Hindawi Publishing CorporationInternational Journal of Antennas and PropagationVolume 2013 Article ID 109145 8 pageshttpdxdoiorg1011552013109145

Research ArticleCharacterizing the Statistical Properties of SAR Clutter byUsing an Empirical Distribution

Gui Gao Gongtao Shi Huanxin Zou 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 3 November 2012 Accepted 30 November 2012

Academic Editor Deren Li

Copyright copy 2013 Gui Gao et alis 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

e performances on the applications of synthetic aperture radar (SAR) data strongly depend on the statistical characteristics ofthe pixel amplitudes or intensities In this paper a new empirical model called simply ℋ119970119970119900119900 has been proposed to characterizethe statistical properties of SAR clutter data over the wide range of homogeneous heterogeneous and extremely heterogeneousreturns of terrain classes A particular case of theℋ119970119970119900119900distribution is the well-known119970119970119900119900 distributions We also derived analyticallythe estimators of the presentedℋ119970119970119900119900 model by applying the ldquomethod of log cumulantsrdquo (MoLCs)e performance of the proposedmodel is veried by using some measured SAR images

1 Introduction

Nowadays synthetic aperture radar (SAR) has become anadvanced tool as compared to optical sensors for monitoringland or sea surfaces because of its advantages regardless ofweather conditions [1 2] e interpretation of statisticalproperties of SAR clutter with various terrain classes is acrucial task for designing ltering [3] detection [4] segmen-tation [5] and classication [6ndash8] of algorithms to optimallyexploit the contents of the processed SAR images eseanalyses result in a growing interest in developing precisemodels for the statistics of the pixel amplitudes or intensities

It is known that SAR images are strongly affected byspeckle noise due to its coherent imaging mechanism [1 2 910]is noise effect degrades the information of SAR imagesand limits the surveillance performances To overcome thisdrawback much work has been done to match amplitudeor intensity statistics [3 5 6 8 11] Among the existingstatistical models the parametric distributions have beenintensively investigated because of their high accuracy andexibility which are usually obtained by two approaches

e rst is to consider the physical mechanism of thebackscattering from the land surfaces e multiplicativemodel [1 2] by combining the speckle noise and the terrainbackscatter is commonly used in this class Generally multi-look SAR speckle noise intensity is assumed to obey a gamma

distribution erefore a central task for establishing thistype of statistical models is to aim at the backscatter mod-eling of different terrain classes [1 2 11] Two importantdistributions consisting of 119974119974 as well as 1199701199700 which in turnin relation to the Gamma and inverse Gamma distributionfor the intensity backscatter have received a great deal ofattention [2 5ndash7 9 11 12] As compared to 119974119974 1199701199700 agreesreasonably better with the heavy tail behavior coming fromthe extremely heterogeneous clutter like the cases of urbanareas or other man-made structures [11]

e second is to conduct the distributions from a purelymathematics view irrespective with physical property ofradar clutter backscatter Some known examples are thelognormal weibull and more recently the Fisher [6] (com-pletely identical to the 1199701199700) distributions e lognormal andweibull t sometimes well with the SAR histogram of someheterogeneous and ocean regions [9] However they tend tooccur a large deviation estimating the histograms with theheavy tail behavior

is paper is devoted to report an empirical model(denoted simply as ℋ1199701199700) for characterizing the statisticalproperties of SAR clutter data to obtain the modeling abilityof more heterogeneous clutter e proposed model hasthe 1199701199700 distribution as a special case Furthermore usingthe second-kind statistics theory developed by Nicolas [13]

2 International Journal of Antennas and Propagation

which relies on the Mellin transform [14] that is ldquomethod-of-log-cumulantsrdquo (MoLC) we derive the parameter estima-tors of the new distribution model

In the rest of this paper the proposed distribution isrst given in Section 2 Section 3 derives the correspondingparameter estimators based on the MoLC We provide theexperimental results of theℋ1199701199700 model using measured SARdata in Section 4 the comparisons with that 1199701199700 ts are alsodiscussed in this sectione last section concludes this paperand give a perspective in the future work

2 The Proposed Distribution

e proposed amplitude distribution is dened as

119901119901119885119885119860119860(119911119911) = minus

2120590120590120590120590119907119907119907119907119899119899119899119899Γ ((119907119907 minus 119899119899) 119907119907)119907119907Γ (119907119907119907119907) Γ (119899119899)

times 1199111199112119899119899minus11989910076501007650120590120590 120582 119899119899119911119911minus211990711990710076661007666(119899119899minus119907119907)119907119907

minus119907119907 minus119907119907 120590120590 120590120590 119899119899 119911119911 120572 0(1)

where 120590120590 119907119907 and 120590120590 are the power shape scale parametersrespectively 119907119907 indicates the stretching parameter Γ(sdot) rep-resents the Gamma function 119899119899 is the number of looks ecorresponding intensity expression of (1) is further given by

119901119901119885119885119868119868(119911119911) = minus

120590120590120590120590119907119907119907119907119899119899119899119899Γ ((119907119907 minus 119899119899) 119907119907)119907119907Γ (119907119907119907119907) Γ (119899119899)

times 119911119911119899119899minus11989910076491007649120590120590 120582 119899119899119911119911minus11990711990710076651007665(119899119899minus119907119907)119907119907 minus119907119907 minus119907119907 120590120590 120590120590 119899119899 119911119911 120572 0120572(2)

We refer to this distribution characterized by (1) or(2) as the ℋ1199701199700 distribution Specically we call the ℋ119970119970119900119900

119868119868distribution and the ℋ119970119970119900119900

119860119860 distribution correspond to (2)and (1) respectively to distinct the intensity statistic as wellas the amplitude statistic Figure 1 gives some examples oftheℋ119970119970119900119900

119860119860 distribution with respect to the various parametersFrom this gure it can be seen clearly that the parameter119907119907 reects the degree of homogeneity for the tested returnswhich implies that the smaller value of |119907119907| obtains the morein-homogeneous (ie larger tails) they are Meanwhile 119907119907corresponds to a stretching of the SAR image amplitudeor intensity showing a strong effect for low values of thereturn Moreover as an independent parameter 120590120590 indicatesthe whole uctuation (contains magnifying or shrinking) ofpower of densities along the vertical axis e parameter 120590120590controls the peak value of the density

Although the probability density functions (PDFs) char-acterized by (1) or (2) are shown as the empirical models theℋ1199701199700 model also can be derived within the structure of themultiplicative model [1 2 11] Herein we account for theintensity distribution as an example the following theoremis established

eorem 1 Letting 119883119883119868119868 and 119884119884119868119868 indicates the backscatteringRCS component and speckle noise one respectively 119885119885119868119868 denotesthe observed intensity of SAR data erefore the relationship

of this three variables is expressed by the multiplicative modelas

119885119885119868119868 = 119883119883119868119868 sdot 119884119884119868119868120572 (3)

If119883119883119868119868 obeys

119901119901119883119883119868119868(119909119909) =

120590120590120590120590119907119907119907119907

Γ (119907119907119907119907)119909119909119907119907minus119899 exp 10076491007649minus12059012059011990911990911990711990710076651007665 minus119907119907 minus119907119907 120590120590 120590120590 119909119909 120572 0

(4)

and the PDF of 119884119884119868119868 is

1199011199011198841198841198681198681007649100764911991011991010076651007665 =

119899119899119899119899

Γ (119899119899)119910119910119899119899minus119899 exp 10076491007649minus119899119899119910119910minus11990711990710076651007665 119910119910 119899119899 minus119907119907 120572 0120572 (5)

en the distribution of the intensity return is characterizedby the density shown in (2) that is 119885119885119868119868 sim ℋ119970119970119900119900

119868119868(119907119907 119907119907 120590120590 120590120590 119899119899)

Proof Combining (4) and (5) via (3) the PDF of 119885119885119868119868 can beeasily derived as

119901119901119885119885119868119868(119911119911) = 10045601004560

infin

0

119899119909119909sdot 119901119901119883119883119868119868

(119909119909) sdot 1199011199011198841198841198681198681007652100765211991111991111990911990910076681007668 119889119889119909119909

=120590120590120590120590119907119907119907119907119899119899119899119899

Γ (119907119907119907119907) Γ (119899119899)119911119911119899119899minus119899 10045601004560

infin

0119909119909119907119907minus119899119899minus119899 exp 1007681100768110076491007649minus120590120590minus119899119899119911119911minus11990711990710076651007665 11990911990911990711990710076971007697 119889119889119909119909120572

(6)

A variable change of 119905119905 = 119899119909119909 leads to 119909119909 = 119899119905119905 and 119889119889119909119909 =minus119905119905minus2119889119889119905119905 (6) turns out be

119901119901119885119885119868119868(119911119911)=

120590120590120590120590119907119907119907119907119899119899119899119899

Γ (119907119907119907119907) Γ (119899119899)119911119911119899119899minus119899 10045601004560

infin

0119905119905minus119907119907120582119899119899minus119899 exp 1007681100768110076491007649minus120590120590minus119899119899119911119911minus11990711990710076651007665 119905119905minus11990711990710076971007697 119889119889119905119905120572

(7)

Applying the integral formula intinfin0 119909119909119886119886minus119899 exp(minus120583120583119909119909119901119901)119889119889119909119909 =(119899119901119901)120583120583minus119886119886119901119901Γ(119886119886119901119901) Re 120583120583 120572 0 Re 119886119886 120572 0 119901119901 120572 0 [15 3478Equation 1] one can easily obtain the result that (7) is equalto (2)

Furthermore ℋ1199701199700 has the charming property that thewell-known 1199701199700 presented by Frery et al [11] to modelhomogeneous heterogeneous and extremely heterogeneousterrains is a special case of this proposed model when 120590120590 = 119899and 119907119907 = minus119899 us the proposed model exhibits higher ttingability as compared to1199701199700

As derived in Appendix A the119898119898th order moments of theℋ119970119970119900119900

119868119868 are given by

119864119864 10076491007649119911119911119898119898119868119868 10076651007665

=120590120590120590120590119907119907119907119907minus(119898119898120582119907119907)119907119907119899119899119899119899120582(119898119898120582119899119899)119907119907Γ (minus (119898119898 120582 119899119899) 119907119907) Γ ((119907119907 120582 119898119898) 119907119907)

1199071199072Γ (119907119907119907119907) Γ (119899119899)120572

(8)

International Journal of Antennas and Propagation 3

0 05 1 15 2 25 3 35 4 45 50

02

04

06

08

1

12

14

Am

pli

tud

e P

DF

= minus 05

= minus 1

= minus 15

= 1 = minus 11 = 4 = 04

(a)

0 05 1 15 2 25 3 35 4 45 50

01

02

03

04

05

06

07

Am

pli

tud

e P

DF

= minus 2

= minus 15

= minus 1

(b)

0 05 1 15 2 25 3 35 4 45 50

10

20

30

40

50

60

70

80

90

Am

pli

tud

e P

DF

= 1

= 2

= 3

(c)

0 05 1 15 2 25 3 35 4 45 50

5

10

15

20

25

30

Am

pli

tud

e P

DF

= 05

= 06

= 07

(d)

F 1 Plots ofℋ119970119970119900119900119860119860 versus different parameters

e119898119898th order moments of the corresponding amplituderandom variable can easily be given by 119864119864119864119864119864119898119898119860119860) = 119864119864119864119864119864

119898119898119898119898119868119868 )

3 The Parameter Estimators ofthe Proposed Distribution

31e Log-Cumulants ofℋ1199701199700 Nicolas [13] has proposed aparametric PDF estimation technique based on theMoLC fora function dened over ℝ+ Given a positive-valued randomvariable 119883119883 with the PDF 119901119901119883119883119864119909119909) the second-ind rst and

second characteristic functions are respectively dened as[13]

120601120601119883119883 119864119904119904) = ℳ10077131007713119901119901119883119883 119864119909119909)10077291007729 119864119904119904) = 10045601004560infin

0119909119909119904119904119904119904119901119901119883119883 119864119909119909) 119889119889119909119909119889

120593120593119883119883 119864119904119904) = log 10076491007649120601120601119883119883 119864119904119904)10076651007665 119889(9)

where ℳ is the Mellin transform operator e 119903119903th orderderivative of 120593120593119883119883119864119904119904) at 119904119904 = 119904 is the log-cumulants of order 119903119903

4 International Journal of Antennas and Propagation

that is

119896119896119903119903 =119889119889119903119903120593120593119883119883 (119904119904)119889119889119904119904119903119903

10092541009254119904119904=119904 (10)

