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Multilinear sparse reconstruction for SAR imaging based on higherorder SVD
EURASIP Journal on Advances in Signal Processing volume 2017, Article number: 44 (2017)
Abstract
This paper focuses on the spotlight synthetic aperture radar (SAR) imaging for point scattering targets based on tensor modeling. In a realworld scenario, scatterers usually distribute in the block sparse pattern. Such a distribution feature has been scarcely utilized by the previous studies of SAR imaging. Our work takes advantage of this structure property of the target scene, constructing a multilinear sparse reconstruction algorithm for SAR imaging. The multilinear block sparsity is introduced into higherorder singular value decomposition (SVD) with a dictionary constructing procedure by this research. The simulation experiments for ideal point targets show the robustness of the proposed algorithm to the noise and sidelobe disturbance which always influence the imaging quality of the conventional methods. The computational resources requirement is further investigated in this paper. As a consequence of the algorithm complexity analysis, the present method possesses the superiority on resource consumption compared with the classic matching pursuit method. The imaging implementations for practical measured data also demonstrate the effectiveness of the algorithm developed in this paper.
Introduction
Synthetic aperture radar (SAR) is an important detection system in many fields for decades, such as remote sensing, environmental monitoring, and ground mapping [1–4]. Imaging algorithm is the key point of SAR application, which extracts the scatter coefficients of targets and reconstructs the image of target scene from the radar echoes [5, 6]. On the azimuth direction, a larger aperture can be synthesized by the coherent processing and relative motions between the antenna and targets. For the range direction, the pulse compression is usually applied to obtain a good range resolution. However, the existing methods encounter several challenges in practical applications. For instance, the overwhelming amount of resource requests for storing and operating in receiver since the high sampling rate; in addition, the conventional imaging method, such as polar format algorithm (PFA), suffers from the sensibility to sidelobe disturbance, which leads to it being unsuitable for highresolution imaging. For solving the problem of high demand on hardware, the compressed sensing (CS) framework has been introduced into SAR imaging [7–11], which can extract necessary information at a lower sampling rate than Nyquist limit [12–14]. Furthermore, the CSbased approach is able to accomplish SAR imaging with the low sidelobe and possess the robustness to noise [15–20]. But the existing methods only utilize the spatial sparsity of the observation without taking advantage of structural features of target scene. In 2016, a sparse reconstruction SAR imaging method was proposed by Zhao et al., which employed structured sparsity constraint based on Bayesian learning framework [21]. But this method requires rigid conditions to ensure model matching; consequently, the imaging quality may be worse if observations cannot perfectly match the model selected.
For practical applications of SAR imaging, the scatterers generally tend to be clustered together as blocks. In this paper, a tensordecompositionbased algorithm is developed to exploit such block sparsity of targets. Tensor decomposition is a kind of higherorder data processing framework which can make use of the multidimensional and highly structured features in signal [22–24]. Recently, combining tensor modeling with sparse reconstruction is becoming the focus of attention in signal processing. In 2010, Lim and Comon proposed a sparse reconstruction method based on lowrank tensor decomposition and analyzed the condition of uniqueness for decomposition [25]. In 2011, Duarte and Eldar introduced the Kronecker structure into dictionary constructing, proposing the KroneckerCS framework [26]. In 2012, Sidiropoulos and Kyrillidis investigated CS theorem with regard to lowrank tensor signal processing, proposing a twostep sparse reconstruction algorithm [27]. In 2013, Caiafa and Cichocki proposed two types of matching pursuit algorithms based on tensor decomposition [28]. However, there has been little work in SAR imaging based on sparse reconstruction combined with tensor decomposition. The higherorder CS approach in hyperspectral imaging enlightens our study in signal modeling [26, 29, 30]. In this paper, an imaging algorithm is developed based on higherorder singular value decomposition (HOSVD) [31, 32] with the Kronecker constraint of reflected waves, which can take advantage of the block sparse feature of target scene to gain more robust highresolution imaging performance. The theoretical analysis and simulation experiments verify that the developed algorithm is superior to the reference methods especially in severe conditions, which demonstrates the practical significances.
The following is the organization of the rest in this paper: Section 2 illustrates the signal model for spotlight SAR; Section 3 introduces the tensor modeling based on HOSVD; Section 4 proposes the algorithm by introducing the multilinear block sparse reconstruction and the preprocessing scheme for dictionary construction; Section 5 demonstrates several numerical simulations to verify the effectiveness of the proposed algorithm; Section 6 concludes this paper.
Problem formulation
First of all, several notions of this paper are stated as follows.

The scalar is indicated by italic letter, e.g., x.

The vector and matrix are indicated by bold italic letters, e.g., x,X.

The tensor is indicated by calligraphic capital letter, e.g., \(\mathcal {X}\).

The index of a set is indicated by script letter, e.g., \(\mathscr {X}\).

∥x∥_{0} is the l _{0}norm which indicates the number of nonzero elements in x.

\(\left \Vert \mathcal {X}\right \Vert _{F}\) is the Frobenius norm of the tensor \(\mathcal {X}\), denoting the square root of the sum of the absolute squares of its elements.

The operator ⊗ denotes the Kronecker product between matrices.

The operator ×_{ n } denotes the moden product between tensor and matrix, where mode is considered as the order of tensor data [33].

