Research Article Ship Detection in High-Resolution Dual ...downloads.hindawi.com/journals/ijap/2013/519296.pdfInternationalJournal of Antennas and Propagation Our goal is to present

Hindawi Publishing CorporationInternational Journal of Antennas and PropagationVolume 2013 Article ID 519296 5 pageshttpdxdoiorg1011552013519296

Research ArticleShip Detection in High-Resolution Dual-Polarization SARAmplitude Images

Gui Gao Gongtao Shi and Shilin Zhou

School of Electronics Science and Engineering National University of Defense Technology Changsha Hunan 410073 China

Correspondence should be addressed to Gui Gao dellar126com

Received 31 October 2012 Accepted 8 January 2013

Academic Editor Deren Li

Copyright copy 2013 Gui Gao et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A constant false alarm rate (CFAR) detecting method for ships in high-resolution dual-polarization synthetic aperture radar (SAR)amplitude images has been proposed in this paper First by the production of amplitude images from two polarimetric channelsa novel detector simply called the PMA detector has been constructed We testified that the PMA detector could improve thesignal-to-clutter ratio (SCR) and make the discrimination of a ship from clutter more easily Second the PMA detectorrsquos statisticalmodel has been described by the well-known G0 distribution when facing complex sea background The experiments performedon measured dual-polarization TerraSAR-X images demonstrate the good performance of the proposed CFAR detecting method

1 Introduction

Ship detection in high-resolution synthetic aperture radar(SAR) images has become an increasing interest during thepast several years [1] It plays an important role for variouspotential applications like marine surveillance vessel trafficcontrol military service and so forth [1ndash3] Additionallywith the development of sensor techniques the advancedpolarimetric SAR systems have also been validated that morecompleted description of target-scattering behavior could beprovided than traditional single-channel SAR systems suchas HH HV and VV SAR which is not sufficient for shipdetection [2 4] Since better discriminating target signalfrom the surrounding clutter can be obtained by polarimetricSAR systems ship detection in multipolarimetric or dual-polarimetric SAR images is receiving intensive attentions inpresent

Nowadays some famous contemporary satellite SARsystems for example TerraSAR-X COSMO-SkyMed andRadarSat-2 support dual-polarization acquisition mode [5]However the available data is mostly the amplitude dataand not the complex-valued data [6] It is an important datamodality because of the image products provided by manysatellite SAR systems for instance ERS Thus we mainly

focus on ship detection using only dual-pol SAR amplitudeimages in this paper

A wide variety of methods have been proposed fordetecting ships in polarimetric SAR images The basic idea isto reduce three channels of polarimetric SAR data to a singledecision criterion [7] Some popular detectors [7] includingthe optimal polarimetric detector (OPD) the polarimetricwhitening filter (PWF) the span detector the power max-imization synthesis (PMS) detector and more recently theidentity likelihood ratio test (ILRT) have been developedand shown that they can perform in a way that targets aremore easily discriminated from clutter Unfortunately otherdetectors mentioned earlier except for the span detector arevery difficult to implement due to the missing of phaseinformation when the only amplitude is considered [7]

The span detector is a widely used processor which isa noncoherent sum of all polarimetric channels and onlymakes use of image intensities or amplitudes It has alsobeen proven that this detector can acquire a better detectionperformance thanHH HV or VV individually Neverthelessthe expected adaptive detection threshold is impossible whenfacing complex clutter background by this detector whichlimits the applications of this detector

2 International Journal of Antennas and Propagation

Our goal is to present a powerful detector for shipdetection in high-resolution dual-polarization SAR ampli-tude images On one hand this detector can improve thesignal-to-clutter ratio (SCR) to enhance the moving targetsor restrain the clutter Meanwhile we wish that a flexible andadaptive constant false alarm rate (CFAR) threshold couldbe derived from this detector Under this consideration thispaper proposed a novel detector similarly with the spandetector simply called the product of multilook amplitudes(PMAs) detector

2 Dual-Polarization SAR Data Description

The fundamental quantities measured by a polarimetric SARare the complex scattering matrix with four elements whichin complete form can be expressed by [8]

S = [119878ℎℎ 119878ℎ119907119878119907ℎ

119878119907119907

] (1)

where 119878119901119902

is the scattering element with 119901 transmit and 119902receive polarizations 119901 and 119902 denote either horizontal (ℎ) orvertical (119907) polarization In a reciprocal medium the cross-polar elements of the scattering are identical that is 119878

