aerospace science and technology - yifei shix. hua et al. / aerospace science and technology 70...

Aerospace Science and Technology 70 (2017) 47–54

Contents lists available at ScienceDirect

Aerospace Science and Technology

www.elsevier.com/locate/aescte

Target detection in sea clutter via weighted averaging filter on the Riemannian manifold

Xiaoqiang Hua ∗, Yongqiang Cheng, Yubo Li, Yifei Shi, Hongqiang Wang, Yuliang Qin

a r t i c l e i n f o a b s t r a c t

Article history:Received 14 October 2016Received in revised form 11 July 2017Accepted 29 July 2017Available online 4 August 2017

Keywords:Target detectionRiemannian manifoldRiemannian meanRiemannian medianWeighted averaging filter

This paper proposes a weighted averaging filter procedure combined with a Riemannian geometry method to carry out a target detection in sea clutter. In particular, the weighted averaging filter, conceived from a philosophy of the bilateral filtering in image denoising, is presented on a Riemannian manifold of Hermitian positive-definite matrix. This filter acts as a clutter suppression procedure in the detection framework of the algorithm proposed in this paper, and can improve the detection performance. The principle of detection is that if a location has enough dissimilarity from the Riemannian mean or median estimated by its neighboring locations, targets are supposed to appear at this location. Numerical experiments and real sea clutter data are given to demonstrate the effectiveness of the proposed target detection algorithm.

© 2017 Elsevier Masson SAS. All rights reserved.

1. Introduction

The detection of targets embedded in sea clutter is an im-portant subject in the field of radar signal processing. The envi-ronment of sea surface is very complicated and changeable. Sea clutter is a complex phenomenon influenced by environmental conditions, parameters of radar systems, and site configurations [1]. In a modern high-resolution radar system, sea clutter usually exhibits four properties of strong non-Gaussian, nonhomogeneous, non-stationary, and time-varying. Under those circumstances, it is difficult to achieve a satisfactory detection performance. Therefore, it is very necessary and meaningful for modern radars to improve their detection performance.

Few pulses can be used for target detection, since the dwell time of beam at target depending on the rotating speed of a scan-ning radar is very short. The classical fast Fourier transform (FFT) based constant false alarm rate (CFAR) detector [2] suffers from severe performance degradation owing to the poor Doppler res-olution as well as the energy spread of the Doppler filter banks. To address those drawbacks, Barbaresco has done much work in the statistical geometry detection, and has proposed a generalized CFAR technique on a Riemannian manifold of Hermitian positive-definite (HPD) matrices, which was referred to as the matrix CFAR detector [3]. In the matrix CFAR detector, the radar received clutter data in each range cell in one coherent processing interval (CPI) is modeled as a HPD matrix. In addition, the Riemannian mean de-

* Corresponding author.E-mail address: [email protected] (X. Hua).

http://dx.doi.org/10.1016/j.ast.2017.07.0421270-9638/© 2017 Elsevier Masson SAS. All rights reserved.

tector [3,4], and the median detector [5] are derived. Moreover, the Riemannian metric was deduced based on this parameterization. The existence and uniqueness of the mean and median had been proven in [6]. The matrix CFAR detector has been used for mon-itoring of wake vortex turbulences [7–9], and target detection in coastal X-band and HF surface wave radars [3]. It has been proven that the matrix CFAR detector has better detection performance than the classical FFT-CFAR detection algorithm [3,4]. Our previ-ous research [10] has explored the matrix CFAR detector based on an alternative measure—the Kullback–Leibler divergence. The experimental results have also shown that the matrix CFAR detec-tor has better detection performance than the classical FFT-CFAR algorithm. Furthermore, the performance of the Kullback Leibler-based matrix CFAR detector is better than that of the Riemannian distance-based matrix CFAR detector. The principle of detection is that targets are supposed to appear at a location if this location has enough dissimilarity from the Riemannian mean or median es-timated by its neighboring locations.