For a thirdorder tensor \(\mathcal {X}\in \mathbb {C}^{M\times N\times P}\), define X _{(1)} as a rectangular matrix combined all P slices of \(\mathcal {X}\) under mode1, i.e., \(\boldsymbol {X}_{(1)}=\left [\boldsymbol {X}_{1},\boldsymbol {X}_{2},...,\boldsymbol {X}_{P}\right ]^{T}\in \mathbb {C}^{(PN)\times M}\); similarly, X _{(2)} and X _{(3)} denote the unfolded matrix of \(\mathcal {X}\) along mode2 and mode3, respectively.
This paper considers spotlight SAR for capturing the structural information of the SAR targets, which is a beamformingbased observation system [34–36]. The radio beam will be pointed to the observation scene continuously while the aircraft carrying SAR flying cross over the target scene. The geometry of such model is shown in Fig. 1. θ and ϕ indicate the azimuth and pitch angles of radio wave, respectively. R(x _{0},y _{0},z _{0}) represents the distance between the antenna and target at (x _{0},y _{0},z _{0}); R _{ c } represents the distance between the antenna and the center of the scene. Defining the speed of light as c, then the corresponding propagation time can be written as τ _{ c }=2R _{ c }/c.
The received signal can be represented as
where g(x,y,z) is the scattering function of the scatterer located at (x,y,z); s(t) is the autocorrelation function of transmitted signal; e(t) is the additive noise. For the simplicity, the noise item is not shown in the following content. After transformation to frequency domain, the phase history of the received signal can be represented as
where \(H_{x,y,z}\left (f,\theta,\phi \right)=j\frac {4\pi }{c}(xf\cos \theta \cos \phi +yf\cos \theta \sin \phi +zf\sin \theta)\) is the phase factor. The constant term C(f)=S(f) exp(−j2π f τ _{ c }) can be ignored in the following discussion, where S(f) is the Fourier transform of s(t).
Because the parameters in the phase factor of the signal model (2) are coupled, it is needed to implement an interpolation approach for decoupling [34]. Thus, the coordinate conversion is done from polarization to Cartesian system [37]. Assume there are M _{1},M _{2},M _{3} grid points in the directions of the axis of x, y, z, respectively, and the numbers of grid points for f,θ,ϕ are P _{1},P _{2},P _{3}, respectively. Then, the steering vector of the scatterer located at (x _{ j },y _{ k },z _{ l }) is
where \(d_{j,k,l}\left (p_{1},p_{2},p_{3}\right)=\exp (j\frac {4\pi }{c}(x_{j}u_{p_{1}}+y_{k}v_{p_{2}}+z_{l}w_{p_{3}}))\). The steering matrix \(\boldsymbol {D}\in \mathbb {C}^{P\times M}\) is defined as \(\left [\boldsymbol {d}_{1,1,1},\ldots,\boldsymbol {d}_{M_{1},M_{2},1},\ldots,\boldsymbol {d}_{1,1,M_{3}},\ldots,\boldsymbol {d}_{M_{1},M_{2},M_{3}}\right ]\), where M=M _{1}×M _{2}×M _{3}, P=P _{1}×P _{2}×P _{3}. Hence, the received signal model (2) can be written as
where \(\phantom {\dot {i}\!}\boldsymbol {s}\!=[\!g\!(x_{1},y_{1},z_{1}\!),\ldots,g(x_{M_{1}},\!y_{M_{2}},z_{1}),\ldots, g(x_{1},y_{1},z_{M_{3}}),\ldots,g(x_{M_{1}},y_{M_{2}},z_{M_{3}})]^{T}\). Without loss of generality, considering the \(\phantom {\dot {i}\!}\left (p_{1}p_{2}p_{3},m_{1}m_{2}m_{3}\right)\)th element of matrix D is \(\phantom {\dot {i}\!}\exp \left (j\frac {4\pi }{c}\left (x_{m_{1}}u_{p_{1}}+y_{m_{2}}v_{p_{2}}+z_{m_{3}}w_{p_{3}}\right)\right)\), thus \(\boldsymbol {D}= \boldsymbol {D}_{3}\otimes \boldsymbol {D}_{2}\otimes \boldsymbol {D}_{1}\phantom {\dot {i}\!}\), by which the Kronecker structure can be shown explicitly. As a consequence, the received signal can be rewritten as
Tensor decomposition for SAR imaging
In signal processing field, a tensor indicates the multilinear data with more than two modes. This paper adopts the HOSVD [24, 31], which can be represented as
where \(\mathcal {Y}\) denotes the received signal tensor. For a HOSVD, the operation criterion is
The onedimensional expansion of a tensor is stacking all elements along the mode1, i.e., y=v e c(Y _{(1)}). Thus, (6) can be rewritten as
The equation above indicates that the HOSVD of a tensor signal is equivalent to a linear representation with Kronecker dictionary. It is defined that a Ksparse representation of signal y is existed on a dictionary D, if the following condition is true:
where K is named as sparsity. Because the rows of dictionary is less than the columns, this is an underdetermined system; hence, no unique solution of s exists. However, if the signal possesses sparsity, a solution may be obtained under the following condition [38],
where \(\mu \left (\boldsymbol {D}\right)=\underset {i\neq j}{\max }\left \left \langle \boldsymbol {d}_{i},\boldsymbol {d}_{j}\right \rangle \right \), representing the coherence coefficient of dictionary D; here, 〈d _{ i },d _{ j }〉 denotes the inner product of d _{ i } and d _{ j }. When the dictionary possesses Kronecker structure, i.e., D=D _{ N }⊗D _{ N−1}⊗…⊗D _{1}, the coherence coefficient should be μ(D)= max{μ _{1},μ _{2},…,μ _{ N }}, with μ _{ n }=μ(D _{ n }), n=1,2,…,N. It is demonstrated that the total coherence depends merely on the factor matrix with the largest coherence coefficient for a Kronecker dictionary [39, 40].
