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Procedía Engineeri ng 29 (2012) 2209 - 2213

Procedía Engineering

www.elsevier.com/Iocate/procedia

2012 International Workshop on Information and Electronics Engineering (IWIEE)

Compressive Radar Imaging Methods Based on Fast Smoothed L0 Algorithm

Liu Jihong*, Xu Shaokun, Gao Xunzhang, Li Xiang

Institute of Space Electronic Technology, National University of Defense Technology, Changsha 410073, China

Abstract

One of the problems that radar imaging technique based on compressed sensing (CS) must confront is the relatively high computation complexity. The sparse representation model of stepped frequency radar echo is established and a 2D joint imaging method based on 2D-SL0 is proposed, which makes the best of the 2D separability of sparse dictionary and compressive measurement, thus has greatly improved efficiency. The performance of CS imaging methods, including SL0, 2D-SL0 under 2D joint model and iterative SL0, MSL0 under 2D decoupled model, are analyzed and compared theoretically. Experiment results verify the validity and superiority of the proposed method.

© 2011 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of Harbin University of Science and Technology

Keywords: Radar imaging; Compressed sensing; 2D separability; Fast reconstruction; Smoothed L0

1. Introduction

The direct information sampling property of compressed sensing (CS) [1] to sparse or compressible signal has great attractions and application perspectives for wideband radar imaging. Radar imaging technique based compressed sensing has great superiority over traditional methods in data acquisition, storage and imaging quality, but must incorporate the highly nonlinear sparsity constraint into the reconstruction process at the cost of software [2, 3]. How to reduce the reconstruction complexity effectively is one of the most important problems that CS-based imaging methods must overcome.

In practice, the structural property of compressive measurement matrix has a substantial impact on the implementation of reconstruction algorithms. As for the two-dimensional (2D) signals that are sparse on

* Corresponding author. Tel.: +8613467574429. E-mail address: ljh63g@163.com.

1877-7058 © 2011 Published by Elsevier Ltd. doi:10.1016/j.proeng.2012.01.289

an overcomplete dictionary composed of separable atoms, a fast algorithm called 2D-SL0 (modified from Smoothed L0) is proposed in [4] and extended to 2D random projection [5], it achieves signal reconstruction in the matrix domain, both benefits of computation and storage are obtained. In [6], a similar CS framework is presented and the application of 2D separable operator to optical imaging is studied. Xie X C et al. [7] introduced traditional SL0 algorithm into CS SAR imaging, but only the range direction is contained and the structural property of sparse dictionary is not considered.

In this paper, the sparse representation dictionary of stepped frequency radar echo is constructed, and then the compressive imaging model based on CS is established. Aiming at the storage and computation problems confronted by CS imaging method under 2D joint model, a 2D joint imaging method based on 2D-SL0 is proposed according to the 2D separability of sparse dictionary and compressive measurement. Besides, the computational superiority of SL0 in multiple sparse recovery is utilized in the 2D decoupled model. The performance and resource requirement of several CS imaging methods are analyzed and compared theoretically, and validated by simulation experiment.

2. Signal representation model of radar echo

For stepped frequency radar, suppose the number of sub-pulses in a burst is N and the number of aspect angles is M . As in [8], the 2D reflectivity distribution 8 = [Spq]PxQ of target can be got by discretizing the 2D target space, where 8pq represents the scattering intensity of the discrete position (xp, yq), xp « pAx, yq« qAy, p = 0, 1,..., P-1, q = 0, 1,..., Q-1, Ax and Ay denotes the discretization intervals of range and crossrange, respectively. In short, the echo data of the mth aspect angle and nth sampling frequency point can be expressed as

Q-1 ( pn V (qm

un,m =LL8pq expI -j2^ PxPI j2n_7T

q=0 p=0 V P J V Q y .

v1 ( v, pn \ Q^z ( qm = L exp I \L8pq exp I j2n_7T

p=0 V P J q = 0 V Q

where n = 0,1,•••,N-1, m = 0,1,•••,M-1, P > N, Q > M .

