Regularized 2-D Complex-Log Spectral Analysis and Subspace Reliability Analysis of Micro-Doppler Signature for UAV Detection

Abstract

Unmanned aerial vehicle (UAV) has become an important radar target recently because of its wide applications and potential security threats. Traditionally, visual features such as spectrogram were often extracted for human operators to identify the micro-Doppler signature (mDS) of UAVs, i.e. sinusoidal modulation. In this paper, the authors aim to design a system for machine automatic classification of UAVs from other targets, particularly from birds as both UAVs and birds are small and slow-moving radar targets. Most existing mDS representations such as spectrogram, cepstrogram and cadence velocity diagram discard the phase spectrum, and only make use of the magnitude spectrum. What’s more, people often take the logarithm of the spectrum to enlarge the weak mDS but without sufficient care, as noise may be enlarged at the same time. The authors thus propose a regularized 2-D complex-log-Fourier transform to address these problems. Furthermore, the authors propose an object-oriented dimension-reduction technique: subspace reliability analysis, which directly removes the unreliable feature dimensions of two class-conditional covariance matrices in two separate subspaces. On the benchmark dataset, the proposed approach demonstrates better performance than the state-of-the-art approaches. More specifically, the proposed approach significantly reduces the equal error rate of the second best approach, cadence velocity diagram, from 6.68% to 3.27%.

Full text

Regularized 2-D Complex-Log Spectral Analysis and
Subspace Reliability Analysis of Micro-Doppler
Signature for UAV Detection
Jianfeng Ren∗
, Xudong Jiang
Electrical & Electronic Engineering, Nanyang Technological University, Nanyang Link,
Singapore 639798.
Abstract
Unmanned aerial vehicle (UAV) has become an important radar target recently
because of its wide applications and potential security threats. Traditionally,
visual features such as spectrogram were often extracted for human operators
to identify the micro-Doppler signature (mDS) of UAVs, i.e. sinusoidal mod-
ulation. In this paper, the authors aim to design a system for machine au-
tomatic classification of UAVs from other targets, particularly from birds as
both UAVs and birds are small and slow-moving radar targets. Most exist-
ing mDS representations such as spectrogram, cepstrogram and cadence veloc-
ity diagram discard the phase spectrum, and only make use of the magnitude
spectrum. What’s more, people often take the logarithm of the spectrum to
enlarge the weak mDS but without sufficient care, as noise may be enlarged
at the same time. The authors thus propose a regularized 2-D complex-log-
Fourier transform to address these problems. Furthermore, the authors propose
an object-oriented dimension-reduction technique: subspace reliability analysis,
which directly removes the unreliable feature dimensions of two class-conditional
covariance matrices in two separate subspaces. On the benchmark dataset, the
proposed approach demonstrates better performance than the state-of-the-art
approaches. More specifically, the proposed approach significantly reduces the
∗Corresponding author. Tel.: +65 6790 5018
Email addresses: jfren@ntu.edu.sg (Jianfeng Ren), exdjiang@ntu.edu.sg
(Xudong Jiang)
Preprint submitted to Pattern Recognition March 15, 2017

equal error rate of the second best approach, cadence velocity diagram, from
6.68% to 3.27%.
Keywords: UAV detection, radar, micro-Doppler signature, 2-D regularized
complex-log-Fourier transform, subspace reliability analysis
1. Introduction1
Unmanned aerial vehicle has become an increasingly important radar target2
because of its low cost, wide applications and potential threats to public security.3
Traditional techniques [1, 2] using combinations of observed kinematic and radar4
cross-section characteristics could not reliably differentiate UAVs from birds,5
as both are small and slow-moving targets. A fundamentally new approach6
is needed for radar to detect small UAVs. In this paper, the authors aim to7
differentiate small UAVs from birds using their radar micro-Doppler signatures.8
A lot of research work has been done in modeling micro-Doppler signature9
of micro-motions [3–9]. Chen V.C. [3] modeled micro-motions such as vibration10
and rotation as sinusoidal modulation in the spectrogram. For example, the11
micro-motion of human gait can be modeled as the vibration of different body12
parts [5, 6], and the micro-motion of birds’ wing beating can be modeled as13
vibration as well [4]. The micro-motion of UAVs is mainly the rotation of14
rotors, which can be naturally modeled as rotation [8]. However, both vibration15
of birds’ wing beating and rotation of UAVs’ rotors share the same mDS model:16
sinusoidal modulation. UAVs cannot be easily differentiated from birds solely17
based on their mDS models. UAV detection is not a simple detection task, but18
above all a discrimination problem.19
In literature, radar micro-Doppler signature has been utilized for many20
automatic-target-recognition tasks, e.g. airplane classification [10–15], ship de-21
tection [16], human detection [17], human gait recognition [18–21], action clas-22
sification [22–25], vehicle classification [26] and many others [27–30]. Some23
researchers also utilized mDS for bird detection [4] and UAV detection [8]. How-24
ever, their primary objective is to provide visual clues for human operators, not25
2

for automatic UAV/bird detection. In fact, many published works only study1
the feasibility of using micro-Doppler signature for target recognition without2
a complete classification solution. To our best knowledge, there is no complete3
automatic classification-based UAV detection framework published to date. In4
this paper, the authors propose a complete system for automatic UAV detection5
using radar micro-Doppler signature.6
Many different kinds of signal representations were proposed in literature7
for mDS analysis, such as spectrogram [3, 4, 19, 24, 31, 32], cepstrogram [8],8
cadence velocity diagram (CVD) [23], and many others [7, 8, 33]. Most of them9
are closely related to spectrogram. For example, the cepstrogram is obtained by10
taking the inverse fast Fourier transform (IFFT) of the log-spectrogram [8], and11
the CVD is obtained by taking the Fourier transform along the time axis of the12
derived spectrogram [23]. Indeed, spectrogram provides useful visual clues for13
the operator to differentiate UAVs from birds. Fig. 1 shows the spectrograms of14
UAVs and birds. When the UAV is close to radar, clear micro-Doppler signature15
of rotor blades, the parallel lines, can be observed as shown in Fig. 1(a). When16
the UAV is far away from radar, it is difficult to visually differentiate the UAV17
shown in Fig. 1(b) from the bird shown in Fig. 1(c), or the group of birds shown18
in Fig. 1(d).19
In this paper, the authors aim to derive a robust signal representation using20
spectral analysis for machine automatic classification. The proposed 2-D com-21
plex spectral analysis is designed to address the following problems of existing22
micro-Doppler signal representations: 1) The existing approaches often generate23
some kind of synthetic image for human operators to visually classify different24
radar targets. But the machine perceives an image differently from the human.25
The visual clues for the human may not be the best feature for the machine.26
The proposed signal representation is designed for machine automatic detec-27
tion of UAVs. 2) The existing approaches discard some important discriminant28
information. Spectrogram is formed by stacking the magnitude spectrums of29
windowed signals, but it discards the phase spectrums. The phase information30
may be also important for classification. The proposed 2-D complex-log-Fourier31
3