Rather than exploring different distance measures used in the matrix CFAR detector, as our previous work [10], in this paper, we extend the framework of the matrix CFAR detector. We com-bine a weighted averaging filter, conceived from a philosophy of the bilateral filtering in image denoising [11], within the detec-tion framework proposed by Barbaresco [3]. The purpose of the image denoising is to reduce the noise, and to enhance the im-age information. Similarly, we propose a weighted averaging filter to reduce the clutter, and to enhance the target signal. Concretely, the radar received clutter data in each cell in one CPI is modeled as a HPD matrix. The information of the target or clutter is repre-sented by this HPD matrix. Then, the weighted averaging filter is

http://dx.doi.org/10.1016/j.ast.2017.07.042http://www.ScienceDirect.com/http://www.elsevier.com/locate/aesctemailto:[email protected]://dx.doi.org/10.1016/j.ast.2017.07.042http://crossmark.crossref.org/dialog/?doi=10.1016/j.ast.2017.07.042&domain=pdf

48 X. Hua et al. / Aerospace Science and Technology 70 (2017) 47–54

imposed on these HPD matrices. The filtered HPD matrix in each cell is a weighted average of HPD matrices of its surrounding cells. Finally, the detection decision is made based on these filtered ma-trices data. As that filter acts as a clutter suppression procedure, the detection performance can be improved.

The rest of this paper is organized as follows. In Section 2, we model the received signal using the HPD matrix. In Section 3, the Riemannian geometry of the space of HPD matrices is presented. The proposed detection algorithm is developed in Section 4. Then, we evaluate the performances of the proposed detection algorithm with different parameters as well as the matrix CFAR detector by simulated data and real sea clutter data in Section 5. Finally, con-clusion is provided in Section 6.

1.1. Notation

A lot of notations are adopted as follows. We use math italic for scalars x, uppercase bold for matrices A, and lowercase bold for vectors x. The conjugate transpose operator is denoted by the symbol (·)H . tr(·) and det(·) are the trace and the determinant of the square matrix argument, respectively. I denotes the identity matrix, and C(n) is the sets of n-dimensional vectors of complex numbers. The Frobenius norm of the matrix A is denoted by ‖A‖F . For any n × n Hermitian matrix A, A > 0 means that A is a HPD matrix, and denoted by P(n). Finally, E(·) denotes statistical expec-tation.

2. Signal modeled using HPD matrix

As mentioned above, the matrix CFAR detector can be illus-trated in Fig. 1. The data R i in the ith range cell is a HPD matrix estimated by the sample data z in the ith range cell in one CPI according to its correlation coefficient. Then, calculating the dis-tance between the covariance matrix R D of the cell under test and the mean matrix R̄ or median matrix R̂ of reference cells around the cell under test. Finally, the detection is made by comparing the distance between R D and R̄ or R̂ with a given threshold γ . The Riemannian distance measure is used when calculating the distance as well as calculating the mean and median matrix, as the geometry of the manifold of HPD matrices is considered. In the following, we will give a brief description about how to con-struct the HPD covariance matrix from the sample data in each cell in one CPI.

For the radar received complex clutter data {z = z1, z2, . . . , zn}in each cell in one CPI, where n is the length of pulses, as-suming z is a complex circular multivariate Gaussian distribution, z ∼ CN(0, R), with zero mean and covariance matrix R . The prob-ability density function p(z; 0, R) is given as follows [3],

p(z;0, R) = 1πn det(R)

exp{−zH R−1z} (1)

where π denotes the circumference ratio. The covariance matrix Ris given by [3],

R = E[zzH ] =⎡⎢⎢⎢⎣

r0 r̄1 · · · r̄n−1r1 r0 · · · r̄n−2...

. . .. . .

