Robust Automatic Target Recognition via HRRP Sequence Based on Scatterer Matching

High resolution range profile (HRRP) plays an important role in wideband radar automatic target recognition (ATR). In order to alleviate the sensitivity to clutter and target aspect, employing a sequence of HRRP is a promising approach to enhance the ATR performance. In this paper, a novel HRRP sequence-matching method based on singular value decomposition (SVD) is proposed. First, the HRRP sequence is decoupled into the angle space and the range space via SVD, which correspond to the span of the left and the right singular vectors, respectively. Second, atomic norm minimization (ANM) is utilized to estimate dominant scatterers in the range space and the Hausdorff distance is employed to measure the scatter similarity between the test and training data. Next, the angle space similarity between the test and training data is evaluated based on the left singular vector correlations. Finally, the range space matching result and the angle space correlation are fused with the singular values as weights. Simulation and outfield experimental results demonstrate that the proposed matching metric is a robust similarity measure for HRRP sequence recognition.


Introduction
High resolution range profile (HRRP)-based automatic target recognition (ATR) has drawn much attention for wideband radars [1][2][3][4][5][6]. As the one-dimensional high-resolution 'image' of a target, HRRP can effectively reflect energy and structure information of multiple scatterers for a distributed target along the slant range direction with respect to a certain radar line-of-sight (RLOS). Furthermore, compared with the two-dimensional high-resolution synthetic aperture radar (SAR) or inverse synthetic aperture radar (ISAR) imaging [7][8][9], HRRP-based ATR has a lower requirement on the RLOS change, lower system complexity and higher computational efficiency. Effective HRRP features and feature extraction approaches have been researched in the literature [10][11][12][13][14]. Although the structure features have excellent performances in HRRP-based ATR, they still mainly face two problems of background interference and aspect sensitivity.
The background noise will inevitably affect the performance for ATR [12][13][14]. Specifically, the training data is usually cooperatively collected in an ideal environment, while the test data is collected non-cooperatively with sheltering of interference, which requires the feature extraction to be robust for ATR in both high signal-to-noise ratio (SNR) and low SNR scenarios. To improve the robustness, structure features including the high-order spectrums and dominant scatterers are explored [15][16][17][18][19]. Compared to the high-order spectrum features with high dimensions, the scatterers are much easier for extraction with a lower computation requirement. In [19], the dominant scatterers provided to demonstrate the effectiveness of the SSM-ANM method. In Section 5, some conclusions are drawn.
Notations: To simplify the presentation, we define the following notations used in this paper. We use bold lower case letters to represent a signal vector, e.g., a ∈ C N , and use bold upper case letters to represent a signal matrix, e.g., A ∈ C M×N . a * b is the convolution of a and b. a, b denotes the inner product. The real inner product is defined as a, b = Re( a, b ). a A is the atomic norm of a under the atom set A. |a| is absolute value of a scaler a. pinv(·) denotes the pseudo-inverse.

