3D Imaging Algorithm for Down-Looking MIMO Array SAR Based on Bayesian Compressive Sensing

Abstract

Down-looking MIMO array SAR can reconstruct 3D images of the observed area in the inferior of the platform of the SAR and has wide application prospects. In this paper, a new strategy based on Bayesian compressive sensing theory is proposed for down-looking MIMO array SAR imaging, which transforms the cross-track imaging process of down-looking MIMO array SAR into the problem of sparse signal reconstruction from noisy measurements. Due to account for additive noise encountered in the measurement process, high quality image can be achieved. Simulation results indicate that the proposed method can provide better resolution and lower sidelobes compared to the conventional method.

1. Introduction

Traditional synthetic aperture radar (SAR) is a microwave sensor which can reconstruct two-dimensional (2D) images of the observed area with weather independence and all-day operation capabilities [1, 2]. However, traditional 2D SAR works in side-looking mode and often meets with shading and lay over effects in urban and mountain areas. Compared with 2D SAR, 3D SAR has distinct advantage in estimation of forest height, 3D digital maps, complex terrain mapping, and so on. Multibaseline SAR tomography is an advanced 3D SAR imaging mode, which forms an additional synthetic aperture in the height direction. Therefore, it has resolving capability along this dimension. Unfortunately, for the current SAR tomography, it is almost impossible to avoid an uneven track distribution in repeat-pass data acquisition, which is just the main reason for the strong ambiguity in height [3, 4].

Down-looking array SAR is an innovative imaging mode, which obtains range resolution by pulse compression, azimuth resolution by virtual aperture synthesis with platform movement, and cross-track resolution by a linear array antenna [5–7]. Down-looking array SAR can overcome restrictions of shading and lay over effects in side-looking SAR and also avoid the height ambiguity problem in SAR tomography caused by the uneven track distribution. However, in order to avoid the grating lobe effect in the cross-track direction, a large number of antenna elements are required, which increase the cost and complexity of the equipment. Therefore, multiple-input-multiple-output (MIMO) antenna array is often used in down-looking array SAR, which can reduce the number of real antenna elements largely for a given size of antenna array [8–10]. The common methods used for down-looking MIMO array SAR imaging are usually based on matched filter, which often suffer from low resolution and high sidelobe interference in the images [7, 8]. Moreover, the cross-track resolution is limited by the length of linear array. Hence, high resolution imaging algorithms are desired.

In recent years, Bayesian compressive sensing (BCS) has caused widespread concern, showing significant advantages to sparse signal reconstruction [11]. BCS methods provide certain improvements compared with norm-based CS methods in low noise level, by exploiting the sparseness prior distribution of the image scene. In addition, the Bayesian framework takes into account the additive noise encountered when implementing compressed sampling. Therefore, some BCS based methods for SAR applications have been concerned about recently [12, 13]. As the 3D illuminated scene contains only a very small strong scattering centers compared with the total space cells, representing strong spatial sparsity in high frequency radar application, we propose a new 3D imaging algorithm for down-looking MIMO array SAR based on BCS.

The rest of the paper is organized as follows. Section 2 presents the geometry and principle of down-looking MIMO array SAR system. In Section 3, a new 3D imaging algorithm for down-looking MIMO array SAR is described in detail. The performance of the method is investigated in Section 4. Finally, Section 5 gives a brief conclusion.