...

rn−1 · · · r1 r0

⎤⎥⎥⎥⎦ , rk = E[zi z̄i+k],

0 ≤ k ≤ n − 1,1 ≤ i ≤ n (2)where rk = E[zn z̄n+k] is called the correlation coefficient and z̄denotes the complex conjugate of z. R is a Toeplitz Hermitian positive-definite matrix with R H = R . According to the ergodicity of a stationary Gaussian process, the correlation coefficient of data z can be calculated by averaging over time instead of its statistical expectation, as

Fig. 1. Matrix CFAR detector [3].

r̂k = 1nn−1−|k|∑

n=0z(n)z̄(n + k), |k| ≤ n − 1 (3)

The pulse data in each cell in one CPI is modeled by Equation (1), and the information of target or clutter can be represented by its HPD matrix. Then, the new observation in each cell is a HPD matrix estimated by equations (2) and (3). These matrices of the range cells are embedded in P(n). According to the parameteriza-tion using the HPD matrix, the radar echo z = {z1, z2, . . . , zn} can be mapped into a n dimensional parameter space.

ψ : C(n) → P(n), z → R ∈ P(n) (4)Here P(n) forms a differentiable Riemannian manifold [12,13]

with non-positive curvature [14,15]. HPD matrix manifold is a closed, self-dual convex cone, and serves as a canonical higher-rank symmetric space [16]. An excellent overview for HPD mani-fold is referred to [17,18].

3. Riemannian geometry of space of HPD matrices

In this Section, we overview some of basic mathematical knowl-edge related to this article, including the Riemannian distance, the geometric mean, and the geometric median. These are necessary for the detector design.

The space of HPD matrices P(n) is a differentiable manifold of dimension n(n + 1)/2. The Riemannian distance between two dif-ferent points R1, R2 on the manifold is defined by [19],

d2R(R1, R2) =∥∥logm(R−1/21 R2 R−1/21 )∥∥2F =

n∑k=1

log2(λk) (5)

where λk is the kth eigenvalue of R−1/21 R2 R

−1/21 , logm(·) is the

logarithm map on the Riemannian manifold of HPD matrices.Based on that distance metric, similar to algebraic mean and

median, the definition of the Riemannian mean or median can be given as follows: Given a set of HPD covariance matrices {R1, R2, . . . , R N }, the average matrix R̄ is the minimum value of the summation of p order distance between R and R i .

R̄ = arg minR

1

N

N∑i=1

dp(R, R i) (6)

where d(·, ·) denotes the Riemannian distance, and p is the order of the distance, when p = 1, R̄ denotes the median; when p = 2, R̄ is the mean.

The Riemannian mean associated with the Riemannian distance (5), of a set of N HPD matrices {R1, R2, . . . , R N}, is given by [3],

R̄t+1 = R̄1/2t expm{

−εt(

N∑logm

(R̄

−1/2t Rk R̄

−1/2t

))}R̄

1/2t (7)

k=1

X. Hua et al. / Aerospace Science and Technology 70 (2017) 47–54 49

where εt is the step size, it varies with t or constant. t is the index of iteration.

Similarly, the Riemannian median related to the Riemannian distance (5), of N HPD matrices {R1, R2, . . . , R N } is given by [5],

R̂t+1 = R̂1/2t expm{εt

( ∑k∈G R̃t

Ct‖Ct‖F

)}R̂

1/2t ,

Ct = logm(

R̂−1/2t Rk R̂

−1/2t

), G R̃t = {k/Rk �= R̂t} (8)

where εt is the step size. expm(·) denotes the exponential map on the Riemannian manifold of HPD matrices.

4. Extended matrix CFAR detector

In general, the performance of a detection algorithm is not very good, when the detection SCR is relatively low (low SCR, low RCS, etc.). A method of improving the detection performance is to carry out a clutter suppression procedure prior to a target detection. As the pulse data is modeled as a HPD matrix, the clutter suppres-sion processing must be imposed on those matrices data. A filter method is conceived from the bilateral filtering in image denois-ing [11]. The principle of image denoising is that a noise-free pixel is estimated as a weighted average of image pixels, where each pixel is weighted according to its neighbouring pixels in an image patch.