(a) The spectrogram of an UAV close to
radar.
(b) The spectrogram of an UAV far away
from radar.
(c) The spectrogram of a bird. (d) The spectrogram of a group of birds.
Figure 1: Spectrograms of micro-Doppler signature of two UAVs and birds are shown in (a)
- (d), respectively. When the UAV is close to radar, we can see the clear micro-Doppler
signature of the rotor blades of the UAV, shown as the parallel lines in (a). However, when
the UAV is far away from radar, the micro-Doppler signature shown in (b) is weak. It is
difficult to visually differentiate the UAV shown in (b) from the bird shown in (c), or the
group of birds shown in (d).
transform makes use of both magnitude and phase spectrums. 3) The log-1
spectrum is often used in signal processing [8] to provide a better view for weak2
mDS, but it may enlarge the noise components as well. If the log-spectrum is3
directly taken as the signal representation, these noise components may ruin the4
classification process due to their large variations in log-scale. To suppress the5
noise, the authors propose a regularization procedure when handling the log-6
spectrum. The proposed regularized 2-D complex-log-Fourier transform better7
4

captures the discriminant information residing in the micro-Doppler signatures1
of UAVs/birds.2
Our second contribution is an object-oriented dimension-reduction tech-3
nique: subspace reliability analysis (SRA). In literature, many dimension-reduction4
techniques have been proposed, e.g. principal component analysis (PCA) [34],5
linear discriminant analysis (LDA) [35], Bayesian maximum likelihood [36], en-6
hanced maximum likelihood [37], dual-space LDA [38], null-space approach [39],7
graph embedding [40], sparse coding [41], asymmetric PCA [42, 43] and eigen-8
feature regularization and extraction approach [44]. Most of them were designed9
for recognition tasks, i.e. to decide the testing sample as one of the many classes.10
UAV detection is a two-class classification problem, i.e. UAVs as the positive11
class and non-UAVs as the negative class. The proposed SRA is specifically12
designed for UAV-detection problem.13
One of the challenges for classification tasks is the small-sample-size prob-14
lem [45], i.e. the within-class scatter matrix Swis often singular. Even in the15
case of full rank, the small eigenvalues of Swmay still cause problems, as the16
inverse of Swis often used to weigh the feature dimensions. Those small and17
unreliable eigenvalues of Swmay impose very large and problematic weights on18
the feature dimensions so that the whole classification process fails. The ob-19
jective of the proposed subspace reliability analysis is to remove the unreliable20
feature dimensions that are harmful to reliable classification. By directly tar-21
geting at the unreliable feature dimensions of two class-conditional covariance22
matrices and removing them separately in two subspaces, the proposed SRA23
delivers better classification performance.24
After deriving the proposed signal representation, subspace reliability anal-25
ysis is applied to remove the unreliable feature dimensions. Then, a minimum-26
Mahalanobis-distance classifier [43, 46] is used for classification.27
The contributions of this paper are summarized as follow: 1) The proposed28
2-D regularized complex-log-Fourier transform well addresses the problems of29
the existing signal representations. The existing approaches are often designed30
for human operator, whereas the proposed one targets at machine automatic31
5

classification. The proposed approach utilizes both magnitude and phase spec-1
trums, whereas the existing approaches often discard the phase spectrum. In2
addition, the proposed regularization procedure greatly suppresses the noise in3
the log-spectrum. 2) The existing subspace approaches could not optimally re-4
move the unreliable feature dimensions of both class-conditional covariance ma-5
trices in one subspace. The proposed subspace reliability analysis thus removes6
them separately in two different subspaces. 3) Lastly, the proposed approach7
is evaluated on a large benchmark dataset, and demonstrates superior perfor-8
mance compared with various feature representations and other state-of-the-art9
approaches.10
2. Proposed Framework of Robust 2-D Complex Spectral Analysis11
2.1. Review of Time-Frequency Analysis on Micro-Doppler Signature12
Many time-frequency representations were proposed for micro-Doppler anal-13
ysis in literature, e.g. spectrogram [3], cepstrogram [8] and cadence velocity14
diagram (CVD) [23].15
Time-frequency analysis such as spectrogram is often used for micro-Doppler16
analysis due to the time-varying nature of mDS [24, 28]. The spectrogram is17
obtained as follow. The signal s(t) is segmented into Moverlapping frames18
{x0,x1,...,xM−1}, where each frame xi={xi[n], n = 0,1,...,N −1}is a19
column vector of length N. These Mframes form a synthetic image X=20
[x0,x1,...,xM−1] of size M×N. Then, the discrete Fourier transform fi=21
[fi,0, fi,1,...,fi,N−1] of xiis computed as:22
fi,k =
N−1
X
n=0
xi[n] exp{−j2πkn
N}, k = 0,1,...,N −1.(1)23
24
Alternatively, people often denote fi=F {xi}as the discrete Fourier transform25
of xi. The magnitude S={|fi,k |2, i = 0,1,...,M −1, k = 0,1,...,N −1}26
forms the spectrogram of size M×N.27
Various micro-Doppler features were extracted from spectrograms and cep-28
strograms in [8] for radar automatic target recognition. The power cepstrum29
6

