## 1. Introduction

Radar, an object detection system using radio waves to detect objects and determine their spatial positions, has been applied in many fields, including radar astronomy, geographical environment surveillance system, and air defense systems. The demand of radar systems for high-resolution and high-speed imaging capabilities is increasing and results in a strong desire for higher bandwidths and more sampling time, which places significant pressure on radar hardware equipment and imaging costs. However, classical time–frequency uncertainty principles based on the Shannon sampling theorem have limited the development of high-resolution and high–speed radar imaging [

1]. Compressive sensing, as an effective approach for direct compressive sampling, has great potential application for radar imaging and is able to solve the problems that radar systems currently face, that is, the mass time requirement of signal sampling and reconstruction for high–resolution imaging [

2]. Meanwhile, a compressive imaging method can simplify the structure of a radar system, allowing elimination of the matched filter for pulse compression at the receiver and thereby reducing the need for analog-to-digital convertors [

3,

4,

5,

6].

Some achievements of radar imaging based on the compressive sensing principle have been reported [

1,

3,

7,

8]. Braniuk and co-workers first applied the compressive sensing theorem in radar imaging systems and confirmed the feasibility of compressive radar imaging by theoretical analysis and numerical experiments [

3]. Zhang and co-workers proposed a framework to realize high–resolution inverse synthetic aperture radar (ISAR) imaging with limited measured data based on the theory of compressed sampling [

4]. Ender presented generic system architectures and implementation considerations to address some further steps to compressive radar imaging and applied the compressive sensing in pulse compression, radar imaging and air space surveillance with array antennas [

9]. An approach that employed pulse accumulation and weighted compressive sensing was also proposed by Zhang and co-workers under low signal-to-noise ratio (SNR) conditions, to realize high-resolution imaging and reduce sensitivity to noise [

10]. Moreover, many reconstruction algorithms of compressive radar images have proposed. For example, Xie and co-workers proposed a smoothed L0 norm (SL0) algorithm to obtain fast radar imaging based on a compressive sensing [

11]; Bhattacharya and co-workers used convex optimization through projection onto convex sets or greedy algorithms to decode compressive synthetic aperture radar (SAR) images [

12,

13]; and Yu and co-workers introduced a turbo-like iterative thresholding algorithm to recover SAR images [

14]. The methods of compressive radar imaging based on these reconstruction algorithms can obtain high-resolution radar images with very small amounts of echo data, and improve the radar imaging speed remarkably, compared with conventional radar imaging methods. However, the radar imaging time is still very long, especially for high-resolution radar imaging, because the conventional strategies for 2D signal reconstruction usually involve stacking a matrix of 2D signals into a huge column vector based on the vector space model, and then recovering the huge vector signal with reconstruction algorithms in the 1D domain [

15,

16]. These approaches exponentially increase the computational complexity involved in recovering 2D sparse signals and the memory requirement for storing the large-bandwidth data of the radar images [

17]. In addition, these approaches ignore the intrinsic spatial structure of 2D signals [

16], especially the coupled range and azimuth information of radar imaging.

To address these drawbacks, some 2D reconstruction algorithms that directly leverage the matrix structure of 2D sparse signals have been proposed recently [

16,

17,

18,

19,

20,

21], and some have been used in radar imaging systems. For example, a fast reconstruction algorithm, called two-dimensional smoothed L0 norm (2D-SL0) algorithm, has been proposed to reduce computational complexity and economize on the memory required by directly utilizes the matrix structure to recover the 2D sparse signals and is designed [

20], but the reconstruction quality of the natural image is poor. Another novel algorithm called the 2D orthogonal matching pursuit (2D-OMP) algorithm, which was extended from 1D-OMP, has been developed to reconstruct 2D sparse signals [

17]. In this algorithm, each atom in the dictionary is a matrix. At each iteration, the best matched matrix atom is selected by projecting the sample matrix onto 2D atoms and the weights for the selected atoms are then updated via the least square. This algorithm significantly reduces the computational complexity with a matrix structure, but it still requires a great deal of memory usage and is suitable only for the square matrix of 2D sparse signals. In addition, an iterative gradient projection algorithm for 2D sparse image reconstruction has been proposed, in which the sparse solution is searched iteratively from the 2D solution space and then updated by gradient descent of the total variation. It recovers the natural image perfectly, being conducive to reduction in both computational complexity and memory requirements for the measurement matrix [

18]; however, the algorithm suffers from the limitation that the sparse signals must be square.