ℎ119907=

119878119907ℎ In this matter the scattering matrix shown in (1) can be

organized into a column vector

X = [119878ℎℎ 119878ℎ119907 119878119907119907]T (2)

where the superscript ldquoTrdquo represents transposeThe equation(2) is called single-look complex scattering vector Basedon the coherent nature of SAR X follows a zero meanmultivariate complex Gaussian distribution The detaileddiscussion about this distribution can be found in [8] and[10]

21 Polarimetric Covariance Matrix for Dual-PolarizationCase For dual-polarization case the single look scatteringvector shown in (2) can be simplified to

u = [1198781 1198782]T (3)

Herein for convenience we use 1198781or 1198782to indicate one of

the scattering elements 119878ℎℎ 119878ℎ119907 and 119878

119907119907in any order as well

as 1198781= 1198782 Additionally to reduce the influence of speckle

SAR data are often multi-look averaged As the polarimetricinformation can also be represented by a covariance matrixthe 119899-look sample covariance matrix is defined as [8 9]

R = 1119899

119899

sum

119896=1

u (119896) u(119896)H = 1119899

119899

sum

119896=1

[

1003816100381610038161003816

1198781(119896)

1003816100381610038161003816

21198781(119896) 1198782(119896)

lowast

1198781(119896)

lowast1198782(119896)

1003816100381610038161003816

1198782(119896)

1003816100381610038161003816

2 ]

(4)

where the superscript lowast means complex conjugate and Hrepresents conjugate complex transpose 119899 is the numberof looks and u(119896) = [119878

1(119896) 1198782(119896)]

T is the 119896th single-look image Assuming statistical ergodicity and constant RCS

background the random matrix R known as the complexWishart distribution [10] is with probability density

119901R (R) =119899

2119899 det (R)119899minus2 exp [minus119899Tr (Cminus1R)]120587Γ (119899) Γ (119899 minus 1) det (C)119899

(5)

where Γ(sdot) is the gamma function and Tr(sdot) indicates thematrix trace The symbol det(sdot) denotes the determinantoperator and the covariance matrix is 2 times 2 complexHermitian written as

C = 119864 [uuH]

=[

[

119864 (

1003816100381610038161003816

1198781

1003816100381610038161003816

2) radic119864 (

1003816100381610038161003816

1198781

1003816100381610038161003816

2) 119864 (

1003816100381610038161003816

1198782

1003816100381610038161003816

2)120588e119895120579

radic119864(

1003816100381610038161003816

1198781

1003816100381610038161003816

2) 119864 (

1003816100381610038161003816

1198782

1003816100381610038161003816

2)120588eminus119895120579 119864 (

1003816100381610038161003816

1198782

1003816100381610038161003816

2)

]

]

(6)

where 120588e119895120579 is the complex correlation coefficient of twocomponents in (3)

22 The Joint Distribution of Two Multilook IntensitiesfromDifferent Polarimetric Channels When only incompletepolarimetric data are available for instance the amplitudeor intensity of copolarized components (ie HH and VV)the joint distribution of intensity or amplitude from twocorrelated polarimetric channels is of importance for con-structing signal processing algorithms such as detection andclassification in this case Lee et al [8] have derived the PDFofjoint normalized multilook intensities by integrating (5) withrespect to the two off-diagonal elements which is modeled as

11990111987711198772

(1198771 1198772)

=

119899

119899+1(11987711198772)

(119899minus1)2 exp (minus119899 (1198771+ 1198772) (1 minus 120588

2))

Γ (119899) (1 minus 120588

2) 120588

119899minus1

times 119868119899minus1(2119899radic119877

11198772

120588

1 minus 120588

2) 119877

1 1198772 119899 gt 0 0 lt 120588 lt 1

(7)

where 119868119899minus1(sdot) is the first type modified Bessel function of

order 119899 minus 1 119877119894 119894 = 1 2 represents normalized multi-look

intensity of 119894th polarimetric channel with the expression 119877119894=

(1119899)sum

119899

119896=1(|119878119894(119896)|

2119864(|119878119894|

2))

Furthermore to facilitate the posterior derivation by thetransforms with the following forms

