A three-step classification framework to handle complex data distribution for radar UAV detection

Unmanned aerial vehicles (UAVs) have been used in a wide range of applications and become an increasingly important radar target. To better model radar data and to tackle the curse of dimensionality, a three-step classi(cid:12)cation framework is proposed for UAV detection. First we propose to utilize the greedy subspace clustering to handle potential outliers and the complex sample distribution of radar data. Parameters of the resulting multi-Gaussian model, especially the covariance matrices, could not be reliably estimated due to insu(cid:14)cient training samples and the high dimensionality. Thus, in the second step, a multi-Gaussian subspace reliability analysis is proposed to handle the unreliable feature dimensions of these covariance matrices. To address the challenges of classifying samples using the complex multi-Gaussian model and to fuse the distances of a sample to di(cid:11)erent clusters at di(cid:11)erent dimensionalities, a subspace-fusion scheme is proposed in the third step. The proposed approach is validated on a large benchmark dataset, which signi(cid:12)cantly outperforms the state-of-the-art approaches.

clustering, multi-Gaussian subspace reliability analysis, subspace fusion 1. Introduction 1 Unmanned aerial vehicles have become an increasingly important radar tar-2 get because of the low cost, wide applications and potential threats to public 3 security. According to Grand View Research [1], the global market for com-4 mercial UAVs will grow by 17% every year. UAVs have been used for many 5 different applications, e.g., package delivery, land surveillance, traffic monitor-6 ing and chasing birds in airport. However, UAVs may impose threats to public 7 security, e.g., UAVs near airport may jeopardize the safety of airplanes [2], or 8 UAVs may carry bombs or dangerous chemicals in a terrorist attack. Thus, it 9 has become increasingly important to reliably detect UAVs using radars.   analysis on the phase term of the radar signal to extract the mDS for UAV 7 detection [18]. Very recently, empirical mode decomposition was employed to 8 extract intrinsic mode functions for UAV classification [19]. Instead of detect-9 ing/classifying one UAV at a time, Zhang and Li detected multiple UAVs by 10 using a k-means classifier on the mean CVD averaged along the Doppler fre-11 quency [15]. Most of these approaches utilized spectrogram or time-frequency 12 representations that are derived from spectrogram, e.g., cepstrogram and CVD. 13 Thus, the proposed approach also utilizes features derived from spectrogram. 14 However, most approaches utilized the magnitude spectrogram only. As shown 15 in [17], both phase and magnitude spectrograms are useful for classifying the 16 radar signal. 17 The authors recently developed an automated UAV-detection system utiliz- 18 ing the regularized 2-D complex-log Fourier transform to extract spectrogram-19 like features and the subspace reliability analysis to remove unreliable feature 20 dimensions [17]. Despite the success, three challenges remain. 1) The com-21 plex sample distribution of radar data. Subspace approaches utilizing up to 22 the second-order statistics work well for Gaussianly distributed data [17,29]. 23 However, the high-dimensional mDS features deviate largely from Gaussian. 2) 24 Outliers in radar data. Due to the poor signal-to-noise ratio of radar signal, 25 it is error-prone for human to label the data, which leads to mislabeled data 26 (outliers). The outliers are harmful for training classifiers.
3) The curse of di-27 mensionality. It is difficult to robustly model the complex data distribution in 28 a high-dimensional feature space. 29 In literature, these three challenges have been partially addressed. To model 30 the complex distribution of radar data, Regev [33] have been developed for 7 dimensionality reduction. In this paper, an integrated three-step classification 8 framework is proposed to address these three challenges.

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In the first step, to handle the complex data distribution (Challenge 1) and 10 the outliers (Challenge 2), the authors propose to utilize a greedy version of the 11 sparse subspace clustering (SSC) algorithm [34,35], the greedy subspace clus-12 tering (GSC) algorithm [36]. Gaussian mixture model (GMM) [37-39] is often 13 used to model the complex data distribution, and the expectation-maximization 14 (EM) algorithm [37] is often used to derive the mixture model. One critical chal-15 lenge of the EM algorithm is that the GMM could not be reliably estimated due 16 to insufficient training samples and the high feature dimensionality.

