A High-Resolution Imaging Method for Strip-Map SAR With Missing Data

Due to the long aperture, the high-resolution imaging for strip-map SAR with missing data is a challenge, in which the range migration correction and phase error correction are challenging. In this paper, a high-resolution imaging method of this type of data based on compressed sensing (CS) is proposed, which divides the strip-map data into several sub-apertures restored by CS and recombined to the strip-map data. The basis matrix and the measurement matrix for CS are deduced. The sub-aperture data is autofocused by the Projection Approximation Subspace Tracking (PAST) algorithm to meet the sparse requirement for the reconstructed image and the intact phase error data is restored by CS in order to combine the sub-apertures. A high-resolution image of the restored data can be obtained by conventional imaging method which performs range migration and autofocus.


I. INTRODUCTION
Synthetic Aperture Radar (SAR) can obtain high-resolution images in day/night and all-weather, so it has been widely used in both the military and the civil application. In practice, the electromagnetic wave transmitted or received by radar is vulnerable to interference, which often results in damaged or missing echo pulses. If we set the damaged or missing data to zero and image the echo data by conventional imaging method, the image quality will degrade to some extent [1], [2]. Therefore, it is significant to improve the image quality of such data [3], [4], which has much practical value in engineering application, for example, the helicopter SAR echo with pulses interrupted by the rotor.
There are many methods to recover the missing data. The first kind is interpolation [5], which can recover the missing data by interpolating based on the good data in the neighborhood of the missing data. But, when the spectrum of data is aliasing or the number of the continuous missing data is large, the result isn't ideal. The second kind is linear prediction and extrapolation [6]- [8]. However, this algorithm is The associate editor coordinating the review of this manuscript and approving it for publication was Qingchun Chen . sensitive to prediction model and signal to clutter ratio (SCR), its recovery ability will exponentially decay when the missing data is large. The third kind is the spectrum estimation (e.g. GAPES [1], [2], [9], [10] and MIAA [11] et al.). It is a nonparametric spectrum estimation method with the advantage of high robustness without affected by model parameter. It has strong restoration ability for 1-D signal or small illuminated scene. Nevertheless, it needs a lot of matrix inversions and multiple iterations which is hard to implement in 2-D SAR data of large size.
Compressed Sensing (CS) [12]- [14] is a novel signal reconstruction algorithm. As long as the signal satisfies the sparse condition at a certain domain, the time domain signal can be sampled at a frequency far below the sampling frequency demanded by the Nyquist Law. The original signal is likely to be restored perfectly using compressed sensing algorithms. When the CS theory is proposed, many scholars study the imaging methods of sparse frequency [15], [16] and azimuth sparse aperture [17]- [20]. In [21], a 2-D sparse sampling method was proposed to reduce the computational burden and the complexity of sparse matrix, which uses sub-block processing in range. After obtaining the subblock images, the complete image was formed by image mosaicking. All the above mentioned imaging methods are based on spotlight mode data, and the current hardware can meet the computational requirements of reconstruction because of the small data matrix in this mode. Literature [22] studied the CS processing method of sparse aperture SAR for strip-map mode, which was based on the R-D imaging algorithm. This method just corrected the range walk without range curvature correction and phase error compensation, leading to degrade image resolution.
At present, the processing difficulties mainly include four aspects for strip-map SAR with missing data. The first is the large amount of data, which leads to the image reconstruction computational complexity far beyond current hardware computational ability. The second is that the range migration correction must be essentially considered because of the long strip-map aperture, otherwise the image resolution will be degraded. The third is the phase error compensation. The pulse interval is non-uniform for the missing data, so the pulses can't be transformed into frequency domain, and the performance of the conventional autofocusing algorithms is limited to a certain extent. Because the SAR signal is a 2-D frequency modulated signal which can't meet the sparse requirements of CS theory as a result of its wide Fourier spectrum. The fourth problem is that we can't restore the missing data directly in the frequency domain.
To solve the above difficulties, this paper proposes a reconstruction method based on CS for missing data. First, the strip-map data is divided into several sub-apertures, which significantly improves the computational efficiency. The subaperture missing data is restored by CS and recombined to the strip-map data. Second, each sub-aperture data is transformed into the image domain by preprocessing and CS is utilized to reconstruct the sub-image. After that, the original data can be restored. Third, each sub-aperture data is autofocused before reconstructing image in order to meet the sparse requirement of CS theory. Fourth, the compensatory phase error signal is restored by CS in order to ensure the sub-aperture recombining without ghost images. Finally, the range migration of the missing data is corrected, and a high-resolution image is obtained by processing the restored data with the conventional imaging method [23] and autofocus [24]. The effectiveness and the practicability of the proposed method are demonstrated by the measured data.
The organization of this paper is as follows: Section II analyzes the data recovery method based on CS. In Section III, the autofocus method of sub-aperture is introduced. In Section IV, the process of restoring the phase error data using CS is researched. Section V presents the signal processing flow of the algorithm. In Section VI, processing of the measured data validates the effectiveness of the proposed method. Finally, we make some conclusions in Section VII.