In this paper, we propose a novel reconstruction algorithm of 2D sparse signals, called the 2D normalized iterative hard thresholding (2D-NIHT) algorithm, which is extended from the normalized iterative hard thresholding (NIHT) algorithm [

22] to reduce the reconstruction time in radar imaging by recovering radar images in the matrix domain directly, and the effectiveness and superiority of the algorithm are theoretically proved and demonstrated by experiments. Moreover, we also present a 2D compressive sampling model of radar imaging systems, which compresses range and azimuth information simultaneously and also ensures that the 2D-NIHT algorithm can be implemented in a compressive radar imaging system.

This paper is organized as follows. In

Section 2, we briefly present the compressive sensing theory and the NIHT algorithm. In

Section 3, we introduce the radar imaging model based on compressive sensing. In

Section 4, we describe the 2D-NIHT algorithm in detail and prove its convergence. In

Section 5, some experiments using randomly synthetic 2D signals and actual radar images are presented. Finally, in

Section 6, we draw conclusions.

## 5. Results

In this section, on the one hand, an experiment with syncretic sparse images was conducted to demonstrate the feasibility of the 2D-NIHT algorithm and its superiority to the NIHT algorithm in reconstruction time. On the other hand, two SAR scenes were used to exam the reconstruction performance of the 2D-NIHT algorithm under different SNR levels and its superiority to other five efficient reconstruction algorithms of sparse signals in compressive radar imaging systems. All data were analyzed in the Matlab R2013a environment using an Intel Core 4, 3.20 GHz processor with 4.0 GB of memory under the Microsoft Windows 7 operating system. The reconstruction time, which is the CPU time, was utilized as an indicator of the computational complexity of signals reconstruction. The probability of exact recovery, which is a crucial criterion for evaluating the practicability of the algorithm, was calculated by the equation $P\left(X\right)=1-{\Vert X-X*\Vert}_{0}/\left({N}_{1}\times {N}_{2}\right)$, where $X\in {R}^{{N}_{1}\times {N}_{2}}$, and $X*\in {R}^{{N}_{1}\times {N}_{2}}$ were original signals and the recovered signals, respectively.

#### 5.1. Experiments on Synthetic Images

#### 5.1.1. Efficiency of the 2D-NIHT Algorithm in 2D Signal Reconstruction

In this experiment, a randomly synthetic 2D sparse signal of

$X$ with size of

$256\times 128$ and a sparsity of 200, as shown in

Figure 2a, was used to validate the efficiency of the 2D-NIHT algorithm. The observation matrix

$Y$ was acquired by the model

$Y=AX{B}^{T}$ with the compressive sampling rate of 0.5 in both row and column, and then the recovered signal

$X*$ was reconstructed using the 2D-NIHT algorithm from

$Y$. As shown in

Figure 2b, it was obvious that the algorithm perfectly recovered the synthetic 2D sparse signal without any information loss. And, as shown in

Figure 2c, it took a finite number of iterations, which was even far less than the sparsity

$K$ of the signal. Moreover, only 0.25 s was spent on the whole process of the sparse signal recovery.

#### 5.1.2. Comparison with the NIHT Algorithm

To demonstrate the superiority of the 2D-NIHT algorithm to the NIHT algorithm, a series of synthetic images were used to test their reconstruction performances. The reconstruction performances of the 2D-NIHT and the NIHT algorithms were comprehensively compared by varying the sparsities and lengths of measurements, respectively.

To test the reconstruction performance of the two algorithms under various sparsities, some randomly synthetic 2D sparse signals

${X}_{i}\left(i=1,2,\dots ,30\right)\in {R}^{100\times 100}$ with different sparsities were used as testing images, and the measurements of a fixed length were acquired with a constant compressive sampling rates of 0.5 for both rows and columns. To test the reconstruction performance of the NIHT algorithm, the 2D signals were also vectorized by

${x}_{i}=vct\left({X}_{i}\right)\in {R}^{10000}$ firstly and the measurements were acquired based on Equation (1) with a constant compressive sampling rate of 0.25 for any signal

${x}_{i}$. The reconstruction performances of the two algorithms, as shown in

Figure 3a, demonstrated that, on one hand, both algorithms could obtain the probability of exact recovery of 1 with a smaller sparsity (≤550), and then their probabilities of exact recovery of both algorithms decreased as the sparsity increased; on the other hand, the reconstruction time of the 2D-NIHT algorithm was far less than that of the NIHT algorithm, and the difference of recovery time between the two algorithms became larger with an increase in the signal sparsity.

