Sea Clutter Distribution Modeling: A Kernel Density Estimation Approach
aa r X i v : . [ s t a t . A P ] S e p Sea Clutter Distribution Modeling: A KernelDensity Estimation Approach
Hongkuan Zhou, Yuzhou Li, and Tao Jiang
School of Electronic Information and Communications, Huazhong University of Science and Technology, Wuhan, China{hongkuanzhou, yuzhouli, taojiang}@hust.edu.cn
Abstract —An accurate sea clutter distribution is crucial fordecision region determination when detecting sea-surface floatingtargets. However, traditional parametric models possibly havea considerable gap to the realistic distribution of sea cluttersdue to the volatile sea states. In this paper, we develop akernel density estimation based framework to model the seaclutter distributions without requiring any prior knowledge. Inthis framework, we jointly consider two embedded fundamentalproblems, the selection of a proper kernel density functionand the determination of its corresponding optimal bandwidth.Regarding these two problems, we adopt the Gaussian, Gamma,and Weibull distributions as the kernel functions, and derive theclosed-form optimal bandwidth equations for them. To deal withthe highly complicated equations for the three kernels, we furtherdesign a fast iterative bandwidth selection algorithm to solvethem. Experimental results show that, compared with existingmethods, our proposed approach can significantly decrease theerror incurred by sea clutter modeling (about two orders of mag-nitude reduction) and improve the target detection probability(up to in low false alarm rate cases).
I. I
NTRODUCTION
An important application for marine surveillance radar is todetect sea-surface small floating targets such as buoys, humandivers, and small boats [1]. When detecting, the received targetsignals at the radar are buried in the strong returned signalsreflected by the sea surface, referred to as sea clutters [2], [3].It is known that better detection performance is achieved ifprior knowledge of sea clutters’ distribution can be acquired,since by this a proper detection threshold can be determinedat the detector [4]–[6]. Therefore, an important question thatarises is how to accurately model the distribution of sea cluttersin the fluctuating sea state for small target detection.By adopting various parametric models [7]–[10], there havebeen extensive works attempting to characterize the distribu-tion of sea clutters to obtain better detection performance.In [7], the authors utilized the Gauss distribution to modelthe amplitude of the sea clutter at a low-resolution radar.With an increase in radar’s spatial resolution, the amplitudedistribution of the sea clutters was further extended from theGaussian to compound-Gaussian probability density functions(PDFs) in [8] for small target detection. Gao et al. in [9]
This work was supported in part by the National Science Foundationof China with Grant Numbers 61601192, 61631015, and 61729101, theYoung Elite Scientists Sponsorship Program by CAST with Grant Number2017QNRC001, the State Key Laboratory of Integrated Services Networks(Xidian University) with Grant Number ISN19-09, and the Fundamental Re-search Funds for the Central Universities with Grant Numbers 2016YXMS298and 2015ZDTD012. adopted the generalized Gamma distribution to describe thestatistical behaviors of sea clutters, and provided a parameterestimation scheme by taking both of estimation precision andapplicable conditions into consideration. In [10], the authorsadopted the Weibull model in the constant false alarm rate(CFAR) detector for radar detection and evaluated the involvedparameter optimization problem.To summarize, the distributions of sea clutters adoptedin [7]–[10] for target detection are generally established asparametric models. However, considering the two followingdefects of the parametric models, a considerable gap possiblyexists between the fitted and realistic distribution of sea clut-ters. Firstly, the parametric models can hardly depict the spikycomponents in sea clutters, which is produced when the high-resolution radar works at a low grazing angle or under dynamicsea states [11]. Secondly, as the distribution of sea cluttersusually varies with different detection environments, assuminga fixed parametric model for it cannot guarantee satisfactoryfitting performance in the varying detection environments andthus degrades the detection performance.Following these insights, the distribution of sea cluttersshould be characterized by sufficiently analyzing the collecteddata instead of assuming a parametric model. Inspired bythis, the kernel density estimation (KDE), a non-parametricapproach for estimating the PDF of a random variable, can beadopted to reveal the distribution of the collected sea clutters.Different from the methods in [7]–[10], the KDE method uti-lizes smooth kernel functions to fit the realistic distribution