2. Down-Looking MIMO Array SAR

2.1. Geometrical Model

The geometry of down-looking MIMO array SAR is shown in Figure 1. 𝑥, 𝑦, and 𝑟 denote the azimuth, cross-track, and slant range direction, respectively. The radar platform flies along the 𝑥-axis corresponding to azimuth direction, with velocity 𝑣 at height 𝐻. The thinned linear antenna array, which contains 𝑀 transmitting antenna elements and 𝑁 receiving antenna elements, is mounted in the cross-track direction along the wings. The transmitting antenna elements are located at the tips with the distance 𝑑, the receiving antenna elements are centered at the 𝑦-axis with spacing 𝑀𝑑/2, and the distance between the transmitting and receiving antenna elements is 𝑑/2. The thinned linear antenna array works in the time division mode. Each time, only one transmitting antenna element transmits signal and all the receiving antenna elements receive echo simultaneously. The transmitting antenna elements work sequentially and an aperture synthesis period is acquired until all the transmitting antenna elements have worked once. According to the principle of equivalent phase center, the thinned linear array formed by the above positions and work mode is equal to a fully distributed virtual uniform linear array [5, 10]. The virtual antenna array is composed of 𝑀𝑁 virtual elements and works in self-transmitting and receiving mode. That is to say, each virtual antenna element transmits and receives signal by itself. These virtual antenna elements are uniform distributed along the wings and centered at the 𝑦-axis. Each virtual antenna element is located at the mean position of a real single transmitting element and a real single receiving element, and the distance between individual virtual antenna elements is 𝑑/2. Figure 2 shows the transmitting and the receiving order of each antenna element for a down-looking MIMO array SAR.

Figure 1 

Geometry of down-looking MIMO array SAR.

Figure 2 

Transmitting and receiving order of down-looking MIMO array SAR.

2.2. Equivalent Phase Error Compensation

Consider the data acquisition shown in Figure 1. At the slow time 𝑡_𝑚, the position of the 𝑖th transmitting antenna element is given by (𝑥, 𝑦_𝑇𝑖, 𝐻), where 𝑥=𝑣𝑡_𝑚 is the azimuth position, and 𝑦_𝑇𝑖 is the cross-track position of the 𝑖th transmitting antenna element. The position of the 𝑗th receiving antenna element is given by (𝑥, 𝑦_𝑅𝑗, 𝐻), where 𝑦_𝑅𝑗 is the cross-track position of the 𝑗th receiving antenna element. For a point scatterer 𝑃 positioned at (𝑥_𝑝, 𝑦_𝑝, 𝑧_𝑝), the transmitting and receiving paths 𝑅_𝑇𝑖 and 𝑅_𝑅𝑗 are given by 𝑅_𝑇𝑖=𝑥−𝑥_𝑝²+𝑦_𝑇𝑖−𝑦_𝑝²+𝐻−𝑧_𝑝²=𝑦_𝑇𝑖−𝑦_𝑝²+𝑅²_𝑃,𝑅_𝑅𝑗=𝑥−𝑥_𝑝²+𝑦_𝑅𝑗−𝑦_𝑝²+𝐻−𝑧_𝑝²=𝑦_𝑅𝑗−𝑦_𝑝²+𝑅²_𝑝, (1) where 𝑅_𝑝=𝑥−𝑥_𝑝²+𝐻−𝑧_𝑝². (2) Then the complete travelling path of the wave from the 𝑖th transmitting antenna element to point scatterer 𝑃 to the 𝑗th receiving antenna element is given by 𝑅_𝑖,𝑗=𝑅_𝑇𝑖+𝑅_𝑅𝑗=𝑦_𝑇𝑖−𝑦_𝑝²+𝑅²_𝑃+𝑦_𝑅𝑗−𝑦_𝑝²+𝑅²_𝑃. (3) From (2) we can get that the complete travelling path contains two square roots, which will lead to complex computation for the following imaging process. According to the principle of equivalent phase center, the above complete travelling path can be equal to the dual echo paths from the virtual antenna element located at 𝑦=(𝑦_𝑇𝑖+𝑦_𝑅𝑗)/2 to point scatterer 𝑃 [5]. And the equivalent echo path can be written as 𝑅=𝑥−𝑥_𝑝²+𝑦−𝑦_𝑝²+𝐻−𝑧_𝑝²=𝑦−𝑦_𝑝²+𝑅²_𝑝. (4) Then, the phase difference between the virtual antenna element and the reality antenna element is given by Δ𝜑=2𝜋𝜆𝑅_𝑖,𝑗=−2𝑅2𝜋𝜆𝑦_𝑇𝑖−𝑦_𝑝²+𝑅²_𝑃+𝑦_𝑅𝑗−𝑦_𝑝²+𝑅²_𝑃−2𝑦−𝑦_𝑝²+𝑅²_𝑝≈𝜋𝑦_𝑇𝑖−𝑦_𝑅𝑗²2𝜆𝑅_𝑝. (5)

When the scatterers are located at the far field, the phase difference Δ𝜑 is approximate to zero. Otherwise, the phase compensation should be implemented before the image processing. From (5) it can be seen that the compensated phase varies with 𝑅_𝑝; in practice we use the center of view field as reference point to compensate the whole view field.