Definition 1
A tensor \(\mathcal {Y}\)is defined as (K _{1},K _{2},…,K _{ N })block sparse, if only K _{ n }(n=1,…,N) columns in the factor matrices of its HOSVD need to be computed, i.e.,
where \(\mathscr {I}_{n}=\left [i_{n}^{1},i_{n}^{2},\ldots,i_{n}^{K_{n}}\right ]\) represents a subset of index for moden (n=1,2,…,N).
The above definition demonstrates that the nonzero elements of core tensor are included in a subtensor \(\mathcal {S}\left (\mathscr {I}_{1},\mathscr {I}_{2},\ldots,\mathscr {I}_{N}\right)\). Here, \(\mathscr {I}_{n}=\left \{ i_{n}^{1},i_{n}^{2},\ldots,i_{n}^{K_{n}}\right \} \) is a subset of the index set for moden (n=1,2,…,N). For instance, given a tensor \(\mathcal {X}\in \mathbb {C}^{5\times 5\times 5}\), \(\mathscr {I}_{1}=\mathscr {I}_{2}=\mathscr {I}_{3}=\left \{ 2,4\right \} \), then \(\mathcal {Y}=\mathcal {X}\left (\mathscr {I}_{1},\mathscr {I}_{2},\mathscr {I}_{3}\right)\) is a third order tensor with a size of 2×2×2. And the first and second elements along each mode of \(\mathcal {Y}\) are the second and fourth elements along each mode of \(\mathcal {X}\), respectively, i.e., \(\mathcal {Y}\) picks the corresponding elements in \(\mathcal {X}\) indexed by \(\left \{ \mathscr {I}_{n}\right \}_{n=1}^{3}\). As a consequence, y is Ksparse with regard to Kronecker dictionary D=(D _{ N }⊗D _{ N−1}⊗…⊗D _{1}), where K=K _{1} K _{2}…K _{ N }. It should be noted that the definition of block sparsehere is different from the conventional definition of block sparse representation for onedimensional signal. For the latter, the data is distributed in adjacent data segments [41–43], which is not necessary at the situation of block sparsity based on tensor frame. In 2009, Eldar and Mishali introduced the concept of block sparsity in onedimensional signal [44]. For an underdetermined system y=Dx, assuming that the coefficient vector x consists of a series data segments with given length \(\left \{ d_{m}\right \}_{m=1}^{M}\) and denoting mth segment as x[m], the segment is called block. Define the block sparse l _{0}norm of x as \(\Vert \boldsymbol {x}\Vert _{0,b}=\sum _{m=1}^{M}f_{b}(\Vert \boldsymbol {x}[m]\Vert _{2}>0)\), where ∥x[m]∥_{2} is the l _{2} norm of the mth data block; f _{ b }(∥x[m]∥_{2}>0) is an indicator function which returns the value 1 if ∥x[m]∥_{2}>0, otherwise returns 0. If ∥x∥_{0,b }≤k, x is called block ksparse. Different from this definition of conventional block sparsity, the multilinear block sparsity considers mainly the addition to the multilinear structural restraint as interpreted in Definition 1.
It has been demonstrated that the recovery method based on Kronecker dictionary has much less strict requirements for coherence than the classic matching pursuit method with the same sparsity in regard to signal reconstruction, and it is deduced that the former has the higher successfully recovery bound [28, 45]. In addition, by utilizing the structural information in each mode, the method based on tensor decomposition can perform robustly under the condition of high coherence [46].
The proposed algorithm
In higherorder signal processing, taking advantage of sparsity is becoming a research trend due to the curse of dimensionality problem [24, 47, 48]. Recently, applying CS scheme is rising in highresolution radar imaging field, which can reconstruct the source signal from the undersampled measurements. The CSbased imaging method must solve an illposed linear inverse problem by using the sparsity of targets [12, 49–51]. However, for the largescale data case, the classic l _{0} algorithm has to encounter many computing problems. For instance, considering a scene of 1024×1024 observation points, the classic sparse reconstruction method must process the data up to 1,048,576 length, which is a largescale linear system demanding numerous computational resources. A more efficient approach is investigated in this paper to ameliorate the above problems. As mentioned previously, the SAR imaging model maintains Kronecker structure and the scatterers in target scene are usually clustered together as blocks. As a result, we can take advantage of the block sparsity feature, building a sparse reconstruction SAR imaging algorithm based on Kronecker dictionary.