represents the rate of change in the spectrum. Formally, after segmenting s(t)1
into overlapping windows, the cepstrum cifor xiis defined as:2
ci=C{xi}3
=|F−1{log(|F {xi}|2)}|2,(2)4
5
where C{∗} denotes the cepstrum and F−1{∗} denotes the inverse fast Fourier6
transform (IFFT). These cepstrums cithen form the cepstrogram [c0,c1,...,cM−1].7
The cepstrogram is often used to determine the spectrogram periodicity.8
In [23], cadence velocity diagram was proposed to extract micro-Doppler9
signatures. The CVD provides a measure on how often the different velocities10
repeat (“cadence frequencies”). Formally, after deriving the spectrogram S, the11
CVD is derived by applying the Fourier transform on the spectrogram along the12
time axis, i.e.13
D=Ft{S},(3)14
15
where Ddenotes the CVD and Ft{∗} denotes the Fourier transform along the16
time axis.17
2.2. Problem Analysis of Existing Signal Representations18
The block diagram of spectrogram, cepstrogram and cadence velocity dia-19
gram is shown in Fig. 2(a). As shown in the figure, the magnitude spectrum20
forms the spectrogram, the IFFT on the log-spectrum forms the cepstrogram,21
and the FFT on the magnitude spectrum along the time axis forms the CVD.22
The block diagram of the proposed robust 2-D complex spectral analysis is23
shown in Fig. 2(b). The proposed approach addresses the following problems of24
the existing approaches:25
1. The existing approaches such as spectrogram [3], cepstrogram [8], and26
cadence velocity diagram [23] do not fully utilize the information of the27
synthetic image X. Only the magnitude of the first Fourier transform is28
utilized and its phase information is discarded. The proposed 2-D complex29
7

(a) Block diagram of previous signal representations.
(b) Block diagram of the proposed signal presentation.
Figure 2: The block diagrams of the existing approaches and the proposed approach are shown
in (a) and (b), respectively. The key difference is that the existing approaches do not utilize
the phase spectrum, but the proposed approach does. In addition, a complex-log-Fourier
transform is proposed to solve the problem of taking the logarithm of the complex spectrum
and a regularization procedure is proposed to suppress the noise in the log-spectrum.
spectral analysis makes use of both phase and magnitude spectrums of the1
first Fourier transform.2
2. The phase spectrum and the magnitude spectrum form a complex spec-3
trum. For cepstrogram, the log-spectrogram is used to enlarge the weak4
micro-Doppler signature. However, it is unclear how to take the logarith-5
m of the complex spectrum. The proposed complex-log-Fourier transform6
achieves two goals simultaneously: to make use of both phase and magni-7
tude spectrums, and to emphasize the weak mDS.8
3. Lastly, by taking the logarithm of the spectrum or spectrogram, it not9
only enhances the weak mDS, but also enlarges the noise components.10
Those noise components may lead to large intra-class variations so that11
two samples of the same class differ significantly. To suppress the noise,12
the authors propose a regularization procedure when taking the logarithm.13
8

2.3. Proposed 2-D Complex Spectral Analysis1
The proposed 2-D complex spectral analysis is equivalent to two-step 1-D2
spectral analysis on X. In the first step, the Fourier transform is applied on3
overlapping windows, and in the second step, the Fourier transform is applied4
along the time axis. Mathematically, the proposed 2-D complex spectral analysis5
F2D{∗} on Xis given as follows:6
F2D{X}=Ft{F{X}},(4)7
8
where F{∗} is the Fourier transform on overlapping windows, and Ft{∗} is the9
Fourier transform along the time axis. Take note that both phase and magni-10
tude of F{X}are utilized in the second Fourier transform Ft{∗}, whereas in11
the existing approaches the phase information is discarded before the second12
Fourier transform. In literature, Oppenheim & Lim showed that for signal re-13
construction, the phase spectrum carries more important information than the14
magnitude spectrum [47]. Recently, Guo et al. utilized the phase spectrum of15
quaternion Fourier transform for salience detection [48]. All these suggest that16
the phase spectrum may carry important discriminant information for classifi-17
cation and hence should be used for UAV detection.18
2.4. Proposed 2-D Complex-Log-Fourier Transform19
To enhance the weak micro-Doppler signature, people often take the loga-20
rithm of the spectrum, e.g. cepstrogram [8] is obtained by taking the inverse21
FFT of the log-spectrogram. Complex cepstrum [49], which is often used to ed-22
it time signals in operational modal analysis, is obtained by taking the inverse23
Fourier transform of the complex logarithm of the complex spectrum. The au-24
thors thus propose a complex-log-Fourier transform to take the advantages of25
the log-spectrogram and to make use of both magnitude and phase spectrums.26
Mathematically, the Fourier transform fi=F{xi}for the i-th window xiis27
a vector of complex numbers, i.e. fi=miexp{jθi}, where miis the magnitude28
spectrum and θiis the phase spectrum. By taking the logarithm of fi, we have29
log{fi}= log{mi}+jθi.(5)30
31
9

For the logarithm of a complex number, the logarithm of the magnitude becomes1
the real part and the phase term becomes the imaginary part. This is very2
different from traditional log-spectrum/log-spectrogram, where only log{mi}is3
used and θiis discarded.4
Eq. (5) can be seen as a way to combine the log-spectrum log{mi}and5
the phase information θi. It is often necessary to normalize the power of the6
spectrogram and hence to change the range of log{mi}. Similarly, the range of7
θineeds to be normalized as well. To balance the effect of log{mi}and θi, the8
authors introduce a weighting factor and hence Eq. (5) is modified as:9
log{fi}= log{mi}+jwθi,(6)10
11
where wis a pre-defined weighting factor, which determines the importance of12
the phase spectrum. In this paper, we set w= 1/π so that the phase term is13
normalized to the range of [−1,1]. For the synthetic image X, the proposed14
2-D complex-log-Fourier transform is hence defined as:15
F2{X}=Ft{log{F{X}}}.(7)16
17
2.5. Proposed Regularized Complex-Log-Fourier Transform18
By taking the logarithm of the spectrum, the small-amplitude frequency19
components are emphasized so that the weak micro-Doppler signature could be20
clearly identified. However, the noise frequency components are enlarged at the21
same time. Fig. 3(a)-(d) show the plots of two successive spectrums of an UAV22
and the corresponding log-spectrums, respectively. From the spectrum, it is d-23
ifficult to see the micro-Doppler signature except the clutter and the main body24
Doppler. After taking the logarithm, more details of the micro-Doppler signa-25
ture can be seen from the log-spectrum, but the noise components are enlarged26
as well. The authors choose these two successive spectrums/log-spectrums with27
50% overlapping to ensure that these two spectrums/log-spectrums are similar28
to each other. The time difference between these two is 1.33 ms only. However,29
due to the noise components, Fig. 3(c) and (d) look different.30
10