1198611= 1198991198771

1198612= 1198991198772

(8)

one can obtain the joint density of 1198611and 119861

2as (see [8]

for details)

11990111986111198612

(1198611 1198612)

=

(11986111198612)

(119899minus1)2 exp (minus (1198611+ 1198612) (1 minus 120588

2))

Γ (119899) (1 minus 120588

2) 120588

119899minus1


11198612

120588

1 minus 120588

2) 119861

1 1198612 119899 gt 0 0 lt 120588 lt 1

(9)

International Journal of Antennas and Propagation 3

3 The PMA Detector

31 PMA Detector In a single-channel SAR image it is usu-ally assumed that strong backscattering comes from targetsA target point will be lost when the backscattering amplitudeor intensity from the target is not large enough comparedwith the clutter background generally due to low signal-to-clutter ratio (SCR) In other words SCR is an essential factorinfluencing the detection performance when only amplitudeor intensity data are available Thus the principle designinga good detector should naturally enable SCR to be improvedthat is enhancing target and restraining clutter

As we know the span detector is a widely used processorwhich is a noncoherent sum of all polarimetric channels andonly makes use of image intensities For dual-polarizationmultilook case the span is given by [7]

span = 1119899

119899

sum

119896=1

1003816100381610038161003816

1198781 (119896)

1003816100381610038161003816

2+

1

119899

119899

sum

119896=1

1003816100381610038161003816

1198782 (119896)

1003816100381610038161003816

2 (10)

This detector can be regarded as the synthetic power of allchannels Consequently some investigations have shown thata lower noise level and a higher SCR can be obtained by thisdetector than HH HV or VV individually This conclusionimplies that the way of synthetic power can arrive at thepurpose of improving SCR so that the targets can be moreeasily discriminated from the clutter comparedwith that onlyarbitrary single-channel information is used Meanwhile it isalso very hard to adaptively give a proper detection thresholdby the span detector due to the unknown knowledge of thecorresponding statistics Motivated by these considerationswe construct a detector in this paper by means of anothersynthetic power that is the product of multilook amplitudesfrom twopolarimetric channels for convenience called PMAdetector which can be defined as

120585 = 12059011205902radic11987711198772= 12059011205902

radic11986111198612

119899

(11)

where 120590119894= 119864(|119878

119894|

2) From an intuitive understanding in the

variable 120585 domain for the targets their amplitudes of twopolarimetric channels are both larger than the surroundingclutter which results in a much faster cumulative speed ofpower for targets verse clutter by multiplying the first imageamplitude by the second image one and hence targetsrsquovalues of 120585 are much more prominent and target signal canbe enhanced

32The CFAR Algorithm of PMADetector For dual-pol SARamplitude data based on the multiplicative model and theassumption that the energy of two channels is balancedthe product of multilook amplitudes from two polarimetricchannels can be denoted as

120577 = radic

1

119899

119899

sum

119896=1

1003816100381610038161003816

11986011198831(119896)

1003816100381610038161003816

2 1

119899

119899

sum

119896=1

1003816100381610038161003816

11986021198832(119896)

1003816100381610038161003816

2

= radic119860

2

1119860

2

2

1

119899

119899

sum

119896=1

1003816100381610038161003816

1198831(119896)

1003816100381610038161003816

2 1

119899

119899

sum

119896=1

1003816100381610038161003816

1198832(119896)

1003816100381610038161003816

2= 11986011198602120585

(12)

where 119860119894represents the backscattering RCS amplitude

component of 119894th receiving polarimetric channel and119883119894(119896) = |119878

119894(119896)|

2119864(|119878119894|

2) Recently Frery et al [11] have

proposed a well-known G0 distribution to model the clutterregions in SAR images and the product of multilook ampli-tudes from two polarimetric channels that is the proposeddetector in this paper employs an intensity expression of thisdistribution with three distinct parameters 120590 119899 and 120572 whenapplying to the modeling of sea background That is

119901120577(120577) =

120590

119899

119861 (119899 minus120572)

120577

119899minus1

(1 + 120590120577)

119899minus120572 120590 119899 minus120572 120577 gt 0 (13)

where 119861(sdot sdot) is the beta function The estimates and corresponded respectively to the parameters 120572 120590 and 119899can be easily obtained with the help of numerical calculationbased on the method of log-cumulants (MoLC) [12] as