After decoupling procedure, the reflected signal can be formulated in the tensor form as explained in (8). Then, the nonzero scatter coefficients need to be recovered from the received signal \(\mathcal {Y}\), i.e., finding a (K _{1},K _{2},…,K _{ N })block sparse representation corresponding to dictionaries D _{ n }(n=1,2,…,N). According to the compressed sampling theory, only a few rows need to be randomly extracted from the dictionary.
To begin with, we adopt a preprocessing approach for constructing the dictionaries from observation data. Let \(\mathcal {E}=\mathcal {Y}\mathcal {S}\times _{1}\boldsymbol {D}_{1}\times _{2}\boldsymbol {D}_{2}\times _{3}\ldots \times _{N}\boldsymbol {D}_{N},\) then we have
thus, (7) can be rewritten as
By utilizing alternating least squares (ALS) algorithm, (12) can be solved in an iterative scheme. Such algorithm fixes (N−1) factor matrices in each iteration and updates the remaining component via least square approach, then repeats this process until the condition of convergence is satisfied.
For instance, fixing \(\left \{ \boldsymbol {D}_{n}\right \}_{n=2}^{N}\) first, the estimation of D _{1} can be obtained:
where \(\mathcal {Z}=\mathcal {Y}\times _{2}\boldsymbol {D}_{2}^{T}\times _{3}\ldots \times _{N}\boldsymbol {D}_{N}^{T}\). Applying SVD on Z _{(1)} and ensuring that the singular values are arranged in descending order, then the first M _{1} left singular vectors of Z _{(1)} give the estimation of D _{1} [52]. Next, fixing \(\left \{ \boldsymbol {D}_{n}\right \}_{n=3}^{N}\) and D _{1}, the estimation of D _{2} can be obtained in the same way. Repeat this procedure until all dictionaries are obtained.
Then, let \(\boldsymbol {B}_{n}=\boldsymbol {D}_{n}\left (:,\mathscr {I}_{n}\right),\:n=1,2,\ldots,N\), then the received signal model can be rewritten as
where \(\boldsymbol {s}_{nz}\in \mathbb {R}^{K}\) denotes the vector of nonzero coefficients. As a consequence, the solution of the problem can be presented as
If B=B _{ N }⊗B _{ N−1}⊗…⊗B _{1}, then the solution of (15) can be deduced as s _{ nz }=(B ^{T} B)^{−1} B ^{T} y. For simplicity, assume that K _{ n }=K _{0},n=1,2,…,N, then a sparse representation can be guaranteed if (K _{0} μ(D))^{N}<2−(1+(K _{0}−1)μ(D))^{N} [28], where D=(D _{ N }⊗D _{ N−1}⊗…⊗D _{1}).
In addition, a more efficient calculation step can be employed. The relationship between \(\mathcal {S}_{nz}\) and \(\mathcal {Y}\) can be described as
where \(\mathcal {S}_{nz}\) is the tensor reshaping form of s _{ nz }. Defining \(\mathcal {Q}=\mathcal {S}_{nz}\times _{1}\boldsymbol {I}\times _{2}\left (\boldsymbol {B}_{2}^{T}\boldsymbol {B}_{2}\right) \times _{3}\cdots \times _{N}\left (\boldsymbol {B}_{N}^{T}\boldsymbol {B}_{N}\right)\), \(\mathcal {R}=\mathcal {Y}\times _{1}\boldsymbol {B}_{1}^{T}\times _{2}\cdots \times _{N}\boldsymbol {B}_{N}^{T}\), then we have \(\boldsymbol {B}_{1}^{T}\boldsymbol {B}_{1}\boldsymbol {Q}_{(1)}=\boldsymbol {R}_{(1)}\). Because \(\boldsymbol {B}_{1}^{T}\boldsymbol {B}_{1}\) is a selfadjoint matrix, there exists a fast algorithm which can obtain the solution of \(\mathcal {Q}\) by utilizing Cholesky decomposition [53]. Next, unfold \(\mathcal {Q}\) to a rectangular matrix along mode2:
by defining \(\mathcal {W}=\mathcal {S}_{nz}\times _{1}\boldsymbol {I}\times _{2}\boldsymbol {I}\times _{3}\left (\boldsymbol {B}_{3}^{T}\boldsymbol {B}_{3}\right)\times _{4}\cdots \times _{N}\left (\boldsymbol {B}_{N}^{T}\boldsymbol {B}_{N}\right)\), similar to the above analysis, then we have \(\boldsymbol {B}_{2}^{T}\boldsymbol {B}_{2}\boldsymbol {W}_{(2)}=\boldsymbol {Q}_{(2)}\). The solution of \(\mathcal {W}\) can be found by utilizing Cholesky decomposition likewise. In accordance with this scheme, after N steps, the estimation of \(\mathcal {S}_{nz}\) will be obtained.
The algorithm flow is shown in Algorithm 1. During iterations, it is necessary to guarantee that the nonzero elements of core tensor are concentrated within a subset of index. When the convergence criterion is achieved, all nonzero elements and the corresponding index set can be obtained as the results, and the expected scatter coefficients will be recovered as well.