Based on this scheme, we extend the detection framework of F. Barbaresco’s researches by exploiting a weighted averaging filter. As illustrated in Fig. 2, the data R i in each cell is a HPD matrix es-timated by the pulse data z in one CPI according to its correlation coefficient. Then, a weighted averaging filter is carried on those matrices. Each filtered matrix is estimated as a weighted average of its neighbouring matrices. The weight is measured by the Rieman-nian distance between the matrix under filter and a matrix around its neighbouring. The filtered matrix also is a HPD. According to those filtered matrices, we carry out the subsequent target detec-tion processing. Calculating the distance between the covariance matrix R D of a cell under test and the mean or median matrix R̄ of reference cells around the cell under test. Finally, the detec-tion is made by comparing the distance between R D and R̄ with a detection threshold γ , which is related to the clutter power level as well as a multiplier corresponding to the desired probability of false alarm P f a . The rule of target detection can be formulated as follows,

d(R D , R̄)target present

≷target absent

γ (9)

where d(R D , R̄) is the test statistic. Since there is not an analyti-cal expression for the threshold γ , the threshold γ is derived by Monte Carlo method in order to maintain the false alarm constant. In the following, we give a detailed description about the weighted averaging filter.

For the pulse data in each cell in one CPI, they can be modeled and represented by a HPD matrix R . For the covariance matrix Rtin the tth range cell, a weighted averaging filter is imposed on it by using its surrounding m covariance matrices {R1, R2, . . . , Rm}. The weighted covariance matrix R wt in the tth range cell can be computed, and the nature constrain must be imposed on these weights, as,

R wt =m∑

i=1wi R i, 0 ≤ wi ≤ 1,

m∑i=1

wi = 1 (10)

where wi is the weight of the ith covariance matrix. In order to preserve more information of the weighted matrix Rt , the weight

Fig. 2. Extended matrix CFAR detector.

wi (i = 1, 2, . . . , m) is in direct proportion to the Riemannian dis-tance between R i and Rt . In this paper, the weight wi is defined as,

wi = 1W

exp{−d2(R i, Rt)/h2} (11)

where d(R i, Rt) is the Riemannian distance between R i and Rt . W = ∑mi=1 exp{−d2(R i, Rt)/h2} is a normalizing factor, and h is a filtering parameter, which controls the exponential decay. When h → 0, the weighted matrix tends to be equal to the original ma-trix; and when h → ∞, the weighted matrix tends to be equal to the arithmetic mean of the input m matrices. It can be noted from (11) that the smaller the distance between R i and Rt , the larger the weight of R i .

5. Performance analysis

In this section, numerical experiments and real sea clutter data are performed to evaluate the detection performance of the pro-posed mean-based extended matrix CFAR detector (EMean detec-tor), and the median-based extended matrix CFAR detector (EMe-dian detector) in comparison with the mean-based matrix CFAR detector (Mean detector) [3], and the median-based matrix CFAR detector (Median detector) [5].

5.1. Numerical experiments

The performances of: 1) Mean detector, and Median detector, 2) EMean detector, and EMedian detector are compared via Monte Carlo simulations. We simulate a target detection environment. A radar transmits a gust of 7 pulses along a direction. The length of pulses Tp is 30 microseconds (μs), and the radar carrier frequency fc is 5 GHz. The radar bandwidth B is 5 MHz, and the Pulse Rep-etition Frequency (PRF) is 1000 Hz. The range resolution is equal to 30 m. A target is located at 16 km with an approximate con-stant velocity v = 5 m/s. The Radar Cross Section (RCS) is equal to 0.6. The radar echoes are sampled and compressed. A total of 40 range cells are considered, and M = 16 range cells are used for averaging. The moving target is located at the 24th range cell. As the Gaussian assumption is no longer met in many situations of practical interest, such as the high-resolution radar and sea clut-ter, a compound-Gaussian model is more appropriate to describe the non-Gaussian radar clutter. The compound-Gaussian clutter can be written as the product of two mutually independent ran-dom processes, where the fast fluctuating ‘speckle’ component is a zero-mean complex Gaussian process and the comparatively slow varying ‘texture’ component is a nonnegative real random process which describes the underlying mean power level of the resultant


Fig. 3. Plots of statistics in each range cell under different SCR.

clutter process [20]. The most common clutter models, such as K-distribution, are compatible with the compound-Gaussian model. In the simulation, the clutter is assumed as the K distribution. The amplitude-pdf of K-distribution is [21],

p(x) =√

2ν/μ

2ν−1Γ (ν)

(√2ν

μx

)νKν−1

(√2ν

μx

), x ≥ 0 (12)

where Γ (·) is the gamma function, and Kν−1(·) is the modified Bessel function of the second kind with order ν − 1.