50 100 150 200 250
0
0.2
0.4
0.6
0.8
1
1.2
1.4
(a) First spectrum.
50 100 150 200 250
0
0.2
0.4
0.6
0.8
1
1.2
1.4
(b) Second spectrum.
50 100 150 200 250
−3.5
−3
−2.5
−2
−1.5
−1
−0.5
0
(c) First log-spectrum.
50 100 150 200 250
−3.5
−3
−2.5
−2
−1.5
−1
−0.5
0
(d) Second log-
spectrum.
50 100 150 200 250
−3.5
−3
−2.5
−2
−1.5
−1
−0.5
0
(e) First regularized
log-spectrum.
50 100 150 200 250
−3.5
−3
−2.5
−2
−1.5
−1
−0.5
0
(f) Second regularized
log-spectrum
Figure 3: Spectrums, log-spectrums and regularized log-spectrums of two successive windows
from an UAV sequence with 50% overlapping. It is difficult to see the micro-Doppler signature
from the spectrums. The log-spectrums emphasize the weak micro-Doppler signature, but also
enlarge the noise. In contrast, the noise is greatly reduced in the regularized log-spectrums.
From the classification point of view, by taking the logarithm of the spec-1
trum, it emphasizes the discriminative information of UAVs and birds so that2
their micro-Doppler signatures are very different in log-scale. However, the3
intra-class variations of the log-spectrums of UAVs/birds are increased as well4
due to the noise components.5
To reduce the intra-class variations, the authors propose a regularization6
procedure for log-spectrum/log-spectrogram. The weak frequency components7
are vulnerable to noise, as shown in Fig. 3 (c) and (d). Their magnitudes change8
largely in log-scale due to the noise. The authors thus propose to add a constant9
11

when taking the logarithm,1
log{fi}= log{mi+Ci}+jwθi,(8)2
3
where Ciis a constant.4
Ciis chosen in such a way that it significantly suppresses the variations5
of the weak frequency components, but does not significantly alter other fre-6
quency components. For the spectrum, most frequency components have weak7
response and only some frequency components have strong response. The weak8
frequency components are easily contaminated by noise in log-scale. Those noise9
components form the noise floor. The target is thus to preserve the frequency10
components above the noise floor, but to suppress the variations of the noise11
components near the noise floor. To estimate the noise floor, the authors choose12
Ci= med{mi}, i.e. the median value of mi. Since most frequency components13
are noise components, the median value of the spectrum could well approximate14
the noise floor.15
By adding such a regularization constant Ciin Eq. (8), the logarithms of16
the strong frequency components above the noise floor will not be significantly17
altered, whereas the logarithms of the weak frequency components below the18
noise floor will be regularized close to log Ci. In such a way, the variations of the19
noise components are greatly reduced. Fig. 3(e) and (f) show the regularized20
versions of log-spectrums in Fig. 3(c) and (d), respectively. It is clear that after21
the regularization, the variations of the noise components are greatly suppressed.22
Now two successive regularized log-spectrums shown in Fig. 3(e) and (f) are23
more similar to each other than the log-spectrums shown in Fig. 3(c) and (d).24
From the spectrums/log-spectrums/regularized log-spectrums of the non-UAV25
samples, we can draw the same conclusion as that of the UAV samples. To26
avoid duplication, we omit the plots of the non-UAV samples.27
12

3. Dimension Reduction by Subspace Reliability Analysis1
3.1. Problem Analysis of Existing Subspace Approaches2
Many subspace approaches have been proposed in literature [34–46, 50–53].3
Among them, principal component analysis (PCA) [34] and linear discriminant4
analysis (LDA) [35, 50] are most popular.5
PCA was originally designed for optimal signal reconstruction [34]. From6
the classification point of view, it improves the generalization performance of7
the novel testing data by removing the subspace spanned by eigenvectors cor-8
responding to the small unreliable eigenvalues of the total scatter matrix St,9
which in turn removes the dimensions corresponding to the small eigenvalues10
of the within-class scatter matrix Swso that the Mahalanobis distance can be11
evaluated reliably [42, 43].12
Linear discriminant analysis aims to find a subspace that maximizes the13
following criterion [35]:14
W∗= argmaxW
|WTSbW|
|WTSwW|,s.t.WTW= 1,(9)15
16
where Sbis the between-class scatter matrix. If Swis full rank, the optimal W17
is obtained by solving the following generalized eigen-decomposition problem,18
SbΦ=SwΦΛ,(10)19
20
where Φand Λare the matrices of eigenvectors and eigenvalues, respectively.21
Due to high dimensionality and the limited number of training samples, Swis22
often singular. Besides, the dimensions corresponding to the small unreliable23
eigenvalues of Swadversely affect reliable classification. Thus, PCA is applied24
on the total scatter matrix Stto solve the singularity problem of Swand to25
remove the unreliable dimensions of Sw. After projecting all samples into the26
subspace, the most discriminative dimensions are derived by solving the gener-27
alized eigen-decomposition problem similarly defined as in Eq. (10).28
The proposed subspace reliability analysis aims to solve the following prob-29
lems of the existing subspace approaches:30
13

1. Most existing subspace approaches are linear subspace approach. Due to1
the limited number of training samples of each class, the class-conditional2
covariance matrix Sioften cannot be reliably estimated, and hence a3
pooled covariance matrix Swis used instead. Using Swimplies linear4
classification. People aim to find a subspace in which a hyperplane is5
defined to separate different classes. But the hyperplane has limited dis-6
criminative power. For UAV detection, there are enough samples for both7
UAV and non-UAV classes. As such, the authors prefer to use Sidirectly8
instead of Sw, which leads to a decision boundary much more complex9
than a hyperplane.10
2. In many existing approaches, the reliability issues of Sware solved by11
applying PCA on Stin the hope of removing the unreliable dimensions of12
Sw. However, it is difficult to derive a subspace in which the unreliable13
feature dimensions of both UAV and non-UAV classes are removed, as the14
unreliable feature dimensions of two classes may be very different. More15
details will be given in the coming section.16
3.2. Proposed Subspace Reliability Analysis17
For Bayes classification, the feature h= [h1, h2,...,hd]T∈Rdis assigned18
to one of the ccategories, i.e. ωi, so that the posteriori probability p(ωi|h)19
is maximized. This maximum a posterior (MAP) rule is a Bayes decision rule20
with the 0/1 loss function. As p(ωi|h) = p(h|ωi)p(ωi)/p(h) and p(h) is not a21
function of ωi, it is equivalent to maximize the following discriminant function:22
gi(h) = ln p(h|ωi) + ln p(ωi).(11)23
24
The natural logarithm is a monotonically increasing function, and hence does25
not affect the decision result but will simplify the evaluation if p(h|ωi) is an26
exponential function.27
Further quantitative analysis requires an analytical form of the class-conditional28
probability function p(h|ωi). It is often assumed to follow multivariate Gaus-29
sian distribution because of the following reasons. Firstly, it is the most natural30
14