Algorithm 1 15 16 17 Higherorder SVDbased SAR imaging algorithm
The key step of Algorithm 1 is estimating the nonzero coefficients by least squares procedure, which needs no more than (2N k(k ^{N}+P+1)+N k ^{2}) operations. By contrast, the orthogonal matching pursuit (OMP) algorithm [38, 54] needs k ^{N}(P ^{N}+3k ^{N}) operations in this step. And in the residual update step, it is required to calculate \(\mathcal {S}_{nz}\times _{1}\boldsymbol {B}_{1}\times _{2}\boldsymbol {B}_{2}\times _{3}\ldots \times _{N}\boldsymbol {B}_{N}\) and subtract it from \(\mathcal {Y}\), which needs P ^{N}(2k+1) operations for the present algorithm, whereas the requirement for OMP algorithm is P ^{N}(2k ^{N}+1). Consequently, for a multilinear (K _{0},K _{0},…,K _{0})block sparse reconstruction problem described in Section 3, the maximum number of iteration of the present algorithm is N K _{0}. By contrast, OMP algorithm requires \(K_{0}^{N}\) iterations for the same condition. The complexity analysis results indicate that the present algorithm can be used to estimate the scatter coefficients with much lower computational resource requirement compared with classic matching pursuit algorithm.
Experiments and discussion
This section will provide several numerical experiments to verify the effectiveness and performance of the proposed algorithm. The experiments include two parts: (1) SAR imaging realizations for ideal point scattering model and practical measured data and (2) performance comparisons between the proposed algorithm and the existing algorithms, including PFA [55, 56], OMP [38], and CoSaMP [57]. The PFA method is a kind of classic SAR imaging method which has good realtime response, although susceptible to the sidelobe effect. OMP and CoSaMP are both sparse reconstruction algorithms which can achieve the superresolution performance, but they need to pay much more computational cost. The hardware configuration for the simulation system is Intel Core i75500U 2.4GHz CPU, 8GB RAM, Windows 10 OS.
SAR imaging realizations
Two groups of numerical experiments are demonstrated in this section. One tests the SAR imaging for ideal point targets, and the other investigates the effectiveness of realworld conditions. The simulation scenario is configured as follows: the center frequency is 9 GHz; the bandwidth is 1 GHz; the frequency resolution is 0.01 GHz; the angle aperture is 5°; the angle resolution is 0.05°; the sampling rate is 50%.
Ideal point targets
Different SNRs
Firstly, the imaging results under different SNRs are presented to confirm the feasibility of the proposed algorithm. Twenty ideal point scatterers are deployed in this simulation scenario, located along the diagonal line of the target scene randomly and clustered as three blocks. Corresponding to the settings of the SNR as 3 and 30 dB, the SAR imaging realizations are shown in Fig. 2.
The simulations under different SNRs show the imaging qualities of the proposed algorithm and the reference algorithms. When SNR is low, the imaging results of classic matching pursuit algorithms exhibit distinct outliers, and PFA shows severe noisy ambiguity at the same time. As a contrast, the proposed algorithm performs much better under the low SNR condition. When SNR is high, the results of all methods indicate the congruent imaging quality.
Different number of scatterers
In this simulation scenario, we consider the influence caused by the number of scatterers. The configuration is the same as the previous section, other than fixing the SNR at 10 dB. When the number of scatterers is set to 30 and 150, the SAR imaging realizations are shown in Fig. 3.
It is shown from the simulations that when the number of scatterers is not large, all the methods are able to achieve an acceptable imaging resolution; contrastively, the proposed algorithm can obtain a neater imaging result. However, the scatterers located near the edge of the scene may obscure as the number of scattering points is increasing. The reconstructed scene for the PFA method show more obvious image noise than the matching pursuit algorithms, and the proposed algorithm can achieve the most accurate images under the same condition.
Practical measured data
In this part, a set of experiments based on practical measured data are demonstrated to verify the practicability of the proposed algorithm. The observation target is a crawlertype vehicle. There are three observation angles of a spotlight SAR:

77.5°∼82.5°, the central angle is 80°

87.5°∼92.5°, the central angle is 90°

97.5°∼102.5°, the central angle is 100°
The center frequency and bandwidth are 9 and 1 GHz. As the results, when the central angles are 80°, 90°, and 100°, the SAR imaging realizations are shown in Figs. 4, 5, and 6, respectively.
The imaging results show the availability and superiority of the proposed algorithm in real SAR imaging. It is demonstrated that all the sparse reconstruction methods can achieve the higher resolution images than PFA. The latter can merely obtain fuzzy images, especially when the observation angle is not perpendicular to the target scene. Although OMP and CoSaMP are able to achieve the highresolution results, there exist some problems with these classic matching pursuit techniques. For instance, when the observation angle is 90°, there is a distinct bright streak that appears on the edge of the target in the imaging result of CoSaMP. As indicated by the arrows in Figs. 4 and 6, CoSaMP shows raster lines over the target image at the central region, which will severely influence the recognition of target. In addition, OMP may result in outliers on the sides of the target, which are marked by white circles in each images. In comparison, the proposed algorithm overcomes such defects and performs much better under the same scenario.
Performance comparison
Two performance indicators, root mean square error (RMSE) and computational cost, are discussed for both the proposed algorithm and the reference algorithms in this section.
