- Research
- Open Access
Feature extraction of SAR scattering centers using M-RANSAC and STFRFT-based algorithm
- Hui Sheng^{1},
- Yesheng Gao†^{1},
- Bingqi Zhu^{1},
- Kaizhi Wang^{1} and
- Xingzhao Liu†^{1}
https://doi.org/10.1186/s13634-016-0345-z
© Sheng et al. 2016
- Received: 2 July 2015
- Accepted: 31 March 2016
- Published: 12 April 2016
Abstract
This paper introduces a modified random sample consensus (M-RANSAC) and short-time fractional Fourier transform (STFRFT)-based algorithm for feature extraction of synthetic aperture radar (SAR) scattering centers. In this algorithm, the range migration curve (RMC) of a scattering center is formulated as a parametric model. By estimating these parameters, the backscattering envelope of scattering center, corresponding to the backscattering variation in synthetic aperture time, is extracted directly from a time-domain range-compressed signal. The estimated parameters can also reconstruct the geographical location and along-track velocity of scattering centers. Thus, even without knowing explicit knowledge of platform velocity and forming a SAR image, this algorithm is capable of realizing feature extraction. To estimate parameters scatter by scatter, M-RANSAC approach is proposed as an implementary method with iterative procedure. In the iterations, fitting precision indicator (FPI) works cooperatively with construction fitness coefficient (CFC) to determine the optimal parameters of different scattering centers. Adapting this method to more general cases, STFRFT is introduced to separate the overlapped trajectories of RMCs of scattering centers. The root mean squared errors (RMSEs) of parameter estimation are close to their Cramér-Rao lower bounds (CRLB). The effectiveness of feature extraction based on the devised algorithm is validated by both simulated and real SAR data.
Keywords
- SAR
- M-RANSAC
- STFRFT
- Parameter estimation
- Feature extraction
1 Introduction
Feature extraction has confirmed its usage in synthetic aperture radar (SAR) target recognition and classification, where a given target is classified as a specific target type by feature matching over the known database [1–5]. In fact, the high-frequency scattering response of a target is well approximated as a sum of response from individual scattering centers [6]. The attributes of these scattering centers, including scattering mechanism, location, and velocity, are physically relevant to those of the target [7]. Thus, to characterize target properties, feature extraction of corresponding scattering centers is a meaningful approach.
Interested attributes for each scattering center generally include backscattering envelope, geographical location, and the relative velocity between radar platform and scattering center. Backscattering envelope indicates the backscattering variation of a scattering center within synthetic aperture time. Illuminated by radar signals, some targets, like metallic surfaces, have a very directive backscattering pattern or can be sensitive only to a singular frequency (anisotropic scatters or dihedral corner reflectors). Oppositely, some targets like trihedral corner reflectors have isotropic patterns. It leads to a stable backscattering during the acquisition. Therefore, the backscattering envelope can be the feature of major concern to characterize target properties, especially when a wide-angle SAR is operated [8]. Moreover, the geographical location and relative velocity are equivalently important, since the location denotes the cross-track and along-track positions while the relative velocity reflects the along-track speed.
To extract the attributes of scattering centers, a family of time-frequency analysis (TFA) approaches has been devised. They use Wigner-Ville decomposition [9], wavelet transforms [10], and Fourier transform [8, 11] to realize feature extraction. Starting with spectrum of SAR imagery, these methods are constrained with knowing explicit knowledge of platform velocity and forming a SAR image first. Free from SAR image formation, another group of approaches can directly extract the feature from the spectrum of raw data. These methods rely on spectral estimation and include parametric [12–14], nonparametric [15–17], and semi-parametric approaches [18]. However, sometimes, the spectrum may wrap around azimuth frequency as a result of ambiguity [19]. Since the aforementioned methods start with the spectrum, it may degrade the effectiveness of feature extraction.