The statistic in each range cell is the Riemannian distance be-tween the covariance matrix of cell under test and its correspond-ing mean or median matrix. After calculating the statistic in each cell, we define the normalized statistic by dividing the maximum statistic. The target cannot be detected and can be detected by all these algorithms when the SCR is less than 5 dB and is larger than 15 dB respectively. It can be learned from Fig. 4 in the next. There-fore, we can give the comparison results when the SCR is between 5 dB and 15 dB. In Fig. 3, the normalized statistic in each cell is plotted with signal to clutter rate (SCR) = 5 dB, 10 dB, and

15 dB respectively. Actually, we can also give a comparison result at any value of the SCR between 5 and 15 dB. The filter parame-ters of the EMean detector and the EMedian detector are m and h, where m is the number of the range cells used for weighted fil-tering, and h denotes the filtering control parameter. m and h are free parameters. Different parameter values lead to different per-formances. In this simulation, we choose m = 11, h = 1; m = 11, h = 1.5; and m = 13, h = 1 respectively. These choices may be not the optimal. The purpose of selection of the parameter values is to enhance the target signal, and to reduce the clutter power simul-taneously.

It can be observed from Fig. 3 that the normalized statistics in range cells with target are much larger than that in range cells without target in these detectors. This means that the target can be detected by all the four detectors correctly. However, the nor-malized statistics of these algorithms in range cells without target are quite different. The smaller the normalized statistics of a detec-tor in range cells without target are, the lower the “clutter energy” reflected by this detector is. It also implies that the “signal energy” reflected by this detector is higher when compared to the “clutter


Fig. 4. Pd versus SCR with different detectors.

Fig. 5. ROC curves of different detectors with SCR = 5, 10 dB.

energy”. Then, this detector has better detection performance. It is clear that the statistics in range cells without target for the EMean and EMedian detectors are much lower than that of the Mean and

Median detectors. It indicates that the EMean and EMedian detec-tors have better detection performance than the Mean and Median detectors. The explanation is that the weighted averaging filter has


Fig. 6. Plots of statistics in each range cell with real sea clutter data.

reduced the clutter power, and the detection performance is im-proved.

Monte Carlo simulations are carried out to give a more accurate comparison of the detection performances of Mean vs EMean, and Median vs EMedian. For the underlying detectors, the detection thresholds are not only related to the background clutter power

but also associated with the threshold multiplier which usually does not exist a closed-form expression. Therefore, it is difficult to determine the detection threshold analytically. In the simula-tion, the detection threshold is obtained via Monte Carlo runs. The test statistic in the absence of target is calculated via 106 Monte Carlo runs, where the detection threshold is determined accord-


Fig. 7. ROC curves of different detectors in real sea clutter.

ing to the given probability of false alarm (P f a). To ensure the detection probability is accurately estimated, we repeat 200 times simulations to estimate the detection probability for different val-ues of SCR, keeping the noise power constant and varying the value of the signal energy. The detection probability is estimated by the relative frequencies. Fig. 4 plots the probability of detection versus SCR under different P f a . The SCR varies from 2 to 20 dB with an interval of 1 dB. The P f a for these plots are 10−5, 10−4respectively. It can be noted from Fig. 4 that the EMean and EMe-dian detectors have better detection performance than the Mean and Median detectors. The detection performances of the EMean and EMedian detectors with different parameter values are close. The parameter values of m and h are chosen as m = 11, h = 1; m = 11, h = 1.5; m = 13, h = 1. Moreover, the detection perfor-mances of the EMean and EMedian detectors are better than the Mean and Median matrix CFAR detectors with 2–4 dB and 1–3 dB SCR improvement when P f a is 10−5, and 10−4 respectively.