Ψ () minus Ψ (minus) minus ln () = 1

119873

119873

sum

119894=1

[ln (119909119894)] =

1

Ψ (1 ) + Ψ (1 minus) =

1

119873

119873

sum

119894=1

[(ln (119909119894) minus

1)

2

]

Ψ (2 ) minus Ψ (2 minus) =

1

119873

119873

sum

119894=1

[(ln (119909119894) minus)

3]

(14)

where Ψ(sdot) represents the digamma function (ie the loga-rithmic derivative of the gamma function) Ψ(119903 sdot) is the 119903thorder polygamma function (ie the 119896th order derivative ofthe digamma function) and 119909

119894 119894 isin [1119873] is a given sample

setGiven the density shown in (13) its cumulative distribu-

tion function (CDF) is written as [11]

119865120577 (119909) =

120590

119899119909

119899

119899119861 (119899 minus120572)

21198651 (119899 minus 120572 119899 119899 + 1 minus120590119909) (15)

where21198651(sdot sdot sdot sdot) is the Gauss hypergeometric function For a

given value of the false alarm probability denoted by 119875fa thecorresponding CFAR threshold 119879 for the distribution shownin (13) can be obtained from

1 minus 119875fa = 119865120577 (119879) =120590

119899119879

119899

119899119861 (119899 minus120572)

21198651(119899 minus 120572 119899 119899 + 1 minus120590119879)

(16)

Considering that 119865120577(119879) is strictly monotonously increas-

ing the threshold 119879 can be accurately calculated via thenumerical solution or a simple bisection method

4 Experimental Results and Analysis

The test dual-pol SAR amplitude data used in this studyare a large TerraSAR-X StripMap mode geocoded sceneover Nanjing China acquired with high-resolution6mtimes 6m (azimuthtimes range) and HH-polarization andVV-polarization Figure 1(a) provides a fake-color image of


(a) (b)

Figure 1 The image of Nanjing (a) TerraSAR-X image (b) SPOT5 optical image

50 100 150 200 250 300

50

100

150

200

250

300

(a)

0 05 1 15 2 25 30

02

04

06

08

1

12

14

16

18

The product of two-channel amplitude

PDF

HistogramThe model fits

(b)

Figure 2 The ship chip (a) the product image (b) the fitting result

this scene The horizontal and vertical axes are the directionsof azimuth and range respectively Meanwhile in orderto make a visible comparison Figure 1(b) gives an optical(ground truth) remote sensing photograph of the test sitefrom SPOT5 satellite

The red rectangle box region shown in Figure 1(a) con-sisting of several ships and sea clutter is our investigatingarea The product image of multilook amplitudes from twopolarimetric channels is shown in Figure 2(a) Figure 2(b)shows the fitting results of the distribution in (13) for theproduct image of the area indicated in the rectangle box inFigure 2(a) The parameters 119899 120572 and 120590 are estimated to =1005744 = minus17851 and = 1207748 respectively As

shown in Figure 2(b) it is clear that the fitting result agreeswell with the theoretical distribution

Furthermore giving the theoretical false alarm probabil-ity119875fa = 10

minus8 the detection results are shown in Figure 3 It iseasy to observe that all ships are detected whilst a false alarmoccurs which proves the effectiveness of CFAR detectionmethod based on PMA detector for ships

5 Conclusion

Aiming at the adaptive detection of a ship when onlyhigh-resolution dual-polarization SAR amplitude data are


50 100 150 200 250 300

50

100

150

200

250

300

Figure 3 The ship detecting result

available a CFAR detecting method has been proposed inthis paper We first design a novel PMA detector whichcan improve the signal-to-clutter ratio (SCR) and make thediscrimination of a ship from clutter more easily Meanwhilethe PMA detectorrsquos statistical model has been describedby the well-known G0 distribution when facing complexsea background The experiments performed on measureddual-polarization TerraSAR-X images demonstrate the goodperformance of the proposed CFAR detecting method

Acknowledgment

The author would like to appreciate the National NaturalScience Foundation of China for the support under Grant no41171316

References

[1] D J Crisp The State-of-the-Art in Ship Detection in Syn-thetic Aperture Radar Imagery DSTO Departement DefenceAustralian Government Canberra Australia Public ReleaseDocument DSTO-RR-0272 2004