HRRP Sequence Model for Recognition
Take the airborne radar for ground target recognition as an example. The geometry model is illustrated in Figure 1 for a single-channel airborne wideband radar. It is known that the echoes from large objects at high frequencies can be modeled as a sum of independent scatterers, and the geometric theory of diffraction (GTD) is widely utilized to describe the scatterer model, by which the system frequency response of the target [40] can be written as where θ is the aspect angle defined as the target-sensor orientation projected on XOY plane, A k (θ) and φ k (θ) correspond to the amplitude and phase with given θ. R k (θ) represents the radial range of kth scatterer to the radar, K denotes the number of scatterers, f 0 is the center frequency and α k is the type parameter corresponding to the kth scatterers. In applications, the scatterers are supposed to be ideal point scatterers with α k = 0 [40]. Then the expression of (1) can be further simplified as The corresponding response in time domain is denoted by where τ k (θ) = 2R k (θ)/c indicates the time delay of the kth scatterer's return. The transmitted signal of the wideband linear frequency modulated (LFM) waveform can be written as where A is the amplitude of signal, T p is the pulse width, and rect(·) is the rectangular function. µ = 2B/T p corresponds to the chirp rate, in which B is the bandwidth. The amplitude spectrum of s(t) is represented by The received signal s r (t, θ) can be regarded as the convolution of the transmitted signal s(t) and the target system response h(t, θ), i.e., After matched filtering the output signal s o (t, θ) can be represented as Sensors 2018, 18, 593 4 of 19 in which the transmitted waveform is also chosen as the matched filter. Thus, its spectrum S o ( f , θ) is derived as By the inverse Fourier transform of S o ( f , θ), the target echo s o (t, θ) is further expressed as where C 0 = A 2 T P . According to (9), as the combination of echoes of different scatterers, the HRRPs vary versus the aspect angle θ. During the CPI, several pulses of echoes are received and constitute the HRRP sequence, where the discrete form and frequency response after digital down-conversion can be written as in which m, l represent the mth time sampling and lth frequency response respectively, where m ∈ {1, · · · , M} and l ∈ {1, · · · , L}. ∆t and ∆ f denote the time and frequency interval respectively. n represents the pulse number in the CPI, and θ n is the aspect angle corresponding to the nth pulse.
. According to (9), as the combination of echoes of different scatterers, the HRRPs vary versus the aspect angle  . During the CPI, several pulses of echoes are received and constitute the HRRP sequence, where the discrete form and frequency response after digital down-conversion can be written as (11) in which , ml represent the mth time sampling and lth frequency response respectively, where t  and f  denote the time and frequency interval respectively. n represents the pulse number in the CPI, and n  is the aspect angle corresponding to the nth pulse. To combine the HRRP sequence in the CPI for recognition, the MTRC caused by the relative motion between target and radar needs to be compensated first. For the ground target whose motion To combine the HRRP sequence in the CPI for recognition, the MTRC caused by the relative motion between target and radar needs to be compensated first. For the ground target whose motion can be omitted in the CPI as it is far smaller than the motion of the airborne radar, the MTRC is compensated directly by the velocity information of the cooperative airborne platform. For the air target, the techniques for range alignment and phase compensation are widely researched [41,42], which will not be discussed here. After motion compensation, fluctuations still remain for the scatterers caused by the aspect angle variation, which can be appropriately applied for recognition in our following discussion in Section 3.

The Proposed HRRP Sequence Matching Method
In this part, a novel method of SVD-based scatterer matching using atomic norm minimization (SSM-ANM) is demonstrated for feature extraction and recognition. Firstly, SVD process for the HRRP sequence is explored in Section 3.1, verifying the effectiveness in noise reduction. Secondly, the whole process for SSM-ANM is illuminated in detail in Section 3.2. Then the implementation of the key procedures in SSM-ANM is introduced in Section 3.3. Lastly, some remarks on the proposed SSM-ANM method are concluded.

Singular Value Decomposition of the HRRP Sequence
Assume that the MTRC effect caused by the relative motion between the platform and target has been compensated as discussed in Section 2. Moreover, the HRRP part corresponding to the target needs to be determined in advance [43]. Then, the received HRRP sequence of the target area in the CPI can be denoted by target, the techniques for range alignment and phase compensation are widely researched [41,42], which will not be discussed here. After motion compensation, fluctuations still remain for the scatterers caused by the aspect angle variation, which can be appropriately applied for recognition in our following discussion in Section 3.

The Proposed HRRP Sequence Matching Method
In this part, a novel method of SVD-based scatterer matching using atomic norm minimization (SSM-ANM) is demonstrated for feature extraction and recognition. Firstly, SVD process for the HRRP sequence is explored in Section 3.1, verifying the effectiveness in noise reduction. Secondly, the whole process for SSM-ANM is illuminated in detail in Section 3.2. Then the implementation of the key procedures in SSM-ANM is introduced in Section 3.3. Lastly, some remarks on the proposed SSM-ANM method are concluded.