Furthermore, the down-looking MIMO array SAR works in the time division mode and the virtual antenna elements obtained from different pulse repetition period are not in a straight line with the movement of the platform. Therefore, in order to obtain a fully distributed virtual uniform linear array, the motion compensation should be implemented. The compensated phase caused by the overtake or lag phases owing to the position difference of the antenna elements is given by Δ𝜑_𝑙,𝑘=4𝜋𝜆𝑀𝑣𝐿PRF𝑙−2+Δ𝑥_𝑘−𝑥_𝑝²+𝑦−𝑦_𝑝²+𝐻−𝑧_𝑝^21/2−𝑀𝑣𝐿PRF𝑙−2−𝑥_𝑝²+𝑦−𝑦_𝑝²+𝐻−𝑧_𝑝^21/2,𝑙=1,2,…,𝐿;𝑘=1,2,…,𝑀, (6) where 𝐿 is the azimuth sample number and Δ𝑥_𝑘 is the move spacing between the 𝑘th and the first transmitting antenna element in the azimuth direction. 𝑣 is the velocity of the platform and PRF is the pulse repetition frequency.

After the process above, the collected data of down-looking MIMO array SAR can be regarded as received by the fully distributed virtual linear array.

3. Three-Dimensional Imaging Algorithm for Down-Looking MIMO Array SAR

Based on the principle of equivalent phase center, the thinned linear array formed by the time division mode can be equal to a virtual linear array, and each virtual antenna element transmits and receives signal by itself. The linear frequency modulated pulse signal transmitted by the 𝑚th virtual antenna element is given by 𝑠𝑡̂𝑡̂=rect𝑇_𝑟exp𝑗2𝜋𝑓_𝑐̂1𝑡+2𝛾𝑡², (7) where 𝑡̂ denotes the fast time, 𝑇_𝑟 denotes the pulse width, 𝑓_𝑐 is the carrier frequency, 𝛾 is the chirp rate, and rect(⋅) is the unit rectangular function.

For an arbitrary point scatterer positioned at 𝑃(𝑥_𝑝,𝑦_𝑝,𝑧_𝑝), the echo signal received by the 𝑚th virtual antenna element can be expressed as 𝑠₁𝑡,̂𝜏,𝑦_𝑚̂=rect𝑡−2𝑅/𝑐𝑇_𝑟rect𝜏−𝜏_𝑝𝑇_𝑎×exp𝑗𝜋𝛾𝑡̂−2𝑅𝑐²−𝑗4𝜋𝜆𝑅, (8) where 𝜏 denotes the azimuth time, 𝜏_𝑝=𝑥_𝑝/𝑣 is the azimuth time of the point scatterer 𝑃, 𝑐 is the light velocity, 𝑇_𝑎 is the observing duration, 𝜆 is the wavelength, and 𝑅 is the instantaneous distance between the 𝑚th virtual antenna element and the point scatterer 𝑅=𝑣𝜏−𝑥_𝑝²+𝑦_𝑚−𝑦_𝑝²+𝐻−𝑧_𝑝²≈𝑅_𝑝+𝑣𝜏−𝑥_𝑝²2𝑅_𝑝+𝑦²_𝑚−2𝑦_𝑚𝑦_𝑝2𝑅_𝑝, (9) where 𝑅_𝑝=(𝐻−𝑧_𝑝)²+𝑦²_𝑝 and 𝑦_𝑚 is the cross-track position of the 𝑚th virtual antenna element.

3.1. Range Compression

Transform the signal expressed in (8) into the range frequency domain 𝑆₁𝑓_𝑟,𝜏,𝑦_𝑚𝑓=rect^𝑟𝑇_𝑟𝛾rect𝜏−𝜏_𝑝𝑇_𝑎×exp−𝑗𝜋𝑓²_𝑟𝛾exp−𝑗4𝜋𝑓_𝑟𝑐𝑅exp−𝑗4𝜋𝜆𝑅, (10) where 𝑓_𝑟 is range frequency.