ofthe observed data without making any assumption on it, whichcan effectively reflect the information of the spiky componentsand flexibly adapt to the varying detection environments.When applying the KDE method, it is of utmost impor-tance to determine two key parameters, namely the kernelfunction and the bandwidth [12]. In [13], the Gaussian kerneland some traditional bandwidth selectors such as the plug-in were adopted in the KDE method, which show goodfitting performance on random sequence samples. However,few works have ever studied how the KDE method worksin the sea-surface target detection. In addition, whether thereare other kernel functions that can achieve better fittingperformance than the Gaussian kernel or not is still unclear.Furthermore, it is also quite challenging to derive the optimalbandwidth for other specialized kernels by the traditionallycomplicated bandwidth selection methods such as the plug-intechnique [13]. These challenges impose restrictions on thepplication of KDE method in estimating the distribution ofsea clutters.In view of these, this paper first develops a KDE-based seaclutter modeling framework that is suitable for different kernelfunctions. In this framework, two embedded fundamentalproblems, the selection of a proper kernel density function andthe determination of its corresponding optimal bandwidth, areneeded to be solved. Considering three kinds of kernels, i.e.,Gaussian, Gamma, and Weibull, we then derive their respectiveclosed-form optimal bandwidth equations and design a fastiterative bandwidth selection algorithm to solve them.The main contributions of this work are as follows: • We propose a KDE-based framework that enfolds ker-nel function selection and bandwidth optimization toprecisely model the sea clutter distribution. Comparedwith traditional parametric methods, this framework cannot only take the information of spiky components intoaccount but also adapt to varying detection environments. • Inspired by parametric sea clutter models, we selectthe Gaussian, Gamma, and Weibull distributions as thekernels in our proposed framework. Particularly, we de-rive closed-form equations of the optimal bandwidth forthese three kernels, which are unlikely to be deducedby adopting traditional bandwidth selection methods suchas the plug-in technique. Due to the high complexity insolving these derived equations, we further design a fastiterative bandwidth selection algorithm to calculate theoptimal bandwidth for each of kernels. • Experimental results exhibit that our proposed approachoutperforms the existing methods in terms of the mod-eling error (about two orders of magnitude reduction).Moreover, applying our modeled sea clutter distributioninto the CFAR detector can significantly improve thedetection probability, especially in low false alarm ratecases (up to 36%).II. S
YSTEM S CENARIOS AND P ROBLEM F ORMULATION
In this section, we first introduce the realistic IntelligentPIxel processing X-band (IPIX) radar datasets, and then for-mulate the asymptotic mean integrated square error (AMISE)minimization problem.
A. IPIX Datasets
In this paper, we adopt the IPIX database, an authoritativeand widely-used database collected at the east coast of Canadain November 1993, to model the distribution of sea clutters.As shown from a website held by Simon Haykin [14], thereare a total of 14 datasets in the collected database. Eachdataset includes 14 separate spatial range cells, with each cellcontaining a length of 131072 time sampling data. These cellscan be divided into three categories. Specifically, the cell withthe target signals is labeled as the primary cell, the adjacentcells affected by the target are labeled as the secondary cells,and the remaining cells are clutter-only cells. For notationalsimplicity, we denote the samples from the primary cells andclutter-only cells as target signals and sea clutters, respectively.
B. Statistical Sea Clutter Distributions
As is known to all, the amplitude of sea clutters usuallyfollows Gaussian distribution for a low-resolution radar. How-ever, it was soon found that the Gaussian distribution exhibitsa poor fitting performance as the spatial resolution of the radarincreases. Moreover, the amplitude distribution of sea cluttersat a high-resolution radar will demonstrate the characteristicsof the compound-Gaussian models. These models, e.g., theGamma and Weibull models, have been widely used forsea clutter distribution modeling and shown a better fittingperformance compared with the Gaussian distribution. In whatfollows, we present the PDFs and fitting performance of theabove distributions.
1) Gaussian distribution:
The PDF of the Gaussian distri-bution is given as f (cid:0) x | µ, σ (cid:1) = 1 √ πσ e − ( x − µ )22 σ (1)where µ is the expectation of the distribution, σ is the standarddeviation, and x denotes the amplitude of the clutter samples.