As can be seen, the exponential term in (1) is 2D separable. For analysis convenience, model (1) can be expressed in matrix form. Let U = [un,m]NxM denote the echo data matrix, Wr = [exp(-j2npn/P]NxP, , ¥d = [exp(j2nqm/Q]MxQ represents the discrete Fourier dictionary of range and azimuth, respectively, then

U = Wr8WTd (2) Generally, the number of equivalent scattering centers is limited, most of the PQ discrete positions have no scatterer, i.e., only a few elements of 8is nonzero, thus the target reflectivity distribution is sparse. Let u = vec(U), c = vec(8), W = Yd where vec(*) means transforming a 2D matrix into a 1D vector by column stacking. Then we get the sparse representation model of 2D IS AR echo data as

u = (Wd ®Wr)c = Wc (3)

3. Compressive radar imaging method based on fast SL0

3.1. 2D joint compressive imaging model

In CS, the measurement process can be viewed as the action of a matrix on the signal of interest. For implementation convenience, we choose to sample randomly at a small number of frequency points and aspect angles, corresponding measurement matrices are constructed by randomly selecting a few rows of identity matrices, and that each aspect takes the uniform sensing matrix.

Suppose the measurement matrices in range and azimuth direction are 0r of size K x N and 0d of size Lx M (K □ N , L □ M), respectively. Let u(t) = [u0j, u\j,..., mn_1j/], W(l) = [ W^-i^+pWe, then u(l) = W(l)a and the measurement model at the lth aspect angle is y(,) = 0ru(l) = 0rW(I)a.

Let y = [y(1);y(2);---;y<L)\ , W' = [W(1); W(2);"-;W(L)] , 0' = diag{1r ,020r} , then the joint compressive sampling model is l

y = 0' W' a = 0a (4)

where 0 = 0 W'. CS theory indicates that, when matrix 0 satisfies the restricted isometry property (RIP) or incoherent condition [1], a can be estimated via nonlinear optimization, and then the 2D radar image can be obtained through realignment.

Considering W' = (0d ® IN )W , 0' = IL ® 0r , W = Wd ® Wr , let 0r = 0rWr , 0d = 0dWd , then 0 = (IL ® 0r)C(0d ® INJ\Wd® Wr) = 0d ®0r ; y = (0d ®0r)a . Suppose YM is the compressed measurement data matrix, namely y = vec(Y), then model (4) can be recast as

Y = 0rS0dT (5)

Generally, 0r and 0d are known, the goal of us is to solve the 2D reflectivity S of radar target from measured data Y as quickly as possible.

3.2. Radar image formation based on 2D-SL0

CS is a theory that highly depends on the optimization algorithm. The SL0 method proposed by Mohimani et al. [9] has a good tradeoff between accuracy and complexity, we can recover radar image based on traditional SL0 algorithm from model (4), but may suffer from storage and computation with respect to large or complicated targets. 2D-SL0 proposed for separable atoms can be directly applied to the 2D sparse decomposition, and has much higher computation efficiency. According to model (5), here we introduce 2D-SL0 into the reconstruction of target reflectivity. Considering the complex-valued character of radar data, the compressive imaging method based on 2D-SL0 can be described as Fig. 1.

• Initialization:

(1) Take the minimum 12 norm solution of Y = 0rS0d as the initial solution S0 , namely S0 = 0r^Y (0d ) .

(2) Set the largest iteration number of times J , choose a suitable decreasing sequence {n, n2,"', nJ} for parameter n . Generally, n1 may be chosen about two to four times of the maximum absolute value of S0 , and n = c0nJ-1 for j > 2 , where c0 is usually between 0.5 and 1.

• External iteration for j = 1,..., J :

1) Let n = nj, S = Sj-1 ■

2) Internal iteration L0 times. Maximize the function = ?exp (_|Spq| ) on the feasible set A = {SY = 0,S0dT} via L0 iterations of the steepest ascent algorithm:

(a) Let A = [mpq \ , where Wpq □ Spq exp (- ^/2n2 ) -

(b) Let S ^S-A , where f> 1 is a small positive constant

(c) Project S back onto the feasible set A : S^S- 0r f (0rS0d H - Y )(0d D)H .

3) Set Sj = S .

• The final answer S = SJ is just the estimation of 2D reflectivity.