The matched filter function for the range compression is given by 𝐻₁𝑓_𝑟𝑓=rect^𝑟𝑇_𝑟𝛾exp𝑗𝜋𝑓²_𝑟𝛾. (11)

After the range compression, the received signal can be written as 𝑠₂𝑡,̂𝜏,𝑦_𝑚=rect𝜏−𝜏_𝑝𝑇_𝑎sinc𝑇_𝑟𝛾𝑡̂−2𝑅𝑐×exp−𝑗4𝜋𝑅𝜆. (12)

3.2. Azimuth Compression

From (9) and (12) we get that the distance between the antenna element and the scatterer varies with the azimuth position of the radar platform, which leads to the coupling of the envelope in the range-azimuth plane. Therefore, the range migration in the azimuth direction should be removed first before the azimuth compression. And the correction function for the range migration can be derived from (9) 𝐻₂𝑓_𝑟,𝜏=exp𝑗2𝜋𝑣𝜏−𝑥_𝑝²𝑐𝑅_𝑝𝑓_𝑟. (13)

Transform the signal expressed in (12) to the range frequency domain and correct the range migration using 𝐻₂(𝑓_𝑟,𝜏). Then, after the range inverse Fourier transform, the signal becomes 𝑠₃𝑡,̂𝜏,𝑦_𝑚=rect𝜏−𝜏_𝑝𝑇_𝑎sinc𝑇_𝑟𝛾𝑡̂−2𝑅_𝑐𝑐×exp−𝑗4𝜋𝑅_𝑝𝜆⋅exp−𝑗2𝜋𝑣²𝜏−𝜏_𝑝²𝜆𝑅_𝑝𝑦×exp−𝑗2𝜋²_𝑚−2𝑦_𝑚𝑦_𝑝𝜆𝑅_𝑝, (14) where 𝑅_𝑐=𝑅_𝑝+(𝑦²_𝑚−2𝑦_𝑚𝑦_𝑝)/2𝑅_𝑝.

An azimuth Fourier transform is then performed on each range gate to transform the data into the range time-azimuth frequency domain, and the signal becomes 𝑆₃𝑡,̂𝑓_𝑎,𝑦_𝑚𝑓=rect^𝑎𝛾_𝑎𝑇_𝑎sinc𝑇_𝑟𝛾𝑡̂−2𝑅_𝑐𝑐𝑦×exp−𝑗2𝜋²_𝑚−2𝑦_𝑚𝑦_𝑝𝜆𝑅_𝑝⋅exp−𝑗4𝜋𝑅_𝑝𝜆×exp𝑗𝜋𝑓²_𝑎𝛾_𝑎exp−𝑗2𝜋𝜏_𝑝𝑓_𝑎, (15) where 𝛾_𝑎=2𝑣²/𝜆𝑅_𝑝. Then a multiplication of the signal with the azimuth matched filter function 𝐻₃(𝑓_𝑎) is performed, where 𝐻₃𝑓_𝑎=exp−𝑗𝜋𝑓²_𝑎𝛾_𝑎. (16) Then, we get the azimuth compressed signal by performing inverse Fourier transform in the azimuth direction 𝑠₄𝑡,̂𝜏,𝑦_𝑚=sinc𝑇_𝑟𝛾𝑡̂−2𝑅_𝑐𝑐sinc𝑇_𝑎𝛾_𝑎𝜏−𝜏_𝑝𝑦⋅exp−𝑗2𝜋²_𝑚−2𝑦_𝑚𝑦_𝑝𝜆𝑅_𝑝exp−𝑗4𝜋𝑅_𝑝𝜆. (17)

3.3. Cross-Track Compression

From (17) it can be seen that the distance between the antenna element and the scatterer also varies with the cross-track position of the antenna element. Therefore, the range migration in the cross-track direction should be removed before the cross-track compression. And the amount of the range migration to be corrected can be given by (9) Δ𝑅_𝑐=𝑦²_𝑚−2𝑦_𝑚𝑦_𝑝2𝑅_𝑝. (18)