Singular Value Decomposition of the HRRP Sequence
Assume that the MTRC effect caused by the relative motion between the platform and target has been compensated as discussed in Section 2. Moreover, the HRRP part corresponding to the target needs to be determined in advance [43]. Then, the received HRRP sequence of the target area in the CPI can be denoted by  X S N (12) where  The singular value decomposition (SVD) of X is given by  T XU ΛV (13) where the columns of  The singular value decomposition (SVD) of X is given by where the columns of U ∈ C N×N and V ∈ C M×M correspond to the LSVs and RSVs of X respectively. Λ ∈ R N×M is a diagonal matrix containing the N SVs of X in descending order that Assume that the background noise in each range cell obeys a complex Gaussian distribution with the mean of 0 and variance of σ 2 n with independent identically distribution (i.i.d). Next, consider the singular vectors as follows. According to (13), the RSVs in V span the orthogonal basis space in the range domain, while the LSVs in U span the basis space in the angle domain [38]. Given the Eckhart and Young theorem [44], the top singular vectors with the largest SVs can well approximate the signal, called the signal subspace, while the tail vectors with smallest SVs span the noise subspace. By reconstructing the signal with the top singular vectors, background interference can be reduced as it corresponds to the tail singular vectors. The energy threshold is set as Thus Q S , the number of SVs to be chosen, can be determined by η. In the work of the state-of-the-art for recognition, the RSVs in V are utilized for ATR while the LSVs are ignored [36][37][38]. However, the information in U is also useful in our HRRP sequence based recognition. Thus, we exploit the SVD-based feature extraction method by jointly using RSVs and LSVs, called the SSM-ANM method which will be discussed in detail in the following part.

The Process of the SSM-ANM Method for Recognition
The training data is denoted by X N×M d,g d,g , where X N×M d,g is the target HRRP sequence in a CPI, in which N is the pulse number and M d,g is the range cell number. d ∈ {1, · · · , D} and g ∈ {1, · · · , G} represent the posture number and the type number of the targets respectively. The test data is denoted by Y N×M , and the recognition task aims to find its right type.
The LSVs, RSVs and SVs for the training data are represented by {u 1 , and {λ 1 , λ 2 , · · · , λ N } d,g , while the LSVs, RSVs and SVs for the test data are As the signal can be well approximated by the singular vectors corresponding to the largest SVs, here we use s, t to represent the number of the dominant SVs for the training and test data, respectively, which can be determined by the energy threshold η.
For all target types g = 1, · · · , G and postures d = 1, · · · , D, loop and calculate the corresponding matching scores. The type g 0 with the largest score is the recognition result. The whole algorithmic recap for the ground target recognition based on SSM-ANM algorithm is demonstrated in Table 1. Table 1. Algorithmic recap of the SSM-ANM method.

Algorithmic Recap of the SSM-ANM Method
with G types of targets and D g aspect-frames for gth target.
The test data Y N×M .
• Choose the t dominant singular values and singular vectors, represented as λ 0 1 , λ 0 2 , · · · , λ 0 t , u 0 1 , · · · , u 0 t and v 0 1 , · · · , v 0 t . • Extract the location and intensity information of the dominant scatterers for v 0 i (i = 1, · · · , t). • Calculate the matching score o d.g for each training data X d.g Choose the s dominant singular values and singular vectors, represented as Extract the location and intensity information of the dominant scatterers by ANM for v d,g j (j = 1, · · · , s), as represented by P. Calculate the ds-HD for v 0 i and v d,g j for each i = 1, · · · , t and j = 1, · · · , s, and form the matching matrix M d,g s×t . Calculate the correlation for u 0 i and u d,g j for each i = 1, · · · , t and j = 1, · · · , s, and denoted by a correlation matrix Z d.g s×t . Get the matching score o d.g according to the following definition (15) End End • Find the largest score o d 0 .g 0 corresponding to the type g 0 with aspect-frame d 0 .
Output: The recognition result g 0 for the test data Y N×M . s×s and Λ t×t are diagonal matrices consisted by their dominant SVs respectively, which are pre-normalized by their traces. Then, the matching score o d.g is fused by the range space matching result and the angle space correlation as follows.
The matching matrix is defined by where (·) is the dot product for the matrix. Equation (14) can be seen as the fusion result of M where max(·) is the operation to get the largest entry of the matrix. o d.g is the output matching score with given type g and posture d. The flowchart of the SSM-ANM is illustrated in Figure 3.
 Get the matching score o according to the following definition (15) End End  Find the largest score 00 . dg o corresponding to the type 0 g with aspect-frame 0 d .