RMSE
The RMSE of spotlight SAR model is defined as
where \(\hat {\boldsymbol {s}}_{l}\) denotes the estimate of scatter coefficient in the lth MonteCarlo independent trial. The configuration of this experiment is the same as Section 5.1.1. Two set of simulations for different number of scatterers are implemented. The SNR range is 3∼30 dB; the numbers of independent trials L is set to 500. The RMSE curves are shown in Fig. 7, where the solid lines and dashed lines indicate the experimental scenarios for 20 scatterers and 100 scatterers, respectively.
The simulation results indicate that the proposed algorithm performs superiorly compared with the reference methods for both the cases about the small and large number of scatterers. It can be seen that even if the SNR is low, the performances of sparse reconstruction schemes are more robust than conventional PFA method. As a result of the sensitivity to noise, the RMSE curve of PFA shows conspicuous precipitous variation as the SNR is less than 18 dB. Considering the classic matching pursuit methods, since there are merely 2k _{ max } atoms to be chosen in each iteration, CoSaMP runs faster than OMP for SAR imaging, which is indicated by the statistical results of the computational cost in Section 5.2.2. However, the simulation results for RMSE show that the performance of CoSaMP is slightly worse than OMP. As a consequence, the existing SAR imaging methods are always difficult to achieve both the high speed and large success ratio of recovery. In contrast, by exploiting the multilinear block sparsity of targets, the proposed algorithm can gain satisfactory performances on the both aspects.
Computing resources requirement
This section investigates the requirements for computational cost. The configuration of this part is the same as Section 5.1.1 and the evaluation is done in terms of CPU time. The statistical results of running time for the case of different SNRs are shown in Table 1.
For the case about different numbers of scatterers, set the configuration to be the same as Section 5.1.1. The statistical results of running time are shown in Table 2.
The statistical data of Tables 1 and 2 indicate that there exist noteworthy differences of computational cost among these algorithms. The fastest two algorithms are PFA and the proposed algorithm, followed by CoSaMP and OMP in sequence. Nevertheless, the PFA cannot provide robust performance, as the RMSE comparison in the previous section. On the other hand, although the classic matching pursuit methods have highresolution performance, the requirements to computing resources are considerable, especially for OMP algorithm. In each iteration, OMP must match every atom, which results in a high time consumption. As a contrast, the proposed algorithm can obtain the optimal RMSE performance and achieve a running speed faster one order of magnitude than CoSaMP, which makes it suitable for the realtime applications.
Conclusions
In a practical scenario, the block sparse distribution of targets is a common assumption for SAR imaging. Nevertheless, the existing methods seldom take advantage of such geometrical feature. In this paper, a Kronecker constrained spotlight SAR imaging algorithm based on tensor decomposition is developed, which exploits the block sparsity of the target scene. HOSVD modeling of the decoupling signal received is employed by introducing a preprocessing scheme for dictionary construction. Then, a multilinear sparse reconstruction algorithm is deduced to obtain the estimation of scatter coefficients. Since the proposed algorithm utilizes the structural information of targets, a more robust performance can be achieved for imaging quality. A group of numerical experiments based on practical measured data are presented to verify the effectiveness of the developed method. In addition, the complexity analysis indicates that the algorithm reduces the demand for computational cost comparing with the classic matching pursuit methods. The statistical results of simulations confirm the superiority of the present algorithm as well, which is vitally important in realtime applications of engineering technique.
References
 1
JC Curlander, RN McDonough, Synthetic Aperture Radar: Systems and Signal Processing (WileyInterscience, New York, 1991).
 2
GC Walter, SG Ron, MM Ronald, Spotlight Synthetic Aperture Radar (Artech House Publishers, Norwood, 1995).
 3
G Franceschetti, R Lanari, Synthetic Aperture Radar Processing (CRC press, Boca Raton, 1999).
 4
A Marino, Synthetic Aperture Radar (Springer, Berlin, 2012).
 5
M Soumekh, Synthetic Aperture Radar Signal Processing with MATLAB Algorithms (WileyInterscience, New York, 1999).
 6
E Mason, IY Son, B Yazici, Passive synthetic aperture radar imaging using lowrank matrix recovery methods. IEEE J. Sel. Topic Signal Process.9(8), 1570–1582 (2015). doi:10.1109/JSTSP.2015.2465361.
 7
R Baraniuk, P Steeghs, in 2007 IEEE Radar Conference. Compressive radar imaging (IEEEBoston, 2007), pp. 128–133.
 8
L Zhang, Qiao Zj, M Xing, Y Li, Z Bao, Highresolution isar imaging with sparse steppedfrequency waveforms. IEEE Trans. Geosci. Remote Sens.49(11), 4630–4651 (2011).
 9
L Zhang, ZJ Qiao, MD Xing, JL Sheng, R Guo, Z Bao, Highresolution isar imaging by exploiting sparse apertures. IEEE Trans. Antennas Propag.60(2), 997–1008 (2012).
 10
B Sun, Y Cao, J Chen, C Li, Z Qiao, Compressive sensing imaging for general synthetic aperture radar echo model based on Maxwell’s equations. EURASIP J. Adv. Signal Process.2014(1), 153 (2014). doi:10.1186/168761802014153.
 11
G Li, Q Hou, S Xu, Z Chen, Multitarget simultaneous isar imaging based on compressed sensing. EURASIP J. Adv. Signal Process.2016(1), 27 (2016). doi:10.1186/s1363401603271.