Hereaer we take the intensity distribution ℋ119970119970119900119900119868119868 as an

example to estimate its parameters 119907119907 120572120572 and 120582120582e processingof the amplitude one is similar Owing to 120601120601119883119883(119904119904) = 119904119904(119883119883119904119904119904119904)consequently via (8) and (9) the second-kind second char-acteristic functions of theℋ119970119970119900119900

119868119868 distribution yields

120593120593ℋ119970119970119900119900119868119868(119904119904) = log (120590120590) + log 10076501007650120582120582119907119907120572120572119904(120572120572119907119907119907)10076661007666

+119904 119904 119904119904119907119907

log (120582120582) + log 10076501007650119899119899119899119899+(119899119899119907119907119907)10076661007666 +119904119904 119904 119904119907119907

log (119899119899)

+ log 10076521007652Γ 10076521007652119904119904119904 119904 119904 + 119899119899

1199071199071007668100766810076681007668 + log 10076521007652Γ 10076521007652

120572120572 + 119904119904 119904 119904119907119907

1007668100766810076681007668

119904 log 100765010076501199071199072Γ (119907119907120572120572) Γ (119899119899)10076661007666(11)

which leads to that the log-cumulants of the ℋ119970119970119900119900119868119868 are

expressed as

119896119896119904 =log (119899119899119907120582120582) 119904 Ψ (119904119899119899119907119907119907) + Ψ (120572120572119907119907119907)

119907119907

119896119896119903119903 =Ψ (119903119903 119904 119904 119904119899119899119907119907119907)

(119904119907119907)119903119903+Ψ (119903119903 119904 119904 120572120572119907119907119907)

119907119907119903119903 119903119903 119903 2

(12)

where Ψ(sdot) represents the digamma function and Ψ(119903119903 sdot) isthe 119903119903th order polygamma function

Given a sample set 119911119911119894119894 119894119894 119894 119894119904119894119894119894 the log-cumulants canbe estimated directly by

10065931006593119896119896119904 =119904119894119894

11989411989410055761005576119894119894=11990410077131007713ln 100764910076491199111199111198941198941007665100766510077291007729

10065931006593119896119896119904 =119904119894119894

11989411989410055761005576119894119894=1199041007715100771510076501007650ln 1007649100764911991111991111989411989410076651007665 119904 1006593100659311989611989611990410076661007666

11990311990310077311007731 119903119903 119903 2

(13)

32 e Parameter Estimators of ℋ1199701199700 We notice that (12)is independent of the parameter 120590120590 and the 119903119903th log-cumulantsshown in (12) are irrespective with the parameter 120582120582 on thecondition that 119903119903 119903 2 In addition we regard the parameter 119899119899as a known constant which can be replaced by the equivalentnumber of looks (ENLs) [4 11] or obtained from some priorknowledge about processed SAR images hence allowing usto divide the parameter estimates to three stages

First the estimates 10065931006593119907119907 and 10065931006593120572120572 of the parameters 119907119907 as well as120572120572 are obtained by solving the following equations resorting tothe numerical computation

100659310065931198961198962 =Ψ (119904 11990411989911989911990710065931006593119907119907) + Ψ (119904 1006593100659312057212057211990710065931006593119907119907)

100659310065931199071199072

100659310065931198961198963 =Ψ (2 1006593100659312057212057211990710065931006593119907119907 ) 119904 Ψ (2 11990411989911989911990710065931006593119907119907 )

100659310065931199071199073

(14)

Second according to the rst equation of (12) theestimate 10065931006593120582120582 of the parameter 120582120582 is

10065931006593120582120582 =119899119899

exp 100765010076501006593100659311990711990710065931006593119896119896119904 + Ψ (11990411989911989911990710065931006593119907119907) 119904 Ψ (1006593100659312057212057211990710065931006593119907119907)10076661007666 (15)

We assume the observed amplitude PDF is 10065931006593119901119901119885119885119860119860(119911119911) from

the actual data which corresponds to the histogram of testeddata e theoretical amplitude PDF denotes as 119901119901119885119885119860119860

(119911119911) Asthe previous analysis since 120590120590 indicates the whole proportionuctuation of 119901119901119885119885119860119860

(119911119911) 119901119901119885119885119860119860(119911119911)119907120590120590 can be calculated by using 10065931006593119907119907

10065931006593120572120572 and 10065931006593120582120582 Given a sample amplitude set 119911119911119894119894 ∣ 119894119894 = 119904119894119894119894 letthe symbol 119902119902119885119885119860119860

(119911119911) be equal to 119901119901119885119885119860119860(119911119911)119907120590120590 then the estimate 10065931006593120590120590

of the parameter 120590120590 is simply given by

10065931006593120590120590 = 1199041199041007655100765510065931006593119901119901119885119885119860119860

1007649100764911991111991111989411989410076651007665119902119902119885119885119860119860

100764910076491199111199111198941198941007665100766510076711007671 119911119911119894119894 119894 ℤ (16)

where ℤ is a subset of 119911119911119894119894 ∣ 119894119894 = 119904119894119894119894 Here in order tofacilitate the stable tting we choose empirically all pointsof histogram within 3 dB bandwidth to calculate 120590120590 such thatℤ = 119911119911119894119894 ∣ 10065931006593119901119901119885119885119860119860

(119911119911119894119894) gt 10065931006593119901119901max1199072 10065931006593119901119901max = max( 10065931006593119901119901119885119885119860119860(119911119911119894119894)) 119894119894 119894

119894119904119894119894119894

4 Experimental Results

In this section we aim at verifying the performance of theproposed ℋ1199701199700 In order to assess how the ℋ1199701199700 performsseveral space-borne TerraSAR-X geocoded scenes with var-ious land-over typologies as examples are reported Figures2(a)ndash2(d) show four selected typical patches from a largeldquoSanchagangrdquo TerraSAR-X image with low resolution whichare the portions of water body drying riverbed mountainand a town returns respectively e four types of scenesare related to the homogeneous and heterogeneous terrainsFor simplicity we denote them as ldquowater-bodyrdquo ldquodryingriverbedrdquo ldquomountainrdquo and ldquotownrdquo Additionally two urbanareas shown in Figures 2(e)-2(f) extracted from a largeldquoBeijingrdquo TerraSAR-X scene with high resolution are furthercarried out to demonstrate the effectiveness of the proposedmodel on the extremely heterogeneous terrains Likewise thetwo urban images are denoted by ldquourban1rdquo and ldquourban2rdquoemain parameters of TerraSAR-X systems for ldquoSanchagangrdquoand ldquoBeijingrdquo data are listed in Table 1

e estimated PDFs of the proposedℋ1199701199700 model for thehistograms of six selected areas indicated in Figure 2 areshown in Figure 3 As the corresponding comparisons thetting results with the intensively used 1199701199700 distribution arealso provided where parameter estimates of this distributionare derived by Tison et al [6] based on theMoLC Herein theestimate 119899119899 inℋ1199701199700 is rst replaced by the ENL that is [4]

10065931006593119899119899 = 119904119904119894119894119899119899 = 119899119899119886119886 + 119899119899119903119903 (17)

where 119899119899119886119886 and 119899119899119903119903 are the effective number of looks in theazimuth and range For the previous ldquoSanchagangrdquo andldquoBeijingrdquo TerraSAR-X images used in this investigation 119899119899119886119886and 119899119899119903119903 are listed in the last two columns of Table 1 according

International Journal of Antennas and Propagation 5

(a) (b)

(c) (d)

(e) (f)

F 2 e Tested TerraSAR-X SAR image (a) ldquoWater bodyrdquo patch with a size of 332 px times 322 px (b) ldquodrying riverbedrdquo image (size 1597times 1734 pixels) (c) ldquomountainrdquo image (size 1996 times 2034 pixels) (d) ldquotownrdquo image (size 525 times 560 pixels) (e) ldquourban1rdquo image (size 1198 times1342 pixels) and (e) ldquourban2rdquo image (size 705 times 728 pixels)

6 International Journal of Antennas and Propagation

T 1 e main parameters of TerraSAR-X space-borne systems for the data in this study

Testing site Acquisitionmode Polarization Resolution

(az times rg) Date Size Azimuthlooks

Rangelooks

Beijing China SpotLight HH 192m times 226m 2008-01-31T1003 UTC 21333 px times 20000 px 119 102Sanchagang China StripMap VV 611m times 598m 2008-01-01T2220 UTC 11909 px times 22363 px 102 267

T 2 Parameter estimations of noted areas in Figure 2

Area ℋ119970119970∘119860119860 (120572120572120572 120572120572120572 120572120572120572 120572120572120572 120572120572120572

ldquoWater bodyrdquo (minus66029 minus11001 63048 10709 3683)ldquoDrying riverbedrdquo (minus60382 minus10555 56813 11897 3683)ldquoMountainrdquo (minus29788 minus05779 24694 01426 3683)ldquoTownrdquo (minus5062 minus07271 45872 04861 3683)ldquoUrban1rdquo (minus11774 minus04895 06905 111945 222)ldquoUrban2rdquo (minus08641 minus07065 03227 23517 222)

to reading their metadata les ext the estimates of allother parameters inℋ1199701199700 are accomplished by the estimatorsderived in Section 3 which are shown at Table 2

Furthermore in order to quantitatively assess the perfor-mances that different distributions t we dene an error ratiofactor (ERF) as

120575120575 120575sum119873119873119894119894120575119894 10092651009265119901119901119888119888 1007649100764911991111991111989411989410076651007665 minus 119902119902 100764910076491199111199111198941198941007665100766510092651009265

2

sum119873119873119894119894120575119894 10092651009265119901119901119887119887 1007649100764911991111991111989411989410076651007665 minus 119902119902 100764910076491199111199111198941198941007665100766510092651009265

2 120572 (18)

where 119911119911119894119894 ∣ 119894119894 120575 119894119894119873119873119894 is a sample set 119901119901119888119888(sdot120572 represents thecompared theoretical PDF 119901119901119887119887(sdot120572 is the basic theoretical PDFand 119902119902(sdot120572 indicates the actual PDF from the observed dataesymbol sdot denotes 2-norm It is obvious that the numeratorand denominator in (18) separately imply the total ttingerrors with the 119901119901119888119888 and 119901119901119887119887 to approximate 119902119902 e better theperformances of 119901119901119888119888 related to 119901119901119887119887 t the smaller 120575120575 is and viceverse Specically 120575120575 120575 119894 if both 119901119901119888119888 and 119901119901119887119887 have the identicalcapability for tting the measured data

Letℋ119970119970119900119900119860119860 represent the basic amplitude PDF and let 119970119970119900119900

119860119860be the compared amplitude one e 120575120575 values of the previoussix areas in this study are given in Figure 3 (see the yellowtextbox) It can be clearly seen that the proposedℋ119970119970119900119900

119860119860 betteragrees with the amplitude histograms of all six terrains thanthe 119970119970119900119900

119860119860 as expected because the 120575120575 values are larger than1 for all six areas e same conclusion can be conrmedfrom a visual point of view and implies the higher ttingprecision using ℋ119970119970119900119900 than using 1199701199700 over homogeneousheterogeneous and extremely heterogeneous regions

5 Conclusion and Perspective

We have developed an empirical model ℋ1199701199700 to exploitthe knowledge of statistical characteristics of SAR ampli-tude or intensity images over the wide terrain classes withhomogeneous heterogeneous and extremely heterogeneousbackscattering properties e parameter estimators of this

model based on the MoLC are also provided Consequentlywe report the performances of different land-over typologieswith ℋ1199701199700 distribution ts e experimental results showthat the ℋ1199701199700 distribution is a more advanced model com-pared with the known 1199701199700 distribution to characterize themultilook processed SAR data

As we know a preliminary statistical analysis of SARclutter data is important for designing signal processing algo-rithms such as speckle ltering target detection buildingextraction image segmentation and classication In futureit is worth expecting to use the ℋ1199701199700 distribution in theseelds Herein we rstly attempt to give an analytical deriva-tion for constructing a constant false alarm rate (CFAR)detector to promote the upcoming studies of target detectionin SAR images

Given the propose amplitude modelℋ119970119970119900119900119860119860 shown in (1)

its cumulative distribution function (CDF) is written as (seeAppendix B)

119865119865ℋ119970119970119900119900119860119860(119909119909120572 120575 minus

120572120572120572120572120572120572120572120572119907(120572120572minus120572120572120572119907120572120572120572120572120572120572minus119894Γ ((120572120572 minus 120572120572120572 119907120572120572120572120572120572Γ (120572120572120572120572120572 Γ (120572120572120572

1199091199092120572120572

times 2119865119865119894 10076521007652120572120572 minus 120572120572120572120572

120572 minus120572120572120572120572 119894 minus

120572120572120572120572 minus120572120572120572120572119909119909minus212057212057210076681007668 120572

minus 120572120572120572 minus120572120572120572 120572120572120572 120572120572120572 120572120572120572 119909119909 120572 0120572