Output:
The recognition result 0 g for the test data NM  Y .
In the SSM-ANM procedures above, for the training data where    is the dot product for the matrix. Equation (14) can be seen as the fusion result of where   max  is the operation to get the largest entry of the matrix.
. dg o is the output matching score with given type g and posture d. The flowchart of the SSM-ANM is illustrated in Figure 3.

Dominant Scatterer Extraction by Atomic Norm Minimization
In this part, the dominant scatterer extraction by ANM is researched. In our former work [23], the dominant scatterers are extracted by single HRRP, which may be unsteady in lower SNR. As the dominant RSVs of the HRRP sequence contain the scatterer information in the range domain with less noise interference, here we utilize ANM to extract the scatterer features based on dominant RSVs instead of the single HRRP. Moreover, it will be demonstrated in Section 4 that the larger RSVs follows the scatterer model which contains the information for more steady scatterers, and the performance of the dominant RSVs is superior to single HRRP in scatterer extraction and recognition. According to [23], the frequency response for scatterers can be denoted by which indicates that the signal in frequency domain can be modeled as the sum of K sinusoid components corresponding to the K dominant scatterers. As the number K is usually far smaller than the sampling frequency numbers, it can be solved by the ANM [20]. Atoms of ANM are constructed by a( f , ϕ) ∈ C, f ∈ [0, 1] and ϕ ∈ [0, 2π) as Then (16) can be represented by the atoms as where The set of atoms is denoted by 2π)}, and the atomic norm · A represents its unit ball with the convex hull of A: The atomic norm minimization problem is where T represents the subset of entries we observe and x # j is the continuous signal. The frequency of f k can be localized by the dual optimal solution [20] introduced as follows. Define the inner product as q, x = x * q, and the real inner product as q, x = Re(x * q). Next, the dual norm of · A is defined as According to [20], the dual atomic norm is equal to the maximum modulus of the polynomial Q(z) = ∑ j∈J q j z −j on the unit circle. The dual problem of (21) can be solved by Therefore, the estimationf k of frequency f k can be determined by Q( f ) = q, a( f , 0) := ∑ j∈J q j e −i2πj f , which satisfies Withf k , the scatterer location estimationR k can be calculated by (19). The corresponding complex amplitude A k can be solved from (16) by which equals to where pinv(·) denotes the pseudo-inverse, and K 1 represents the scatterer number, which may be larger or smaller than the real scatterer number K. The amplitude and phase estimationÂ k ,φ k can be obtained from A k . The output of ANM dominant scatterer extraction is a set denoted by the location and intensity of each scatterer P k , expressed by the feature set as P = P 1 , P 2 , · · · , P K 1 = R 1 ,Â 1 , R 2 ,Â 2 , · · · , R K 1 ,Â K 1 (28) Compared to the scatterer extraction by OMP, ANM is a sparse recovery method without a prebuild discrete dictionary, which can get rid of the off-grid problem. Thus, the estimation precision can be increased by ANM, which will be verified in Section 4.1.
The output scatterer set P will be used to compute the dominant-scatterers' Hausdorff distance [19] between the training data and test data, which is introduced in detail in the Section 3.3.2.

