THz-SAR Vibrating Target Imaging via the Bayesian Method

Target vibration bears important information for target recognition, and terahertz, due to significant micro-Doppler effects, has strong advantages for remotely sensing vibrations. In this paper, the imaging characteristics of vibrating targets with THz-SAR are at first analyzed. An improved algorithm based on an excellent Bayesian approach, that is, the expansion-compression variancecomponent (ExCoV) method, has been proposed for reconstructing scattering coefficients of vibrating targets, which provides more robust and efficient initialization and overcomes the deficiencies of sidelobes as well as artifacts arising from the traditional correlation method. A real vibration measurement experiment of idle cars was performed to validate the range model. Simulated SAR data of vibrating targets and a tank model in a real background in 220GHz show good performance at low SNR. Rapidly evolving high-power terahertz devices will offer viable THz-SAR application at a distance of several kilometers.


Introduction
Vibration is one of the commonest motions in nature, such as the vibration of a car engine, the bumping of a car on a rude road, the chest movement caused by heart beat and breath, and the arm swing of a walking terrorist.The characteristics of vibration differ greatly for different targets.For example, the vibration amplitude of different types of tanks or armored cars ranges from several millimeters to several centimeters, and the vibration frequency ranges from several hertz to several decades hertz.The exact values are closely related to the vehicle type, size, and structure.In addition, the remote sensing of vibration is helpful for gathering the road condition information [1].One can see that vibration contains subtle and stable information of targets.The vibration parameters, once extracted, can play an important role in target recognition, which will open up a new range of applications in battlefield reconnaissance, precision seeking, traffic surveillance, and antiterrorist tasks.
Detecting vibrating targets on the ground with synthetic aperture radar (SAR) is an emerging subject in the remote sensing community.Direct SAR imaging for vibrating targets, however, will result in ghosts in the image due to the periodic micro-Doppler modulation [2], which cannot be overcome by the SAR ground moving target indication (SAR/GMTI) techniques.Relevant researches on SAR target vibration have been carried out internationally since 1995 [3], and our group started the study early in 2006 in China and defined it, together with rotating target detection and imaging, as SAR/micromotion target indication (SAR/MMTI) [4].Recent works on it are mainly focused on the vibrating target phenomenology or vibration's influence, vibrating target detection, micro-Doppler analysis as well as extraction, and real vibration sensing experiments.On the phenomenology, main conclusion is that vibration gives rise to high frequency phase and paired echoes in the time domain, to micro-Doppler in the frequency domain and to ghosts [2], fence [5], bowknot [6], or fuzzballs [7] in the image domain.On the detection, the main methods include the generalized likelihood ratio test and cyclostationary spectrum density method [8][9][10] or the multiantenna based displaced phase center antenna (DPCA) and along-track interferometry (ATI) methods [11][12][13][14].On the micro-Doppler analysis and extraction, time-frequency analysis [12] and the fractional Fourier transform (FrFT) [15] are predominant and the pulse-repetition-interval transform [16] can also provide good results.On the experiments, X-band APY-6 [17], Kuband Lynx [18] and mmW MEMPHIS [12] radar systems have been utilized.The basic viability of vibration sensing has been proven, but two main problems have yet to be addressed.
(1) Experimental data show that minor vibrations with the amplitude under several millimeters cannot be detected even 2 International Journal of Antennas and Propagation by mmW radars [12].(2) Few researches on vibrating target imaging have been done.Although the phase compensation method and the Doppler keystone transform [19] can be used for imaging, they are mainly useful to point scattering targets, and the range cell migration of vibrating targets is still difficult to be corrected.The MLE imaging method [20] proposed previously by us can directly model the echo signal and then estimate scattering centers of vibrating targets but is essentially of the high-dimensional nonlinear optimization problem with ill condition, and it is impossible to obtain the global minimum without using prior information.
To solve problem (1), we introduce terahertz wave into SAR for vibration sensing.Due to the shorter wavelength, terahertz wave is more sensitive to micromotions, making it easier to detect vibrating targets.The emission power of solidstate and vacuum electronics terahertz devices approaches to several decade Watts according to the DARPA terahertz electronics program.As there is serious atmosphere attenuation, the emission power is sufficient to meet the needs of airborne terahertz radar for operating range of several kilometers at the frequencies of 220 GHz or 330 GHz atmosphere windows.In 2012, the US DARPA proposed a project called Video SAR (ViSAR) [21] and the frequency band around 230 GHz was chosen, prophesying realization of THz-SAR.To solve problem (2), we convert vibrating target imaging to the problem of sparse representation, and the expansion-compression variance-component (ExCoV) method is utilized to obtain the scattering coefficients of vibrating targets.The ExCoV method is of sparse Bayesian learning (SBL) [22], but it is more effective and has less parameters than the traditional SBL and is promising to efficiently solve the high-dimensional optimization problems.In this paper, we will modify the initial iteration in ExCoV, that is, superseding pseudoinverse by correlation processing, to improve efficiency and stability.The simulated data of vibrating point targets and a tank model at 220 GHz will validate the performance of the proposed algorithm.