Fig. 5 shows the ROC curves of these detectors with different SCR. It can be observed from Fig. 5 that the detection performances of different parameter values of m and h are close. Furthermore, the detection performances of the EMean and EMedian detectors outperform that of the Mean and Median detectors. These results can prove the superiority of the extended matrix CFAR detector.

5.2. Real sea clutter data

The real sea clutter data that we use is from the McMaster Uni-versity IPIX radar, which is collected at the Osborne Head Gunnery Range (OHGR), Dartmouth, Nova Scotia, Canada [22]. There are more than 300 data files altogether, but not all data files are pro-vided. Moreover, the four data files, 19931107_135603_starea.cdf (#17), 19931107_141630_starea.cdf (#18), 19931107_145028_starea.cdf (#19), and 19931108_213827_starea.cdf (#25), are often used to verify the performance of a detection algorithm in prac-tice. In our real data experiments, we use these four data files to evaluate the detection performances of the Mean, EMean, Median and EMedian detectors. Table 1 lists the cell where the target is strongest (‘primary’), and neighboring range cells where the tar-get may also be visible (‘secondary’). The weak static target is a spherical block of Styrofoam, wrapped with wire mesh. The IPIX radar has polarimetric information; shown results correspond to horizontal polarization (HH) only. For the four complex data sets, the fast time or range dimension consists of 14 samples, and the range resolution is 30 m. The number of transmitted pulses, the number of samples in the slow-time dimension, is 131072 with a pulse repetition frequency of 1000 Hz. The average SCR varies in the range cell is 0–6 dB.

The observation data was previously processed by the program ‘ipixload.m’. This program consists of two steps: Remove mean and

Table 1Real target parameters with IPIX radar.

Filename Primary range cell

Secondary range cells

Range cells

Polarization SCR

#17 9 8–11 14 HH 0∼6 dB#18 9 8–11 14 HH 0∼6 dB#19 9 8–11 14 HH 0∼6 dB#25 7 6–8 14 HH 0∼6 dB

standard deviation from the I and Q channels separately; and re-move the phase imbalance. Then, the preprocessed data in each range cell is modeled and described as a HPD matrix. More de-tailed information about this program is referred to [23]. By com-paring the statistics of Mean vs EMean, and Median vs EMedian, we can investigate in which range cell targets are visible in Fig. 6. Here, the filter parameters of the EMean and EMedian detectors are also m = 11, h = 1; m = 11, h = 1.5; m = 13, h = 1, respec-tively.

It is clear from Fig. 6 that the primary target cells can be well detected by these four matrix CFAR detectors. Moreover, the nor-malized statistics in range cells without target are close, since the average SCR is very low, as 0–6 dB. There is no enough data to generate ROC curves for the real target. Actually, if the real tar-get is located in one range cell, then, we need numbers of sample data without targets in this range cell to determine the thresh-old under different P f a . In real clutter data, there are no sample data which not contain the target signal in the cell where the real target is located. Thus, in order to compare the detection perfor-mances of these detectors in real sea data, we exclude the cells that contain the real target in #19 dataset, and add a synthetic tar-get with the method presented in numerical experiments. We use the first 70000 pulses to estimate the threshold according to P f a . The HPD matrix in each cell is square matrix of order 7 ·104 Monte Carlo runs to calculate the test statistics in the absence of target. 10 reference cells are used for computing the mean and median matrices. We use the followed 1400 pulses to carry out the target detection in the presence of target, and 200 times experiments are repeated to estimate the detection probability with SCR = 10 dB. The detection probability is estimated by the relative frequencies. ROC curves for different detectors are given in Fig. 7. From Fig. 7, we can conclude that our proposed extended matrix CFAR detector with different parameter values has better performance than the matrix CFAR detector.