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The sparse subspace clustering [34,35] handles the complex distribution by 18 clustering data according to the underlying subspace structure, which leads to 19 a multi-Gaussian model if each cluster of samples follow the Gaussian distri-20 bution. The SSC algorithm is robust to outliers owing to the l 1 optimization 21 when building the similarity matrix. As the SSC is slow, the authors propose to 22 utilize the greedy subspace clustering [36]. Instead of the time-consuming l 1 op-23 timization in the SSC, the GSC algorithm utilizes a nearest-subspace-neighbor 24 algorithm to sequentially find the nearest neighbors to form linear subspaces.

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The neighborhood matrix is then used as the similarity matrix for subsequen-26 t spectral clustering. Similar outliers may form a cluster. Thus, a drop-off 27 technique is proposed to remove samples in the smallest cluster as outliers.

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In the second step, to tackle the curse of dimensionality (Challenge 3), a 29 multi-Gaussian subspace reliability analysis (MGSRA) is proposed to remove 30 the unreliable feature dimensions of the multi-Gaussian model derived in the first 31 step. The model cannot be reliably estimated due to insufficient samples in each 1 cluster and the high dimensionality, especially the dimensions corresponding to 2 the small eigenvalues of covariance matrices. As the inverse of covariance matrix 3 is used to weigh the feature dimensions, those small eigenvalues will impose very 4 large and problematic weights to the corresponding dimensions [17,29,40]. 5 Thus, the MGSRA algorithm is proposed to handle those unreliable feature 6 dimensions separately at different subspaces.

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The proposed MGSRA is different from previous approaches [17,41] in the 8 following aspects: 1) Most subspace approaches are designed based on a uni-9 Gaussian model, whereas the MGSRA is built on a multi-Gaussian model, which 10 could better model the distribution of radar data. 2) Most subspace approaches 11 aim to find one linear subspace that meets a certain optimization criterion, 12 whereas the proposed MGSRA aims to find a set of linear subspaces separately 13 for each class. A problem thus arises naturally: how to optimally combine the 14 results from different subspaces?

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In the third step, a subspace-fusion scheme is proposed to combine these 16 results. More specifically, the Mahalanobis distances of a sample to each cluster 17 center at a set of given feature dimensionalities are calculated. The rational of 18 choosing multiple dimensionalities is that it is difficult to determine the optimal 19 feature dimensionality for subspace approaches. Thus, a range of dimension-20 alities covering the optimal one are sampled and the Mahalanobis distances at 21 these dimensionalities are evaluated. Then, the distances of a sample to differ-22 ent cluster centers of different classes at different subspace dimensionalities are 23 treated as a feature vector, and a support vector machine is trained to combine 24 these distances. The proposed subspace fusion works better than traditional 25 approaches in which the distances are merged as a posterior probability, and 26 evaluated only at some fixed dimensionality for each class [37,38].

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The contributions of this study are summarized as follow: 1) Three chal-28 lenges for radar UAV detection are identified: the complex data distribution, 29 the outliers and the curse of dimensionality. 2) A three-step classification frame-30 work is proposed to address these challenges, i.e. a) the greedy subspace clus-31 tering is utilized to handle the complex distribution and the outliers of radar 1 data; b) a multi-Gaussian subspace reliability analysis is proposed to tackle the 2 unreliable feature dimensions of the derived model; c) a subspace-fusion scheme 3 is proposed to combine the subspace distances. 3) The proposed approach is 4 systematically evaluated on a large benchmark dataset, and demonstrates a 5 superior performance compared with the state-of-the-art approaches.   In this study, the authors find that it is insufficient to use a Gaussian distri-19 bution to model either the UAV class or the non-UAV class, as shown in Fig. 3 20 later in Section 7.2. This is primarily due to the following: 1) There are many 21 different types of UAVs, e.g., helicopter, tricopter, quadcopter, hexacopter, oc-   1 The radar micro-Doppler signatures are weak, much weaker than the main 2 body Doppler. In addition, the thermal noise in a circuit and the noise/interference 3 to radar receiver may contaminate radar signals. All these make it difficult to 4 label radar targets. The labeling errors may come from different sources: 1)