2) Gamma distribution:
The PDF of the Gamma distribu-tion is presented as f ( x | α, β ) = β α Γ ( α ) x α − e − βx (2)where α > and β > are the shape parameter and the rateparameter of the Gamma distribution, respectively. In addition, Γ ( α ) represents the complete gamma function.
3) Weibull distribution:
The PDF of the Weibull distribu-tion can be described as f ( x | c, s ) = ( sc (cid:0) xc (cid:1) s − e − ( x/c ) s x ≥ x < (3)where s > and c > are the shape parameter and the scaleparameter of the Weibull distribution, respectively.To obtain the best combination of parameters for the afore-mentioned three functions, we first utilize (1), (2), and (3) tomodel the distribution of sea clutter data based on the meansquared error (MSE) criterion. Under the best parameter set-tings, we then calculate their corresponding PDFs, which areplotted in Fig. 1. From the figure, the fitting performance of theGamma and Weibull distributions are better than the Gaussiandistribution in the case when the normalized amplitude is morethan 0. However, there still exists considerable bias betweenthe curves of statistical PDFs and the practical sea clutterPDF, especially in the cases of low normalized amplitude.This phenomenon implies that the distribution of sea cluttersshould be estimated by tracking and characterizing the instantchanges instead of traditionally assuming a parametric model. C. Problem Formulation
We firstly utilize the kernel density estimation, a nonpara-metric approach for estimating the probability density functionof a random variable, to model the distribution of sea clutters.By adopting the kernel function and bandwidth, the KDEapproach assigns a height curve to each observation point. x P D F Actual dataGaussian distributionGamma distributionWeibull distribution
Fig. 1. Comparison among the three PDFs produced by individually applyingthe Gaussian, Gamma, and Weibull distributions to estimate the actual PDFof the IPIX data, where σ = 1 . and µ = 0 . are selected for the Gaussiandistribution, α = 1 . and β = 0 . for the Gamma distribution, and c = 0 . and s = 2 for the Weibull distribution. Each curve needs to be normalized first and then summed upby the kernel estimator function to estimate the density of seaclutters. The expression of KDE can be written as follows [12] ˆ f ( x ) = 1 N h N X i =1 K (cid:18) x − x i h (cid:19) (4)where K is the kernel density function, h is the bandwidth ofthe KDE method, N is the number of sample points, and x i is the i -th sample point.Generally, the kernel density function and the bandwidth aretwo key factors that determine the estimation performance. Fora given kernel density function, there exists an optimal band-width that achieves the best estimation accuracy, and largeror smaller bandwidth will lead to worse fitting performance.In what follows, we denote f ( x ) as the density function ofsea clutters and demonstrate the procedure of the theoreticalderivation of the optimal bandwidth.To derive the expression of optimal bandwidth, we thenintroduce a useful criterion, referred to as the AMISE, toevaluate the fitting performance of the distribution estimation ˆ f ( x ) , given by [15]AMISE n ˆ f ( x ) o = 1 N h R ( K ) + 14 h µ ( K ) R ( f ′′ ) (5)where R ( K ) = R R K ( x ) dx , µ ( K ) = R R x K ( x ) dx , and f ′′ is the second derivative of the density f . Then the optimalbandwidth can be straightforward derived by an optimizationproblem as follows h AMISE = arg min h AMISE n ˆ f ( x ) o . (6)The solution of (6) is illustrated in the following theorem. Theorem 1.
For each kernel density function to estimate the unknown density f , there exists a general expression for theoptimal bandwidth, given by h opt = " R ( K ) µ ( K ) R ( f ′′ ) N / . (7) Proof:
Proof of this theorem can be found in [15].Theorem 1 indicates that R ( K ) , µ ( K ) , and R ( f ′′ ) shouldbe precalculated when determining the optimal bandwidth ofthe KDE method. Among these variables, although the R ( K ) and µ ( K ) can be easily derived if the kernel function isgiven, the value of R ( f ′′ ) is difficult to obtain as the density f is still unknown to us. To solve this problem, the estimationof R ( f ′′ ) is used to transform R ( f ′′ ) into an easy-to-calculateexpression, which will be illustrated in the next section.III. D ERIVATION OF O PTIMAL B ANDWIDTHS FOR D IFFERENT K ERNELS
In this section, we propose an analytical approach tosolve problem (6) in Section II-C, where the derived optimalbandwidth will be varied with different kinds of kernels.Interestingly, statistical models, e.g., Gaussian, Gamma, andWeibull distributions, usually can reveal the physical natureof sea clutters [1]. Inspired by this, we take the Gaussian,Gamma, and Weibull kernels as examples to evaluate the fittingperformance of the KDE-based method.