(19)

where 2119865119865119894(sdot120572 sdot sdot sdot120572 is the Gauss hypergeometric function For agiven value of the false alarm probability denoted by 119875119875119891119891119891119891 thecorresponding CFAR threshold 119879119879 for theℋ119970119970119900119900

119860119860 distributioncan be obtained from

119894 minus 119875119875119891119891119891119891 120575 119865119865ℋ119970119970119900119900119860119860(119879119879120572

120575 minus120572120572120572120572120572120572120572120572119907(120572120572minus120572120572120572119907120572120572120572120572120572120572minus119894Γ ((120572120572 minus 120572120572120572 119907120572120572120572

120572120572Γ (120572120572120572120572120572 Γ (1205721205721205721198791198792120572120572

times 2119865119865119894 10076521007652120572120572 minus 120572120572120572120572

120572 minus120572120572120572120572 119894 minus

120572120572120572120572 minus120572120572120572120572119879119879minus212057212057210076681007668

(20)

Considering 119865119865ℋ119970119970119900119900119860119860(119879119879120572 is strictly monotonously increas-

ing the threshold119879119879 can be accurately calculatedwith the helpof the numerical solution or a simple bisection method [16]

Our future work will focus on demonstrating the perfor-mances of the proposed CFAR detector using somemeasuredSAR data and investigating how the ℋ1199701199700 distribution per-forms when extending it to other application elds

International Journal of Antennas and Propagation 7

0 05 1 15 2 25 3 350

02

04

06

08

1

12

14

Amplitude

PD

FWater body

= 10042

(a)

0 05 1 15 2 25 3 350

05

1

15

Amplitude

PD

F

Drying riverbed

= 35351

(b)

0 05 1 15 2 25 3 350

02

04

06

08

1

12

14

Amplitude

PDF

Mountain

= 23306

(c)

0 05 1 15 2 25 3 350

02

04

06

08

1

12

14

Amplitude

PDF

Town

= 26756

(d)

0 05 1 15 2 25 3 350

02

04

06

08

1

12

14

Amplitude

PD

F

Urban1

= 1944

Histogram

The proposed model

(e)

0 05 1 15 2 25 3 350

02

04

06

08

1

12

14

Amplitude

PD

F

Urban2

= 13562

Histogram

The proposed model

(f)

F 3 Plots of aplitude istogras and of te estiated PFs (a)ndash(f) are corresponded to te tting results for te scenes son inFigures 2(a)ndash2(f) respectively

8 International Journal of Antennas and Propagation

Appendices

A The Derivation of119898119898th OrderMoments oftheℋ119970119970119900119900

119868119868 Distribution

Via (2) the119898119898th order moments of theℋ119970119970119900119900119868119868 are expressed as

119864119864 10076491007649119911119911119898119898119868119868 10076651007665 = 10045601004560infin

0119911119911119898119898119901119901119885119885119868119868

(119911119911) 119889119889119911119911

= minus120590120590120590120590119907119907119907119907119899119899119899119899Γ ((119907119907 minus 119899119899) 119907119907)

119907119907Γ (119907119907119907119907) Γ (119899119899)

times 120590120590(119899119899minus119907119907)119907119907 10045601004560infin

0119911119911119898119898119898119899119899minus11989810076521007652119898 119898

119899119899120590120590119911119911minus11990711990710076681007668

(119899119899minus119907119907)119907119907119889119889119911119911119889

(A1)

A variable replacement 119909119909 = 119911119911minus119907119907 leads to 119911119911 = 119909119909minus119898119907119907 and119889119889119911119911 = minus(119898119907119907)119909119909minus(119898119907119907)minus119898119889119889119909119909 thus (A1) can be rewritten as

119864119864 10076491007649119911119911119898119898119868119868 10076651007665 = minus120590120590120590120590119907119907119907119907119899119899119899119899Γ ((119907119907 minus 119899119899) 119907119907)

1199071199072Γ (119907119907119907119907) Γ (119899119899)

times 120590120590(119899119899minus119907119907)119907119907 10045601004560infin

0119909119909((119898119898119898119899119899)119907119907)minus11989810076521007652119898 119898

11989911989912059012059011990911990910076681007668

(119899119899minus119907119907)119907119907119889119889119909119909119889

(A2)

According to the integral formula intinfin0 119909119909120583120583minus119898119889119889119909119909(119898119898119889119889119909119909)119907119907 =119889119889minus120583120583Γ(120583120583)Γ(119907119907minus120583120583)Γ(119907119907) | arg 119889119889| 120573 120573120573Re 119907119907 119907 Re 120583120583 119907 0 [15 3194Equation 3] one can obtain (8)

B The Derivation of the CumulativeDistribution Function ofℋ119970119970119900119900

119860119860

Based on the denition of the CDF the CDF ofℋ119970119970119900119900119860119860 is

119865119865ℋ119970119970119900119900119860119860(119909119909) = 10045601004560

119909119909

minusinfin119901119901119885119885119860119860

(119911119911) 119889119889119911119911 = 10045601004560119909119909

0119901119901119885119885119860119860

(119911119911) 119889119889119911119911

= minus2120590120590120590120590119907119907119907119907119899119899119899119899Γ ((119907119907 minus 119899119899) 119907119907)

119907119907Γ (119907119907119907119907) Γ (119899119899)

times 120590120590(119899119899minus119907119907)119907119907 10045601004560119909119909

01199111199112119899119899minus11989810076521007652119898 119898

119899119899120590120590119911119911minus211990711990710076681007668

(119899119899minus119907119907)119907119907119889119889119911119911119889

(B1)

Similarly utilizing a symbol change 119905119905 = 119911119911minus2119907119907 (B1) turnsout to be

119865119865ℋ119970119970119900119900119860119860(119909119909) =

120590120590120590120590119907119907119907119907119899119899119899119899Γ ((119907119907 minus 119899119899) 119907119907)1199071199072Γ (119907119907119907119907) Γ (119899119899)

times 120590120590(119899119899minus119907119907)119907119907 10045601004560119909119909minus2119907119907

0119905119905(minus119899119899119907119907)minus11989810076521007652119898 119898

11989911989912059012059011990511990510076681007668

(119899119899minus119907119907)119907119907119889119889119905119905119889

(B2)

Likewise applying the integral formula int1199061199060 119909119909120583120583minus119898119889119889119909119909(119898 119898 119889119889119909119909)119907119907

= (119906119906120583120583120583120583)2119865119865119898(119907119907119907 120583120583119907 119898 119898 120583120583119907 minus119889119889119906119906) | arg(119898 119898 119889119889119906119906)| 120573 120573120573 Re 120583120583 119907 0[15 3194 Equation 1] we arrive at (19) by simplifying (B2)

References

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

[2] G Gao ldquoStatistical modeling of SAR images a surveyrdquo Sensorsvol 10 no 1 pp 775ndash795 2010

[3] A Achim E E Kuruolu and J Zerubia ldquoSAR image lteringbased on the heavy-tailed rayleigh modelrdquo IEEE Transactionson Image Processing vol 15 no 9 pp 2686ndash2693 2006

[4] T Esch M iel A Schenk A Roth A Muumlller and S DechldquoDelineation of Urban footprints from TerraSAR-X data byanalyzing speckle characteristics and intensity informationrdquoIEEE Transactions on Geoscience and Remote Sensing vol 48no 2 pp 905ndash916 2010

[5] 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

[6] C Tison J M Nicolas F Tupin and H Maicirctre ldquoA newstatistical model for Markovian classication of Urban areas inhigh-Resolution SAR imagesrdquo IEEE Transactions on Geoscienceand Remote Sensing vol 42 no 10 pp 2046ndash2057 2004

[7] A C Frery A H Correia and C D C Freitas ldquoClassify-ing multifrequency fully polarimetric imagery with multiplesources of statistical evidence and contextual informationrdquoIEEE Transactions on Geoscience and Remote Sensing vol 45no 10 pp 3098ndash3109 2007

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

[9] Y Delignon R Garello and A Hillion ldquoStatistical modeling ofocean SAR imagerdquo IEE Proceedings Radar Sonar and Naviga-tion vol 144 no 6 pp 348ndash354 1997

[10] Y Delignon and W Pieczynski ldquoModeling non-Rayleighspeckle distribution in SAR imagesrdquo IEEE Transactions onGeoscience and Remote Sensing vol 40 no 6 pp 1430ndash14352002

[11] A C Frery H-J Muumlller C D C F Yanasse and S J SSantrsquoAnna ldquoAmodel for extremely heterogeneous clutterrdquo IEEETransactions on Geoscience and Remote Sensing vol 35 no 3pp 648ndash659 1997

[12] K L P Vasconcellos A C Frery and L B Silva ldquoImprovingestimation in speckled imageryrdquo Computational Statistics vol20 no 3 pp 503ndash519 2005

[13] 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

[14] S N Annsen and T Elto ldquoApplication of the Matrix-variatemellin transform to analysis of polarimetric radar imagesrdquo IEEETransactions on Geoscience and Remote Sensing vol 49 no 6pp 2281ndash2295 2011

[15] S I Gradshteyn and I M Ryzhik Table of Integrals Series andProducts Academic Press San Diego Calif USA 7th edition2007

[16] G Moser J Zerubia and S B Serpico ldquoSAR amplitudeprobability density function estimation based on a generalizedGaussian modelrdquo IEEE Transactions on Image Processing vol15 no 6 pp 1429ndash1442 2006

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DistributedSensor Networks

International Journal of

2 International Journal of Antennas and Propagation

which relies on the Mellin transform [14] that is ldquomethod-of-log-cumulantsrdquo (MoLC) we derive the parameter estima-tors of the new distribution model

In the rest of this paper the proposed distribution isrst given in Section 2 Section 3 derives the correspondingparameter estimators based on the MoLC We provide theexperimental results of theℋ1199701199700 model using measured SARdata in Section 4 the comparisons with that 1199701199700 ts are alsodiscussed in this sectione last section concludes this paperand give a perspective in the future work

2 The Proposed Distribution

e proposed amplitude distribution is dened as

119901119901119885119885119860119860(119911119911) = minus

2120590120590120590120590119907119907119907119907119899119899119899119899Γ ((119907119907 minus 119899119899) 119907119907)119907119907Γ (119907119907119907119907) Γ (119899119899)

times 1199111199112119899119899minus11989910076501007650120590120590 120582 119899119899119911119911minus211990711990710076661007666(119899119899minus119907119907)119907119907

minus119907119907 minus119907119907 120590120590 120590120590 119899119899 119911119911 120572 0(1)

where 120590120590 119907119907 and 120590120590 are the power shape scale parametersrespectively 119907119907 indicates the stretching parameter Γ(sdot) rep-resents the Gamma function 119899119899 is the number of looks ecorresponding intensity expression of (1) is further given by

119901119901119885119885119868119868(119911119911) = minus

120590120590120590120590119907119907119907119907119899119899119899119899Γ ((119907119907 minus 119899119899) 119907119907)119907119907Γ (119907119907119907119907) Γ (119899119899)

times 119911119911119899119899minus11989910076491007649120590120590 120582 119899119899119911119911minus11990711990710076651007665(119899119899minus119907119907)119907119907 minus119907119907 minus119907119907 120590120590 120590120590 119899119899 119911119911 120572 0120572(2)

We refer to this distribution characterized by (1) or(2) as the ℋ1199701199700 distribution Specically we call the ℋ119970119970119900119900

119868119868distribution and the ℋ119970119970119900119900

119860119860 distribution correspond to (2)and (1) respectively to distinct the intensity statistic as wellas the amplitude statistic Figure 1 gives some examples oftheℋ119970119970119900119900

119860119860 distribution with respect to the various parametersFrom this gure it can be seen clearly that the parameter119907119907 reects the degree of homogeneity for the tested returnswhich implies that the smaller value of |119907119907| obtains the morein-homogeneous (ie larger tails) they are Meanwhile 119907119907corresponds to a stretching of the SAR image amplitudeor intensity showing a strong effect for low values of thereturn Moreover as an independent parameter 120590120590 indicatesthe whole uctuation (contains magnifying or shrinking) ofpower of densities along the vertical axis e parameter 120590120590controls the peak value of the density

Although the probability density functions (PDFs) char-acterized by (1) or (2) are shown as the empirical models theℋ1199701199700 model also can be derived within the structure of themultiplicative model [1 2 11] Herein we account for theintensity distribution as an example the following theoremis established

eorem 1 Letting 119883119883119868119868 and 119884119884119868119868 indicates the backscatteringRCS component and speckle noise one respectively 119885119885119868119868 denotesthe observed intensity of SAR data erefore the relationship

of this three variables is expressed by the multiplicative modelas

119885119885119868119868 = 119883119883119868119868 sdot 119884119884119868119868120572 (3)