Transform the signal expressed in (17) into the range frequency domain, and correct the range migration in the range frequency domain. Then the signal becomes 𝑠₄𝑡,̂𝜏,𝑦_𝑚=sinc𝑇_𝑟𝛾𝑡̂−2𝑅_𝑝𝑐sinc𝑇_𝑎𝛾_𝑎𝜏−𝜏_𝑝⋅exp−𝑗2𝜋𝑦²_𝑚𝜆𝑅_𝑝exp𝑗4𝜋𝑦_𝑚𝑦_𝑝𝜆𝑅_𝑝exp−𝑗4𝜋𝑅_𝑝𝜆. (19) The first phase term in (19) represents a quadratic distortion, which can be compensated by 𝐻₄𝑦_𝑚=exp𝑗2𝜋𝑦²_𝑚𝜆𝑅_𝑝. (20)

In high frequency radar application, the interest scene can be modeled by a limited number of strong scattering centers reflecting impinging electromagnetic waves isotropically to all receivers, representing strong spatial sparsity [14]. Therefore, the cross-track imaging process of down-looking MIMO array SAR can be transformed into the problem of sparse signal reconstruction from noisy measurements. After the 2D imaging process in the range and azimuth directions, the signal in the range-azimuth cell corresponding to (𝑡̂=2𝑅_𝑝/𝑐,𝜏=𝜏_𝑝) by neglecting the constant phase term of (19) can be written as 𝑠₅𝑡,̂𝜏,𝑦_𝑚=^𝑃_𝑝=1𝜎_𝑝𝑡,̂𝜏exp𝑗2𝜋𝑓_𝑝𝑦_𝑚, (21) where 𝜎_𝑝(𝑡,̂𝜏) and 𝑓_𝑝=2𝑦_𝑝/𝜆𝑅_𝑝 denote the backward scattering coefficient and the frequency of the 𝑝th point scatterer, respectively.

For numerical analysis, (21) can be described by discrete system model 𝐬=𝚽𝝈, (22) where 𝐬=[𝑠₅(𝑡,̂𝜏,𝑦₁),𝑠₅(𝑡,̂𝜏,𝑦₂),…,𝑠₅(𝑡,̂𝜏,𝑦_𝑀𝑁)]^𝑇 is the signal vector corresponding to the 𝑀𝑁 virtual antenna elements, 𝝈=[𝜎₁(𝑡,̂𝜏),𝜎₂(𝑡,̂𝜏),…,𝜎_𝑃(𝑡,̂𝜏)]^𝑇is the complex-valued scatter coefficient vector in the cross-track direction, and the matrix 𝚽 can be constructed as 𝚽=𝝓₁,𝝓₂,…,𝝓_𝑝,…,𝝓_𝑃, (23) where 𝝓_𝑝=exp𝑗2𝜋𝑓_𝑝𝑦₁,exp𝑗2𝜋𝑓_𝑝𝑦₂,…,exp𝑗2𝜋𝑓_𝑝𝑦_𝑀𝑁^𝑇, (24) where (⋅)^𝑇 represents the transpose operation.

In the more realistic case some noise is added on the measurements 𝐬=𝚽𝝈+𝐧 (25) with 𝐧 a complex Gaussian vector with zero mean and power 𝜎²_𝑛.

Then, the Bayesian compressive sensing method is employed to estimate the 𝝈. From the aspect of denoising, Laplace distribution is often used as the sparseness prior [15]. Hence, the probability distribution function of 𝝈 can be given by 𝑝𝝈∣𝜎_𝛿=^𝑃_𝑖=1𝑝𝜎_𝑖∣𝜎_𝛿=^𝑃_𝑖=11√2𝜎_𝛿−√exp2𝜎_𝑖𝜎_𝛿, (26) where 𝜎_𝛿 is the scale parameter of the Laplace distribution.

And the probability density function of Gaussian noise 𝐧 is given by 𝑝𝐧∣𝜎²_𝑛=12𝜋𝜎²_𝑛^𝑀𝑁−‖𝐧‖exp²²2𝜎²_𝑛, (27) where ‖⋅‖²₂ denotes the 𝑙₂ norm.