A. Gaussian Kernel Density Function
According to the Gaussian distribution (1), the kernel den-sity function K can be expressed as K ( x ) = 1 √ πσ e − ( x − µ )22 σ . (8)In order to derive the optimal bandwidth h opt in (7), R ( K ) and µ ( K ) should be derived in the first place, which will bequantified in the following lemma. Lemma 1.
For the Gaussian kernel, R ( K ) and µ ( K ) canbe expressed as R ( K ) = Z ∞−∞ K ( x ) dx = 12 √ πσ (9) µ ( K ) = Z ∞−∞ x K ( x ) dx = σ + µ . (10)Compared with R ( K ) and µ ( K ) , it is more difficult tocalculate R ( f ′′ ) due to the lack of the prior knowledge of theunknown density f . Our idea is to first reshape R ( f ′′ ) in aneasy-to-calculate form and then estimate it via the estimatorof the second derivative of the density. The following theoremquantifies the optimal bandwidth for the Gaussian kernel. Theorem 2.
The AMISE gets its minimum for the Gaussiankernel density function when h opt = ( σ + µ ) Q ( h opt ) √ πσ N , where Q ( h opt ) is a function of h opt .Proof: As concluded above, the AMISE gets its minimumwhen the optimal bandwidth h opt is adopted in (4). Substitut-ng h opt into ˆ f ( x ) , the optimal estimation distribution ˆ f opt ( x ) of an unknown density f is given by ˆ f opt ( x ) = 1 N h opt N X i =1 K (cid:18) x − x i h opt (cid:19) . (11)Based on (11), the m -order derivatives of ˆ f opt ( x ) with respectto x is calculated as [16] ˆ f ( m )opt ( x ) = 1 N h m +1opt N X i =1 K ( m ) (cid:18) x − x i h opt (cid:19) (12)where K ( m ) is the m -th derivative of the kernel K .To transform the integral of the squared second derivative of f , i.e., R ( f ′′ ) , into an easy-to-calculate expression, we utilizethe function ˆ f ( m )opt ( x ) in (12) to estimate it, given by R (cid:0) f ′′ (cid:1) ≈ Z (cid:16) ˆ f (2)opt ( x ) (cid:17) dx = 12 π (cid:0) Nh σ (cid:1) · Z N X i =1 (cid:18) ( z i − µ ) σ − (cid:19) exp (cid:18) − ( z i − µ ) σ (cid:19)! dx (13)where z i = x − x i h opt . (14)For notational simplicity, we set P i = ( z i − µ ) σ − ! exp − ( z i − µ ) σ ! . (15)Rearranging R ( f ′′ ) in (13) yields R ( f ′′ ) = 12 π (cid:0) N h σ (cid:1) Z N X i =1 P i ! dx. (16)Substituting the variable z i (14) and P i (15) into (16), it isclearly seen that (16) is related to h opt if the parameters σ , µ ,and N are fixed.Let Q ( h opt ) = Z N X i =1 P i ! dx. (17)Substituting (9), (10), (16), and (17) into (7), we obtain theoptimal bandwidth h opt based on Lemma 1, given by h opt = (cid:0) σ + µ (cid:1) Q ( h opt ) √ πσ N . (18)
B. Gamma Kernel Density Function
For the case of Gamma distribution, we first deduce the R ( K ) and µ ( K ) in the following lemma. Lemma 2.