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As the micro-Doppler signatures are weak, it is error-prone to manually la-6 bel the data by analyzing the radar recordings; 2) The radar may capture the 7 micro-motions of background objects, which will distort the radar signals of the    where F{·} denotes the discrete Fourier transform. 1 Secondly, the 2-D complex Fourier transform of S is derived, which is equiv-2 alent to two 1-D spectral analysis on S:

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where m i is the magnitude spectrum and θ i is the phase spectrum. To balance 15 the effects of log{m i } and θ i , a weighting factor w is introduced: w is simply set as w = 1/π so that the phase term is normalized to [−1, 1]. 1 Fourthly, a regularization term is introduced to Eqn. (4) to reduce the noise, 2 because taking the logarithm not only enhances the weak micro-Doppler signa-3 ture, but also enlarges the noise. Finally, the regularized 2-D complex-log-Fourier transform is derived as: 14 15 where log{F{S}} is calculated according to Eqn. (5). probability density function (PDF) of the likelihood function is defined as: This PDF is a weighted linear combination of M Gaussian densities p i (x), each 24 parameterized by a mean vector µ i ∈ R D and a covariance matrix that the discrimination power of the GMM is greatly reduced. To address this 10 problem, the authors propose to utilize the greedy subspace clustering [36]. Given data points 18 19 where 26 27 is a vector whose nonzero entries correspond to the points in Yî lying in the 28 same subspace as y i . By inserting a zero entry at the i-th row of c i , it becomes 29 an N -dimensional vectorc i ∈ R N . The l 1 optimization is repeated for every y i , 1 i = 1, 2, . . . , N . Then, the following coefficient matrix is obtained: 3 4 which can be seen as the similarity matrix for Y . Then, the spectral clustering 5 algorithm [36] is applied on C to segment the data.

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The sparse subspace clustering is robust to noise and outliers owning to the 7 l 1 optimization, but l 1 optimization is slow. On the collected UAV-detection 8 dataset consisting of more than 10,000 training samples of 7236 dimensions, 9 it takes more than 300 seconds for the l 1 optimization of one sample. The 10 total execution time for all samples is about 35 days, which is too long. In 11 addition, memory of a few gigabytes is required for each l 1 optimization, and 12 hence parallel computing using a graphic card is not a feasible option. These 13 are the motivations of using the greedy subspace clustering [36].
, the number of expected neighbors K, and maximum subspace dimension k max .
◃ Normalized to unit variance.  (D 2 +k 3 +kDN ). The time complexity for the whole algorithm   which naturally leads to a multi-Gaussian model.

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The derived model could not be reliably estimated due to the curse of di-1 mensionality. For the i-th cluster of the j-th class, the PDF of the Gaussian 2 model is given as: where µ ij ∈ R D and Σ ij ∈ R D×D are the mean vector and the covariance matrix 6 for the i-th cluster of the j-th class, respectively. The key issue here is to reliably 7 estimate the covariance matrices Σ ij ∈ R D×D so that the Mahalanobis distance Denote the Mahalanobis distance of x to the i-th cluster of the j-th class as 18 19 The targets are to remove the small eigenvalues of Σ ij so that d ij (x) could be 20 evaluated reliably, and to preserve the discriminant information among different 21 classes, which mainly resides in the between-class scatter matrix decomposition is applied on S ij = Σ ij + Σ b as:

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where Φ ij and Λ ij are the eigenvector and eigenvalue matrices of S ij , respective-1 ly. Then, the eigenvectors are chosen corresponding to the leading m eigenvalues 2 of S ij , i.e., Φ ijm , as the projection matrix. The Mahalanobis distance d ij (x) in 3 the projected m-dimensional subspace becomes:

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The optimal feature dimensionality cannot be easily determined. Thus, 7 many subspace approaches report the classification accuracies at different di-8 mensionalities to show how the accuracies vary with the dimensionality, without 9 determining the optimal dimensionality in advance. In the proposed approach, 10 the distances are evaluated at a range of dimensionalities that probably will 11 cover the optimal one. As these distances are evaluated in different subspaces, 12 their scalings are different, and they should be properly weighted before fusion.