If119883119883119868119868 obeys

119901119901119883119883119868119868(119909119909) =

120590120590120590120590119907119907119907119907

Γ (119907119907119907119907)119909119909119907119907minus119899 exp 10076491007649minus12059012059011990911990911990711990710076651007665 minus119907119907 minus119907119907 120590120590 120590120590 119909119909 120572 0

(4)

and the PDF of 119884119884119868119868 is

1199011199011198841198841198681198681007649100764911991011991010076651007665 =

119899119899119899119899

Γ (119899119899)119910119910119899119899minus119899 exp 10076491007649minus119899119899119910119910minus11990711990710076651007665 119910119910 119899119899 minus119907119907 120572 0120572 (5)

en the distribution of the intensity return is characterizedby the density shown in (2) that is 119885119885119868119868 sim ℋ119970119970119900119900

119868119868(119907119907 119907119907 120590120590 120590120590 119899119899)

Proof Combining (4) and (5) via (3) the PDF of 119885119885119868119868 can beeasily derived as

119901119901119885119885119868119868(119911119911) = 10045601004560

infin

0

119899119909119909sdot 119901119901119883119883119868119868

(119909119909) sdot 1199011199011198841198841198681198681007652100765211991111991111990911990910076681007668 119889119889119909119909

=120590120590120590120590119907119907119907119907119899119899119899119899

Γ (119907119907119907119907) Γ (119899119899)119911119911119899119899minus119899 10045601004560

infin

0119909119909119907119907minus119899119899minus119899 exp 1007681100768110076491007649minus120590120590minus119899119899119911119911minus11990711990710076651007665 11990911990911990711990710076971007697 119889119889119909119909120572

(6)

A variable change of 119905119905 = 119899119909119909 leads to 119909119909 = 119899119905119905 and 119889119889119909119909 =minus119905119905minus2119889119889119905119905 (6) turns out be

119901119901119885119885119868119868(119911119911)=

120590120590120590120590119907119907119907119907119899119899119899119899

Γ (119907119907119907119907) Γ (119899119899)119911119911119899119899minus119899 10045601004560

infin

0119905119905minus119907119907120582119899119899minus119899 exp 1007681100768110076491007649minus120590120590minus119899119899119911119911minus11990711990710076651007665 119905119905minus11990711990710076971007697 119889119889119905119905120572

(7)

Applying the integral formula intinfin0 119909119909119886119886minus119899 exp(minus120583120583119909119909119901119901)119889119889119909119909 =(119899119901119901)120583120583minus119886119886119901119901Γ(119886119886119901119901) Re 120583120583 120572 0 Re 119886119886 120572 0 119901119901 120572 0 [15 3478Equation 1] one can easily obtain the result that (7) is equalto (2)

Furthermore ℋ1199701199700 has the charming property that thewell-known 1199701199700 presented by Frery et al [11] to modelhomogeneous heterogeneous and extremely heterogeneousterrains is a special case of this proposed model when 120590120590 = 119899and 119907119907 = minus119899 us the proposed model exhibits higher ttingability as compared to1199701199700

As derived in Appendix A the119898119898th order moments of theℋ119970119970119900119900

119868119868 are given by

119864119864 10076491007649119911119911119898119898119868119868 10076651007665

=120590120590120590120590119907119907119907119907minus(119898119898120582119907119907)119907119907119899119899119899119899120582(119898119898120582119899119899)119907119907Γ (minus (119898119898 120582 119899119899) 119907119907) Γ ((119907119907 120582 119898119898) 119907119907)

1199071199072Γ (119907119907119907119907) Γ (119899119899)120572

(8)

International Journal of Antennas and Propagation 3

0 05 1 15 2 25 3 35 4 45 50

02

04

06

08

1

12

14

Am

pli

tud

e P

DF

= minus 05

= minus 1

= minus 15

= 1 = minus 11 = 4 = 04

(a)

0 05 1 15 2 25 3 35 4 45 50

01

02

03

04

05

06

07

Am

pli

tud

e P

DF

= minus 2

= minus 15

= minus 1

(b)

0 05 1 15 2 25 3 35 4 45 50

10

20

30

40

50

60

70

80

90

Am

pli

tud

e P

DF

= 1

= 2

= 3

(c)

0 05 1 15 2 25 3 35 4 45 50

5

10

15

20

25

30

Am

pli

tud

e P

DF

= 05

= 06

= 07

(d)

F 1 Plots ofℋ119970119970119900119900119860119860 versus different parameters

e119898119898th order moments of the corresponding amplituderandom variable can easily be given by 119864119864119864119864119864119898119898119860119860) = 119864119864119864119864119864

119898119898119898119898119868119868 )

3 The Parameter Estimators ofthe Proposed Distribution

31e Log-Cumulants ofℋ1199701199700 Nicolas [13] has proposed aparametric PDF estimation technique based on theMoLC fora function dened over ℝ+ Given a positive-valued randomvariable 119883119883 with the PDF 119901119901119883119883119864119909119909) the second-ind rst and

second characteristic functions are respectively dened as[13]

120601120601119883119883 119864119904119904) = ℳ10077131007713119901119901119883119883 119864119909119909)10077291007729 119864119904119904) = 10045601004560infin

0119909119909119904119904119904119904119901119901119883119883 119864119909119909) 119889119889119909119909119889

120593120593119883119883 119864119904119904) = log 10076491007649120601120601119883119883 119864119904119904)10076651007665 119889(9)

where ℳ is the Mellin transform operator e 119903119903th orderderivative of 120593120593119883119883119864119904119904) at 119904119904 = 119904 is the log-cumulants of order 119903119903

4 International Journal of Antennas and Propagation

that is

119896119896119903119903 =119889119889119903119903120593120593119883119883 (119904119904)119889119889119904119904119903119903

10092541009254119904119904=119904 (10)

Hereaer we take the intensity distribution ℋ119970119970119900119900119868119868 as an

example to estimate its parameters 119907119907 120572120572 and 120582120582e processingof the amplitude one is similar Owing to 120601120601119883119883(119904119904) = 119904119904(119883119883119904119904119904119904)consequently via (8) and (9) the second-kind second char-acteristic functions of theℋ119970119970119900119900

119868119868 distribution yields

120593120593ℋ119970119970119900119900119868119868(119904119904) = log (120590120590) + log 10076501007650120582120582119907119907120572120572119904(120572120572119907119907119907)10076661007666

+119904 119904 119904119904119907119907

log (120582120582) + log 10076501007650119899119899119899119899+(119899119899119907119907119907)10076661007666 +119904119904 119904 119904119907119907

log (119899119899)

+ log 10076521007652Γ 10076521007652119904119904119904 119904 119904 + 119899119899

1199071199071007668100766810076681007668 + log 10076521007652Γ 10076521007652

120572120572 + 119904119904 119904 119904119907119907

1007668100766810076681007668

119904 log 100765010076501199071199072Γ (119907119907120572120572) Γ (119899119899)10076661007666(11)

which leads to that the log-cumulants of the ℋ119970119970119900119900119868119868 are

expressed as

119896119896119904 =log (119899119899119907120582120582) 119904 Ψ (119904119899119899119907119907119907) + Ψ (120572120572119907119907119907)

119907119907

119896119896119903119903 =Ψ (119903119903 119904 119904 119904119899119899119907119907119907)

(119904119907119907)119903119903+Ψ (119903119903 119904 119904 120572120572119907119907119907)

119907119907119903119903 119903119903 119903 2

(12)

where Ψ(sdot) represents the digamma function and Ψ(119903119903 sdot) isthe 119903119903th order polygamma function

Given a sample set 119911119911119894119894 119894119894 119894 119894119904119894119894119894 the log-cumulants canbe estimated directly by

10065931006593119896119896119904 =119904119894119894

11989411989410055761005576119894119894=11990410077131007713ln 100764910076491199111199111198941198941007665100766510077291007729

10065931006593119896119896119904 =119904119894119894

11989411989410055761005576119894119894=1199041007715100771510076501007650ln 1007649100764911991111991111989411989410076651007665 119904 1006593100659311989611989611990410076661007666

11990311990310077311007731 119903119903 119903 2

(13)

32 e Parameter Estimators of ℋ1199701199700 We notice that (12)is independent of the parameter 120590120590 and the 119903119903th log-cumulantsshown in (12) are irrespective with the parameter 120582120582 on thecondition that 119903119903 119903 2 In addition we regard the parameter 119899119899as a known constant which can be replaced by the equivalentnumber of looks (ENLs) [4 11] or obtained from some priorknowledge about processed SAR images hence allowing usto divide the parameter estimates to three stages

First the estimates 10065931006593119907119907 and 10065931006593120572120572 of the parameters 119907119907 as well as120572120572 are obtained by solving the following equations resorting tothe numerical computation

100659310065931198961198962 =Ψ (119904 11990411989911989911990710065931006593119907119907) + Ψ (119904 1006593100659312057212057211990710065931006593119907119907)

100659310065931199071199072

100659310065931198961198963 =Ψ (2 1006593100659312057212057211990710065931006593119907119907 ) 119904 Ψ (2 11990411989911989911990710065931006593119907119907 )

100659310065931199071199073

(14)

Second according to the rst equation of (12) theestimate 10065931006593120582120582 of the parameter 120582120582 is

10065931006593120582120582 =119899119899

exp 100765010076501006593100659311990711990710065931006593119896119896119904 + Ψ (11990411989911989911990710065931006593119907119907) 119904 Ψ (1006593100659312057212057211990710065931006593119907119907)10076661007666 (15)

We assume the observed amplitude PDF is 10065931006593119901119901119885119885119860119860(119911119911) from

the actual data which corresponds to the histogram of testeddata e theoretical amplitude PDF denotes as 119901119901119885119885119860119860

(119911119911) Asthe previous analysis since 120590120590 indicates the whole proportionuctuation of 119901119901119885119885119860119860

(119911119911) 119901119901119885119885119860119860(119911119911)119907120590120590 can be calculated by using 10065931006593119907119907

10065931006593120572120572 and 10065931006593120582120582 Given a sample amplitude set 119911119911119894119894 ∣ 119894119894 = 119904119894119894119894 letthe symbol 119902119902119885119885119860119860

(119911119911) be equal to 119901119901119885119885119860119860(119911119911)119907120590120590 then the estimate 10065931006593120590120590

of the parameter 120590120590 is simply given by

10065931006593120590120590 = 1199041199041007655100765510065931006593119901119901119885119885119860119860

1007649100764911991111991111989411989410076651007665119902119902119885119885119860119860

100764910076491199111199111198941198941007665100766510076711007671 119911119911119894119894 119894 ℤ (16)

where ℤ is a subset of 119911119911119894119894 ∣ 119894119894 = 119904119894119894119894 Here in order tofacilitate the stable tting we choose empirically all pointsof histogram within 3 dB bandwidth to calculate 120590120590 such thatℤ = 119911119911119894119894 ∣ 10065931006593119901119901119885119885119860119860

(119911119911119894119894) gt 10065931006593119901119901max1199072 10065931006593119901119901max = max( 10065931006593119901119901119885119885119860119860(119911119911119894119894)) 119894119894 119894

119894119904119894119894119894

4 Experimental Results

In this section we aim at verifying the performance of theproposed ℋ1199701199700 In order to assess how the ℋ1199701199700 performsseveral space-borne TerraSAR-X geocoded scenes with var-ious land-over typologies as examples are reported Figures2(a)ndash2(d) show four selected typical patches from a largeldquoSanchagangrdquo TerraSAR-X image with low resolution whichare the portions of water body drying riverbed mountainand a town returns respectively e four types of scenesare related to the homogeneous and heterogeneous terrainsFor simplicity we denote them as ldquowater-bodyrdquo ldquodryingriverbedrdquo ldquomountainrdquo and ldquotownrdquo Additionally two urbanareas shown in Figures 2(e)-2(f) extracted from a largeldquoBeijingrdquo TerraSAR-X scene with high resolution are furthercarried out to demonstrate the effectiveness of the proposedmodel on the extremely heterogeneous terrains Likewise thetwo urban images are denoted by ldquourban1rdquo and ldquourban2rdquoemain parameters of TerraSAR-X systems for ldquoSanchagangrdquoand ldquoBeijingrdquo data are listed in Table 1

e estimated PDFs of the proposedℋ1199701199700 model for thehistograms of six selected areas indicated in Figure 2 areshown in Figure 3 As the corresponding comparisons thetting results with the intensively used 1199701199700 distribution arealso provided where parameter estimates of this distributionare derived by Tison et al [6] based on theMoLC Herein theestimate 119899119899 inℋ1199701199700 is rst replaced by the ENL that is [4]