To test the reconstruction performance of the two algorithms with measurements of various lengths, 2D sparse signals

${X}_{i}\left(i=1,2,\dots ,12\right)\in {R}^{100\times 100}$ with a fixed sparsity of

$K=100$ were generated for testing and the measurements were obtained with different lengths. For the sparse signal

${X}_{i}$, the lengths of the measurement for the 2D-NIHT algorithm were set as

${M}_{1i}={M}_{2i}=10+5i$ and the lengths of measurement for NIHT algorithm was set as

${M}_{i}={M}_{1i}\times {M}_{2i}$ accordingly. The reconstruction performances of the two algorithms, as shown in

Figure 3b, indicated that both algorithms could acquire a very high probability of exact recovery with measurements of large size (>40 × 40), while the reconstruction time of the NIHT algorithm was far greater than that of the 2D-NIHT algorithm.

Overall, the 2D-NIHT algorithm and the NIHT algorithm displayed a similar tendency in the probability of exact recovery with respect to the sparsity of signals and the size of measurements, indicating that they had consistent performances in the reconstruction convergence and accuracy, and the reconstruction time of the 2D-NIHT algorithm was much less than that of the NIHT algorithm, illustrating that the proposed 2D-NIHT algorithm dramatically reduced computational complexity in signal reconstruction.

#### 5.2. Experiments on Actual Radar Images

In this section, two SAR scenes were used to verify the efficiency and superiority of the 2D-NIHT algorithm in compressive radar imaging systems, as shown in

Figure 4. One was a simple scene that acquired 11 point targets (

Figure 4a), and the other was a helicopter acquired by a SAR (

Figure 4b). The SAR parameters were set such that the carrier frequency was 2 GHz, the working frequency was 9.5 GHz–10.5 GHz with a step size 20 MHz, the sampling number was

${N}_{1}=64$, the observation azimuth varied within −3.2°~3.0°, and corresponding sampling number was

${N}_{2}=64$. In addition, it was unavoidable that there were noises in the actual radar imaging system, so, the noises of different SNR levels were also added in the echo signals to explore the influence of the noise on the reconstruction performance of the 2D-NIHT algorithm.

To test radar imaging performance using the 2D-NIHT algorithm in the presence of noise, Gaussian white noises with different SNR levels (noise free, 30 dB and 20 dB) were added into the echo signals.

Figure 5 shows the imaging results of the 11 point targets (

Figure 5a) and the helicopter (

Figure 5b) with different SNR levels of noises using the 2D-NIHT algorithm. The compressive sampling rates were 0.5 in the range and azimuth information for the two SAR scenes. The imaging results illustrated that the actual target positions and amplitudes of the both scenes were reconstructed without any information loss at the higher SNR level (noise free and 30 dB). At the low SNR level (20 dB), the imaging results contained some false values at the target positions for both scenes. Moreover, it was obvious that the reconstructed image using the 2D-NIHT algorithm was clean without any noise, because the nonlinear operation process of

${{\rm H}}_{K}\left[X\right]$ in the 2D-NIHT algorithm was capable of setting

${N}_{1}\times {N}_{2}-K$ elements of

$X$ as 0 and these elements contained most of the noises. Therefore, the 2D-NIHT algorithm is an effective method to remove background noise used in compressive radar imaging systems.

In addition, four other algorithms, including CoSaMP, SL0, block-based compressive sensing (BCS) [

31], and NIHT algorithms, were used to recover the two scenes without noises, and their reconstruction performances were compared with the 2D-NIHT algorithms in terms of reconstruction time and probability of exact recovery. The simulation for each SAR scene was repeated 20 times with every algorithm, and the statistical average values and standard deviations in reconstruction time and probability of exact recovery for the five algorithms were achieved, as shown in

Table 1. The results demonstrate that, compared with the other four algorithms, the 2D-NIHT algorithm could acquire the reconstructed images with ultrahigh probability of exact recovery, and its performance in reconstruction time was only a little inferior to the BCS model, which, however, had the worst performance for the probability of exact recovery. Therefore, the 2D-NIHT is capable of significantly reducing the computational complexity and reconstruction time for recovering radar images and has an overall performance superior to the other four algorithms.