A. ZERO PADDING AND MOTION COMPENSATION
For the missing data of strip-map SAR, we first fill the missing data by zero padding prior to the following processing.
For the moving platform, there often exists motion error, which causes serious degradations in the final images [25], so the trajectory deviations must be compensated before imaging. The phase error due to the trajectory deviations in x and y directions can be written as [26].
In (1), x (t) and y (t) are the trajectory deviations in x and y directions respectively, and θ represents the incidence angle, which is written as shown in (2) where H represents the height of the plane above the topography, R 0 is the range to the reference point, and R is the range bin size.

B. SUB-APERTURE SEGMENTATION
The pulse number of strip-map data is far beyond the data recovery capacity of CS and the computational complexity is far beyond current hardware computational ability, so it is the next step to divide the strip-map data into several nonoverlapping sub-apertures which are processed by algorithms of the spotlight mode. According to the data characteristics of the strip-map and the spotlight [27], the sub-aperture data segmentation scheme is shown in Fig. 1. where L is the pulse number of the whole aperture, the number of sub-apertures is J = L/W , W is the pulse number of the sub-aperture, which range is normally as In (3), PRF is the pulse repetition frequency, and K a is the azimuth chirp rate. In the following section, in order to demonstrate the advantage for the division of the aperture, the computational complexity after sub-aperture segmentation and original data is compared.
Suppose that S is an arbitrary k-sparse signal in C N and the sparse measurement vector has M samples. The complexity of OMP (the reconstruction algorithm in this paper) is around O (kMN ) [13].
Assume that the azimuth pulse number of the sub-aperture is N, the available pulse number is M(M<N), and the subaperture is a k-sparse signal, so the original data has JN components, the available pulse number is JM and the original 5962 VOLUME 8, 2020 aperture is a Jk-sparse signal. The total cost of sub-aperture segmentation is O (JkMN ) and that of the original aperture is O J 3 kMN .

C. SUB-APERTURE PREPROCESSING
For each sub-aperture, we can not directly restore the missing data using CS, where preprocessing is needed. For each subaperture, we perform reconstruction in the image domain. The processing is as follows.
The radar transmits a Linear Frequency Modulated (LFM) signal with chirp rate k, the 2-D echo signal reflected from the targets is where rect(·) represents the time window, which is a rectangular function, t is the slow (azimuth) time, T a is the azimuth aperture time of sub-aperture, τ is the fast (range) time, c is the light velocity, T r is pulse duration, f c is carrier frequency, R a is the distance from the target to the antenna phase center, and σ is the target reflection coefficient. Performing range Fourier transform, matching filter and motion compensating on Eq. (4), we can obtain [27], [28] where R 0 is the distance between the antenna phase center and the sub-aperture imaging center, and B = KT r is the bandwidth of the transmitted signal.
Performing range inverse Fourier transform on Eq. (5), we can obtain After range compression, we can reconstruct the image in azimuth based on CS.