[2] C J Oliver and S Quegan Understanding Synthetic ApertureRadar Images Artech House Norwood Mass USA 1998

[3] I C Sikaneta Detection of ground moving objects with syntheticaperture radar [PhD thesis] University of Ottawa 2002

[4] F T Ulaby and C Elachi Radar Polarimetric for GeoscienceApplication Artech House Boston Mass USA 1990

[5] C E Livingstone and A A Thompson ldquoThe moving objectdetection experiment on RADARSAT-2rdquo Canadian Journal ofRemote Sensing vol 30 no 3 pp 355ndash368 2004

[6] V A Krylov GMoser S B Serpico and J Zerubia ldquoSupervisedhigh-resolution dual-polarization SAR image classification byfinite mixtures and copulasrdquo IEEE Journal on Selected Topics inSignal Processing vol 5 no 3 pp 554ndash566 2011

[7] RDChaneyMC Bud andLMNovak ldquoOn the performanceof polarimetric target detection algorithmsrdquo IEEE Aerospaceand Electronic Systems Magazine vol 5 no 11 pp 10ndash15 1990

[8] J S Lee K W Hoppel S A Mango and A R MillerldquoIntensity and phase statistics of multilook polarimetric and

interferometric SAR imageryrdquo IEEE Transactions on Geoscienceand Remote Sensing vol 32 no 5 pp 1017ndash1028 1994

[9] C H Gierull ldquoStatistics of SAR interferograms with applicationto moving target detectionrdquo Tech Rep 2001-045 DefenseResearch Establishment Ottawa Department of NationalDefense Ottawa Canada 2001

[10] N R Goodman ldquoStatistical analysis based on a certainmultivariate complex gaussian distribution (an introduction)rdquoAnnals of Mathematical Statistics vol 34 no 152 pp 152ndash1801963

[11] A C Frery J Jacobo-Berlles J Gambini and M E MejailldquoPolarimetric SAR image segmentation with B-splines and anew statistical modelrdquo Multidimensional Systems and SignalProcessing vol 21 no 4 pp 319ndash342 2010

[12] J M Nicolas ldquoIntroduction to second kind statistic applicationof log-moments and log-cumulants to SAR image law analysisrdquoTraitement du Signal vol 19 no 3 pp 139ndash167 2002

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Our goal is to present a powerful detector for shipdetection in high-resolution dual-polarization SAR ampli-tude images On one hand this detector can improve thesignal-to-clutter ratio (SCR) to enhance the moving targetsor restrain the clutter Meanwhile we wish that a flexible andadaptive constant false alarm rate (CFAR) threshold couldbe derived from this detector Under this consideration thispaper proposed a novel detector similarly with the spandetector simply called the product of multilook amplitudes(PMAs) detector

2 Dual-Polarization SAR Data Description

The fundamental quantities measured by a polarimetric SARare the complex scattering matrix with four elements whichin complete form can be expressed by [8]

S = [119878ℎℎ 119878ℎ119907119878119907ℎ

119878119907119907

] (1)

where 119878119901119902

is the scattering element with 119901 transmit and 119902receive polarizations 119901 and 119902 denote either horizontal (ℎ) orvertical (119907) polarization In a reciprocal medium the cross-polar elements of the scattering are identical that is 119878

ℎ119907=

119878119907ℎ In this matter the scattering matrix shown in (1) can be

organized into a column vector

X = [119878ℎℎ 119878ℎ119907 119878119907119907]T (2)

where the superscript ldquoTrdquo represents transposeThe equation(2) is called single-look complex scattering vector Basedon the coherent nature of SAR X follows a zero meanmultivariate complex Gaussian distribution The detaileddiscussion about this distribution can be found in [8] and[10]

21 Polarimetric Covariance Matrix for Dual-PolarizationCase For dual-polarization case the single look scatteringvector shown in (2) can be simplified to

u = [1198781 1198782]T (3)

Herein for convenience we use 1198781or 1198782to indicate one of

the scattering elements 119878ℎℎ 119878ℎ119907 and 119878

119907119907in any order as well

as 1198781= 1198782 Additionally to reduce the influence of speckle

SAR data are often multi-look averaged As the polarimetricinformation can also be represented by a covariance matrixthe 119899-look sample covariance matrix is defined as [8 9]