Fig. 1. CS radar imaging method based on 2D-SL0

It worth mentioning that, in light of the 2D separable property of the echo model, we can perform the reconstruction of range and azimuth direction separately under the 2D decoupled model. The routine approach is to solve the target range profiles individually by SL0 algorithm, and then recover S based on the reconstructed range profiles. Since every aspect angle adopts the uniform measurement matrix, the effective matrices for reconstruction are identical, so the reconstruction of range profiles subjects to multiple sparse recovery issue. As all the steps of SL0 are in matrix form, it can be directly run on the whole data matrices, called MSL0 (SL0 for multiple sparse recovery). Because of the speed of the current matrix multiplication algorithms, this will result in an increased speed in the total reconstruction process. Similarly, the azimuth reconstruction of all range cells can also be solved at a single step. Therefore, the 2D reflectivity distribution can be reconstructed trough just twice MSL0 operation under the 2D decoupled model.

3.3. Performance analysis

In essence, the CS-based imaging methods shift the complexity of imaging system from hardware to smart recovery algorithms. The structural prior information of sparse dictionary is utilized to improve the efficiency of CS imaging methods based on SL0. The resource requirements of four CS imaging methods, i.e., SL0, 2D-SL0 under 2D joint model and iterative SL0, MSL0 under 2D decoupled model, are analyzed from the perspectives of memory space and computation complexity, as listed in Table 1. Apparently, under the same condition of measurements, the proposed imaging methods save a great deal of memory space and computation cost.

Table 1. Resources requirement of CS imaging methods

Item SL0 2D-SL0 Iteration SL0 MSL0

Memory space 0( KLPQ) O(KP) + 0(LQ) O(KP) + 0( LQ) O(KP) + 0(LQ)

Running time 0((PQ)2) 0((K + L)PQ) 0(LP2) + 0(PQ2) 0(LP1376) + 0(PQ1 376 )

4. Simulations and analysis

Suppose radar transmits stepped frequency signal with bandwidth 1GHz, the work frequency is 9.5GHz-10.5GHz with step size 10MHz, sampling number N = 101, and the observation azimuth varies within -2.8°-2.9°, corresponding sampling number M = 101. The simulated target is a warhead model composed of 7 scattering centers, as shown in Fig. 2(a), radar echoes are generated according to the point scattering model and 15dB Gaussian complex noises are added. Define the measurement compression ratios in range and azimuth direction as N/K and M/L, respectively, here both are specified as 4.

Fig. 2. (a) Reflectivity distribution of simulated target; (b) FFT imaging result; (c) 2D-SL0 imaging result; (d) MSL0 imaging result

Fig. 2(b)-(d) presents the imaging results of FFT, 2D-SL0 and MSL0, respectively. Due to space limitation, the results of traditional SL0 are not given here. As can be seen, compared to FFT method, CS-based imaging methods can get better imaging results from fewer measurements, the reflectivity distribution with reduced sidelobe and clutters is much clearer. Since the 2D decoupled model ignores the coupling effect between rows and columns of radar data, the imaging quality of MSL0 is slightly worse than 2D-SL0, yet can be improved by increasing measurements.

The CPU running time of CS-based imaging methods under different compression ratios are given in Table 2, where r represents the compression ratios of both range and azimuth, 2D joint SL0 method is out of memory when r = 2. Obviously, the running time of every algorithm increases with the decrease of compression ratio, and the 2D joint SL0 method which is the most storage consuming increases remarkably. Compared to traditional SL0 imaging methods, 2D-SL0 and MSL0 have considerable improvement on the operation speed, and the latter reveals the superiority of SL0 in multiple sparse recovery, this is consistent with the theoretical analysis.

Table 2. Running time of CS imaging methods under different compression ratios (s)

Compression ratio SL0 2D-SL0 Iteration SL0 MSL0

r= 4 32.7656 0.1563 0.4844 0.1875

r= 3 103.9844 0.1563 0.7500 0.2188

r= 2 -- 0.2188 1.4688 0.2656

5. Conclusions

In light of the sparsity of target reflectivity, this paper established and analyzed the sparse representation model of stepped frequency radar echo, and then proposed a 2D joint imaging method based on 2D-SL0 according to the 2D separable property of sparse dictionary, meanwhile introduced MSL0 into the 2D decoupled imaging. Compared to traditional CS algorithms, the proposed methods have higher efficiency and lower resource requirement, which are beneficial to practical implementation.

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