D. SUB-APERTURE RECONSTRUCTION BASED ON CS
If the radar echo reflection intensities of some patches are much stronger than other patches in the imaging area, or the targets occupy a tiny fraction of the whole imaging area, we assume that the scene is sparse [17], [29].
Because we perform imaging in the azimuth frequency domain, which meets the sparse requirement, this paper utilizes the Fourier Basis as the Base Matrix.
Assume that the azimuth pulse number of the sub-aperture is N and the available pulse number is M (M < N ), the missing pulse number is N − M . Firstly, we establish the azimuth basis matrix ϕ = {ϕ 0 , ϕ 1 , . . . , ϕ N −1 } N ×N : where ϕ n is the column vector, f a = PRF N is the azimuth frequency interval, and t a = 1 PRF is the pulse repetition interval.
The M × N azimuth measurement matrix A is made up of M lines from a N ×N unit matrix. The location of the M lines corresponds to the location of the available echo pulse. The azimuth measurement vector of each range is expressed as: In (8), θ a is the azimuth sparse vector, namely the image reconstructed in azimuth frequency domain for each range unit. CS reconstruction can replace the Fourier transform of Eq. (6).
Carrying out the reconstruction processing of each range unit with the OMP algorithm iteratively, we get the reconstructed image of each sub-aperture.
After performing azimuth inverse Fourier transform on the reconstructed image, the range compressed data is obtained. So far, the missing pulse data has been restored and the azimuth data has been restored completely. Then range Fourier transform is performed on the range compressed data, and multiply the data with the following formula (9). We can obtain the complete data of each sub-aperture after performing range inverse Fourier transform.

E. SUB-APERTURE RECOMBINING
Because there is no overlap between the sub-apertures, the integrated strip-map data can be directly obtained by recombining all sub-apertures in sequential order.

III. SUB-APERTURE AUTOFOCUS
There often exist phase error, which leads to defocus in azimuth, and fail to satisfy the sparse requirement of CS theory. Even if the scene is sparse, it is impossible to reconstruct the image. Therefore before reconstructing the sub-aperture image based on CS, we should perform autofocus on each sub-aperture data. Because the missing data is sparse in azimuth, if we transform the missing data into frequency domain by setting the damaged or missing data to zero, the obtained images are greatly defocused. The performance of PGA [30] and ROPE [31] algorithms will not be ideal. The EMMLE algorithm [32] can get better focusing performance without a window, but it is a computational expensive task and also limits the engineering application. This paper selects the PAST algorithm [33] to estimate the phase error of the sub-aperture, which avoids the procedures of covariance matrix estimation and eigenvector decomposition of the EMMLE algorithm. VOLUME 8, 2020 Literature [33] introduces the PAST algorithm of full data in details. For the missing data, we need to do some improvement on PAST algorithm, the processing is as follows.
Step 1: Remove the data of zero padding position and calculate the corresponding eigenvector of the largest eigenvalue by the PAST algorithm.
Step 2: Extract the phase part (phase error) of the eigenvector according to formula (10).
where Error phase is the phase error, Vector Max presents the corresponding eigenvector of the largest eigenvalue.
Step 3: Compensate the phase error of the missing data according to formula (11).
In (11), Data before and Data after are the sub-aperture data before and after autofocus, and conj(·) represents taking the conjugate of the content in bracket.
Step 4: Get an initial image by inserting zeroes and performing Fourier transform for the azimuth missing data. Perform inverse Fourier transform after moving the strongest scattering point to the Doppler zero point for each range gate.
Step 5: Transform the image back into the data domain, and remove the data of zero padding position.
Step 6: Extract the corresponding eigenvector of the largest eigenvalue by the PAST algorithm and extract the phase part of the eigenvector and compensate the phase error of the missing data.
Step 7: Repeat the third to the sixth step, until it meets the end condition of the iteration.
After performing the above steps, the phase error is compensated. The accuracy of reconstructed image based on CS is guaranteed.