In this paper, we propose an innovative algorithm to realizes feature extraction. Starting with a time-domain range-compressed signal, this algorithm establishes its main contribution as the signal-level ambiguity-free feature extraction of scattering centers. The realization of feature extraction without knowing explicit knowledge of platform velocity and forming a SAR image provides additional novelty of this algorithm. The procedure of this algorithm is detailed as follows. First, a parametric model is presented to describe the range migration curve (RMC) of scattering center in a range-compressed signal of SAR raw data. Then, using the points extracted from the contour of the range-compressed signal, an modified random sample consensus (M-RANSAC)-based algorithm is developed to estimate the parameters scatter by scatter. Within the method, fitting precision indicator (FPI) works cooperatively with construction fitness coefficient (CFC) to determine the optimal parameters of different scattering centers through iterations. Given the estimated parameters, the backscattering envelopes can be extracted from the range-compressed signal. Along with the backscattering envelopes, geographical location and relative velocity can also be reconstructed. However, the performance of M-RANSAC-based algorithm may be degraded when the trajectories of RMCs are overlapped in the range-compressed signal. To guarantee the effectiveness in more general cases, a trajectories separation method based on STFRFT [20] is proposed, further improving the M-RANSAC-based algorithm in feature extraction.
This paper is organized as follows. Section 2 reviews the mathematical expression of received signal and models the RMC of scattering center. Section 3 describes the M-RANSAC-based algorithm for feature extraction of SAR scattering centers. Section 4 introduces a STFRFT-based trajectories separation method. An enhanced M-RANSAC algorithm embedded with this STFRFT-based method is also detailed in this section. Section 5 discusses the root mean squared error and Cramér-Rao bounds of the parameter estimation. Section 6 presents the experimental results to validate the performance of the algorithm in feature extraction and demonstrates the usage of extracted feature in target recognition and classification. In the end, Section 7 concludes this paper.
2 Mathematical model
Since we assume both the exposure time and the squint angle are moderate, the terms up to quadratic order in (5) are sufficient to model a RMC precisely.
In this proposed algorithm, \({\vec {\mu }} = \{{A},{\kern 3pt} {B},{\kern 3pt}{C}\}\), which parameterizes the RMC of an individual scattering center, is estimated scatter by scatter. Applying the estimated \({\vec {\mu }}\), the backscattering envelope σ can be extracted from range-compressed signal. Along with it, the geographical information R _{0} and η _{0} and the relative velocity V _{ r } will be reconstructed. The process will be detailed in the next section.
3 M-RANSAC-based feature extraction algorithm
In the step of parameter estimation, the observed data is extracted from the contour of the range-compressed signal. It is a mix set of “inliers” and “outliers”, indicating the trajectories of RMCs. The inliers can be explained by the parameter set \({\vec {\mu } }\) of current scattering center, while the outliers do not fit the model and may come from other scattering centers’ RMCs or noise.
To separate the inliers from the outliers and obtain the current optimal fitting RMC with parameterized representation \({\vec {\mu }}\), RMC construction and performance measure are implemented iteratively in this algorithm. The iterative procedure of M-RANSAC-based approach continues until the points within observed data set are classified according to their corresponding RMCs, thus scattering centers. Along with the classified points, the overall number of scattering centers M and a set of \({\vec {\mu }}\) corresponding to different scattering centers are obtained.
Then, the step of feature extraction starts with these classified points and the estimated \({\vec {\mu }}\). The location and relative velocity of scattering centers can be directly reconstructed by \({\vec {\mu }}\). The backscattering scattering envelopes will be extracted from the range-compressed signal. Thus, feature extraction of M scattering centers are accomplished. The details of parameter estimation and feature extraction are summarized in the following subsections.
3.1 RMC construction with hypothetical inliers
Here, η _{ s } and R _{ s } are the minimum slow time and slant range of the given raw data, respectively.
The accurate construction mainly depends on the accuracy of the selected points to solve (11). Only when \({\vec {\psi }_{1}}\), \({\vec {\psi }_{2}}\), and \({\vec {\psi }_{3}}\) come from the same RMC, this constructed \({\vec {\mu }_{c}}\) can be the parametric representation of a scattering center. However, the randomly chosen points might belong to different RMCs or be just noise points. Therefore, to assess the performance of this constructed RMC, a measure needs to be established in the iterations.
3.2 Performance measure establishment based on quadratic orthogonal distance
In this subsection, a double-measure system is developed to evaluate the performance of a candidate RMC. To deal with the situation that selected points come from different RMCs or are noise points, CFC, which denotes the number of points in observed data set can be explained by the candidate RMC with \({\vec {\mu }_{c}}\), is introduced. Another measure, called FPI, is proposed to assure that a more precise RMC will be chosen when two candidate RMCs share the same CFC.
in which ε means unit step function. FPI, which is the negative overall QOD of inliers, is known as the accuracy of fitting. It works cooperatively with CFC to locate the optimal candidate RMC with the largest number of inliers and best fitting precision. Conventional RANSAC-based algorithm [22] only considers CFC as measure without applying weighting for inliers’ QOD and the maximum likelihood estimation sample consensus (MLESAC)-based method [23] obtains the overall error with a computationally complicated process. They fail in either accuracy or efficiency. The double-measure system of CFC and FPI in this algorithm steps out of this dilemma and achieves a balance between precision and efficiency.