R = 1119899

119899

sum

119896=1

u (119896) u(119896)H = 1119899

119899

sum

119896=1

[

1003816100381610038161003816

1198781(119896)

1003816100381610038161003816

21198781(119896) 1198782(119896)

lowast

1198781(119896)

lowast1198782(119896)

1003816100381610038161003816

1198782(119896)

1003816100381610038161003816

2 ]

(4)

where the superscript lowast means complex conjugate and Hrepresents conjugate complex transpose 119899 is the numberof looks and u(119896) = [119878

1(119896) 1198782(119896)]

T is the 119896th single-look image Assuming statistical ergodicity and constant RCS

background the random matrix R known as the complexWishart distribution [10] is with probability density

119901R (R) =119899

2119899 det (R)119899minus2 exp [minus119899Tr (Cminus1R)]120587Γ (119899) Γ (119899 minus 1) det (C)119899

(5)

where Γ(sdot) is the gamma function and Tr(sdot) indicates thematrix trace The symbol det(sdot) denotes the determinantoperator and the covariance matrix is 2 times 2 complexHermitian written as

C = 119864 [uuH]

=[

[

119864 (

1003816100381610038161003816

1198781

1003816100381610038161003816

2) radic119864 (

1003816100381610038161003816

1198781

1003816100381610038161003816

2) 119864 (

1003816100381610038161003816

1198782

1003816100381610038161003816

2)120588e119895120579

radic119864(

1003816100381610038161003816

1198781

1003816100381610038161003816

2) 119864 (

1003816100381610038161003816

1198782

1003816100381610038161003816

2)120588eminus119895120579 119864 (

1003816100381610038161003816

1198782

1003816100381610038161003816

2)

]

]

(6)

where 120588e119895120579 is the complex correlation coefficient of twocomponents in (3)

22 The Joint Distribution of Two Multilook IntensitiesfromDifferent Polarimetric Channels When only incompletepolarimetric data are available for instance the amplitudeor intensity of copolarized components (ie HH and VV)the joint distribution of intensity or amplitude from twocorrelated polarimetric channels is of importance for con-structing signal processing algorithms such as detection andclassification in this case Lee et al [8] have derived the PDFofjoint normalized multilook intensities by integrating (5) withrespect to the two off-diagonal elements which is modeled as

11990111987711198772

(1198771 1198772)

=

119899

119899+1(11987711198772)

(119899minus1)2 exp (minus119899 (1198771+ 1198772) (1 minus 120588

2))

Γ (119899) (1 minus 120588

2) 120588

119899minus1


11198772

120588

1 minus 120588

2) 119877

1 1198772 119899 gt 0 0 lt 120588 lt 1

(7)

where 119868119899minus1(sdot) is the first type modified Bessel function of

order 119899 minus 1 119877119894 119894 = 1 2 represents normalized multi-look

intensity of 119894th polarimetric channel with the expression 119877119894=

(1119899)sum

119899

119896=1(|119878119894(119896)|

2119864(|119878119894|

2))

Furthermore to facilitate the posterior derivation by thetransforms with the following forms

1198611= 1198991198771

1198612= 1198991198772

(8)

one can obtain the joint density of 1198611and 119861

2as (see [8]

for details)

11990111986111198612

(1198611 1198612)

=

(11986111198612)

(119899minus1)2 exp (minus (1198611+ 1198612) (1 minus 120588

2))

Γ (119899) (1 minus 120588

2) 120588

119899minus1


11198612

120588

1 minus 120588

2) 119861

1 1198612 119899 gt 0 0 lt 120588 lt 1

(9)


3 The PMA Detector



span = 1119899

119899

sum

119896=1

1003816100381610038161003816

1198781 (119896)

1003816100381610038161003816

2+

1

119899

119899

sum

119896=1

1003816100381610038161003816

1198782 (119896)

1003816100381610038161003816

2 (10)


120585 = 12059011205902radic11987711198772= 12059011205902

radic11986111198612

119899

(11)

where 120590119894= 119864(|119878

119894|




120577 = radic

1

119899

119899

sum

119896=1

1003816100381610038161003816

11986011198831(119896)