10065931006593119899119899 = 119904119904119894119894119899119899 = 119899119899119886119886 + 119899119899119903119903 (17)

where 119899119899119886119886 and 119899119899119903119903 are the effective number of looks in theazimuth and range For the previous ldquoSanchagangrdquo andldquoBeijingrdquo TerraSAR-X images used in this investigation 119899119899119886119886and 119899119899119903119903 are listed in the last two columns of Table 1 according

International Journal of Antennas and Propagation 5

(a) (b)

(c) (d)

(e) (f)

F 2 e Tested TerraSAR-X SAR image (a) ldquoWater bodyrdquo patch with a size of 332 px times 322 px (b) ldquodrying riverbedrdquo image (size 1597times 1734 pixels) (c) ldquomountainrdquo image (size 1996 times 2034 pixels) (d) ldquotownrdquo image (size 525 times 560 pixels) (e) ldquourban1rdquo image (size 1198 times1342 pixels) and (e) ldquourban2rdquo image (size 705 times 728 pixels)

6 International Journal of Antennas and Propagation

T 1 e main parameters of TerraSAR-X space-borne systems for the data in this study

Testing site Acquisitionmode Polarization Resolution

(az times rg) Date Size Azimuthlooks

Rangelooks

Beijing China SpotLight HH 192m times 226m 2008-01-31T1003 UTC 21333 px times 20000 px 119 102Sanchagang China StripMap VV 611m times 598m 2008-01-01T2220 UTC 11909 px times 22363 px 102 267

T 2 Parameter estimations of noted areas in Figure 2

Area ℋ119970119970∘119860119860 (120572120572120572 120572120572120572 120572120572120572 120572120572120572 120572120572120572

ldquoWater bodyrdquo (minus66029 minus11001 63048 10709 3683)ldquoDrying riverbedrdquo (minus60382 minus10555 56813 11897 3683)ldquoMountainrdquo (minus29788 minus05779 24694 01426 3683)ldquoTownrdquo (minus5062 minus07271 45872 04861 3683)ldquoUrban1rdquo (minus11774 minus04895 06905 111945 222)ldquoUrban2rdquo (minus08641 minus07065 03227 23517 222)

to reading their metadata les ext the estimates of allother parameters inℋ1199701199700 are accomplished by the estimatorsderived in Section 3 which are shown at Table 2

Furthermore in order to quantitatively assess the perfor-mances that different distributions t we dene an error ratiofactor (ERF) as

120575120575 120575sum119873119873119894119894120575119894 10092651009265119901119901119888119888 1007649100764911991111991111989411989410076651007665 minus 119902119902 100764910076491199111199111198941198941007665100766510092651009265

2

sum119873119873119894119894120575119894 10092651009265119901119901119887119887 1007649100764911991111991111989411989410076651007665 minus 119902119902 100764910076491199111199111198941198941007665100766510092651009265

2 120572 (18)

where 119911119911119894119894 ∣ 119894119894 120575 119894119894119873119873119894 is a sample set 119901119901119888119888(sdot120572 represents thecompared theoretical PDF 119901119901119887119887(sdot120572 is the basic theoretical PDFand 119902119902(sdot120572 indicates the actual PDF from the observed dataesymbol sdot denotes 2-norm It is obvious that the numeratorand denominator in (18) separately imply the total ttingerrors with the 119901119901119888119888 and 119901119901119887119887 to approximate 119902119902 e better theperformances of 119901119901119888119888 related to 119901119901119887119887 t the smaller 120575120575 is and viceverse Specically 120575120575 120575 119894 if both 119901119901119888119888 and 119901119901119887119887 have the identicalcapability for tting the measured data

Letℋ119970119970119900119900119860119860 represent the basic amplitude PDF and let 119970119970119900119900

119860119860be the compared amplitude one e 120575120575 values of the previoussix areas in this study are given in Figure 3 (see the yellowtextbox) It can be clearly seen that the proposedℋ119970119970119900119900

119860119860 betteragrees with the amplitude histograms of all six terrains thanthe 119970119970119900119900

119860119860 as expected because the 120575120575 values are larger than1 for all six areas e same conclusion can be conrmedfrom a visual point of view and implies the higher ttingprecision using ℋ119970119970119900119900 than using 1199701199700 over homogeneousheterogeneous and extremely heterogeneous regions

5 Conclusion and Perspective

We have developed an empirical model ℋ1199701199700 to exploitthe knowledge of statistical characteristics of SAR ampli-tude or intensity images over the wide terrain classes withhomogeneous heterogeneous and extremely heterogeneousbackscattering properties e parameter estimators of this

model based on the MoLC are also provided Consequentlywe report the performances of different land-over typologieswith ℋ1199701199700 distribution ts e experimental results showthat the ℋ1199701199700 distribution is a more advanced model com-pared with the known 1199701199700 distribution to characterize themultilook processed SAR data

As we know a preliminary statistical analysis of SARclutter data is important for designing signal processing algo-rithms such as speckle ltering target detection buildingextraction image segmentation and classication In futureit is worth expecting to use the ℋ1199701199700 distribution in theseelds Herein we rstly attempt to give an analytical deriva-tion for constructing a constant false alarm rate (CFAR)detector to promote the upcoming studies of target detectionin SAR images

Given the propose amplitude modelℋ119970119970119900119900119860119860 shown in (1)

its cumulative distribution function (CDF) is written as (seeAppendix B)

119865119865ℋ119970119970119900119900119860119860(119909119909120572 120575 minus

120572120572120572120572120572120572120572120572119907(120572120572minus120572120572120572119907120572120572120572120572120572120572minus119894Γ ((120572120572 minus 120572120572120572 119907120572120572120572120572120572Γ (120572120572120572120572120572 Γ (120572120572120572

1199091199092120572120572

times 2119865119865119894 10076521007652120572120572 minus 120572120572120572120572

120572 minus120572120572120572120572 119894 minus

120572120572120572120572 minus120572120572120572120572119909119909minus212057212057210076681007668 120572

minus 120572120572120572 minus120572120572120572 120572120572120572 120572120572120572 120572120572120572 119909119909 120572 0120572

(19)

where 2119865119865119894(sdot120572 sdot sdot sdot120572 is the Gauss hypergeometric function For agiven value of the false alarm probability denoted by 119875119875119891119891119891119891 thecorresponding CFAR threshold 119879119879 for theℋ119970119970119900119900

119860119860 distributioncan be obtained from

119894 minus 119875119875119891119891119891119891 120575 119865119865ℋ119970119970119900119900119860119860(119879119879120572

120575 minus120572120572120572120572120572120572120572120572119907(120572120572minus120572120572120572119907120572120572120572120572120572120572minus119894Γ ((120572120572 minus 120572120572120572 119907120572120572120572

120572120572Γ (120572120572120572120572120572 Γ (1205721205721205721198791198792120572120572

times 2119865119865119894 10076521007652120572120572 minus 120572120572120572120572

120572 minus120572120572120572120572 119894 minus

120572120572120572120572 minus120572120572120572120572119879119879minus212057212057210076681007668

(20)

Considering 119865119865ℋ119970119970119900119900119860119860(119879119879120572 is strictly monotonously increas-

ing the threshold119879119879 can be accurately calculatedwith the helpof the numerical solution or a simple bisection method [16]

Our future work will focus on demonstrating the perfor-mances of the proposed CFAR detector using somemeasuredSAR data and investigating how the ℋ1199701199700 distribution per-forms when extending it to other application elds

International Journal of Antennas and Propagation 7

0 05 1 15 2 25 3 350

02

04

06

08

1

12

14

Amplitude

PD

FWater body

= 10042

(a)

0 05 1 15 2 25 3 350

05

1

15

Amplitude

PD

F

Drying riverbed

= 35351

(b)

0 05 1 15 2 25 3 350

02

04

06

08

1

12

14

Amplitude

PDF

Mountain

= 23306

(c)

0 05 1 15 2 25 3 350

02

04

06

08

1

12

14

Amplitude

PDF

Town

= 26756

(d)

0 05 1 15 2 25 3 350

02

04

06

08

1

12

14

Amplitude

PD

F

Urban1

= 1944

Histogram

The proposed model

(e)

0 05 1 15 2 25 3 350

02

04

06

08

1

12

14

Amplitude

PD

F

Urban2

= 13562

Histogram

The proposed model

(f)

F 3 Plots of aplitude istogras and of te estiated PFs (a)ndash(f) are corresponded to te tting results for te scenes son inFigures 2(a)ndash2(f) respectively

8 International Journal of Antennas and Propagation

Appendices

A The Derivation of119898119898th OrderMoments oftheℋ119970119970119900119900

119868119868 Distribution

Via (2) the119898119898th order moments of theℋ119970119970119900119900119868119868 are expressed as

119864119864 10076491007649119911119911119898119898119868119868 10076651007665 = 10045601004560infin

0119911119911119898119898119901119901119885119885119868119868

(119911119911) 119889119889119911119911

= minus120590120590120590120590119907119907119907119907119899119899119899119899Γ ((119907119907 minus 119899119899) 119907119907)

119907119907Γ (119907119907119907119907) Γ (119899119899)

times 120590120590(119899119899minus119907119907)119907119907 10045601004560infin

0119911119911119898119898119898119899119899minus11989810076521007652119898 119898

119899119899120590120590119911119911minus11990711990710076681007668

(119899119899minus119907119907)119907119907119889119889119911119911119889

(A1)

A variable replacement 119909119909 = 119911119911minus119907119907 leads to 119911119911 = 119909119909minus119898119907119907 and119889119889119911119911 = minus(119898119907119907)119909119909minus(119898119907119907)minus119898119889119889119909119909 thus (A1) can be rewritten as

119864119864 10076491007649119911119911119898119898119868119868 10076651007665 = minus120590120590120590120590119907119907119907119907119899119899119899119899Γ ((119907119907 minus 119899119899) 119907119907)

1199071199072Γ (119907119907119907119907) Γ (119899119899)

times 120590120590(119899119899minus119907119907)119907119907 10045601004560infin

0119909119909((119898119898119898119899119899)119907119907)minus11989810076521007652119898 119898

11989911989912059012059011990911990910076681007668

(119899119899minus119907119907)119907119907119889119889119909119909119889

(A2)

According to the integral formula intinfin0 119909119909120583120583minus119898119889119889119909119909(119898119898119889119889119909119909)119907119907 =119889119889minus120583120583Γ(120583120583)Γ(119907119907minus120583120583)Γ(119907119907) | arg 119889119889| 120573 120573120573Re 119907119907 119907 Re 120583120583 119907 0 [15 3194Equation 3] one can obtain (8)

B The Derivation of the CumulativeDistribution Function ofℋ119970119970119900119900

119860119860

Based on the denition of the CDF the CDF ofℋ119970119970119900119900119860119860 is

119865119865ℋ119970119970119900119900119860119860(119909119909) = 10045601004560

119909119909

minusinfin119901119901119885119885119860119860

(119911119911) 119889119889119911119911 = 10045601004560119909119909

0119901119901119885119885119860119860

(119911119911) 119889119889119911119911

= minus2120590120590120590120590119907119907119907119907119899119899119899119899Γ ((119907119907 minus 119899119899) 119907119907)

119907119907Γ (119907119907119907119907) Γ (119899119899)

times 120590120590(119899119899minus119907119907)119907119907 10045601004560119909119909

01199111199112119899119899minus11989810076521007652119898 119898

119899119899120590120590119911119911minus211990711990710076681007668

(119899119899minus119907119907)119907119907119889119889119911119911119889

(B1)

Similarly utilizing a symbol change 119905119905 = 119911119911minus2119907119907 (B1) turnsout to be

119865119865ℋ119970119970119900119900119860119860(119909119909) =

120590120590120590120590119907119907119907119907119899119899119899119899Γ ((119907119907 minus 119899119899) 119907119907)1199071199072Γ (119907119907119907119907) Γ (119899119899)

times 120590120590(119899119899minus119907119907)119907119907 10045601004560119909119909minus2119907119907

0119905119905(minus119899119899119907119907)minus11989810076521007652119898 119898

11989911989912059012059011990511990510076681007668

(119899119899minus119907119907)119907119907119889119889119905119905119889

(B2)

Likewise applying the integral formula int1199061199060 119909119909120583120583minus119898119889119889119909119909(119898 119898 119889119889119909119909)119907119907

= (119906119906120583120583120583120583)2119865119865119898(119907119907119907 120583120583119907 119898 119898 120583120583119907 minus119889119889119906119906) | arg(119898 119898 119889119889119906119906)| 120573 120573120573 Re 120583120583 119907 0[15 3194 Equation 1] we arrive at (19) by simplifying (B2)