Scatterer Matching for the Dominant RSVs by Dominant-Scatterers' Hausdorff Distance
According to Section 3.3.2, the scatterer feature set of the test target extracted by ANM based on the sum of K sinusoid components model is denoted by P 0 . Similarly, the scatterer feature sets of the training data are denoted by {P} d,g for g = 1, . . . , G and d = 1, . . . , D. As the Hausdorff measure is widely used for calculating the distance between two point sets, here, we utilize the Hausdorff distance proposed in [19], called dominant-scatterers' Hausdorff distance(ds-HD) for the scatterer matching of the dominant RSVs. Given two point sets A = {a 1 , a 2 , · · · , a N } and B = {b 1 , b 2 , · · · , b Q }, the ds-HD is expressed as h ds (A, B) = 1 P 1 ∑ a n 1 ∈A min a n 1 ∈A d b q 1 , a n 1 (29) where A = A 1 ∪ {a P 1 } = {a 1 , a 2 , · · · , a P 1 −1 } ∪ {a P 1 }, A ⊂ A, P 1 < N, and B = B 1 ∪ {b P 2 } = {b 1 , b 2 , · · · , b P 2 −1 } ∪ {b P 2 }, B ⊂ B, P 2 < Q. Let A and B be the feature sets corresponding to one dominant RSV of the training data and the test data respectively. The d(·, ·) is calculated from the Mahalanobis distance d a n 1 , b q 1 = a n 1 − b q 1 T Σ −1 a n 1 − b q 1 (30) where Σ is a diagonal matrix with diagonal entries consisted by the measurement error variance for each feature of the scatterers, which can be estimated by the training data. Then, the ds-HD represents the mean distance from one point set to another. According to (28), two features extracted by ANM are chosen, one is the location index of the scatterer and the other is its intensity. Finally, the ds-HD used here is utilized to evaluate the distance between the training and test data as

Angle Space Correlation by the LSVs
Moreover, the basis space for each RSV can be spanned in the angle domain by the corresponding LSV. Therefore, the dominant LSVs are exploited to obtain the angle space correlation between the training and test data. For better demonstration, take s = t = 3 as example. As illustrated in Figure 4 spans. The larger angle between them, the bigger difference their signal spaces have, which should be included in the recognition process. Therefore, we define the angle space correlation matrix as Z = z i,j s×t where corr(·) is the correlation operation. Considering that v 0 1 , v 0 2 , v 0 3 have different energy according to their SVs λ 0 1 , λ 0 2 , λ 0 3 , they should be given with different weight to decide the recognition result, as used in [38]. Similarly, the weight for v ,,  , they should be given with different weight to decide the recognition result, as used in [38]. Similarly, the weight for

Some Remarks on the Proposed SSM-ANM Method
Remark 1. In the SSM-ANM method, the number of the dominant singular vectors for the training data can be different from that for the test data, which is more flexible than the traditional pattern recognition methods. Moreover, the number is determined according to the environment noise and the aspect angle variation during the CPI. With higher SNR and smaller aspect angle variation, the fewer number of singular vectors contains the signal subspace, making the SSM-ANM method adaptive to various cases.

Remark 2.
In the Section 3.3.1, we firstly extract the scatterers on dominant RSVs by ANM and then calculate the ds-HD between the training and test data. Compared with the template matching method which calculates the RSVs' correlation between the training and test data directly, the scatterer extraction procedure abandons the noise-only range cells and, thus, improves the performance stability in lower SNR without information loss as the structure information is mostly determined by the dominant scatterers.
Remark 3. The paper mainly focuses on the feature extraction stage of ATR, thus, only the scatterer matching by ds-HD is used as the classifier to compare the performances of the SSM-ANM method with the state-of-the-art literature in feature extraction. In our future work, more classifiers will be tested such as support vector machine (SVM), random forest (RF) or others for possible better classification performance.

Numerical and Outfield Experiments
In this section, the results of both numerical simulations and outfield experiments are given to evaluate the performance of the proposed SSM-ANM recognition accordance with different scenarios.