1003816100381610038161003816

2 1

119899

119899

sum

119896=1

1003816100381610038161003816

11986021198832(119896)

1003816100381610038161003816

2

= radic119860

2

1119860

2

2

1

119899

119899

sum

119896=1

1003816100381610038161003816

1198831(119896)

1003816100381610038161003816

2 1

119899

119899

sum

119896=1

1003816100381610038161003816

1198832(119896)

1003816100381610038161003816

2= 11986011198602120585

(12)



119894(119896)|

2119864(|119878119894|



119901120577(120577) =

120590

119899

119861 (119899 minus120572)

120577

119899minus1

(1 + 120590120577)

119899minus120572 120590 119899 minus120572 120577 gt 0 (13)



119873

119873

sum

119894=1

[ln (119909119894)] =

1

Ψ (1 ) + Ψ (1 minus) =

1

119873

119873

sum

119894=1

[(ln (119909119894) minus

1)

2

]


1

119873

119873

sum

119894=1

[(ln (119909119894) minus)

3]

(14)





119865120577 (119909) =

120590

119899119909

119899

119899119861 (119899 minus120572)

21198651 (119899 minus 120572 119899 119899 + 1 minus120590119909) (15)



1 minus 119875fa = 119865120577 (119879) =120590

119899119879

119899

119899119861 (119899 minus120572)

21198651(119899 minus 120572 119899 119899 + 1 minus120590119879)

(16)






(a) (b)


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(b)







5 Conclusion



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Acknowledgment


References
















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










3 The PMA Detector



span = 1119899

119899

sum

119896=1

1003816100381610038161003816

1198781 (119896)

1003816100381610038161003816

2+

1

119899

119899

sum

119896=1

1003816100381610038161003816

1198782 (119896)

1003816100381610038161003816

2 (10)


120585 = 12059011205902radic11987711198772= 12059011205902

radic11986111198612

119899

(11)

where 120590119894= 119864(|119878

119894|




120577 = radic

1

119899

119899

sum

119896=1

1003816100381610038161003816

11986011198831(119896)

1003816100381610038161003816

2 1

119899

119899

sum

119896=1

1003816100381610038161003816

11986021198832(119896)

1003816100381610038161003816

2

= radic119860

2

1119860

2

2

1

119899

119899

sum

119896=1

1003816100381610038161003816

1198831(119896)

1003816100381610038161003816

2 1

119899

119899

sum

119896=1

1003816100381610038161003816

1198832(119896)

1003816100381610038161003816

2= 11986011198602120585

(12)



119894(119896)|

2119864(|119878119894|



119901120577(120577) =

120590

119899

119861 (119899 minus120572)

120577

119899minus1

(1 + 120590120577)

119899minus120572 120590 119899 minus120572 120577 gt 0 (13)



119873

119873

sum

119894=1

[ln (119909119894)] =

1

Ψ (1 ) + Ψ (1 minus) =

1

119873

119873

sum

119894=1

[(ln (119909119894) minus

1)

2

]


1

119873

119873

sum

119894=1

[(ln (119909119894) minus)

3]

(14)





119865120577 (119909) =

120590

119899119909

119899

119899119861 (119899 minus120572)

21198651 (119899 minus 120572 119899 119899 + 1 minus120590119909) (15)



1 minus 119875fa = 119865120577 (119879) =120590

119899119879

119899

119899119861 (119899 minus120572)

21198651(119899 minus 120572 119899 119899 + 1 minus120590119879)

(16)






(a) (b)


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100

150

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250

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(a)

0 05 1 15 2 25 30

02

04

06

08

1

12

14

16

18


PDF


(b)







5 Conclusion



50 100 150 200 250 300

50

100

150

200

250

300



Acknowledgment


References
















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










(a) (b)


50 100 150 200 250 300

50

100

150

200

250

300

(a)

0 05 1 15 2 25 30

02

04

06

08

1

12

14

16

18


PDF


(b)







5 Conclusion



50 100 150 200 250 300

50

100

150

200

250

300



Acknowledgment


References
















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










50 100 150 200 250 300

50

100

150

200

250

300



Acknowledgment


References
















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Documents

Research Article Ship Detection in High-Resolution Dual ...downloads.hindawi.com/journals/ijap/2013/519296.pdfInternationalJournal of Antennas and Propagation Our goal is to present