Therefore, the likelihood function of the received data 𝐬 is 𝑝𝐬∣𝝈,𝜎²_𝑛=12𝜋𝜎²_𝑛^𝑀𝑁−‖‖exp𝐬−𝚽𝝈²₂2𝜎²_𝑛. (28)

Based on Bayesian theory, the maximum a posteriori (MAP) estimator is used to estimate 𝝈 as 𝑝(𝑝𝝈=argmax𝝈∣𝐬)=argmax𝐬∣𝝈,𝜎²_𝑛⋅𝑝𝝈∣𝜎_𝛿. (29)

Apparently, the MAP solution of 𝝈 can be estimated by maximizing the log posterior of 𝝈 as 𝝈=argmax𝐽(𝝈)=argmaxln𝑝𝐬∣𝝈,𝜎²_𝑛+ln𝑝𝝈∣𝜎_𝛿−1=argmax2𝜎²_𝑛‖𝐬−𝚽𝝈‖²₂−√2𝜎_𝛿^𝑃_𝑖=1𝜎_𝑖=argmin‖𝐬−𝚽𝝈‖²₂+𝜇‖𝝈‖₁, (30) where 𝜇=2√2𝜎²_𝑛/𝜎_𝛿.

Then, the quasi-Newton iterative method with Hessian update scheme is used to solve the optimization problem [16]. The gradient of the objective function 𝐽(𝝈)with respect to 𝝈 is given by ∇𝐽(𝝈)=𝐻(𝝈)𝝈−2𝚽^𝐻𝐬, (31) where 𝐻(𝝈)=2𝚽^𝐻(𝝈),𝚲(𝝈)𝝈)𝚽+𝜇𝚲=diag(_𝑖²+𝜀^−1/2. (32)

Here 𝐻(𝝈) is used as an approximation to the Hessian, and 𝝈 can be obtained from the following quasi-Newton iteration: 𝝈^(𝑛+1)=𝝈^(𝑛)−𝛾𝐻𝝈^(𝑛)−1∇𝐽𝝈^(𝑛), (33) where 𝛾 is the step size. After substituting (31) into (33), we obtain the following iterative algorithm: 𝐻𝝈^(𝑛)𝝈^(𝑛+1)=1−𝛾𝐻𝝈^(𝑛)𝝈^(𝑛)+2𝛾𝚽^𝐻𝐬. (34) The iteration is stopped when ‖𝝈^(𝑛+1)−𝝈^(𝑛)‖²₂<𝜁, where 𝜁 is a small positive constant.

Then, the 3D image of down-looking MIMO array SAR can be obtained until all the range-azimuth cells have been processed using the same procedure.

In conclusion, the 3D imaging processing flow of the proposed method for down-looking MIMO array SAR can be shown in Figure 3.

Figure 3 

The flowchart of proposed method for down-looking MIMO array SAR imaging.

3.4. Estimation of the Parameters 𝜎²_𝑛 and 𝜎_𝛿

It is clear that the imaging performance of the proposed method is related to the correct selection of the noise level 𝜎²_𝑛. Therefore, the noise level must be estimated accurately. Because the noise always distributes evenly and there are many range cells containing noise only in down-looking SAR imaging, the noise level estimation is available by setting an energy-based threshold to select the noise cells [17]. The threshold is given by thres=𝐸_𝑚+^𝑃𝑄_𝑖=1𝐸_𝑖−𝐸_𝑚²𝑃𝑄^1/2, (35) where 𝐸_𝑖 is the energy of the 𝑖th range cell and 𝐸_𝑚 denotes the mean energy of all range cells. If the energy of a range cell is below the threshold, this range cell is selected as noise cell. Then, the energy of all the selected cells can be used as the noise level 𝜎²_𝑛.

In addition, the maximum likelihood estimate method is used to choose the objective statistical parameter 𝜎_𝛿. According to (26), the log-likelihood function for the Laplace distribution can be written as ℓ𝜎_𝛿∣𝜎₁,𝜎₂,…,𝜎_𝑃=−𝑃ln√2𝜎_𝛿−^𝑃_𝑖=1√2𝜎_𝑖𝜎_𝛿. (36) Maximizing the log likelihood function (36) with respect to 𝜎_𝛿 produces the following equation: 𝜎_𝛿=1𝑃^𝑃_𝑖=1√2𝜎_𝑖. (37) From (37) it is clear that we can average the estimates of all pixel values to obtain the estimation of the statistical parameter 𝜎_𝛿.