References

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

[2] G Gao ldquoStatistical modeling of SAR images a surveyrdquo Sensorsvol 10 no 1 pp 775ndash795 2010

[3] A Achim E E Kuruolu and J Zerubia ldquoSAR image lteringbased on the heavy-tailed rayleigh modelrdquo IEEE Transactionson Image Processing vol 15 no 9 pp 2686ndash2693 2006

[4] T Esch M iel A Schenk A Roth A Muumlller and S DechldquoDelineation of Urban footprints from TerraSAR-X data byanalyzing speckle characteristics and intensity informationrdquoIEEE Transactions on Geoscience and Remote Sensing vol 48no 2 pp 905ndash916 2010

[5] 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

[6] C Tison J M Nicolas F Tupin and H Maicirctre ldquoA newstatistical model for Markovian classication of Urban areas inhigh-Resolution SAR imagesrdquo IEEE Transactions on Geoscienceand Remote Sensing vol 42 no 10 pp 2046ndash2057 2004

[7] A C Frery A H Correia and C D C Freitas ldquoClassify-ing multifrequency fully polarimetric imagery with multiplesources of statistical evidence and contextual informationrdquoIEEE Transactions on Geoscience and Remote Sensing vol 45no 10 pp 3098ndash3109 2007

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

[9] Y Delignon R Garello and A Hillion ldquoStatistical modeling ofocean SAR imagerdquo IEE Proceedings Radar Sonar and Naviga-tion vol 144 no 6 pp 348ndash354 1997

[10] Y Delignon and W Pieczynski ldquoModeling non-Rayleighspeckle distribution in SAR imagesrdquo IEEE Transactions onGeoscience and Remote Sensing vol 40 no 6 pp 1430ndash14352002

[11] A C Frery H-J Muumlller C D C F Yanasse and S J SSantrsquoAnna ldquoAmodel for extremely heterogeneous clutterrdquo IEEETransactions on Geoscience and Remote Sensing vol 35 no 3pp 648ndash659 1997

[12] K L P Vasconcellos A C Frery and L B Silva ldquoImprovingestimation in speckled imageryrdquo Computational Statistics vol20 no 3 pp 503ndash519 2005

[13] 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

[14] S N Annsen and T Elto ldquoApplication of the Matrix-variatemellin transform to analysis of polarimetric radar imagesrdquo IEEETransactions on Geoscience and Remote Sensing vol 49 no 6pp 2281ndash2295 2011

[15] S I Gradshteyn and I M Ryzhik Table of Integrals Series andProducts Academic Press San Diego Calif USA 7th edition2007

[16] G Moser J Zerubia and S B Serpico ldquoSAR amplitudeprobability density function estimation based on a generalizedGaussian modelrdquo IEEE Transactions on Image Processing vol15 no 6 pp 1429ndash1442 2006

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

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International Journal of Antennas and Propagation 3

0 05 1 15 2 25 3 35 4 45 50

02

04

06

08

1

12

14

Am

pli

tud

e P

DF

= minus 05

= minus 1

= minus 15

= 1 = minus 11 = 4 = 04

(a)

0 05 1 15 2 25 3 35 4 45 50

01

02

03

04

05

06

07

Am

pli

tud

e P

DF

= minus 2

= minus 15

= minus 1

(b)

0 05 1 15 2 25 3 35 4 45 50

10

20

30

40

50

60

70

80

90

Am

pli

tud

e P

DF

= 1

= 2

= 3

(c)

0 05 1 15 2 25 3 35 4 45 50

5

10

15

20

25

30

Am

pli

tud

e P

DF

= 05

= 06

= 07

(d)

F 1 Plots ofℋ119970119970119900119900119860119860 versus different parameters

e119898119898th order moments of the corresponding amplituderandom variable can easily be given by 119864119864119864119864119864119898119898119860119860) = 119864119864119864119864119864

119898119898119898119898119868119868 )

3 The Parameter Estimators ofthe Proposed Distribution

31e Log-Cumulants ofℋ1199701199700 Nicolas [13] has proposed aparametric PDF estimation technique based on theMoLC fora function dened over ℝ+ Given a positive-valued randomvariable 119883119883 with the PDF 119901119901119883119883119864119909119909) the second-ind rst and

second characteristic functions are respectively dened as[13]

120601120601119883119883 119864119904119904) = ℳ10077131007713119901119901119883119883 119864119909119909)10077291007729 119864119904119904) = 10045601004560infin

0119909119909119904119904119904119904119901119901119883119883 119864119909119909) 119889119889119909119909119889

120593120593119883119883 119864119904119904) = log 10076491007649120601120601119883119883 119864119904119904)10076651007665 119889(9)

where ℳ is the Mellin transform operator e 119903119903th orderderivative of 120593120593119883119883119864119904119904) at 119904119904 = 119904 is the log-cumulants of order 119903119903

4 International Journal of Antennas and Propagation

that is

119896119896119903119903 =119889119889119903119903120593120593119883119883 (119904119904)119889119889119904119904119903119903

10092541009254119904119904=119904 (10)

Hereaer we take the intensity distribution ℋ119970119970119900119900119868119868 as an

example to estimate its parameters 119907119907 120572120572 and 120582120582e processingof the amplitude one is similar Owing to 120601120601119883119883(119904119904) = 119904119904(119883119883119904119904119904119904)consequently via (8) and (9) the second-kind second char-acteristic functions of theℋ119970119970119900119900

119868119868 distribution yields

120593120593ℋ119970119970119900119900119868119868(119904119904) = log (120590120590) + log 10076501007650120582120582119907119907120572120572119904(120572120572119907119907119907)10076661007666

+119904 119904 119904119904119907119907

log (120582120582) + log 10076501007650119899119899119899119899+(119899119899119907119907119907)10076661007666 +119904119904 119904 119904119907119907

log (119899119899)

+ log 10076521007652Γ 10076521007652119904119904119904 119904 119904 + 119899119899

1199071199071007668100766810076681007668 + log 10076521007652Γ 10076521007652

120572120572 + 119904119904 119904 119904119907119907

1007668100766810076681007668

119904 log 100765010076501199071199072Γ (119907119907120572120572) Γ (119899119899)10076661007666(11)

which leads to that the log-cumulants of the ℋ119970119970119900119900119868119868 are

expressed as

119896119896119904 =log (119899119899119907120582120582) 119904 Ψ (119904119899119899119907119907119907) + Ψ (120572120572119907119907119907)

119907119907

119896119896119903119903 =Ψ (119903119903 119904 119904 119904119899119899119907119907119907)

(119904119907119907)119903119903+Ψ (119903119903 119904 119904 120572120572119907119907119907)

119907119907119903119903 119903119903 119903 2

(12)

where Ψ(sdot) represents the digamma function and Ψ(119903119903 sdot) isthe 119903119903th order polygamma function

Given a sample set 119911119911119894119894 119894119894 119894 119894119904119894119894119894 the log-cumulants canbe estimated directly by

10065931006593119896119896119904 =119904119894119894

11989411989410055761005576119894119894=11990410077131007713ln 100764910076491199111199111198941198941007665100766510077291007729

10065931006593119896119896119904 =119904119894119894

11989411989410055761005576119894119894=1199041007715100771510076501007650ln 1007649100764911991111991111989411989410076651007665 119904 1006593100659311989611989611990410076661007666

11990311990310077311007731 119903119903 119903 2

(13)

32 e Parameter Estimators of ℋ1199701199700 We notice that (12)is independent of the parameter 120590120590 and the 119903119903th log-cumulantsshown in (12) are irrespective with the parameter 120582120582 on thecondition that 119903119903 119903 2 In addition we regard the parameter 119899119899as a known constant which can be replaced by the equivalentnumber of looks (ENLs) [4 11] or obtained from some priorknowledge about processed SAR images hence allowing usto divide the parameter estimates to three stages

First the estimates 10065931006593119907119907 and 10065931006593120572120572 of the parameters 119907119907 as well as120572120572 are obtained by solving the following equations resorting tothe numerical computation

100659310065931198961198962 =Ψ (119904 11990411989911989911990710065931006593119907119907) + Ψ (119904 1006593100659312057212057211990710065931006593119907119907)

100659310065931199071199072

100659310065931198961198963 =Ψ (2 1006593100659312057212057211990710065931006593119907119907 ) 119904 Ψ (2 11990411989911989911990710065931006593119907119907 )

100659310065931199071199073

(14)

Second according to the rst equation of (12) theestimate 10065931006593120582120582 of the parameter 120582120582 is

10065931006593120582120582 =119899119899

exp 100765010076501006593100659311990711990710065931006593119896119896119904 + Ψ (11990411989911989911990710065931006593119907119907) 119904 Ψ (1006593100659312057212057211990710065931006593119907119907)10076661007666 (15)

We assume the observed amplitude PDF is 10065931006593119901119901119885119885119860119860(119911119911) from

the actual data which corresponds to the histogram of testeddata e theoretical amplitude PDF denotes as 119901119901119885119885119860119860

(119911119911) Asthe previous analysis since 120590120590 indicates the whole proportionuctuation of 119901119901119885119885119860119860

(119911119911) 119901119901119885119885119860119860(119911119911)119907120590120590 can be calculated by using 10065931006593119907119907

10065931006593120572120572 and 10065931006593120582120582 Given a sample amplitude set 119911119911119894119894 ∣ 119894119894 = 119904119894119894119894 letthe symbol 119902119902119885119885119860119860

(119911119911) be equal to 119901119901119885119885119860119860(119911119911)119907120590120590 then the estimate 10065931006593120590120590

of the parameter 120590120590 is simply given by

10065931006593120590120590 = 1199041199041007655100765510065931006593119901119901119885119885119860119860

1007649100764911991111991111989411989410076651007665119902119902119885119885119860119860

100764910076491199111199111198941198941007665100766510076711007671 119911119911119894119894 119894 ℤ (16)

where ℤ is a subset of 119911119911119894119894 ∣ 119894119894 = 119904119894119894119894 Here in order tofacilitate the stable tting we choose empirically all pointsof histogram within 3 dB bandwidth to calculate 120590120590 such thatℤ = 119911119911119894119894 ∣ 10065931006593119901119901119885119885119860119860

(119911119911119894119894) gt 10065931006593119901119901max1199072 10065931006593119901119901max = max( 10065931006593119901119901119885119885119860119860(119911119911119894119894)) 119894119894 119894

119894119904119894119894119894

4 Experimental Results

In this section we aim at verifying the performance of theproposed ℋ1199701199700 In order to assess how the ℋ1199701199700 performsseveral space-borne TerraSAR-X geocoded scenes with var-ious land-over typologies as examples are reported Figures2(a)ndash2(d) show four selected typical patches from a largeldquoSanchagangrdquo TerraSAR-X image with low resolution whichare the portions of water body drying riverbed mountainand a town returns respectively e four types of scenesare related to the homogeneous and heterogeneous terrainsFor simplicity we denote them as ldquowater-bodyrdquo ldquodryingriverbedrdquo ldquomountainrdquo and ldquotownrdquo Additionally two urbanareas shown in Figures 2(e)-2(f) extracted from a largeldquoBeijingrdquo TerraSAR-X scene with high resolution are furthercarried out to demonstrate the effectiveness of the proposedmodel on the extremely heterogeneous terrains Likewise thetwo urban images are denoted by ldquourban1rdquo and ldquourban2rdquoemain parameters of TerraSAR-X systems for ldquoSanchagangrdquoand ldquoBeijingrdquo data are listed in Table 1

e estimated PDFs of the proposedℋ1199701199700 model for thehistograms of six selected areas indicated in Figure 2 areshown in Figure 3 As the corresponding comparisons thetting results with the intensively used 1199701199700 distribution arealso provided where parameter estimates of this distributionare derived by Tison et al [6] based on theMoLC Herein theestimate 119899119899 inℋ1199701199700 is rst replaced by the ENL that is [4]

10065931006593119899119899 = 119904119904119894119894119899119899 = 119899119899119886119886 + 119899119899119903119903 (17)

where 119899119899119886119886 and 119899119899119903119903 are the effective number of looks in theazimuth and range For the previous ldquoSanchagangrdquo andldquoBeijingrdquo TerraSAR-X images used in this investigation 119899119899119886119886and 119899119899119903119903 are listed in the last two columns of Table 1 according

International Journal of Antennas and Propagation 5

(a) (b)

(c) (d)

(e) (f)