 12
Z Zhang, Y Xu, J Yang, X Li, D Zhang, A survey of sparse representation: algorithms and applications. IEEE Access. 3:, 490–530 (2015). doi:10.1109/ACCESS.2015.2430359.
 13
B Chen, J Wang, H Zhao, N Zheng, JC Príncipe, Convergence of a fixedpoint algorithm under maximum correntropy criterion. IEEE Signal Process. Lett.22(10), 1723–1727 (2015).
 14
W Ma, H Qu, G Gui, L Xu, J Zhao, B Chen, Maximum correntropy criterion based sparse adaptive filtering algorithms for robust channel estimation under nongaussian environments. J. Frankl. Inst.352(7), 2708–2727 (2015).
 15
J Fang, Z Xu, B Zhang, W Hong, Y Wu, Fast compressed sensing SAR imaging based on approximated observation. IEEE J. Sel. Topic Appl. Earth Obs. Remote Sens.7(1), 352–363 (2014).
 16
L Zhang, J Duan, ZJ Qiao, MD Xing, Z Bao, Phase adjustment and ISAR imaging of maneuvering targets with sparse apertures. IEEE Trans. Aerosp. Electron. Syst.50(3), 1955–1973 (2014).
 17
J Zhang, Y Ban, D Zhu, G Zhang, Random filtering structurebased compressive sensing radar. EURASIP J. Adv. Signal Process.2014(1), 94 (2014). doi:10.1186/16876180201494.
 18
X Zhang, G Liao, S Zhu, D Yang, W Du, Efficient compressed sensing method for movingtarget imaging by exploiting the geometry information of the defocused results. IEEE Geosci. Remote Sens. Lett.12(3), 517–521 (2015).
 19
X Cong, G Gui, X Li, G Wen, X Huang, Q Wan, Objectlevel sar imaging method with canonical scattering characterisation and intersubdictionary interferences mitigation. IET Radar Sonar Navig. 10(4), 784–790 (2016).
 20
L Zhang, H Wang, Qiao Zj, Resolution enhancement for isar imaging via improved statistical compressive sensing. EURASIP J. Adv. Signal Process.2016(1), 80 (2016). doi:10.1186/s1363401603792.
 21
L Zhao, L Wang, G Bi, S Li, L Yang, H Zhang, Structured sparsitydriven autofocus algorithm for highresolution radar imagery. Signal Process.125:, 376–388 (2016).
 22
R Bro, Parafac. tutorial and applications. Chemometr. Intell. Lab. Syst.38(2), 149–171 (1997).
 23
ND Sidiropoulos, R Bro, GB Giannakis, Parallel factor analysis in sensor array processing. IEEE Trans. Signal Process.48(8), 2377–2388 (2000).
 24
A Cichocki, D Mandic, L De Lathauwer, G Zhou, Q Zhao, C Caiafa, HA Phan, Tensor decompositions for signal processing applications: from twoway to multiway component analysis. IEEE Signal Process. Mag.32(2), 145–163 (2015).
 25
LH Lim, P Comon, Multiarray signal processing: tensor decomposition meets compressed sensing. Comptes Rendus Mecanique. 338(6), 311–320 (2010).
 26
MF Duarte, YC Eldar, Structured compressed sensing: from theory to applications. IEEE Trans. Signal Process.59(9), 4053–4085 (2011).
 27
ND Sidiropoulos, A Kyrillidis, Multiway compressed sensing for sparse lowrank tensors. IEEE Signal Process. Lett.19(11), 757–760 (2012).
 28
CF Caiafa, A Cichocki, Multidimensional compressed sensing and their applications. Wiley Interdiscip. Rev. Data Min. Knowl. Disc.3(6), 355–380 (2013). doi:10.1002/widm.1108.
 29
S Friedland, Q Li, D Schonfeld, Compressive sensing of sparse tensors. IEEE Trans. Image Process.23(10), 4438–4447 (2014). doi:10.1109/TIP.2014.2348796.
 30
E Li, MJ Shafiee, F Kazemzadeh, A Wong, Sparse reconstruction of compressive sensing multispectral data using an interspectral multilayered conditional random field model. IEEE Access. 4:, 5540–5554 (2016). doi:10.1109/ACCESS.2016.2598320.
 31
M Haardt, F Roemer, G Del Galdo, Higherorder SVDbased subspace estimation to improve the parameter estimation accuracy in multidimensional harmonic retrieval problems. IEEE Trans. Signal Process.56(7), 3198–3213 (2008).
 32
F Roemer, Advanced algebraic concepts for efficient multichannel signal processing (PhD thesis, Ilmenau University of Technology, Ilmenau, 2013).
 33
TG Kolda, BW Bader, Tensor decompositions and applications. SIAM Rev.51(3), 455–500 (2009). doi:10.1137/07070111X.
 34
DH Vu, Advanced techniques for synthetic aperture radar image reconstruction (PhD thesis, University of Florida, Gainesville, 2012).
 35
W Xu, P Huang, Y Deng, Efficient sliding spotlight SAR raw signal simulation of extended scenes. EURASIP J. Adv. Signal Process.2011(1), 52 (2011). doi:10.1186/16876180201152.
 36
L Zhang, Li Hl, Qiao Zj, Xing Md, Z Bao, Integrating autofocus techniques with fast factorized backprojection for highresolution spotlight SAR imaging. IEEE Geosci. Remote Sens. Lett.10(6), 1394–1398 (2013).