For the Gamma kernel, R ( K ) and µ ( K ) can beexpressed as R ( K ) = (cid:18) β α Γ ( α ) (cid:19) Γ (2 α − β ) α − (19) TABLE IO
PTIMAL BANDWIDTH FOR DIFFERENT KERNEL DENSITY FUNCTIONS .Kernel functions Optimal bandwidthGaussian h opt = ( σ + µ ) Q ( h opt ) √ πσ N = ψ ( h opt ) Gamma h opt = (2 β ) α − Γ ( α +2) W ( h opt ) Γ(2 α − ( α ) N = ψ ( h opt ) Weibull h opt = − s Γ ( s +1 ) sV ( h opt ) Γ ( − s ) cN = ψ ( h opt ) µ ( K ) = Γ ( α +2)Γ ( α ) . (20)Then, we derive the h opt for the Gamma kernel basedon Lemma 2. Similar to the proof of Theorem 2, we firstutilize (12) to estimate R ( f ′′ ) for the Gamma kernel, and thenobtain W ( h opt ) and h opt (corresponding to Q ( h opt ) and h opt in (17) and (18), respectively). Related results are summarizedin the following theorem and we omit its proof for brevity. Theorem 3.
The AMISE gets its minimum for the Gammakernel density function when h opt = (2 β ) α − Γ ( α +2) W ( h opt )Γ(2 α − ( α ) N ,where W ( h opt ) = R (cid:16)P Ni =1 G i (cid:17) dx , z i = x − x i h opt , and G i = e − βz i (cid:26) ( α −
1) ( α − z α − i − β ( α − z α − i + β z α − i (cid:27) .C. Weibull Kernel Density Function As for Weibull kernel density function, we first deduce the R ( K ) and µ ( K ) in the following lemma. Lemma 3.
For the Weibull kernel, R ( K ) and µ ( K ) can beexpressed as R ( K ) = Γ (cid:0) − s (cid:1) s − s c (21) µ ( K ) = c Γ (cid:18) s + 1 (cid:19) . (22)Based on Lemma 3, we then quantify the optimal bandwidthfor the Weibull kernel in the following theorem, the proof ofwhich is similar to those of Theorem 2 and 3. Theorem 4.
The AMISE gets its minimum for the Weibullkernel density function when h opt = − s Γ ( s +1 ) sV ( h opt )Γ ( − s ) cN ,where V ( h opt ) = R (cid:16)P Ni =1 L i (cid:17) dx , L i = e − ( zic ) s (cid:0) z i c (cid:1) s − · (cid:26) ( s −
1) ( s − − s ( s − (cid:0) z i c (cid:1) s + s (cid:0) z i c (cid:1) s (cid:27) , z i = x − x i h opt . Remark 1.
Based on Theorems 2, 3, and 4, we have rig-orously derived the closed-form equations about the optimalbandwidths for the three different kernels with their involvedparameters in Table I. It is observed from the table that, theoptimal bandwidths are all determined once the parametersof the selected kernel density functions are given. Moreover,the expressions of optimal bandwidths are all in the form ofthe fixed-point equations since the Q ( h opt ) , W ( h opt ) , and V ( h opt ) are the functions of h opt . This phenomenon motivatesus to apply the fixed-point iterative algorithm to obtain h opt . lgorithm 1 Bandwidth selection algorithm. Initialization • Set the started approximation bandwidth h l, and thetolerance TOL. • Set the maximum number of iteration N max and theiteration index i = 1 . Denote ψ , ψ , and ψ as the optimal bandwidth functionsfor the Gaussian, Gamma, and Weibull kernels, which arereferred to in Table I. while ( i ≤ N max ) do Set h l = ψ l ( h l, ) , where l =
1, 2, or 3. if (cid:12)(cid:12)(cid:12) h l − h l, h l (cid:12)(cid:12)(cid:12) < TOL then Set H l = h l, . break. else Set i = i + 1 and h l, = h l . end if end while Output the optimal bandwidth H l . D. Algorithms for the Optimal Bandwidth Selection
In this subsection, we calculate the optimal bandwidths forthe three kernels based on equations in Table I. However,it is difficult to obtain a straightforward expression for theoptimal bandwidth as the Q ( h opt ) , W ( h opt ) , and V ( h opt ) are the functions of h opt , which are complicated and unsolv-able. Furthermore, traditional numerical analysis such as theNewton-Raphson method that is based on the derivation of thetarget equation also increases the computation complexity andreduces the efficiency of the KDE.In view of these, we adopt the fixed-point theory to ef-ficiently solve the equations in Table I, where the optimalbandwidths for the three kernels can be obtained by derivingthe fixed-points of the equations. To determine these fixed-points efficiently, we further design a fast iterative bandwidthselection algorithm, which is described in Algorithm 1.IV. E XPERIMENTAL R ESULTS AND A NALYSIS