Effectiveness of ANM in Dominant Scatterer Extraction
To validate the effectiveness of the SVD-based dominant scatterer extraction using ANM, a simple scenario is designed as Figure 1. The airborne radar is flying towards the target with radial range R 0 = 10 km and aspect angle θ = 30 • , in which the target is consisted by six dominant scatterers from A to F. The relative radial range of the six scatterers as well as their intensities are listed in Table 2, where C and D have the largest and stable intensity and the other smaller ones have fluctuate intensity following Rayleigh distribution. The signal is modeled as (8)-(11), and the system parameters are listed as Table 3. During the flight, the aspect angle varies with the relative positions between target and radar. HRRPs are synthesized and the MTRC between them is compensated with SNR = 30 dB. Figure 5 shows the HRRP sequence in the CPI, where the six dominant scatterers are noted. As the scatterers A, B, E and F have fluctuate intensity, their profiles vary seriously. After SVD, the SVs are plotted in Figure 6a, where most energy falls on the first three largest ones, and the smaller SVs only contain the noise. Figure 6b-d give the RSVs corresponding to the 1st, 2nd, 3rd largest SVs. It can be seen that the 1st RSV is consisted by the most stable components of the HRRP sequence, thus the scatterer C and D have more energy in it. The 2nd singular value contains more fluctuate components where scatterer F, E, B and A are more prominent than C and D. As to the 3rd singular value, C and D are even weaker than the others, nearly submerged by the noise.
HRRPs are synthesized and the MTRC between them is compensated with SNR = 30 dB. Figure 5 shows the HRRP sequence in the CPI, where the six dominant scatterers are noted. As the scatterers A, B, E and F have fluctuate intensity, their profiles vary seriously. After SVD, the SVs are plotted in Figure 6a, where most energy falls on the first three largest ones, and the smaller SVs only contain the noise. Figure 6b-d give the RSVs corresponding to the 1st, 2nd, 3rd largest SVs. It can be seen that the 1st RSV is consisted by the most stable components of the HRRP sequence, thus the scatterer C and D have more energy in it. The 2nd singular value contains more fluctuate components where scatterer F, E, B and A are more prominent than C and D. As to the 3rd singular value, C and D are even weaker than the others, nearly submerged by the noise.  The scatterer extraction results on the largest RSV by ANM, OMP and BP are illustrated in Figure 7, where ANM is more accurate on the estimations of scatterer location and intensity than OMP and BP. The scatterer extraction results on the largest RSV by ANM, OMP and BP are illustrated in Figure 7, where ANM is more accurate on the estimations of scatterer location and intensity than OMP and BP.
(c) (d) The scatterer extraction results on the largest RSV by ANM, OMP and BP are illustrated in Figure 7, where ANM is more accurate on the estimations of scatterer location and intensity than OMP and BP. The simulations above verify that, after SVD, the noise is reduced in the dominant RSVs, which increases the accuracy of scatterer extraction. Moreover, the most stable components of HRRPs are focused on the RSVs corresponding to the largest SVs. Moreover, ANM shows superiority in dominant scatterer extraction than OMP and BP, which is free of the off-grid problem. The simulations above verify that, after SVD, the noise is reduced in the dominant RSVs, which increases the accuracy of scatterer extraction. Moreover, the most stable components of HRRPs are focused on the RSVs corresponding to the largest SVs. Moreover, ANM shows superiority in dominant scatterer extraction than OMP and BP, which is free of the off-grid problem.

The Effectiveness of the LSVs for Target Recognition
In the following discussion, we evaluate the effectiveness of the LSVs in feature extraction and recognition. For the comparison convenience, we define the matching matrix and matching score calculated without the angle space correlation matrix by the LSVs as HRRP sequences are generated corresponding to four types of military vehicles with SNR = 20 dB. The professional software 3DS MAX and FEKO are used to generate the scatterer model as well as the echoes of four types of military vehicles. Furthermore, the system parameters of the following simulations are listed in Table 3, which also act as the system parameters for the outfield experiments in Section 4.2.3. Then, the training HRRP sequences for the four types are synthesized in aspect angle centers 0, 1 • , 2 • , . . . , 180 • , with angle variation of 1 • in each CPI. Thus, we have 181 training samples for each target. The test HRRP sequences are generated with random aspect angle centers in 0-180 • , with angle variation of 0.5 • , 1 • and 2 • .
Two hundred Monte Carlo simulations are carried out to calculate the matching score between the training and test HRRP sequences as defined in (15) and (35). In Table 4, the average maximum matching score in each simulation for arbitrary two types are listed, where the matching score (35) is compared with the matching score (15) in bold font. The results indicate that by utilizing LSVs, the matching scores among different types are decreased, which effectively improves the robustness to the aspect-sensitivity in a degree.