In the next subsection, M-RANSAC approach will integrate the measures and RMC construction into the iterative process of parameter estimation.
3.3 Iterative procedure of parameter estimation in M-RANSAC-based approach
To begin with, the minimum iterative times min, maximum iterative times max, the threshold QOD to define an inlier rho_thr and the threshold number of inliers to confirm a scattering center N_thr should be preestablished. What is more, we should initialize maximum non-updating times m to infinite, the number of scattering center M to zero, and the iteration proceeding factor CONT to 1.
A point-level iteration starts with randomly selecting three points from D_set to construct a candidate RMC. The \(\vec {\mu }_{c}\) of this RMC is computed with (9) and (11). Then, according to subsection 3.2, the QOD between every point in the D_set and this candidate RMC are calculated and denoted by rho. The points whose rho stay no more than rho_thr are defined as inliers and stored in \(\text {set}(\vec {\mu }_{c})\). Then, CFC \(N(\vec {\mu }_{c})\) and FPI \(\chi (\vec {\mu }_{c})\) of this candidate RMC can be computed by (20) and (21).
This RMC can be regarded as the current optimal one in two cases. The CFC \(N(\vec {\mu }_{c})\) exceeds that of the former optimal BestN, or the FPI \(\chi (\vec {\mu }_{c})\) goes over that of former optimal Bestχ under the circumstance that \(N(\vec {\mu }_{c})\) equals BestN. When the conditions are satisfied and the current optimal is renewed, not only Bestμ, BestN, Bestχ, and Bestset are updated in line with the values of current optimal RMC but also the maximum non-updating times m will be recalculated. The point-level iteration stops when iterative times iteration exceed max or non-updating times non_upd surpass m. An additional minimum iteration times min is used to remain the stability.
In scatter-level iterations, a new scattering center will be confirmed when the output of the point-level iterations BestN goes beyond N_thr. At this time, the number of recovered scattering centers M is updated. Bestμ and Bestset are stored in setμ(M) and set(M). The idea of CLEAN technique [24, 25] are taken, and the points in Bestset will be subtracted from D_set. Another point-level iterations will be processed to locate the next RMC. Oppositely, if the point-level iteration fails to locate a scattering center, the remaining points in observed data set are considered as noise points. Thus, the scatter-level iterations stop by setting CONT=0.
After the two-level iterations, the total number of scattering centers M is determined, the points of inliers are classified in set, and the parametric representation μ of scattering centers are estimated and saved in setμ. These data will help to realize feature extraction for dominant scattering centers of the targets in the next subsection.
3.4 Feature extraction based on estimated parameters
Knowing R _{0i }, η _{0i }, and V _{ ri }, compressed range envelope p _{ r }, azimuth beam pattern w _{ a }, and the phase information related to instantaneous slant range R _{ i } are calculated. By locating inliers set(i) in the range-compressed signal, the complex values along the RMC of scattering center can be extracted. According to (2), we obtain the complex backscattering envelope σ _{ i }(η−ζ _{ i }) by eliminating the influence of the aforementioned components in the extracted complex values. The vector set \(\left \{ {{{\vec {T}_{1}}},{\kern 3pt} {{\vec {T}_{2}}}, \ldots {\kern 3pt} {{\vec {T}_{M}}}} \right \}\) are calculated scatter by scatter.
It is worth noting that, the process of M-RANSAC-based algorithm does not need the explicit parameters (e.g., platform velocity). However, for conventional methods of feature extraction based on SAR image formation, the platform velocity works as a crucial parameter of realizing range cell migration correction (RCMC) and azimuth matched filtering. Thus, compared with the conventional approach, M-RANSAC-based algorithm can be utilized in a more flexible way. Moreover, the proposed algorithm extracts the features directly from a range-compressed signal. Without forming SAR image, we may realize target recognition and classification directly in a signal level rather than in an image level.