In this section, we present experimental results to exhibitthe fitting and detection performance of our proposed methodusing the realistic IPIX radar datasets.Consider that the background sea clutter samples are neededto train the KDE model in Eq. (4), the clutter-only cells inIPIX radar dataset, e.g., the 14-th cell in dataset 17, are thusadopted to be served as training sets. Based on the trainingsets, we obtain the best combination of parameters of thethree kernel density functions under the MSE criterion, whichare as follows: σ = 1 . , µ = 0 . , α = 1 . , β = 0 . , c = 0 . , s = 2 , and N = 2048 . Then, we utilize Algorithm 1to search for the optimal bandwidths for different kernelfunctions. The experimental results show that our proposedalgorithm converges very fast (within 5 iterations) and theoptimal bandwidths are equal to 0.08, 0.05, and 0.06 for theGaussian, Gamma, and Weibull kernel functions, respectively. x P D F Gaussian kernelActual dataGamma kernelWeibull kernel
Fig. 2. Comparison among the three PDFs produced by individually applyingthe Gaussian, Gamma, and Weibull kernels to estimate the actual PDF of theIPIX data, where σ = 1 . , µ = 0 . , and h opt = 0 . are selected for theGaussian kernel, α = 1 . , β = 0 . , and h opt = 0 . for the Gamma kernel,and c = 0 . , s = 2 , and h opt = 0 . for the Weibull kernel. Based on the derived optimal bandwidths, we firstly com-pare the PDF fitting performance of our proposed KDEmethods under different kernels in Fig. 2. From the figure,it is obtained that, compared with the Gaussian kernel, thecurves of Gamma and Weibull kernels are much closer to thedistribution of sea clutters. This phenomenon indicates that,if the optimal bandwidths are adopted for the three kernels,the kernel density function itself will play the dominant role inestimation. Thus, it may be a better choice to select the Weibulland Gamma distributions rather than the Gaussian distributionas kernel functions.Secondly, we compare the complementary cumulative distri-bution function (CCDF) fitting performance of our proposedKDE methods for different kernels. As depicted in Fig. 3,the CCDF curves of our proposed KDE methods are muchcloser to the sea clutter than those of the traditional parametricmodels, especially in high normalized amplitude. Furthermore,the curves of the Gamma kernel and Weibull kernel almostoverlap the curve of sea clutter, except for the Gaussiankernel which shows more or less difference from the others.In particular, the obtained data shows that, compared withtraditional parametric models, the Gaussian, Gamma, andWeibull kernels can reduce the MSE from the magnitude of − to the magnitudes of − , − , and − , respectively.Therefore, the Gamma and Weibull kernels are more preferredwhen applying the KDE methods to detect sea-surface targets.Finally, we introduce the CFAR detector to test the detectionperformance ( P d ) of our proposed KDE method. The CFARdetector works in a two-step process, namely the training andtesting steps. In the training step, estimate the distribution ofthe clutter-only cell data and calculate the background levelof sea clutters. Then, determine the threshold at a given falsealarm rate ( P fa ) by multiplying the background level with a Normalized amplitude -2 -1 CCD F Actual dataGaussian kernelGamma kernelWeibull kernelGaussian distributionGamma distributionWeibull distribution
Fig. 3. Comparison among the six CCDFs produced by individually applyingthe Gaussian, Gamma, and Weibull kernels and the Gaussian, Gamma, andWeibull distributions to estimate the actual CCDF of the IPIX data, where theexperimental parameters are the same as those in Fig. 2. -3 -2 -1 False alarm rate, P fa D e t e c t i on p r obab ili t y , P d Gaussian kernelGamma kernelWeibull kernelGaussian distributionGamma distributionWeibull distribution