distribution and the sum of many independent random variables, irrespective1
of their original distribution, obeys Gaussian distribution. Given mean and2
variance, it has the maximum entropy of all distributions. Secondly, Gaussian3
mixture model, which is the weighed sum of a number of Gaussian distribu-4
tions, in theory can closely approximate arbitrarily shaped distribution. Lastly,5
dimension-reduction techniques such as principal component analysis, linear dis-6
criminant analysis and many others utilize up to the second-order statistics, and7
so does the Gaussian distribution. Under Gaussian assumption,8
p(h|ωi) = 1
(2π)d/2|Σi|1/2exp[−1
2(h−¯
hi)TΣ−1
i(h−¯
hi)],(12)9
10
where ¯
hiis the class mean and Σiis the class-conditional covariance matrix.11
The discriminant function Eq. (11) becomes12
gi(h) = −1
2(h−¯
hi)TΣ−1
i(h−¯
hi) + bi.(13)13
14
In practice, biis a threshold to control the error rate of class ωiat the price15
of others. Eq. (13) is basically a minimum-Mahalanobis-distance classifier. As16
the covariance matrix Σiof individual class may not be reliably estimated in17
many applications, a pooled covariance matrix Σwcalled the within-class scatter18
matrix is often used instead in Eq. (13), which is calculated as:19
Σw=1
N
c
X
i=1
NiΣi(14)20
21
where Niis the number of samples in class ωiand Nis the total number of22
samples. The decision rule defined in Eq. (13) is modified as:23
gi(h) = −1
2(h−¯
hi)TΣ−1
w(h−¯
hi) + bi.(15)24
25
If eigen-decomposition is applied on Σw, i.e. Σw=ΦwΛwΦT
w, the decision rule26
is further simplified to:27
gi(h) =
d
X
j=1
(yj−¯yi,j )2
λw
j
+bi,(16)28
29
where yj=φw
jThand ¯yi,j =φw
jT¯
hiare the projections of hand ¯
hion φw
j,30
respectively. φw
jis the j-th eigenvector of Σwand λw
jis the j-th eigenvalue of31
15

Σw. Unfortunately, the Mahalanobis distance cannot be reliably evaluated in1
the high-dimensional space due to the limited number of training samples, i.e.2
the small eigenvalues λw
jare unreliable and many of them are zero. In literature,3
many subspace approaches [34–46] have been developed to address this issue.4
Specifically for UAV-detection problem, two discriminant functions g1(h) for5
the UAV class and g2(h) for the non-UAV class are defined as follow:6
g1(h) = (h−¯
h1)TΣ−1
1(h−¯
h1) + b1,(17)7
g2(h) = (h−¯
h2)TΣ−1
2(h−¯
h2) + b2,(18)8
9
where ¯
h1,¯
h2,Σ1,Σ2, b1and b2are the mean vectors, the class-conditional co-10
variance matrices and the thresholds for the UAV class and the non-UAV class,11
respectively. Take note that Σ1and Σ2are used instead of the pooled co-12
variance matrix Σw. A straight-forward way to define the decision rule is: if13
D(h) = g1(h)−g2(h)< b,hbelongs to the UAV class, otherwise it belongs to14
the non-UAV class. bis a threshold to trade-off the error rate of one class at15
the price of the other.16
As the inverses of Σ1and Σ2are used to weigh the feature dimensions, the17
unreliable feature dimensions of both Σ1and Σ2should be removed so that18
g1(h) and g2(h) can be evaluated reliably. However, the existing approaches19
could not achieve this.20
In PCA [34], the unreliable feature dimensions of Σtis removed, in the hope21
of removing the unreliable dimensions of Σ1and Σ2. However, the unreliable22
feature dimensions of Σ1and Σ2may be very different from that of Σt. By23
definition, Σt=αΣ1+(1−α)Σ2+Σb, where Σbis the between-class covariance24
matrix, α=N1/(N1+N2), N1and N2are the number of samples in the UAV25
class and the non-UAV class, respectively. By removing the feature dimensions26
corresponding to the small eigenvalues of Σt, it may not be feasible to remove the27
unreliable feature dimensions of Σ1and Σ2. Particularly, it cannot guarantee28
that in the resulting subspace the unreliable feature dimensions of both Σ1
29
and Σ2are removed. For example, the remaining subspace may have some30
dimensions in which the eigenvalues of Σ1are large, but the eigenvalues of Σ2
31
16

are small. In another word, some feature dimensions of Σ1or Σ2are still small1
and unreliable.2
Asymmetric PCA [42, 43] tackles this problem by assigning a larger weight3
to the less reliable class-conditional covariance matrix when building the total4
scatter matrix, e.g. a larger αvalue if Σ1is less reliable than Σ2. However, the5
aforementioned problem still remains.6
In many existing subspace approaches, people aim to find one subspace in7
which some form of objective function is optimized [34–41, 44, 45]. They do8
not target at removing the unreliable feature dimensions, and hence improving9
the classification accuracy. What’s more, it is very difficult to find one subspace10
in which we could remove all the unreliable feature dimensions of both Σ1
11
and Σ2. It may be necessary to remove excessively more feature dimensions,12
including some feature dimensions corresponding to small eigenvalues of one13
class-conditional covariance matrix but the large eigenvalues of the other, if14
targeting at removing all unreliable feature dimensions. Apparently by doing15
so, a large amount of discriminant information will be lost.16
In this paper, the authors solve the problem by removing the unreliable17
feature dimensions of Σ1and Σ2separately in two different subspaces. More18
specifically, the authors aim to remove the unreliable feature dimensions of Σ1
19
by removing the feature dimensions corresponding to the small eigenvalues of20
S1=Σ1+Σb.(19)21
22
The between-class covariance matrix Σbcontains the discriminant feature di-23
mensions that could well separate two classes. Thus, this information is pre-24
served in the projected subspace. Then, the eigen-decomposition is applied on25
S1as:26
S1=Φ1Λ1ΦT
1.(20)27
28
The dimensions corresponding to the leading m1eigenvalues of S1, i.e. Φ1,m1,29
are chosen as the projection matrix. The discriminant function g1(h) in the30
17