6. Conclusion

In this paper, we have extended the framework of the ma-trix CFAR detector by combining with a weighted averaging filter, which is conceived from the philosophy of the bilateral filtering in


image denoising. Concretely, the filtered matrix in each cell is a weighted average of covariance matrices of its surrounding cells. The filter processing can enhance target signal, and reduce the clutter power. It acts as the clutter suppression procedure. Thus, the detection performance can be improved. At the analysis stage, we have assessed the detection performance by giving plots of Pd vs SCR, and ROC curves in simulated experiments and real sea clut-ter. These results have shown that our proposed extended matrix CFAR detector has better performance than the matrix CFAR detec-tor.

Conflict of interest statement

There is no conflict of interests.

References

[1] Y. Kwon, R.M. Narayanan, M. Rangaswamy, A multi-target detector using mu-tual information for noise radar systems in low SNR regimes, in: Waveform Diversity and Design Conference, 2010.

[2] M.A. Richards, Fundamentals of Radar Signal Processing, second edition, McGraw–Hill, 2014.

[3] J. Lapuyade-Lahorgue, F. Barbaresco, Radar detection using Siegel distance be-tween autoregressive processes, application to HF and X-band radar, in: 2008 IEEE Radar Conference, 2008, pp. 1–6.

[4] F. Barbaresco, New Foundation of radar Doppler signal processing based on advanced differential geometry of symmetric spaces: Doppler matrix CFAR and radar application, in: International Radar Conference, Radar’09, Bordeaux, France, 2009.

[5] F. Barbaresco, Robust statistical Radar Processing in Fréchet metric space: OS-HDR-CFAR and OS-STAP Processing in Siegel homogeneous bounded domains, in: 2011 12th International Radar Symposium, IRS, 2011, pp. 639–644.

[6] M. Arnaudon, F. Barbaresco, L. Yang, Medians and means in Riemannian geom-etry: existence, uniqueness and computation, in: Matrix Information Geometry, 2011, pp. 169–197.

[7] Z. Liu, F. Barbaresco, Doppler Information Geometry for Wake Turbulence Mon-itoring, Springer Berlin Heidelberg, 2013.

[8] F.S. Barbaresco, Method for Radar Monitoring of Wake Turbulence, US, 2012.[9] F. Barbaresco, U. Meier, Radar monitoring of a wake vortex: electromagnetic

reflection of wake turbulence in clear air, C. R. Phys. 11 (2010) 54–67.[10] Y. Cheng, X. Hua, H. Wang, Y. Qin, X. Li, The geometry of signal detection with

applications to radar signal processing, Entropy 18 (2016) 381–397.[11] C. Tomasi, R. Manduchi, Bilateral filtering for gray and color images, in: ICCV,

1998, pp. 839–846.[12] R. Bhatia, Positive Definite Matrices, Princeton University Press, Princeton, NJ,

2007.[13] F. Hiai, D. Petz, Riemannian metrics on positive definite matrices related to

means, Linear Algebra Appl. 430 (2015) 2117–2136.[14] M.R. Bridson, A. Haefliger, Metric Spaces of Non-Positive Curvature, Springer,

1999.[15] S. Sra, Positive Definite Matrices and the Symmetric Stein Divergence, 2011.[16] A. Terras, Harmonic Analysis on Symmetric Spaces and Applications II,

Springer-Verlag, 2014.[17] C.R. Johnson, Positive definite matrices, Am. Math. Mon. 77 (1970) 69–75.[18] H. Wolkowicz, R. Saigal, L. Vandenberghe, Handbook of Semidefinite Program-

ming: Theory, Algorithms and Applications, Kluwer, 2000.[19] S. Lang, Fundamentals of Differential Geometry, Springer, 1999.[20] L.P. Roy, R.V. Raja Kumar, A GLRT detector in partially correlated texture based

compound-Gaussian clutter, in: Communications, 2010, pp. 1–5.[21] S. Watts, Radar detection prediction in K-distributed sea clutter and thermal

noise, IEEE Trans. Aerosp. Electron. Syst. 23 (1987) 40–45.[22] S. Haykin, C. Krasnor, T.J. Nohara, B.W. Currie, A coherent dual-polarized radar

for studying the ocean environment, IEEE Trans. Geosci. Remote Sens. 29 (1991) 189–191.