F 2 e Tested TerraSAR-X SAR image (a) ldquoWater bodyrdquo patch with a size of 332 px times 322 px (b) ldquodrying riverbedrdquo image (size 1597times 1734 pixels) (c) ldquomountainrdquo image (size 1996 times 2034 pixels) (d) ldquotownrdquo image (size 525 times 560 pixels) (e) ldquourban1rdquo image (size 1198 times1342 pixels) and (e) ldquourban2rdquo image (size 705 times 728 pixels)

6 International Journal of Antennas and Propagation

T 1 e main parameters of TerraSAR-X space-borne systems for the data in this study

Testing site Acquisitionmode Polarization Resolution

(az times rg) Date Size Azimuthlooks

Rangelooks

Beijing China SpotLight HH 192m times 226m 2008-01-31T1003 UTC 21333 px times 20000 px 119 102Sanchagang China StripMap VV 611m times 598m 2008-01-01T2220 UTC 11909 px times 22363 px 102 267

T 2 Parameter estimations of noted areas in Figure 2

Area ℋ119970119970∘119860119860 (120572120572120572 120572120572120572 120572120572120572 120572120572120572 120572120572120572

ldquoWater bodyrdquo (minus66029 minus11001 63048 10709 3683)ldquoDrying riverbedrdquo (minus60382 minus10555 56813 11897 3683)ldquoMountainrdquo (minus29788 minus05779 24694 01426 3683)ldquoTownrdquo (minus5062 minus07271 45872 04861 3683)ldquoUrban1rdquo (minus11774 minus04895 06905 111945 222)ldquoUrban2rdquo (minus08641 minus07065 03227 23517 222)

to reading their metadata les ext the estimates of allother parameters inℋ1199701199700 are accomplished by the estimatorsderived in Section 3 which are shown at Table 2

Furthermore in order to quantitatively assess the perfor-mances that different distributions t we dene an error ratiofactor (ERF) as

120575120575 120575sum119873119873119894119894120575119894 10092651009265119901119901119888119888 1007649100764911991111991111989411989410076651007665 minus 119902119902 100764910076491199111199111198941198941007665100766510092651009265

2

sum119873119873119894119894120575119894 10092651009265119901119901119887119887 1007649100764911991111991111989411989410076651007665 minus 119902119902 100764910076491199111199111198941198941007665100766510092651009265

2 120572 (18)

where 119911119911119894119894 ∣ 119894119894 120575 119894119894119873119873119894 is a sample set 119901119901119888119888(sdot120572 represents thecompared theoretical PDF 119901119901119887119887(sdot120572 is the basic theoretical PDFand 119902119902(sdot120572 indicates the actual PDF from the observed dataesymbol sdot denotes 2-norm It is obvious that the numeratorand denominator in (18) separately imply the total ttingerrors with the 119901119901119888119888 and 119901119901119887119887 to approximate 119902119902 e better theperformances of 119901119901119888119888 related to 119901119901119887119887 t the smaller 120575120575 is and viceverse Specically 120575120575 120575 119894 if both 119901119901119888119888 and 119901119901119887119887 have the identicalcapability for tting the measured data

Letℋ119970119970119900119900119860119860 represent the basic amplitude PDF and let 119970119970119900119900

119860119860be the compared amplitude one e 120575120575 values of the previoussix areas in this study are given in Figure 3 (see the yellowtextbox) It can be clearly seen that the proposedℋ119970119970119900119900

119860119860 betteragrees with the amplitude histograms of all six terrains thanthe 119970119970119900119900

119860119860 as expected because the 120575120575 values are larger than1 for all six areas e same conclusion can be conrmedfrom a visual point of view and implies the higher ttingprecision using ℋ119970119970119900119900 than using 1199701199700 over homogeneousheterogeneous and extremely heterogeneous regions

5 Conclusion and Perspective

We have developed an empirical model ℋ1199701199700 to exploitthe knowledge of statistical characteristics of SAR ampli-tude or intensity images over the wide terrain classes withhomogeneous heterogeneous and extremely heterogeneousbackscattering properties e parameter estimators of this

model based on the MoLC are also provided Consequentlywe report the performances of different land-over typologieswith ℋ1199701199700 distribution ts e experimental results showthat the ℋ1199701199700 distribution is a more advanced model com-pared with the known 1199701199700 distribution to characterize themultilook processed SAR data

As we know a preliminary statistical analysis of SARclutter data is important for designing signal processing algo-rithms such as speckle ltering target detection buildingextraction image segmentation and classication In futureit is worth expecting to use the ℋ1199701199700 distribution in theseelds Herein we rstly attempt to give an analytical deriva-tion for constructing a constant false alarm rate (CFAR)detector to promote the upcoming studies of target detectionin SAR images

Given the propose amplitude modelℋ119970119970119900119900119860119860 shown in (1)

its cumulative distribution function (CDF) is written as (seeAppendix B)

119865119865ℋ119970119970119900119900119860119860(119909119909120572 120575 minus

120572120572120572120572120572120572120572120572119907(120572120572minus120572120572120572119907120572120572120572120572120572120572minus119894Γ ((120572120572 minus 120572120572120572 119907120572120572120572120572120572Γ (120572120572120572120572120572 Γ (120572120572120572

1199091199092120572120572

times 2119865119865119894 10076521007652120572120572 minus 120572120572120572120572

120572 minus120572120572120572120572 119894 minus

120572120572120572120572 minus120572120572120572120572119909119909minus212057212057210076681007668 120572

minus 120572120572120572 minus120572120572120572 120572120572120572 120572120572120572 120572120572120572 119909119909 120572 0120572

(19)

where 2119865119865119894(sdot120572 sdot sdot sdot120572 is the Gauss hypergeometric function For agiven value of the false alarm probability denoted by 119875119875119891119891119891119891 thecorresponding CFAR threshold 119879119879 for theℋ119970119970119900119900

119860119860 distributioncan be obtained from

119894 minus 119875119875119891119891119891119891 120575 119865119865ℋ119970119970119900119900119860119860(119879119879120572

120575 minus120572120572120572120572120572120572120572120572119907(120572120572minus120572120572120572119907120572120572120572120572120572120572minus119894Γ ((120572120572 minus 120572120572120572 119907120572120572120572

120572120572Γ (120572120572120572120572120572 Γ (1205721205721205721198791198792120572120572

times 2119865119865119894 10076521007652120572120572 minus 120572120572120572120572

120572 minus120572120572120572120572 119894 minus

120572120572120572120572 minus120572120572120572120572119879119879minus212057212057210076681007668

(20)

Considering 119865119865ℋ119970119970119900119900119860119860(119879119879120572 is strictly monotonously increas-

ing the threshold119879119879 can be accurately calculatedwith the helpof the numerical solution or a simple bisection method [16]

Our future work will focus on demonstrating the perfor-mances of the proposed CFAR detector using somemeasuredSAR data and investigating how the ℋ1199701199700 distribution per-forms when extending it to other application elds

International Journal of Antennas and Propagation 7

0 05 1 15 2 25 3 350

02

04

06

08

1

12

14

Amplitude

PD

FWater body

= 10042

(a)

0 05 1 15 2 25 3 350

05

1

15

Amplitude

PD

F

Drying riverbed

= 35351

(b)

0 05 1 15 2 25 3 350

02

04

06

08

1

12

14

Amplitude

PDF

Mountain

= 23306

(c)

0 05 1 15 2 25 3 350

02

04

06

08

1

12

14

Amplitude

PDF

Town

= 26756

(d)

0 05 1 15 2 25 3 350

02

04

06

08

1

12

14

Amplitude

PD

F

Urban1

= 1944

Histogram

The proposed model

(e)

0 05 1 15 2 25 3 350

02

04

06

08

1

12

14

Amplitude

PD

F

Urban2

= 13562

Histogram

The proposed model

(f)

F 3 Plots of aplitude istogras and of te estiated PFs (a)ndash(f) are corresponded to te tting results for te scenes son inFigures 2(a)ndash2(f) respectively

8 International Journal of Antennas and Propagation

Appendices

A The Derivation of119898119898th OrderMoments oftheℋ119970119970119900119900

119868119868 Distribution

Via (2) the119898119898th order moments of theℋ119970119970119900119900119868119868 are expressed as

119864119864 10076491007649119911119911119898119898119868119868 10076651007665 = 10045601004560infin

0119911119911119898119898119901119901119885119885119868119868

(119911119911) 119889119889119911119911

= minus120590120590120590120590119907119907119907119907119899119899119899119899Γ ((119907119907 minus 119899119899) 119907119907)

119907119907Γ (119907119907119907119907) Γ (119899119899)

times 120590120590(119899119899minus119907119907)119907119907 10045601004560infin

0119911119911119898119898119898119899119899minus11989810076521007652119898 119898

119899119899120590120590119911119911minus11990711990710076681007668

(119899119899minus119907119907)119907119907119889119889119911119911119889

(A1)

A variable replacement 119909119909 = 119911119911minus119907119907 leads to 119911119911 = 119909119909minus119898119907119907 and119889119889119911119911 = minus(119898119907119907)119909119909minus(119898119907119907)minus119898119889119889119909119909 thus (A1) can be rewritten as

119864119864 10076491007649119911119911119898119898119868119868 10076651007665 = minus120590120590120590120590119907119907119907119907119899119899119899119899Γ ((119907119907 minus 119899119899) 119907119907)

1199071199072Γ (119907119907119907119907) Γ (119899119899)

times 120590120590(119899119899minus119907119907)119907119907 10045601004560infin

0119909119909((119898119898119898119899119899)119907119907)minus11989810076521007652119898 119898

11989911989912059012059011990911990910076681007668

(119899119899minus119907119907)119907119907119889119889119909119909119889

(A2)

According to the integral formula intinfin0 119909119909120583120583minus119898119889119889119909119909(119898119898119889119889119909119909)119907119907 =119889119889minus120583120583Γ(120583120583)Γ(119907119907minus120583120583)Γ(119907119907) | arg 119889119889| 120573 120573120573Re 119907119907 119907 Re 120583120583 119907 0 [15 3194Equation 3] one can obtain (8)

B The Derivation of the CumulativeDistribution Function ofℋ119970119970119900119900

119860119860

Based on the denition of the CDF the CDF ofℋ119970119970119900119900119860119860 is

119865119865ℋ119970119970119900119900119860119860(119909119909) = 10045601004560

119909119909

minusinfin119901119901119885119885119860119860

(119911119911) 119889119889119911119911 = 10045601004560119909119909

0119901119901119885119885119860119860

(119911119911) 119889119889119911119911

= minus2120590120590120590120590119907119907119907119907119899119899119899119899Γ ((119907119907 minus 119899119899) 119907119907)

119907119907Γ (119907119907119907119907) Γ (119899119899)

times 120590120590(119899119899minus119907119907)119907119907 10045601004560119909119909

01199111199112119899119899minus11989810076521007652119898 119898

119899119899120590120590119911119911minus211990711990710076681007668

(119899119899minus119907119907)119907119907119889119889119911119911119889

(B1)

Similarly utilizing a symbol change 119905119905 = 119911119911minus2119907119907 (B1) turnsout to be

119865119865ℋ119970119970119900119900119860119860(119909119909) =

120590120590120590120590119907119907119907119907119899119899119899119899Γ ((119907119907 minus 119899119899) 119907119907)1199071199072Γ (119907119907119907119907) Γ (119899119899)

times 120590120590(119899119899minus119907119907)119907119907 10045601004560119909119909minus2119907119907

0119905119905(minus119899119899119907119907)minus11989810076521007652119898 119898

11989911989912059012059011990511990510076681007668

(119899119899minus119907119907)119907119907119889119889119905119905119889

(B2)

Likewise applying the integral formula int1199061199060 119909119909120583120583minus119898119889119889119909119909(119898 119898 119889119889119909119909)119907119907

= (119906119906120583120583120583120583)2119865119865119898(119907119907119907 120583120583119907 119898 119898 120583120583119907 minus119889119889119906119906) | arg(119898 119898 119889119889119906119906)| 120573 120573120573 Re 120583120583 119907 0[15 3194 Equation 1] we arrive at (19) by simplifying (B2)

References

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

[2] G Gao ldquoStatistical modeling of SAR images a surveyrdquo Sensorsvol 10 no 1 pp 775ndash795 2010

[3] A Achim E E Kuruolu and J Zerubia ldquoSAR image lteringbased on the heavy-tailed rayleigh modelrdquo IEEE Transactionson Image Processing vol 15 no 9 pp 2686ndash2693 2006

[4] T Esch M iel A Schenk A Roth A Muumlller and S DechldquoDelineation of Urban footprints from TerraSAR-X data byanalyzing speckle characteristics and intensity informationrdquoIEEE Transactions on Geoscience and Remote Sensing vol 48no 2 pp 905ndash916 2010