projected subspace becomes:1
g1,m1(h) = (h−¯
h1)TΦ1,m1(ΦT
1,m1Σ1Φ1,m1)−1ΦT
1,m1(h−¯
h1).(21)2
3
Now g1,m1(h) can be evaluated reliably as its unreliable feature dimensions are4
removed by removing the subspace corresponding to the small eigenvalues of5
S1.6
Similarly, the authors remove the unreliable feature dimensions of Σ2by7
removing the subspace corresponding to the small eigenvalues of S2,8
S2=Σ2+Σb.(22)9
10
Then, the eigen-decomposition is applied on S2,11
S2=Φ2Λ2ΦT
2.(23)12
13
The dimensions corresponding to the leading m2eigenvalues of S2are chosen14
as the project matrix, and the discriminant function g2(h) in the projected15
subspace becomes:16
g2,m2(h) = (h−¯
h2)TΦ2,m2(ΦT
2,m2Σ2Φ2,m2)−1ΦT
2,m2(h−¯
h2).(24)17
18
g1,m1(h) and g2,m2(h) are evaluated in two different subspaces. To take ac-19
count of the potential scaling effect in these two subspaces, the authors propose20
the following criterion to fuse g1,m1(h) and g2,m2(h):21
g(h) = g1,m1(h)
g2,m2(h).(25)22
23
Intuitively, if h∈ω1, the Mahalanobis distance of hto the class center of24
class 1 in the pro jected subspace, g1,m1(h), should be small and the Mahalanobis
25
distance of hto the class center of class 2 in the projected subspace, g2,m2(h),
26
should be large. As such, g(h) should be small. Thus, the decision rule becomes:27
If g(h)< b,h∈ω1, otherwise h∈ω2, where bis the threshold to control the28
error rates of these two classes.29
18

3.3. Discussion on the Proposed Subspace Reliability Analysis1
The proposed subspace reliability analysis is different from existing ap-2
proaches in the following aspects:3
1. The proposed approach is specifically designed for detection problem. Dif-4
ferent from multi-class classification problem, in which a pooled covariance5
matrix Σwis used to weigh different feature dimensions, in the proposed6
approach the feature dimensions of each class are weighed by the inverse7
of its own class-conditional covariance matrices.8
2. The proposed approach is an object-oriented dimension-reduction tech-9
nique, which studies the reliability issues of class-conditional covariance10
matrices, and aims to optimally remove the unreliable feature dimensions11
of these matrices, towards the objective of improving the classification ac-12
curacy. Most existing approaches aim to find one subspace in which some13
optimization criterion is met. However, it is difficult to find one subspace14
in which the unreliable feature dimensions of two class-conditional covari-15
ance matrices could be optimally removed. The proposed SRA removes16
them separately in two subspaces.17
3. When removing the unreliable feature dimensions of Σi, the between-18
class scatter matrix is added into consideration as defined in Eq. (19) and19
Eq. (22), in order to preserve the discriminative information between two20
classes. This is different from PCA in which unreliable feature dimensions21
of Stare removed.22
4. Lastly, two discriminant functions are fused by Eq. (25), which is very23
different from existing approaches.24
By utilizing the proposed approach, the authors optimally remove the unre-25
liable feature dimensions of two class-conditional covariance matrices separately26
in two different subspaces. This offers much more flexibility of improving the27
reliability of these class-conditional covariance matrices, and hence greatly in-28
creases the classification accuracy.29
19

Figure 4: The block diagram of the proposed UAV-detection system.
3.4. The Proposed UAV-Detection System1
The block diagram of the proposed UAV-detection system is shown in Fig. 4.2
In the training stage, the proposed 2-D regularized complex-log-Fourier trans-3
form is used to extract the feature vectors, and then the proposed subspace4
reliability analysis is applied to remove the unreliable feature dimensions and5
to obtain the projection matrix Φ1,m1for the UAV class and Φ2,m2for the
6
non-UAV class. In the testing stage, after extracting the feature vector for the7
testing sample, the discriminant functions g1,m1(h) and g2,m2(h) are evaluated
8
according to Eq. (21) and Eq. (24), respectively. Then, the discriminant func-9
tion g(h) is evaluated according to Eq. (25). Finally, g(h) is compared with a10
user-defined threshold b, which is used to control the error rates of two classes.11
If g(h)< b,hbelongs to the UAV class, otherwise hbelongs to the non-UAV12
class.13
4. Experimental Evaluation14
4.1. Experimental Setup15
The original UAV-detection dataset was provided by Thales. A low-power16
continuous-wave radar operating at X-band was used to acquire measurements.17
20

In total, there are 854 seconds of recording for UAVs, and 204 seconds of record-1
ing for non-UAVs, where non-UAVs mainly contain birds. As a longer duration2
is needed for dynamic time warping (DTW) approach [28], the recordings are3
chopped into 1-second samples when evaluating DTW. For all other approaches,4
the recordings are chopped into 50-ms samples. As a result, there are in total5
1058 samples when evaluating DTW, and 21160 samples when evaluating oth-6
ers. The authors randomly partition half of the dataset as the training samples,7
and the other half as the testing samples. The experiments are repeated 208
times, and the average performance is reported.9
The authors report two evaluation criteria: equal error rate (EER) and10
false acceptance rate (FAR) at false rejection rate (FRR) of 1% (denoted as11
F ARF RR=1%). False rejection rate is defined as the percentage of the UAV12
samples being falsely classified as non-UAVs, and false acceptance rate is defined13
as the percentage of non-UAV samples being falsely classified as UAVs. By14
varying threshold b, there are different combinations of FAR and FRR. When15
these two error rates are the same, it is defined as the equal error rate. The16
authors report the performance in terms of these two criteria because: 1) EER17
is commonly used in many verification tasks. 2) The authors would like to18
evaluate how the system performs at a low missing detection rate (i.e. a low19
FRR) of UAVs, and hence report F ARF RR=1%.20
The sampling rates of all sequences are normalized to 96kHz. For all the21
spectrograms, the spectrum utilizes 256 data points and the windows have 50%22
overlapping. After removing the clutter and some unreliable high-frequency23
components, the initial feature vector has 201 ×36 = 7236 dimensions.24
This paper contains two main contributions: the proposed 2-D regularized25
complex-log-Fourier transform and the proposed subspace reliability analysis.26
In the following sections, the performance improvement caused by these two27
will be evaluated one by one. Finally, we compare the proposed approach with28
the state-of-the-art approaches.29
21