When platform velocity is known and SAR image is formed, the backscattering envelope extracted by the proposed algorithm may classify targets which are similar in the gray-level SAR image. Moreover, given the platform velocity and the relative velocity between radar platform and the dominant scattering center, the along-track velocity of target can be computed. Thus, even if SAR image is formed, this feature extraction algorithm may help us to better understand the target.
4 Trajectories separation based on STFRFT
where, PRI is the pulse repetition interval and N _{ a } denotes the length of azimuth sample. In the real application, N _{ a }PRI^{2} is used as the factor of coordinate transformation in digitalized computation [26, 27]. Back to (29), the STFRFT of an individual scattering center is decided by \(\text {STF}{_{{K_{a}}}}\left ({\eta,u} \right)\). The matched STFRFT of (30) will locate the spectrogram line of a scattering center parallel to η. For multiple scattering centers, the shift Δu=−2πf _{ i } sinα along u axis in (29) separates their energy according to their different azimuth locations. As shown in Fig. 7 b, a simple spatial filter using a rectangle window will separate the energy of one scattering center from the others. After inverse STFRFT, the trajectories of scattering centers with similar range position but different azimuth location are separated (e.g., Fig. 7 c).
- 1.
Select the trajectory of an isolated scattering center and estimate K _{ a } using the points extracted from it.
- 2.
Calculate α based on (31).
- 3.
Execute α-angle STFRFT for a range bin.
- 4.
Implement spatial filtering using rectangle windows.
- 5.
Realize trajectories separation with inverse STFRFT.
- 6.
Repeat steps 3 to 5 until the last range bin is processed.
- 7.
For every sub-patch range compressed signal, estimate M, setμ, and set using M-RANSAC approach.
- 8.
Use the estimated M, setμ, and set to compute the vector set \(\left \{ {{{\vec {T}_{1}}},{\kern 3pt} {{\vec {T}_{2}}}, \ldots {\kern 3pt} {{\vec {T}_{M}}}} \right \}\).
5 CRLB and RMSE of parameter estimation
The parameter estimation of \({\vec {\mu }}\) lays the foundation for feature extraction in this algorithm. In this section, Monte-Carlo tests are conducted to obtain the root mean squared errors (RMSEs) of the estimates. To evaluate the accuracy of estimation, these RMSEs of estimators compared their theoretical minimal errors, named Cramér-Rao lower bound (CRLB). We start this section with computing the CRLBs according to observation.
Here, ω _{0}(n) denotes a complex white noise with zero mean and the variance of σ _{0}. The estimator vector \({\vec \phi _{e}} = [\hat A, \hat B, \hat C]\) contains three parameters waiting to be estimated.
where the Λ is the average amplitude of backscattering envelope σ{n·PRI} and azimuth beam pattern w _{ a }{n·PRI}. Θ _{1}=N _{ a }(N _{ a }−1)(N _{ a }−4), \({\Theta _{2}} = {N_{a}}\left (- {N_{a}^{3}} - {N_{a}^{2}} + 4{N_{a}} + 4\right)\) and Θ _{3}=N _{ a }(N _{ a }+1)(N _{ a }+2). According to (35), the CRLBs of estimated parameters are the diagonal elements of inverse matrix. Thus, the CRLBs of \(\hat A\), \(\hat B\), and \(\hat C\) are equal to \(\left [ {I({{\vec \phi }_{e}})} \right ]_{11}^{- 1}\), \(\left [ {I({{\vec \phi }_{e}})} \right ]_{22}^{- 1}\) and \(\left [ {I({{\vec \phi }_{e}})} \right ]_{33}^{- 1}\), respectively.