 37
L Greengard, J Lee, Accelerating the nonuniform fast fourier transform. SIAM Rev.46(3), 443–454 (2004).
 38
JA Tropp, Greed is good: algorithmic results for sparse approximation. IEEE Trans. Inf. Theory. 50(10), 2231–2242 (2004).
 39
S Jokar, V Mehrmann, Sparse solutions to underdetermined Kronecker product systems. Linear Algebra Appl.431(12), 2437–2447 (2009).
 40
S Jokar, in 2010 44th Annual Conference on Information Sciences and Systems (CISS). Sparse recovery and kronecker products, (2010), pp. 1–4. doi:10.1109/CISS.2010.5464722.
 41
YC Eldar, M Mishali, in 2009 16th International Conference on Digital Signal Processing. Block sparsity and sampling over a union of subspaces, (2009), pp. 1–8. doi:10.1109/ICDSP.2009.5201211.
 42
YC Eldar, P Kuppinger, H Bolcskei, Blocksparse signals: uncertainty relations and efficient recovery. IEEE Trans. Signal Process.58(6), 3042–3054 (2010).
 43
RG Baraniuk, V Cevher, MF Duarte, C Hegde, Modelbased compressive sensing. IEEE Trans. Inf. Theory. 56(4), 1982–2001 (2010).
 44
YC Eldar, M Mishali, Robust recovery of signals from a structured union of subspaces. IEEE Trans. Inf. Theory. 55(11), 5302–5316 (2009). doi:10.1109/TIT.2009.2030471.
 45
MF Duarte, RG Baraniuk, Kronecker compressive sensing. IEEE Trans. Image Process.21(2), 494–504 (2012).
 46
YF Gao, L Zou, Q Wan, A twodimensional arrival angles estimation for lshaped array based on tensor decomposition. AEU  Int. J. Electron. Commun.69(4), 736–744 (2015). doi:10.1016/j.aeue.2015.01.001.
 47
DL Donoho, et al., Highdimensional data analysis: the curses and blessings of dimensionality. AMS Math Chall. Lect., 1–32 (2000).
 48
IV Oseledets, EE Tyrtyshnikov, Breaking the curse of dimensionality, or how to use SVD in many dimensions. SIAM J. Sci. Comput.31(5), 3744–3759 (2009).
 49
B Sun, H Gu, M Hu, Z Qiao, in SPIE Sensing Technology+ Applications. Compressive sensing for a general sar imaging model based on maxwell’s equations (International Society for Optics and PhotonicsSan Diego, 2015), pp. 948402–948402.
 50
Y Li, Y Wang, Sparse SMNLMS algorithm based on correntropy criterion. Electron. Lett.52(17), 1461–1463 (2016).
 51
Y Li, C Zhang, S Wang, Lowcomplexity nonuniform penalized affine projection algorithm for sparse system identification. Circ. Syst. Signal Process.35(5), 1611–1624 (2016).
 52
L De Lathauwer, B De Moor, J Vandewalle, On the best rank1 and rank(r 1, r 2,…, rn) approximation of higherorder tensors. SIAM J. Matrix Anal. Appl.21(4), 1324–1342 (2000).
 53
R Rubinstein, M Zibulevsky, M Elad, Efficient implementation of the KSVD algorithm using batch orthogonal matching pursuit. Technical Report 8, Computer Science Department, Technion  Israel Institute of Technology (2008).
 54
JA Tropp, SJ Wright, Computational methods for sparse solution of linear inverse problems. Proc. IEEE. 98(6), 948–958 (2010).
 55
Y Yuan, J Sun, S Mao, PFA algorithm for airborne spotlight SAR imaging with nonideal motions. IEE Proc. Radar Sonar Navig. 149(4), 174–182 (2002).
 56
BD Rigling, RL Moses, Polar format algorithm for bistatic SAR. IEEE Trans. Aerosp. Electron. Syst.40(4), 1147–1159 (2004).
 57
D Needell, JA Tropp, Cosamp: Iterative signal recovery from incomplete and inaccurate samples. Appl. Comput. Harmon. Anal.26(3), 301–321 (2009).
Acknowledgements
This work was supported in part by the National Natural Science Foundation of China (NSFC) under Grant U1533125 and National Science and technology major project under Grant 2016ZX03001022. The authors would like to thank the anonymous reviewers for their valuable comments that significantly improved the quality of this paper.
Authors’ contributions
YFG conceived and verified the feasibility of SAR imaging by introducing tensor modeling and developed the multilinear sparse target reconstruction algorithm based on tensor decomposition; YFG, XCC, and YY performed the experiments; GG and QW contributed to the design of numerical simulations; YFG wrote this paper; XCC, YY, YBZ, and GG checked the manuscript and contributed to the rearrangement of the materials. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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Gao, Y., Gui, G., Cong, X. et al. Multilinear sparse reconstruction for SAR imaging based on higherorder SVD. EURASIP J. Adv. Signal Process. 2017, 44 (2017). https://doi.org/10.1186/s1363401704797
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Keywords
 Spotlight SAR imaging
 Kronecker constraint
 Compressed sensing
 HOSVD
 Alternating least squares