sparse methods, we adopt the split Bregman framework [2] to achieve fast convergence and better numerical stability. Proper parameters and stopping criteria for the split Bregman algorithm were selected. Also, we chose the sparse transform as Haar wavelet transform in (5). All the conventional methods and sparse methods were implemented by MATLAB R2011a (x86) on a computer with Intel(R) Core(TM)2 Quad CPU Q9400 @ 2.66 GHz and 8.00 GB RAM. The SSIM [12] has been considerably used for 2-D image quality assessment, and is considered more consistent with human eye perception than the PSNR. Hence, we adopt the mean SSIM (MSSIM) averaged over 3-D windows other than 2-D windows to measure the similarity between the 3-D reconstructed image and ground truth image. In the extreme case, if the reconstructed image is exactly the same as the ground truth image, then the MSSIM would be 1. In other words, a larger MSSIM index corresponds to better reconstruction quality, and vice versa.

Figure 2 shows the comparison of the ground truth image and the reconstructed images from 40% undersampled measurements using the conventional methods and the sparse methods. Left, middle, right columns are 3-D view, top view, and side view, respectively. The ground truth image in Fig. 2(a) shows the geometric features and reflectivity of the three objects, and Fig. 2(b) and Fig. 2(c) show the directly reconstructed images using the Stolt-RMA and the NUFFT-RMA, respectively. Without applying the image denoising and CS techniques, the shadow of the targets and artifacts caused by the RMAs and random undersampling are dominant in the 3-D images. Specifically, the Stolt-RMA blurred the SUT to a large extent, and the NUFFT-RMA disrupted the shape of the square-pad. In contrast, Fig. 2(d) to Fig. 2(g) show the cleaner reconstructed images when using the sparse methods. Not surprisingly, the sparse methods based on the NUFFT-RMA recovered the 3-D images of the SUT with better resolution and less background noise than those based on the Stolt-RMA. Specifically, the Stolt-DN had worse denoising capability than the Stolt-CS, and the NUFFT-CS recovered the image with even less horizontal shadow of the cross-profile and circle-profile than the NUFFT-DN. When comparing to the ground truth image, however, both NUFFT-DN and NUFFT weakened the reflectivity of the square-pad and failed to remove the vertical shadow of the SUT. Since the vertical resolution of the 3-D SAR image depends on the bandwidth of the imaging system [13], the sparse methods demonstrated incapability to make up the bandwidth deficiency. The running time for each method with 40% undersampling rate is also given in Fig. 2. Note that in our simulations, since the undersampling rate has little influence on the running time, we omit showing the running time for each method with different undersampling


Fig. 2. Comparison of the ground truth image and the reconstructed images from 40% undersampled measurements using the conventional methods and sparse methods. Left, middle, right columns are 3-D view, top view, and side view, respectively. The running time for each method is also given.

rates for brevity. The NUFFT-RMA took five times the running time of the Stolt-RMA, while the four sparse methods increased the running time by one to two orders due to their iterative reconstructions. Since the RMA only served as preprocessing for the image denoising approach, the Stolt-DN and NUFFT-DN took almost the same running time. Involved the forward and backward RMAs, the CS approach indeed brought larger computational load than the image denoising approach. Interestingly, the running time of the NUFFT-CS was only slightly more than that of the Stolt-CS. Thanks to the better accuracy of the NUFFT-RMA, the NUFFT-CS took much less iterations to reach the stopping criteria than the Stolt-CS . Therefore, the low-complexity Stolt transform actually cannot decrease the overall computational load for the sparse methods due to its inaccuracy.