IV. PHASE ERROR SIGNAL RECOVERY
When the sub-aperture is autofocused, there is unknown linear phase [24]. It will cause ghost images for the stripmap data to recombine directly the recovered sub-aperture data after autofocus. Therefore before combining the subaperture data, it is necessary to recover the phase error signal compensated by autofocus, in order to ensure the accuracy and integrity.
The phase error signal of the original missing data is noncontinuous, as shown in Fig. 2.
From Fig. 2, the phase error is non-continuous because of the non-continuous for the missing pulse position in raw data.
We have recovered the missing data entirely by CS, so the non-continuous phase error signal can not be used to compensate the continuous data. This paper proposes a method which recovers the phase error signal by CS to turn non-continuous phase error signal into continuous signal. The process is as follows.
First, we perform Fourier transform on the same range gate signals before and after PAST autofocus for each   sub-aperture, then conjugate multiplication will be applied to the signal. The phase error signal of missing data will be obtained. It is shown in (12) = fft(S 1 ) · conj(fft(S 2 )) (12) where S 1 and S 2 are the corresponding signals before and after autofocus of the same range gate respectively, fft(·) represents performing Fourier transform on the content in bracket. still is a sparse signal in frequency domain, as shown in Fig. 3.
The phase error signal is reconstructed sparsely in the frequency domain with the basis matrix ϕ and the measurement matrix A introduced in section II.D.
= Aϕφ (13) where, φ is the frequency domain sparse vector of . After sparsely reconstructing with the OMP algorithm, we perform inverse Fourier transform for φ. The complete phase error signal can be obtained, as shown in Fig. 4.
To compare the Fig. 2 and 4, we find that the phase error signal is changed from non-continuous to continuous after CS recovery.

V. ALGORITHM FLOW
The algorithm flow of the high-resolution imaging method presented in this paper is shown in Fig. 5. 5964 VOLUME 8, 2020

VI. THE MEASURED DATA PROCESSING RESULTS AND ANALYSIS
The paper firstly takes the measured data of a certain type of helicopter airborne SAR as an example in order to analyze the imaging results of the proposed algorithm. The major radar parameters are as follows: radar carrier frequency f c = 10GHz, signal band width B = 1GHz, range sampling rate f s = 1.2GHz, pulse width τ = 15µs, and pulse repetition frequency PRF = 2000Hz. The pulse number of the entire aperture is 16,384, and the pulse number of the sub-aperture is 2048. As this type of helicopter radar is located at the top of the helicopter rotor, it is subject to rotor blocking effect, and the received echoes are disturbed, which results in missing echo data.
The original missing radar echo data of the helicopter SAR is shown in Fig. 6.   In Fig. 6, there are dark stripes. In order to demonstrate the dark stripes of the interfered data distinctly, the zoomed in original data of the above marked areas is shown. It is found from Fig. 6 that the power of the interfered data is lower than that of the undisturbed data. The dark stripes in the strips are the interfered pulses. The power summations of each pulse are shown in Fig. 7.
It can be seen from Fig. 7 that the disturbed pulse power is significantly weaker than the available pulse. The positions of the disturbed pulses can be distinguished according to the difference of the power, which is below the threshold. The threshold is defined as In (14), Thre presents the value of the threshold, Power Max is the max value of the power summations, Power Min is the min value.
The sub-aperture image reconstructed using CS from Fig. 6 is shown in Fig. 8.
The sub-aperture data recovered using CS is shown in Fig. 9.
It can be seen from Fig. 9 that there is no damaged pulse data in the sub-aperture data, and all the data are recovered. The strip-map data can be obtained by recombining the subaperture data, which is shown in Fig.10.
It can be seen through the comparison of Figs. 10 and 6 that the original missing data has been fully recovered.   The imaging results of the original missing data, interpolation recovery data, GAPES recovered data and the proposed method are shown in Fig.11.   . 11 (a) shows the image of the original missing data. Fig. 11 (b) shows the image of the interpolation recovery data with Sinc interpolation. Fig. 11 (c) shows the image of the GAPES recovery data using the 100 pulses in the neighborhood of the missing data. Fig. 11 (d) shows the image of the missing pulse data recovered with the proposed method. In the following section, through the zoomed in images of the above marked areas, comparison is made among these methods.
By comparing the four images in Figs. 11 and 12, it can be seen that, due to the loss of the pulse data, the image quality of the original data degrades significantly. The ghost images are very apparent. Although the interpolation method somewhat improves the result, the obtained image is smeared with the ghost images. Due to the fact that the GAPES recovery method adopts the 100 pulses in the neighborhood of the missing data to recover the missing data, its computational quantity is acceptable. However, the spectral width of the 100 pulses is far narrower than that of the entire aperture, the GAPES recovered data does not include all of the spectral information of the missing data. Although the imaging quality is improved, the ghost images are still present. The proposed method overcomes the ghost images of the original the missing data, improves the image quality, and achieves a high-resolution image with excellent focusing.
In the following section, in order to show the performance of the proposed algorithm, the image quality is quantitatively evaluated by the image entropy values.
The entropy of the 2-D SAR image is defined as follows: where Na is the azimuth number of the image, Nr is the range number of the image, and P (i, j) is the scattering intensity density of the image, which is defined as: where Q(S) is the total energy of the image, which is defined as where S (i, j) is the reflection intensity of each point in the image.
Regarding the imaging result of the same data, the clearer the image is, the smaller the corresponding entropy value is, and the more blurred the image is, the greater the entropy value is. The entropy values of the four whole images in Fig.11 are shown in Table 1.
To compare of the entropy values of the four images, we can find that the entropy value of the SAR image obtained by the proposed algorithm is significantly smaller than those of the other three images, thus it is proved that the proposed algorithm greatly enhances the imaging quality of the stripmap missing data.
For further quantifying the comparison results, another evaluation indicator is applied i.e., the contrast of image.
The contrast of the 2-D SAR image is defined as follows: where where S (i, j) and S r (i, j) is the reflection intensity of each point and the four points around it, p δ (i, j) is the distribution probability which gray difference between adjacent pixels is δ (i, j). The contrasts of the four whole images in Fig.11 are shown in Table 2.
From the table 2, the excellent performance of the proposed method is further demonstrated.
In order to further validate our method, we provided the performance of another group measured data with stronger targets.
The major parameters of the radar are as follows: radar carrier frequency f c = 10GHz, signal band width B = 300MHz, range sampling rate f s = 400MHz, pulse width τ = 7.5µs, and pulse repetition frequency PRF = 500Hz. The pulse number of the entire aperture is 8192, and the pulse number of sub-aperture is 1024.
The imaging results of the four methods are shown in Fig. 13.
The entropy values of the four whole images in Fig. 13 are shown in Table 3.
The contrasts of the four whole images in Fig. 13 are shown in Table 4.
The results of Fig. 13, Table 3 and Table 4 demonstrate the excellent performance of the proposed method against other three methods with stronger targets. Fig. 13 shows that the proposed method has better image quality compared with the other three methods.