Dim
0 100 200 300 400 500
EER
4
5
6
7
8
9
10
11
12
13
Spectrogram
CVD
Cepstrogram
Proposed Feature
(a) EER
dim
0 100 200 300 400 500
FARFRR=1%
4
10
20
40 Spectrogram
CVD
Cepstrogram
Proposed Feature
(b) F ARF RR=1%
Figure 5: Error rates vs. feature dimensions for spectrogram, cepstrogram, CVD and the
proposed signal representation. The proposed signal representation significantly outperforms
others at all feature dimensions in terms of both EER and FARF RR=1% .
4.2. Performance Evaluation of the Proposed 2-D Regularized Complex-Log-1
Fourier Transform2
To evaluate the performance improvement caused by the proposed signal3
representation, the authors compare it with spectrogram [3], cepstrogram [8]4
and cadence velocity diagram [23]. The authors treat spectrogram, cepstrogram,5
CVD and the proposed signal representation as the initial feature vector, and6
apply principal component analysis to remove the unreliable feature dimensions.7
A minimum-Mahalanobis-distance classifier is then used for classification.8
The EER and F ARF RR=1% vs. feature dimensions are shown in Fig. 5.9
It can be seen that for all different feature dimensions, the proposed signal10
representation significantly outperforms others in terms of both error rates. It11
can also be seen that although the original feature dimensionality is high (more12
than 7000), the optimal feature dimensionality is relatively low, e.g. a few13
dozen. After 100 feature dimensions, the error rates continuously increase as the14
feature dimensionality increases. The proposed approach achieves its optimal15
performance at 90 dimensions, and the performance does not drop significantly16
in a wide range of feature dimensions. The proposed approach could achieve a17
near-optimum performance in a wide range of feature dimensions.18
22

The error rates of different approaches at the optimal dimensionality are1
summarized in Table 1. The feature dimensionality is chosen in the way that2
F ARF RR=1% is minimum. Take note that all the compared approaches do not
Feature EER F ARF RR=1%
Spectrogram 7.75% 27.00%
CVD 6.68% 28.41%
Cepstrogram 10.17% 48.07%
Proposed repre-
sentation
3.98% 4.50%
Table 1: Comparison of different signal representations in terms of EER and F ARF RR=1%
at the optimal feature dimensionality.
3
utilize the phase spectrum except the proposed approach. Among the compared4
approaches, the CVD is most similar to the proposed approach in the sense that5
both approaches conduct some sort of 2-D Fourier transform. Indeed, the CVD6
performs the second best in terms of EER. The proposed signal representation7
consistently and significantly outperforms all the compared features. Compared8
with the second best performed methods, the proposed signal representation9
reduces EER by 2.7% and F ARF RR=1% by 22.50%.10
4.3. Performance Evaluation of the Proposed Subspace Reliability Analysis11
To study the effectiveness of the proposed subspace reliability analysis, the12
authors compare it with standard PCA. The proposed 2-D regularized complex-13
log-Fourier transform is used to extract the initial feature vector. The authors14
aim to study the reliability of the class-conditional covariance matrices and re-15
move those feature dimensions that are harmful to the reliable classification. For16
those unreliable feature dimensions, the training samples could not well repre-17
sent the testing samples, as the statistics of the testing samples differ largely18
from those of the training samples.19
The authors thus study whether the unreliable feature dimensions are prop-20
erly removed after dimension reduction. The authors perform so-called eigen-21
23

0 20 40 60 80 100
100
101
102
Dim
Eigenvalues
Eigenspectrum of training data
Eigenspectrum of testing data
(a) eigen-spectrums of non-UAV class, PCA
0 20 40 60 80 100
100
101
102
Dim
Eigenvalues
Eigenspectrum of training data
Eigenspectrum of testing data
(b) eigen-spectrums of UAV class, PCA
0 20 40 60 80 100
100
101
102
Dim
Eigenvalues
Eigenspectrum of training data
Eigenspectrum of testing data
(c) eigen-spectrums of non-UAV class, SRA
0 20 40 60 80 100
100
101
102
Dim
Eigenvalues
Eigenspectrum of training data
Eigenspectrum of testing data
(d) eigen-spectrums of UAV class, SRA
Figure 6: Eigen-spectrum plots for standard PCA and the proposed SRA. Eigen-spectrums
of non-UAV and UAV class for the subspace derived using PCA are shown in (a) and (b),
respectively. Eigen-spectrums of non-UAV and UAV class for the proposed SRA are shown in
(c) and (d), respectively.
spectrum analysis [42, 43]. The eigen-spectrums of the subspace (100 dimension-1
s) derived using PCA and the proposed SRA are shown in Fig. 6. More specif-2
ically, for the non-UAV class, the authors compute the variance of the train-3
ing/testing data in each dimension. Those variances form the eigen-spectrum4
of the training/testing data for the non-UAV class. Similarly, we can derive the5
eigen-spectrum for the UAV class. If the eigen-spectrum of the testing samples6
is well matched with that of the training samples, it indicates a good gener-7
alization performance. More details on eigen-spectrum analysis can be found8
in [42, 43].9
24