System parameter of simulation and real data
Parameter | Simulation test | Real data experiment |
---|---|---|
Squint angle | 0^{∘} | −1.584^{∘} |
Signal bandwidth | 150 MHz | 30.111 MHz |
Sample frequency | 200 MHz | 32.317 MHz |
Pulse duration | 5.12 μs | 41.74 μs |
PRI | 1.7 ms | 0.7956 ms |
Platform velocity | 153.3 m/s | 7062 m/s |
Range to scene center | 7500 m | 998263 m |
6 Experimental results
To validate the performance of this algorithm, a series of experiments, named performance test, simulation test, and real data test, respectively, are presented in this section. In the performance test, we generate raw data of a single target when the broadside airborne SAR system operates. This raw data is added with various Gaussian white noise and then taken as the input of M-RANSAC algorithm to estimate the location and velocity of dominant scattering center. RMSE of the estimated parameters are listed corresponding to different input SNR and iterative times. Then, the scenario of multiple targets is considered in the simulation test. In the illuminated scene, three targets with different backscattering envelopes and along-track velocities are introduced. The features of dominant scattering centers are extracted from the generated data by M-RANSAC-based algorithm and compared with the theoretical ones. In addition, the potential usage of these extracted features in target recognition and classification are fully considered. In the end, the real data of RADARSAT-1 are processed to validate the performance of the proposed algorithm when the trajectories of targets are overlapped. The features of the dominant scattering centers, including locations, relative velocities, and backscattering envelopes, are extracted using both STFRFT-based trajectories separation and M-RANSAC-based feature extraction method. To verify the effectiveness of feature extraction, the reconstructed locations and velocities are compared with those obtained using conventional methods. To confirm the potential usage of these extracted features in target classification, the backscattering envelopes are used to further interpret the ships in English Bay which is located in the city of Vancouver, Canada (see Fig. 13 a).
6.1 Performance test
In this subsection, raw data of a single target are simulated to evaluate the estimation accuracy. This target is a stationary one with the geographical location R _{0}=7500 m and η _{0}=0.8717 s. In the simulation, the system parameters are listed in the middle column of Table 1 and beam width in azimuth dimension is set to be 0.059 rad. The generated raw data are added with Gaussian white noise when the input SNR =−10, −5, 0, 5, and 10 dB. For each input SNR, we generate 150 sets of random Gaussian noise; thus, in total, 750 sets of observed raw data are obtained.
- (1)
The estimation errors of location and relative velocity are quite limited especially when input SNR is 5 and 10 dB. Thus, We can expect a high estimation accuracy in a high-SNR case.
- (2)
The estimation accuracy may decrease along with the input SNR. The reasonable explanation is that a higher-level noise will impact the precision of inliers in a larger degree and thus decrease the estimation accuracy. In the performance test, this phenomenon becomes obvious when the low-SNR data is implemented.
- (3)
The estimation accuracy will be continuously enhanced with the increasing of iterative times until it converges. When input SNR is high, the estimation error converges fast. We can expect a high-precision output with a small number of iterative numbers. However, under low-SNR scenario, the estimation error converges slowly. max can be considerably increased to obtain relatively high-accuracy estimators. Unfortunately, there exists no SNR-related closed-form expression of max. The initial parameter max is an empirical parameter in this paper.
6.2 Simulation test
Original and estimated parameters in both simulation test and real data test
Type | Scattering center | R _{ 0 } [m] | η _{ 0 } [s] | V _{ r }[m/s] | \(\boldsymbol {\hat R_{0}}\)[m] | \(\boldsymbol {\hat \eta _{0}}\)[s] | \(\boldsymbol {\hat V_{r}}\) [m/s] |
---|---|---|---|---|---|---|---|
Simulation | T1 | 7500 | 0.8717 | 153.3 | 7500.7 | 0.8703 | 153.56 |
T2 | 7462.5 | 0.8717 | 147.8 | 7462.2 | 0.8721 | 146.62 | |
T3 | 7537.5 | 0.8717 | 153.3 | 7538.7 | 0.8719 | 153.48 | |
Real data | T1 | 997743 | 1.3143 | 7062.4 | 997744.3 | 1.3127 | 7062.9 |
T2 | 997226 | 1.4202 | 7061.8 | 997226.8 | 1.4212 | 7061.9 | |
T3 | 997375 | 1.7073 | 7061.7 | 997374.1 | 1.7097 | 7061.2 | |
T4 | 997582 | 1.8665 | 7062.2 | 997582.5 | 1.8670 | 7061.7 |
Without knowing the explicit knowledge of platform velocity and forming a SAR image, the extracted backscattering envelopes can label T1 and T2 as azimuth invariant targets and T3 as an azimuth variant target. Thus, a rough target classification can be achieved. \(\boldsymbol {\hat R_{0}}\) and \(\boldsymbol {\hat \eta _{0}}\) present the geographical locations of dominant scattering centers. To visualize the extracted information, we map \(\boldsymbol {\hat R_{0}}\) and \(\boldsymbol {\hat \eta _{0}}\) of scattering centers into image domain. The amplitudes of them are obtained by averaging their backscattering envelopes. As shown in Fig. 12 d, the image is free from the impact of sidelobes. Realizing the target classification and location, M-RANSAC-based algorithm help us to comprehend the targets without forming a SAR image.