Fig. 4. Comparison of the detection probability between our proposedmethod and traditional parametric modeling methods. In this figure, the testingsamples are collected from dataset 17, and the CFAR detector is adopted fortarget detection. given factor T . In the testing step, compare the amplitudeof the testing samples with the obtained threshold to decidewhether they are targets or not. We refer the readers to [1] onmechanisms for the detection and we omit it for brevity.Fig. 4 depicts how the detection probability of our proposedmethod and traditional parametric methods varies with thefalse alarm rate. From the figure, it is obtained that althoughthe detection probabilities of these methods all increase withthe false alarm rate, our proposed KDE method alwaysachieves better detection performance than the others either inhigh or low false alarm rate cases. For example, our proposedWeibull kernel KDE method improves the P d by , ,and compared with the Weibull, Gamma, and Gaussian distribution based methods, respectively, when the P fa is 0.001.V. C ONCLUSIONS
In this paper, we have put forward a KDE-based seaclutter modeling framework that is suitable for different kernelfunctions. In this framework, we have firstly derived theclosed-form optimal bandwidth equations for the Gaussian,Gamma, and Weibull kernels and then designed a fast iterativebandwidth selection algorithm to solve them. Experimentalresults have exhibited that, compared with existing methods,our proposed approach can significantly decrease the error in-curred by sea clutter modeling (about two orders of magnitudereduction) and improve the target detection probability (up to in low false alarm rate cases).R
EFERENCES[1] W. Zhou, J. Xie, G. Li, and Y. Du, “Robust CFAR detector with weightedamplitude iteration in nonhomogeneous sea clutter,”
IEEE Trans. Aero.Elec. Sys. , vol. 53, no. 3, pp. 1520–1535, Jun. 2017.[2] G. Gao and G. Shi, “CFAR ship detection in nonhomogeneous sea clutterusing polarimetric SAR data based on the notch filter,”
IEEE Trans.Geosci. Remote , vol. 55, no. 8, pp. 4811–4824, Aug. 2017.[3] Y. Li, Y. Zhang, W. Li, and T. Jiang, “Marine wireless Big Data: Effi-cient transmission, related applications, and challenges,”
IEEE WirelessCommun. , vol. 25, no. 1, pp. 19–25, Feb. 2018.[4] S. Haykin and T. K. Bhattacharya, “Modular learning strategy for signaldetection in a nonstationary environment,”
IEEE Trans. Signal Process. ,vol. 45, no. 6, pp. 1619–1637, Jun. 1997.[5] Y. Li, T. Jiang, M. Sheng, and Y. Zhu, “QoS-aware admission controland resource allocation in underlay device-to-device spectrum-sharingnetworks,”
IEEE J. Sel. Areas Commun. , vol. 34, no. 11, pp. 2874–2886,Nov. 2016.[6] Y. Li, M. Sheng, Y. Sun, and Y. Shi, “Joint optimization of BS operation,user association, subcarrier assignment, and power allocation for energy-efficient HetNets,”
IEEE J. Sel. Areas Commun. , vol. 34, no. 12, pp.3339–3353, Dec. 2016.[7] F. T. Ulaby and M. C. Dobson,
Handbook of Radar Scattering Statisticsfor Terrain.
Norwood, MA: Artech House, 1989.[8] K. Ward, “Compound representation of high resolution sea clutter,”
Electron. Lett. , vol. 17, no. 16, pp. 561–563, Aug. 1981.[9] G. Gao, K. Ouyang, Y. Luo, S. Liang, and S. Zhou, “Scheme of param-eter estimation for generalized Gamma distribution and its applicationto ship detection in SAR images,”
IEEE Trans. Geosci. Remote , vol. 55,no. 3, pp. 1812–1832, Mar. 2017.[10] D. Schleher, “Radar detection in Weibull clutter,”
IEEE Trans. Aero.Elec. Sys. , vol. AES-12, no. 6, pp. 736–743, Nov. 1976.[11] Y. Wei, L. Guo, and J. Li, “Numerical simulation and analysis of thespiky sea clutter from the sea surface with breaking waves,”
IEEE Trans.Antenn. Propag. , vol. 63, no. 11, pp. 4983–4994, Nov. 2015.[12] A. Qahtan, S. Wang, and X. Zhang, “KDE-Track: An efficient dynamicdensity estimator for data streams,”
IEEE Trans. Knowl. Data En. ,vol. 29, no. 3, pp. 642–655, Mar. 2017.[13] M. C. Jones, “A brief survey of bandwidth selection for densityestimation,”
J. Am. Stat. Assoc. , vol. 91, no. 433, pp. 401–407, Jun.1996.[14] [Online]. Available: http://soma.ece.mcmaster.ca/ipix/.[15] B. W. Silverman,