VII. CONCLUSION
This paper proposes a high-resolution imaging method for strip-map SAR with missing data based on compressed sensing. First, the strip-map missing data is divided into several sub-apertures in order to reduce the data amount and the computational complexity, then the intact sub-aperture data is recovered with CS. The full strip-map data are obtained by combining the sub-aperture data. Through the sparse PAST algorithm, autofocus is performed on each sub-aperture, thus ensuring the sparse requirement of the reconstructed image. The missing phase error signal is recovered using CS, in order to ensure the correctness of the sub-apertures combining. The results of the measured data indicate that the method detailed in this paper ensures the correctness of the data recovery and resolves the ghost images for strip-map SAR with missing data. The entropy value and the contrast of the image from the proposed method are markedly better than those of the other three imaging results, which proves that the proposed method enhances the imaging quality of the strip-map missing data, and also proves that the proposed method is feasible and effective.
HUAJUN DUAN was born in Shandong, China, in 1976. He received the B.S degree in communication engineering, in 2001. He is currently pursuing the Ph.D. degree with NUAA.
His research interests include radar sparse imaging and radar imaging algorithms.
DAIYIN ZHU was born in Wuxi, China, in 1974. He received the B.S. degree in electronic engineering from Southeast University, Nanjing, China, in 1996, and the M.S. and Ph.D. degrees in electronics from the Nanjing University of Aeronautics and Astronautics (NUAA), Nanjing, in 1998 and 2002, respectively.
From 1998 to 1999, he was a Guest Scientist with the Institute of Radio Frequency Technology, German Aerospace Center, Oberphaffenhofen, where he worked in the field of SAR interferometry. In 1998, he joined the Department of Electronic Engineering, NUAA, where he is currently a Professor. He has developed algorithms for several operational airborne SAR systems. His current research interests include radar imaging algorithms, SAR/ISAR autofocus techniques, SAR ground moving target indication (SAR/GMTI), and SAR interferometry.
TIANSHUN XIANG was born in Shannxi, China, in 1990. He received the B.S degree in communication engineering from Anhui University, Hefei, China, in 2012. He is currently pursuing the Ph.D. degree with NUAA. His research interests include radar imaging algorithms and motion compensation algorithms. VOLUME 8, 2020