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Most importantly, a proper classification scheme needs to be developed for the 14 derived multi-Gaussian model. To address these challenges, a subspace-fusion 15 scheme is proposed as illustrated in the next section.   The measurement data were collected by Thales using a low-power continuous-

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wave radar operating at X-band (9.7 GHz radio frequency). Some signals were acceptance rate (FAR) and the false rejection rate (FRR) are defined as follow:  clustering results using the GSC algorithm with 3 and 4 clusters for the UAV 1 class and the non-UAV class are plotted respectively in the subspace built from 2 the first two principal components, as shown in Fig. 3. Take note that the results 3 are plotted in two different subspaces, as they utilize the first two principal 4 components of UAV samples and non-UAV samples, respectively.

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The following can be observed from the plots: the Gaussian. This is consistent with the previous analysis that neither 1 UAV nor non-UAV samples follow the Gaussian distribution.    Table 1.

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The feature vector is obtained by averaging the spectrogram over time. The 8 minimum covariance determinant (MCD) estimator implemented using "rrcov" 9 package in R programming is used to remove the outliers. PCA is then used to 10 reduce the feature dimensionality. Finally, the feature vectors are normalized to 11 zero mean with unit variance, and classified by a linear support vector machine 12 with the cost parameter C = 40.

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The error rates at various dimensionalities are shown in Fig. 4. These three 14 figures follow the same trend, i.e., the error rates at very low dimensionality are 15 high, drop with increasing dimensionality, and stabilize at high dimensionality.

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The lowest error rates in these three figures are achieved at 80 dimensions. The 17 error rates at this optimal dimensionality are shown in Table 2. The robust 18 PCA performs better than the DTW.

F AR F RR=1% F AR F RR=0.1%
7.46% 41.30% 90.24%   Table 4.   The performance comparisons to the state-of-the-art approaches are summa-20 rized in Table 5. The proposed approach significantly outperforms the others.

Method
where P n is the power of the injected Gaussian noise and P x is the power of the 7 radar signal after removing the clutter. Gaussian noise is used as it is one of 8 the most common noise types. The clutter is removed before injecting the noise 9 as it is not relevant to the radar target but much stronger than the Doppler 10 signatures. Note that the main body Doppler is much stronger than the micro-

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Doppler signatures. Thus, the actual SNR w.r.t. mDS is much lower than the 12 reported SNR. The error rates for different SNRs are summarized in Table 6.
13 Table 6 shows that when the noise is small or even comparable to the micro-14 Doppler signatures, the proposed approach achieves a fairly good performance.

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The error rates do not change significantly when the noise level is low. The

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proposed approach is shown robust to noise. Even when the noise level is high, 17 the error rates of the proposed approach remain at a reasonable level.   radar target can be obtained, and hence a higher classification accuracy can be 1 achieved, but the extracted feature vector will become larger.

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In the previous experiments, the observation duration is set as 50 ms, as 3 suggested by Thales. In this experiment, the system is evaluated for the obser-     Once the demo starts execution, the user will be prompted to choose a radar 5 recording for analysis. The overlapping ratio of 50% is preset in the demo, as Dell PC with Intel Xeon Silver 4108 CPU @1.80 GHz. This demo shows that 12 the proposed system can detect UAVs reliably in real time.  system is compared with existing approaches on a large benchmark dataset, and 13 significantly outperforms the state-of-the-art approaches.
14 The proposed three-step classification framework could well handle the com-15 plex distribution of radar data. However, a potential problem here is that the 16 model in the early stage is optimized without considering the later ones. The 17 future plan is to integrate these three steps as one unified algorithm, e.g., con- plan is to explore other ways to construct the model, or extend this research to 25 other pattern-recognition tasks, e.g., from UAV detection to UAV classification.

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Lastly, as a new dataset has been collected using SQUIRE radar (a FMCW 27 radar) from Thales, the authors will explore the feasibility of not only detecting 28 UAV, but also determining the direction and the distance of the UAV to the 29 radar.