Models of THz-SAR Vibrating Targets
As illustrated in Figure 1, the radar moves at velocity   .Then at slow time  it moves to We could see that  has the similar meaning as slow time , and thus the symbol   may be more appropriate; nevertheless it is abandoned for brevity.Now considering an arbitrary moving target, let vector  represent the target motion parameters, such as the initial position (, ), velocity, and vibration frequency. can thus be regarded as the identification code of a target.Suppose that when the radar moves to   =    or  for simplicity, the target of  moves to ( , ,  , ), and then the target range model is where the approximation is a result of plane waves; that is, Spotlight SAR echoes of the target could then be represented in the wavenumber domain as where  = 4/ = 4/ is the two way wavenumber,  is the speed of light,  is the wavelength,  is the frequency,   = 4/  = 4  / is the center wavenumber,   is the carrier wavelength,   is the carrier frequency, and () is the Fourier transform of the transmitted signal.Suppose multiple targets exit, recall that  is the identification code of a target, and thus we use () as the scattering coefficient of the target with respect to .Then the total echoes of all targets can be expressed in terms of integration as (5) Note ( 5) is a Fredholm integral equation of the first kind.When the target experiences vibration, the real vehicle vibration frequencies have complex components and the vibration frequencies also differ for its parts such as the roof and the bumper [12].However, it can be regarded as a synthesis of simple harmonic vibrations, as measured for two idle cars which can be seen later in the experiment section.Considering only one frequency component for simplicity and without loss of generality, we have where vibration parameters compose a parameter vector and (, ) is the center of the vibration axis,  is the effective vibration amplitude,   is the vibration frequency, and  0 is the initial phase at  = 0.By the way, ( 6) can also be used to model target rotations.Substituting ( 6) into ( 5), discretizing the result, and adding noise yield where  is the number of vibrating targets (it can also include stationary body points by just setting  or   to zero and can include no target by setting   to zero),   denotes the scattering coefficient of the th target, and During the derivation of ( 9),  =   tan /  from (1) and 2    /  ≫ 1 have been used.
In (9), we can clearly see that there is an additional exponential component, that is, the last term, representing target vibration in (9) when compared with the stationary scattering center model.Note  and  can also be discretized into  and  values, respectively, and therefore (8) can be expressed in a matrix form as where From (10), the imaging or scattering coefficient reconstruction problem can be regarded as an inverse problem of solving linear equations.
What deserves special mention is that the unknown  has also been discretized into  1 ,  2 , . . .,   , which are located at a high-dimensional cubic grid, and therefore Φ contains no unknowns.

The Influence of Target Vibration on THZ-SAR Image
It is well known that vibrating targets cannot be focused in microwave SAR images and a ghost image will generally result according to the paired echo principle.We now try to analyze THz-SAR image characteristics of vibrating targets.
The space between ghosts is a constant [2] For THz-SAR, Δ will decrease by a commensurate amount compared with microwave SAR since   is significantly smaller than the wavelength at X-band or mmW band.
Note the azimuth resolution   =     /2    , where   is the coherent processing interval (CPI), and the space in pixel or resolution cell can be expressed as It is apparent that the space in pixel or resolution cell has no relation with the frequency band.However,   needed for the same resolution as microwave SAR is reduced for THz-SAR.For typical parameter values, Δ/  ≤ 1, which states that the ghosts are not resolvable and line segment results.
According to Carson's rule [22], the ghost number plus the real target is From ( 12) and ( 14), the length of the ghost image or the line segment in azimuth is International Journal of Antennas and Propagation Equation ( 15) makes it clear that Δ will decrease when the frequency moves to the terahertz band.