Recognition Performance Comparisons
The recognition performance for the proposed SSM-ANM method is compared with the SVD-based scatterer matching by OMP (SSM-OMP), the HRRP-based scatterer matching by OMP (HSM-OMP) proposed in [18], as well as the method using the matching score O d,g r in (35) without LSVs (denoted by RSM-ANM). We generate echoes of the four targets in the same way discussed in Section 4.2.1, with 200 times of Monte Carlo simulations carried out. We investigate the recognition performance in different SNRs. Figure 8 compares the recognition rates of the four types of targets for the four methods, with test data varying from 0 to 20 dB and training data of 20 dB. The result indicates the advantage of the SSM-ANM in HRRP-based feature extraction and recognition. Next, we consider the performances of the four methods in extended CPI, where Figure 9 demonstrates the recognition results with CPI = 20. The SSM-ANM method still has good performances, as it can adaptively determine the number of dominant SVs when the CPI extends to more pulses and energy spreads to more SVs. But the computation complexity will increase and the data rate will decrease for recognition when the CPI is too large. In our system, CPI in the range of 10-20 pulses is a better choice for the ATR task. Next, we consider the performances of the four methods in extended CPI, where Figure 9 demonstrates the recognition results with CPI = 20. The SSM-ANM method still has good performances, as it can adaptively determine the number of dominant SVs when the CPI extends to more pulses and energy spreads to more SVs. But the computation complexity will increase and the data rate will decrease for recognition when the CPI is too large. In our system, CPI in the range of 10-20 pulses is a better choice for the ATR task.
Next, we consider the performances of the four methods in extended CPI, where Figure 9 demonstrates the recognition results with CPI = 20. The SSM-ANM method still has good performances, as it can adaptively determine the number of dominant SVs when the CPI extends to more pulses and energy spreads to more SVs. But the computation complexity will increase and the data rate will decrease for recognition when the CPI is too large. In our system, CPI in the range of 10-20 pulses is a better choice for the ATR task.  The computational complexity of the four methods is listed in Table 5. Although the SSM-ANM method has the largest computational complexity, some fast approaches for SVD and ANM are researched to reduce the complexity further [45][46][47]. Table 5. Computational complexity of different methods.

Methods
Computational Complexity

Outfield Real Data Experiment
Outfield real data experiments are carried out with the same parameters in Table 3. The echoes for the four typical vehicles in different aspect angles are processed and their HRRPs are generated for recognition, as illustrated in Figure 10. CPI is set as ten pulses and 500 HRRP sequences in each aspect angle center are obtained, with five angle centers from 0 • to 180 • with step of 45 • and about 1 • angle variation. Every fifth HRRP sequence is used for training and the rest are used for testing. Thus, we have 100 sequences for training and 400 sequences for testing in each aspect angle center. The SNR for the training and test data is about 30 dB. The radical range is about 5 km. The recognition results are listed in Tables 6-9, which verified the effectiveness of the SSM-ANM method in scatterer matching and recognition. aspect angle center are obtained, with five angle centers from 0° to 180° with step of 45° and about 1° angle variation. Every fifth HRRP sequence is used for training and the rest are used for testing. Thus, we have 100 sequences for training and 400 sequences for testing in each aspect angle center. The SNR for the training and test data is about 30 dB. The radical range is about 5 km. The recognition results are listed in Tables 6-9, which verified the effectiveness of the SSM-ANM method in scatterer matching and recognition.

Conclusions
In this paper, we proposed a scatterer matching method of SSM-ANM by HRRP sequence to get the dominant scatterer features. Firstly, the HRRP sequence in a CPI is first decomposed by SVD. Then, the dominant RSVs are processed by ANM to extract the dominant scatterer information, while the corresponding LSVs are utilized to estimate the angle space correlation between the training and test data. Moreover, the singular vectors are allocated with different weights according to their corresponding singular values. The ds-HD is used to compute the scatterer distance and the matching degree between the training and test data. Finally, the results of both numerical experiments and real measured outfield experiments are provided, which demonstrate the effectiveness and robustness of the proposed SSM-ANM method.
In our future work, we will explore the situation that the target is occluded, where the performance of the SSM-ANM method will be affected, especially when the occlusion is severe. We will make further improvement to increase the robustness to occlusion by adapting the weight of the occluded scatterers in recognition. Besides, the open set recognition problem will be considered if the object is exclusive in the training data. Thus, a classifier with rejection capability needs to be exploited.