4. Simulation Results

In this section, point target simulation is carried out to verify the validity of the proposed imaging algorithm. The main parameters used for simulation are listed in Table 1.

Table 1 

Parameters used for simulation.

Suppose that there are five point targets located at the scene with the azimuth-range-cross track values equal to (0, 485, 0), (8, 495, 20), (8, 495, −20), (−8, 495, 20), and (−8, 495, −20), respectively. The distributions of the five point targets are shown in Figure 4(a). After raw data generation and 3D imaging processing by using the proposed algorithm, the surfaces of the final 3D image are plotted at −20 dB in Figure 4(b). As expected, the image is reconstructed in 3D space, and the whole space structure is very consistent with the real situation in Figure 4(a). Figure 5 shows three selected sections of the final 3D image of down-looking MIMO array SAR. Figure 5(a) shows the 2D image of the selected range-cross track section corresponding to azimuth position 8 m. Figure 5(b) shows the 2D image of the selected azimuth-cross track section corresponding to range position 495 m. Figure 5(c) shows the 2D image of the selected azimuth-range section corresponding to cross-track position −20 m. The below imaging results show that the point scatterers are well focused in three directions, confirming the validity of the proposed algorithm.

(a) Real spatial position

(b) Final 3D image

Figure 4 

Real spatial position and final 3D image.

(a) Selected range-cross track section

(b) Selected azimuth-cross track section

(c) Selected azimuth-range section

Figure 5 

2D image of selected sections.

In order to analyze the performance of the proposed method, the imaging result obtained by Fourier transform is given for comparison. Supposed that there are two targets located at the azimuth-range-cross track unit of (8, 0.18, 495) and (8, −0.18, 495). Figure 6(a) shows the range-cross track reconstruction result obtained by Fourier transform, and Figure 6(b) shows the range-cross track reconstruction result obtained by the proposed method. As the cross-track distance of the two targets is 0.36 m, which is less than the cross-track resolution of 0.4 m, the two targets can not be distinguished in the image obtained by Fourier based method. However, the proposed method can improve the spatial resolution and distinguish the two targets clearly.

(a) Imaging result obtained by Fourier transform

(b) Imaging result obtained by proposed method

Figure 6 

Comparison of the range-cross track imaging results of down-looking MIMO array SAR.

In the following experiment, we take into account two targets located at the same azimuth and range positions and with the cross-track values equal to −2 m and 2 m, respectively. Moreover, additional Gaussian distributed complex noise is added to generate measurements with SNR of 5 dB. Figures 7(a) and 7(b) show the cross-track distribution of the two targets obtained by Fourier transform and the proposed method, respectively. By comparing the imaging results, it can be seen that the proposed method is more robust to noise and it recovers target image together with suppressing the noise components.

(a) Cross-track distribution obtained by Fourier transform

(b) Cross-track distribution obtained by proposed method

Figure 7 

Comparison of the cross-track distribution obtained by Fourier transform and the proposed method.

5. Conclusions

Down-looking MIMO array SAR can reconstruct 3D images of the observed area and overcome restrictions of shading and lay over effects in side-looking SAR. Therefore, down-looking MIMO array SAR has challenging potential for 3D digital maps, complex terrain mapping, and so on. However, the cross-track resolution of down-looking MIMO array SAR is limited by the length of linear array. In this paper, a novel 3D imaging strategy is proposed for down-looking MIMO array SAR. Exploiting the spatial sparsity of the interest scene, we transform the cross-track imaging process of down-looking MIMO array SAR into the problem of sparse signal reconstruction from noisy measurements. Raw data of down-looking MIMO array SAR in Ka-band is simulated and the 3D image is achieved. The results of the simulated data confirm the effectiveness of the proposed method.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant 61201390, Grant 61201389 and Grant 11201120, and the Natural Science Foundation for Education Department of Henan Province under Grant 13A510184.