Vibrating Target Imaging via the Improved ExCoV Method
In order to solve (8), we model the pdf of THz-SAR echo vector g given a and  2 using the standard additive Gaussian noise model where C is a known positive definite symmetric matrix of size  × ,  2 is an unknown noise-variance parameter, and  2 C is the noise covariance matrix.The SBL employs a Gaussian prior on the signal with a distinct variance component on each signal element However, we have known, according to the sparse recovery theory and a priori knowledge, that in most real-world cases only a few elements of a have significant magnitudes and that the remaining elements are either strictly zeros or close to zeros [23].Therefore the prior distribution (17) did not capture the signal key feature, that is, sparsity.On the contrary, the ExCoV method assigns distinct variance components to the candidates for significant signal elements and uses only one common variance-component parameter to account for the variability of the rest of signal coefficients [24].Let Λ be the full index set in the signal and let the size be .Denote A as the set of indices of the signal elements with distinct variance components and the size as  A , and define the complementary index set B = Λ/A with cardinality  B =  −  A .Accordingly, we partition Φ and a into submatrices Φ A and Φ B and subvectors a A anda B .Then the following prior model for the signal coefficients is adopted: where A, A } and  B ( 2 ) =  2 I  B are the signal covariance matrices and the variance components  2 A,1 ,  2 A,2 , . . .,  2 A, A for a A are distinct while the common variance  2 accounts for the variability of a B .Assume that the signal variance components  A and  2 are unknown and define the set of all unknowns  = (A,  A ,  2 ,  2 ), and then the marginal pdf of the observations g given  is [24]  (g | ) = ∫  (g | a,  2 ) ⋅  (a |  A ,  2 ) da =  (g; 0 ×1 ,  −1 ()) , (19) where () is the inverse covariance matrix of g given .
Clearly, (g | ) provides generalized maximum-likelihood (GML) estimation of .The GML estimation of (g | a,  2 )(a |  A ,  2 ) proceeds for obtaining the only unknown a as the image.The whole estimation process is iteratively realized via applying the ExCoV algorithm, and the iteration will terminate when A does not change between two consecutive cycles [24].
ExCoV has demonstrated an excellent algorithm for solving linear equations arising from the real world especially from the radar field [25].Compared to SBL (SBL-EM, SBL-Type II, VBSBL, and ESBL), SpaRSA, and CoSaMP, it has the smallest mean-square error (MSE) in the low signal-to-noise ratio (SNR) and the second highest computation efficiency (not better than CoSaMP whose MSE is large).It begins with an initialization followed by complex iterations.The initialization provides the starting point for the following iterations and thus plays an important part for fast convergence.The original ExCoV utilizes Moore-Penrose pseudoinverse to initialize the parameters; that is, where  denotes complex conjugation.The first  (0) A largest elements of a (0) are found to initialize A.
However, ΦΦ  is usually rank deficient for real radar problems and hence ΦΦ  is not inversible, resulting in wrong estimation of a (0) .An example is shown in Figure 2(a).We recognize that ΦΦ  denotes autocorrelation and crosscorrelation of the measurement matrix and will give a sincshaped system response.(ΦΦ  ) −1 is just the inverse sinc response, which is an identity matrix in ideal condition, to realize super-resolution estimation of a. Thus we abandon this term and obtain a more robust, efficient, and simpler estimation than (20) as which can also be called correlation imaging, back projection imaging, or multidimensional matched filtering imaging.The estimation is displayed in Figure 2(b).

Vibration Measurement Results of Two Idle
Cars.We at first measured the vibration of two cars, that is, a Mengshi car and a Nissan car, whose engines were running idly, by a frequency-modulated continuous wave (FMCW) radar, as Figures 3(a  the vibration by the synthesis of simple harmonic vibrations.In the paper, we only use one component for simplicity and without loss of generality.

THz-SAR Imaging Results of Simulated Vibrating Targets.
We proceed to conduct the following numerical experiment, as real experiments are right now not available due to low power of our THz source available, to validate the algorithm proposed.THz-SAR system parameters are given in Table 1.The experiment includes comparison of vibrating target images in microwave and terahertz bands and imaging of vibrating point targets as well as a tank model.