From these eigen-spectrums shown in Fig. 6, it is clear that the proposed1
SRA is better than PCA. 1) PCA does not effectively remove all the unreli-2
able feature dimensions. The optimal feature dimensionality for PCA-based3
approach is 90, as shown in Fig. 5. However, from Fig. 6(a) and (b), many4
feature dimensions in the PCA subspace are unreliable, i.e. For both UAV and5
non-UAV classes, the eigen-spectrum of the testing samples deviates largely6
from that of the training samples in the PCA subspace. 2) Comparing SRA7
with PCA in the projected subspace, for the non-UAV class the difference be-8
tween eigen-spectrum of the testing samples and that of the training samples in9
the SRA subspace is much smaller than eigen-spectrum difference in the PCA10
subspace. In another word, the proposed SRA better removes the unreliable11
feature dimensions so that in the SRA subspace, the eigen-spectrum difference12
is smaller than that in the PCA subspace. 3) For the eigen-spectrums of the13
UAV class shown in Fig. 6(b) and (d), neither PCA nor SRA could well remove14
the unreliable feature dimensions at 100 dimensions. However, SRA has the15
flexibility of choosing different dimensionalities for two classes, but PCA does16
not. SRA could further reduce the feature dimensionality to a much lower num-17
ber so that those unreliable feature dimensions of the UAV class are properly18
removed. In contrast, if PCA further reduces the dimensionality, many reliable19
feature dimensions of the non-UAV class will be removed at the same time.20
The proposed subspace reliability analysis achieves the optimal performance21
when the dimensionality is 10 for the UAV class and 100 for the non-UAV class.22
We now show why the optimal feature dimensionalities of two classes are very23
different using the eigen-spectrum plots. For the non-UAV class, the eigen-24
spectrum of the testing samples matches well with that of the training samples,25
as shown in Fig. 6(c). We thus should utilize 100 feature dimensions of the26
non-UAV class. On the other hand, many feature dimensions are unreliable27
for the first 100 dimensions of the UAV class, as the eigen-spectrum of the28
testing samples does not match well with that of the testing samples, as shown29
in Fig. 6(d). To remove these unreliable feature dimensions, we thus should30
reduce the feature dimensionality of the UAV class to 10.31
25

The performance comparison between PCA and the proposed SRA is shown1
in Table 2. We can see that the proposed SRA outperforms PCA in terms of2
both error rates.
Table 2: Comparison between PCA and the proposed SRA in terms of EER and FARF RR=1% .
Method EER F ARF RR=1%
Proposed signal repre-
sentation + PCA
3.98% 4.50%
Proposed signal repre-
sentation + SRA
3.27% 3.89%
3
4.4. Comparison to the State-of-the-Art Approaches4
As no complete solution was published for UAV-detection problem, the au-5
thors implement two state-of-the-art techniques designed for other mDS recog-6
nition tasks and adapt them for UAV detection. The proposed approach and7
the compared approaches are implemented using Matlab R2015a [54–59].8
Dynamic time warping: dynamic time warping (DTW) [28] was originally9
designed to classify vehicles and human. The optimal global path derived by10
DTW 1is treated as the distance between two samples. Then, the distances11
between one sample and all training samples are treated as the feature vector,12
and classified by a linear support vector machine (SVM). The cost parameter13
of SVM is set as C= 40.14
Robust PCA: Robust PCA [24] was originally designed for action classifica-15
tion. The feature vector is obtained by averaging the spectrogram over time.16
The Minimum Covariance Determinant (MCD) estimator is then used to remove17
the outlier. “rrcov” package in R programming is used to implement the MCD18
estimator. PCA is then applied to reduce the feature dimensionality. After fea-19
ture vectors are normalized to zero mean with unit variance, they are classified20
1The matlab code of DTW can be downloaded from http://labrosa.ee.columbia.edu/
matlab/dtw/.
26

by a linear support vector machine. The cost parameter is C= 40.1
The authors summarize the performance comparison between the proposed2
approach and the state-of-the-art approaches in Table 3. Based on these exper-
Table 3: Comparison between the proposed approach and the state-of-the-art approaches.
Method EER F ARF RR=1%
DTW [28] 8.04% 47.42%
Robust PCA [24] 8.19% 54.16%
Spectrogram + PCA 7.75% 27.00%
CVD + PCA 6.68% 28.41%
Cepstrogram + PCA 10.17% 48.07%
Proposed feature rep-
resentation + PCA
3.98% 4.50%
Proposed feature rep-
resentation + SRA
3.27% 3.89%
3
imental results, we could see the following:4
1. Neither DTW nor robust PCA can well solve the UAV-detection problem.5
The error rates of both approaches are very high. These two approaches6
were originally designed for other applications, and hence they are not7
good at solving the UAV-detection problem.8
2. The performance of simply taking the spectrogram as features is not sig-9
nificantly better than DTW or robust PCA. Cepstrogram shows its ef-10
fectiveness in visually recognizing micro-Doppler signature. However, our11
experiment results show that cepstrogram-based approach does not yield12
good performance for machine-automatic UAV detection.13
3. By utilizing the proposed robust spectral analysis, the error rates are14
greatly reduced compared with other signal representations. The proposed15
feature utilizes not only the amplitude, but also the phase information of16
the first FFT. What’s more, the proposed regularized complex logarithm17
resolves the noise-amplification problem of the log-spectrogram, and hence18
27

significantly improves the performance.1
4. Lastly, the proposed subspace reliability analysis well resolves the relia-2
bility issues of two class-conditional covariance matrices in this two-class3
classification problem. More specifically, the proposed SRA better removes4
the unreliable feature dimensions of two covariance matrices separately in5
two different subspaces, which further improves the classification perfor-6
mance.7
5. Conclusion8
In this paper, the authors propose a robust signal representation - 2-D9
regularized complex-log-Fourier transform and an object-oriented dimension-10
reduction technique - subspace reliability analysis. The proposed signal repre-11
sentation addresses the problems of the existing feature representations by mak-12
ing full use of both magnitude and phase information of the first Fourier trans-13
form, enlarging the weak micro-Doppler signature and suppressing the noise in14
the log-spectrogram. The proposed subspace reliability analysis is specifically15
designed for UAV-detection problem. It removes the unreliable feature dimen-16
sions of both UAV and non-UAV classes separately in two different subspaces.17
The proposed approach is compared with the state-of-the-art approaches. It18
significantly outperforms others on the benchmark dataset.19
Acknowledgment20
This research is supported by Singapore Future Systems and Technology21
Directorate (FSTD) under pro ject reference: MINDEF-NTU-DIRP/2014/01.22
Thank Thales Solutions Asia for providing the data. Special thanks to Mr.23
Adriaan Smits, the Director of Centre of Excellence for Radar & Integrated24
Sensors, Thales Solutions Asia, for the valuable discussion and advice.25
28

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