When the explicit platform velocity is given, SAR image (see Fig. 12 f) can be formed by chirp-scaling algorithm [30, 31]. Figure 12 e presents the imaging result with eight times interpolation. In this image, T1 is well-focused while T2 and T3 are defocused. From the perspective of SAR image, T2 and T3 may be mistakenly classified into the same type. However, defocus only indicates the mismatch of azimuth matched filter. It may result from either the along-track motion (T2) or the invariant azimuth envelope (T3). Since the two cases are hardly distinguished directly from SAR image, the importance of feature extraction is proved. The extracted envelopes of dominant scattering centers in Fig. 11 a clearly reveals the backscattering feature of targets. Thus, we can label T1 and T2 as azimuth invariant targets and T3 an azimuth variant one. The backscattering envelope of target’s dominant scattering center can be complementary to the SAR image in the application of target classification. Moreover, given the explicit platform velocity, we confirm T2 as a moving target according to column \(\boldsymbol {\hat V_{r}}\) of Table 2.
6.3 Real data test
In this subsection, RADARSAT-1 raw data included in the CD of [19] is applied in feature extraction. The key system parameters are listed in Table 1. As shown in Fig. 13 a, the SAR image of English Bay is formed using the chirp-scaling algorithm. Then, the region of four ships, marked with a white rectangle, are truncated from this SAR image. This patch of complex image is converted to a range-compressed signal (use inverse chirp-scaling algorithm and range matched filter).
The M-RASNAC-based feature extraction algorithm starts with the range-compressed signal in Fig. 13 c. The scale factor 𝜗= 5800 in (10), the lower bound of iterative times min = 30, and the upper bound max = 300, the threshold of quadratic orthogonal distance rho_thr = 0.0016, and the threshold number of inliers N_thr=0.85·N _{ a }. After estimating \({\vec {\mu }}\) for the dominant scattering centers of each ship, the geographical locations and velocity information are reconstructed based on (22) (see columns \(\boldsymbol {\hat R_{0}}\), \(\boldsymbol {\hat \eta _{0}}\), and \(\boldsymbol {\hat V_{r}}\) of Table 2). To verify the performance of parameters construction, we define the dominant scattering center of a ship as the point with maximum amplitude in SAR image. Their geographical locations are listed in columns R _{ 0 } and η _{ 0 } of Table 2. The micro along-track velocity of dominant scattering centers are estimated using fractional Fourier transform (FRFT)-based method introduced in [32]. The relative velocity between radar platform and these scattering centers are then listed in column \(\boldsymbol {\hat V_{r}}\) of Table 2. Since two groups of data are in good agreements, the reconstructed errors are quite limited.
Then, to better understand the targets, the backscattering envelopes of dominant scattering centers are extracted from the range-compressed signal. As shown in Fig. 13 d, T1 is much brighter than the others which may indicate the a relatively higher radar cross section (RCS) [29]. Moreover, T2 and T3 are azimuth variant while T1 and T4 are nearly azimuth invariant. It means the illuminated regions of T2 and T3 are more “flat” than those of T1 and T4.
7 Conclusions
An M-RANSAC and STFRFT-based technique is introduced to extract feature of SAR dominant scattering centers in this paper. Starting with the time-domain range-compressed signal, this algorithm provides an ambiguity-free signal-level approach. Meanwhile, this algorithm requires no explicit knowledge of platform velocity. It can conduct feature extraction without forming a SAR image. Within the extracted features, the backscattering envelope is promising to classify the target type in signal level, the geographical location indicates the target position relative to SAR platform, and the relative velocity denotes the along-track motion of illuminated target.
Experiments are conducted to illustrate the performance of this algorithm. In the tests, the estimation errors of location and relative velocity are quite limited when SNR is relatively high. The normalized extracted backscattering envelopes express their theoretical ones well. Moreover, these extracted features validate their usage in target recognition and classification. Without forming a SAR image, these extracted features will help us roughly understand and classify the illuminated targets. When SAR image is formed by conventional methods, the extracted backscattering envelopes can be complementary to SAR image in the application of target recognition and classification.
Declarations
Acknowledgements
This work has been supported by key project of the National Natural Science Foundation (NNSF) of China (nos.61132005). The authors also want to express gratitude to editors and anonymous reviewers who give the helpful comments and suggestions to his paper.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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