Comparison of Vibrating Target Images in Microwave
and Terahertz Bands. Figure 4 provides a good representation of original image characteristics of a vibrating target in different bands.For the X-band, the resolution is 0.15 m and 0.28 m in range and azimuth, respectively.Only a few ghosts are present in the image as Figure 4(a) shows, and the ghosts bear strong resemblance to stationary point targets or their sidelobes, hence difficult to discriminate between vibration and stillness.In contrast, the vibrating target takes on a line segment in the THz-SAR image as Figure 4(b) depicts.Clearly, terahertz can enforce the vibrating target image feature and therefore is expected to be used as the best band for vibration sensing.In addition, the line segment in Figure 4(b) also consists of ghosts, while the spacing between which is so near in the terahertz regime that is not resolved, as (13) predicts.

Vibrating Point Target Imaging.
For quantitatively analyzing the performance of the algorithm proposed, we use three point targets in the experiment herein, whose parameters are shown in Table 1.The images formulated by the polar format algorithm (PFA) are displayed in Figure 5, where the stationary target is well focused at the bottom while the two vibrating ones are blurred into segments.Now we employ the correlation method and the improved ExCoV to reconstruct vibrating target scattering coefficients, hence obtaining their refocused images.The imaged size is just set 0.8 m × 0.8 m both with 0.04 m grid spacing for the memory as well as computation time costs, and the sparse sampling rate is 40%, implying that compressive sensing is utilized together with ExCoV.Multidimensional images are obtained with the dimension in accord with  def = [,,,  ,  0 ]  , and parts of their slices along the vibration frequency   and amplitude  are shown in Figures 6 and 7. Clearly, the ExCoV method provides better imaging results, while the image by the correlation method shows high sidelobes and generates some artifacts for the stationary target.From the location of the focused image slices in Figure 7, one can readily estimate the vibration parameters.
We define the root MSE (RMSE) between the reconstructed scattering coefficient and the true values, that is, RMSE = ‖ã − â‖/‖ã‖, where ã denotes the true values, to test the algorithm performance.100 times Monte Carlo simulations are conducted for each SNR value to obtain Figure 8, indicating that RMSE drops sharply with SNR.
Excellent performance can be realized in low SNR and Figure 7 is an example of −10 dB SNR.

Vibrating Tank Model Imaging.
A real SAR image around the Isleta Lake in New Mexico (Figure 9(b)) is downloaded from the Sandia Cooperation website and is used to generate the scene echoes [26].Then a simulated vibrating tank model (Figure 9 is displayed in Figure 10, and the location of the slice indicates the correct vibration frequency and amplitude.In reality, the vibrating parameters will take on estimation error due to parameter grid mismatch, which remains to be explored in future investigations.

Conclusion
Terahertz has significant advantages of remote vibration sensing over microwave bands.In this paper, the THz-SAR echo model of vibrating targets has been established.Their imaging characteristics with THz-SAR are analyzed and the main features of vibrating point targets are segments consisting of unresolvable ghosts.An improved algorithm based on ExCoV, with more robust and efficient initialization, has been proposed in the sparse Bayesian framework for reconstructing scattering coefficients.Data of vibrating targets and a tank model in a real background has been simulated at 220 GHz and the reconstructed images show good performance at low SNR.The algorithm proposed can also overcome the deficiencies of sidelobes and artifacts arising from traditional correlation (or back projection) imaging method, as compared in Figures 6 and 7.In fact, the algorithm begins with correlation and can be treated as combination of ExCoV and correlation.Terahertz, together with the sparse Bayesian theory, shows a viable and most promising alternative to remote sensing of minor vibrations in particular with rapidly evolving of solid-state and high-power electric-vacuum terahertz devices.Although the Bayesian method provides a fine solution to vibrating target imaging, algorithms with higher efficiency to realize video-rate THz-SAR imaging are still to be studied in future, especially when multiple vibration components are taken into account.

Figure 2 :
Figure 2: Initial imaging of a vibrating target at (0, 0) by different methods (the arrow points to the maxima and the true maximum is located at (0, 0)): (a) Moore-Penrose pseudoinverse and (b) correlation.

Figure 10 :
Figure 10: THz-SAR image slice of the vibrating tank model on the Isleta Lake background, refocused by the ExCoV method.