[23] IPIX radar file, IPIX radar dataset files in Dartmouth, http://soma.ece.mcmaster.ca/ipix/dartmouth/cdfhowto.html, 1993.

http://refhub.elsevier.com/S1270-9638(16)30850-1/bib32s1http://refhub.elsevier.com/S1270-9638(16)30850-1/bib32s1http://refhub.elsevier.com/S1270-9638(16)30850-1/bib33s1http://refhub.elsevier.com/S1270-9638(16)30850-1/bib33s1http://refhub.elsevier.com/S1270-9638(16)30850-1/bib33s1http://refhub.elsevier.com/S1270-9638(16)30850-1/bib35s1http://refhub.elsevier.com/S1270-9638(16)30850-1/bib35s1http://refhub.elsevier.com/S1270-9638(16)30850-1/bib35s1http://refhub.elsevier.com/S1270-9638(16)30850-1/bib36s1http://refhub.elsevier.com/S1270-9638(16)30850-1/bib36s1http://refhub.elsevier.com/S1270-9638(16)30850-1/bib36s1http://refhub.elsevier.com/S1270-9638(16)30850-1/bib37s1http://refhub.elsevier.com/S1270-9638(16)30850-1/bib37s1http://refhub.elsevier.com/S1270-9638(16)30850-1/bib38s1http://refhub.elsevier.com/S1270-9638(16)30850-1/bib39s1http://refhub.elsevier.com/S1270-9638(16)30850-1/bib39s1http://refhub.elsevier.com/S1270-9638(16)30850-1/bib3130s1http://refhub.elsevier.com/S1270-9638(16)30850-1/bib3130s1http://refhub.elsevier.com/S1270-9638(16)30850-1/bib3131s1http://refhub.elsevier.com/S1270-9638(16)30850-1/bib3131s1http://refhub.elsevier.com/S1270-9638(16)30850-1/bib3132s1http://refhub.elsevier.com/S1270-9638(16)30850-1/bib3132s1http://refhub.elsevier.com/S1270-9638(16)30850-1/bib3133s1http://refhub.elsevier.com/S1270-9638(16)30850-1/bib3133s1http://refhub.elsevier.com/S1270-9638(16)30850-1/bib3134s1http://refhub.elsevier.com/S1270-9638(16)30850-1/bib3134s1http://refhub.elsevier.com/S1270-9638(16)30850-1/bib3135s1http://refhub.elsevier.com/S1270-9638(16)30850-1/bib3136s1http://refhub.elsevier.com/S1270-9638(16)30850-1/bib3136s1http://refhub.elsevier.com/S1270-9638(16)30850-1/bib3137s1http://refhub.elsevier.com/S1270-9638(16)30850-1/bib3138s1http://refhub.elsevier.com/S1270-9638(16)30850-1/bib3138s1http://refhub.elsevier.com/S1270-9638(16)30850-1/bib3139s1http://refhub.elsevier.com/S1270-9638(16)30850-1/bib3230s1http://refhub.elsevier.com/S1270-9638(16)30850-1/bib3230s1http://refhub.elsevier.com/S1270-9638(16)30850-1/bib3231s1http://refhub.elsevier.com/S1270-9638(16)30850-1/bib3231s1http://refhub.elsevier.com/S1270-9638(16)30850-1/bib3232s1http://refhub.elsevier.com/S1270-9638(16)30850-1/bib3232s1http://refhub.elsevier.com/S1270-9638(16)30850-1/bib3232s1http://soma.ece.mcmaster.ca/ipix/dartmouth/cdfhowto.htmlhttp://soma.ece.mcmaster.ca/ipix/dartmouth/cdfhowto.html

Target detection in sea clutter via weighted averaging ﬁlter on the Riemannian manifold1 Introduction1.1 Notation

2 Signal modeled using HPD matrix3 Riemannian geometry of space of HPD matrices4 Extended matrix CFAR detector5 Performance analysis5.1 Numerical experiments5.2 Real sea clutter data

6 ConclusionConﬂict of interest statementReferences

aerospace science and technology - yifei shix. hua et al. / aerospace science and technology 70...

Documents