[5] 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

[6] C Tison J M Nicolas F Tupin and H Maicirctre ldquoA newstatistical model for Markovian classication of Urban areas inhigh-Resolution SAR imagesrdquo IEEE Transactions on Geoscienceand Remote Sensing vol 42 no 10 pp 2046ndash2057 2004

[7] A C Frery A H Correia and C D C Freitas ldquoClassify-ing multifrequency fully polarimetric imagery with multiplesources of statistical evidence and contextual informationrdquoIEEE Transactions on Geoscience and Remote Sensing vol 45no 10 pp 3098ndash3109 2007

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

[9] Y Delignon R Garello and A Hillion ldquoStatistical modeling ofocean SAR imagerdquo IEE Proceedings Radar Sonar and Naviga-tion vol 144 no 6 pp 348ndash354 1997

[10] Y Delignon and W Pieczynski ldquoModeling non-Rayleighspeckle distribution in SAR imagesrdquo IEEE Transactions onGeoscience and Remote Sensing vol 40 no 6 pp 1430ndash14352002

[11] A C Frery H-J Muumlller C D C F Yanasse and S J SSantrsquoAnna ldquoAmodel for extremely heterogeneous clutterrdquo IEEETransactions on Geoscience and Remote Sensing vol 35 no 3pp 648ndash659 1997

[12] K L P Vasconcellos A C Frery and L B Silva ldquoImprovingestimation in speckled imageryrdquo Computational Statistics vol20 no 3 pp 503ndash519 2005

[13] 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

[14] S N Annsen and T Elto ldquoApplication of the Matrix-variatemellin transform to analysis of polarimetric radar imagesrdquo IEEETransactions on Geoscience and Remote Sensing vol 49 no 6pp 2281ndash2295 2011

[15] S I Gradshteyn and I M Ryzhik Table of Integrals Series andProducts Academic Press San Diego Calif USA 7th edition2007

[16] G Moser J Zerubia and S B Serpico ldquoSAR amplitudeprobability density function estimation based on a generalizedGaussian modelrdquo IEEE Transactions on Image Processing vol15 no 6 pp 1429ndash1442 2006

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DistributedSensor Networks

International Journal of

4 International Journal of Antennas and Propagation

that is

119896119896119903119903 =119889119889119903119903120593120593119883119883 (119904119904)119889119889119904119904119903119903

10092541009254119904119904=119904 (10)

Hereaer we take the intensity distribution ℋ119970119970119900119900119868119868 as an

example to estimate its parameters 119907119907 120572120572 and 120582120582e processingof the amplitude one is similar Owing to 120601120601119883119883(119904119904) = 119904119904(119883119883119904119904119904119904)consequently via (8) and (9) the second-kind second char-acteristic functions of theℋ119970119970119900119900

119868119868 distribution yields

120593120593ℋ119970119970119900119900119868119868(119904119904) = log (120590120590) + log 10076501007650120582120582119907119907120572120572119904(120572120572119907119907119907)10076661007666

+119904 119904 119904119904119907119907

log (120582120582) + log 10076501007650119899119899119899119899+(119899119899119907119907119907)10076661007666 +119904119904 119904 119904119907119907

log (119899119899)

+ log 10076521007652Γ 10076521007652119904119904119904 119904 119904 + 119899119899

1199071199071007668100766810076681007668 + log 10076521007652Γ 10076521007652

120572120572 + 119904119904 119904 119904119907119907

1007668100766810076681007668

119904 log 100765010076501199071199072Γ (119907119907120572120572) Γ (119899119899)10076661007666(11)

which leads to that the log-cumulants of the ℋ119970119970119900119900119868119868 are

expressed as

119896119896119904 =log (119899119899119907120582120582) 119904 Ψ (119904119899119899119907119907119907) + Ψ (120572120572119907119907119907)

119907119907

119896119896119903119903 =Ψ (119903119903 119904 119904 119904119899119899119907119907119907)

(119904119907119907)119903119903+Ψ (119903119903 119904 119904 120572120572119907119907119907)

119907119907119903119903 119903119903 119903 2

(12)

where Ψ(sdot) represents the digamma function and Ψ(119903119903 sdot) isthe 119903119903th order polygamma function

Given a sample set 119911119911119894119894 119894119894 119894 119894119904119894119894119894 the log-cumulants canbe estimated directly by

10065931006593119896119896119904 =119904119894119894

11989411989410055761005576119894119894=11990410077131007713ln 100764910076491199111199111198941198941007665100766510077291007729

10065931006593119896119896119904 =119904119894119894

11989411989410055761005576119894119894=1199041007715100771510076501007650ln 1007649100764911991111991111989411989410076651007665 119904 1006593100659311989611989611990410076661007666

11990311990310077311007731 119903119903 119903 2

(13)

32 e Parameter Estimators of ℋ1199701199700 We notice that (12)is independent of the parameter 120590120590 and the 119903119903th log-cumulantsshown in (12) are irrespective with the parameter 120582120582 on thecondition that 119903119903 119903 2 In addition we regard the parameter 119899119899as a known constant which can be replaced by the equivalentnumber of looks (ENLs) [4 11] or obtained from some priorknowledge about processed SAR images hence allowing usto divide the parameter estimates to three stages

First the estimates 10065931006593119907119907 and 10065931006593120572120572 of the parameters 119907119907 as well as120572120572 are obtained by solving the following equations resorting tothe numerical computation

100659310065931198961198962 =Ψ (119904 11990411989911989911990710065931006593119907119907) + Ψ (119904 1006593100659312057212057211990710065931006593119907119907)

100659310065931199071199072

100659310065931198961198963 =Ψ (2 1006593100659312057212057211990710065931006593119907119907 ) 119904 Ψ (2 11990411989911989911990710065931006593119907119907 )

100659310065931199071199073

(14)

Second according to the rst equation of (12) theestimate 10065931006593120582120582 of the parameter 120582120582 is

10065931006593120582120582 =119899119899

exp 100765010076501006593100659311990711990710065931006593119896119896119904 + Ψ (11990411989911989911990710065931006593119907119907) 119904 Ψ (1006593100659312057212057211990710065931006593119907119907)10076661007666 (15)

We assume the observed amplitude PDF is 10065931006593119901119901119885119885119860119860(119911119911) from

the actual data which corresponds to the histogram of testeddata e theoretical amplitude PDF denotes as 119901119901119885119885119860119860

(119911119911) Asthe previous analysis since 120590120590 indicates the whole proportionuctuation of 119901119901119885119885119860119860

(119911119911) 119901119901119885119885119860119860(119911119911)119907120590120590 can be calculated by using 10065931006593119907119907

10065931006593120572120572 and 10065931006593120582120582 Given a sample amplitude set 119911119911119894119894 ∣ 119894119894 = 119904119894119894119894 letthe symbol 119902119902119885119885119860119860

(119911119911) be equal to 119901119901119885119885119860119860(119911119911)119907120590120590 then the estimate 10065931006593120590120590

of the parameter 120590120590 is simply given by

10065931006593120590120590 = 1199041199041007655100765510065931006593119901119901119885119885119860119860

1007649100764911991111991111989411989410076651007665119902119902119885119885119860119860

100764910076491199111199111198941198941007665100766510076711007671 119911119911119894119894 119894 ℤ (16)

where ℤ is a subset of 119911119911119894119894 ∣ 119894119894 = 119904119894119894119894 Here in order tofacilitate the stable tting we choose empirically all pointsof histogram within 3 dB bandwidth to calculate 120590120590 such thatℤ = 119911119911119894119894 ∣ 10065931006593119901119901119885119885119860119860

(119911119911119894119894) gt 10065931006593119901119901max1199072 10065931006593119901119901max = max( 10065931006593119901119901119885119885119860119860(119911119911119894119894)) 119894119894 119894

119894119904119894119894119894

4 Experimental Results

In this section we aim at verifying the performance of theproposed ℋ1199701199700 In order to assess how the ℋ1199701199700 performsseveral space-borne TerraSAR-X geocoded scenes with var-ious land-over typologies as examples are reported Figures2(a)ndash2(d) show four selected typical patches from a largeldquoSanchagangrdquo TerraSAR-X image with low resolution whichare the portions of water body drying riverbed mountainand a town returns respectively e four types of scenesare related to the homogeneous and heterogeneous terrainsFor simplicity we denote them as ldquowater-bodyrdquo ldquodryingriverbedrdquo ldquomountainrdquo and ldquotownrdquo Additionally two urbanareas shown in Figures 2(e)-2(f) extracted from a largeldquoBeijingrdquo TerraSAR-X scene with high resolution are furthercarried out to demonstrate the effectiveness of the proposedmodel on the extremely heterogeneous terrains Likewise thetwo urban images are denoted by ldquourban1rdquo and ldquourban2rdquoemain parameters of TerraSAR-X systems for ldquoSanchagangrdquoand ldquoBeijingrdquo data are listed in Table 1

e estimated PDFs of the proposedℋ1199701199700 model for thehistograms of six selected areas indicated in Figure 2 areshown in Figure 3 As the corresponding comparisons thetting results with the intensively used 1199701199700 distribution arealso provided where parameter estimates of this distributionare derived by Tison et al [6] based on theMoLC Herein theestimate 119899119899 inℋ1199701199700 is rst replaced by the ENL that is [4]

10065931006593119899119899 = 119904119904119894119894119899119899 = 119899119899119886119886 + 119899119899119903119903 (17)

where 119899119899119886119886 and 119899119899119903119903 are the effective number of looks in theazimuth and range For the previous ldquoSanchagangrdquo andldquoBeijingrdquo TerraSAR-X images used in this investigation 119899119899119886119886and 119899119899119903119903 are listed in the last two columns of Table 1 according

International Journal of Antennas and Propagation 5

(a) (b)

(c) (d)

(e) (f)

F 2 e Tested TerraSAR-X SAR image (a) ldquoWater bodyrdquo patch with a size of 332 px times 322 px (b) ldquodrying riverbedrdquo image (size 1597times 1734 pixels) (c) ldquomountainrdquo image (size 1996 times 2034 pixels) (d) ldquotownrdquo image (size 525 times 560 pixels) (e) ldquourban1rdquo image (size 1198 times1342 pixels) and (e) ldquourban2rdquo image (size 705 times 728 pixels)

6 International Journal of Antennas and Propagation

T 1 e main parameters of TerraSAR-X space-borne systems for the data in this study

Testing site Acquisitionmode Polarization Resolution

(az times rg) Date Size Azimuthlooks

Rangelooks

Beijing China SpotLight HH 192m times 226m 2008-01-31T1003 UTC 21333 px times 20000 px 119 102Sanchagang China StripMap VV 611m times 598m 2008-01-01T2220 UTC 11909 px times 22363 px 102 267

T 2 Parameter estimations of noted areas in Figure 2

Area ℋ119970119970∘119860119860 (120572120572120572 120572120572120572 120572120572120572 120572120572120572 120572120572120572

ldquoWater bodyrdquo (minus66029 minus11001 63048 10709 3683)ldquoDrying riverbedrdquo (minus60382 minus10555 56813 11897 3683)ldquoMountainrdquo (minus29788 minus05779 24694 01426 3683)ldquoTownrdquo (minus5062 minus07271 45872 04861 3683)ldquoUrban1rdquo (minus11774 minus04895 06905 111945 222)ldquoUrban2rdquo (minus08641 minus07065 03227 23517 222)

to reading their metadata les ext the estimates of allother parameters inℋ1199701199700 are accomplished by the estimatorsderived in Section 3 which are shown at Table 2

Furthermore in order to quantitatively assess the perfor-mances that different distributions t we dene an error ratiofactor (ERF) as

120575120575 120575sum119873119873119894119894120575119894 10092651009265119901119901119888119888 1007649100764911991111991111989411989410076651007665 minus 119902119902 100764910076491199111199111198941198941007665100766510092651009265

2

sum119873119873119894119894120575119894 10092651009265119901119901119887119887 1007649100764911991111991111989411989410076651007665 minus 119902119902 100764910076491199111199111198941198941007665100766510092651009265

2 120572 (18)

where 119911119911119894119894 ∣ 119894119894 120575 119894119894119873119873119894 is a sample set 119901119901119888119888(sdot120572 represents thecompared theoretical PDF 119901119901119887119887(sdot120572 is the basic theoretical PDFand 119902119902(sdot120572 indicates the actual PDF from the observed dataesymbol sdot denotes 2-norm It is obvious that the numeratorand denominator in (18) separately imply the total ttingerrors with the 119901119901119888119888 and 119901119901119887119887 to approximate 119902119902 e better theperformances of 119901119901119888119888 related to 119901119901119887119887 t the smaller 120575120575 is and viceverse Specically 120575120575 120575 119894 if both 119901119901119888119888 and 119901119901119887119887 have the identicalcapability for tting the measured data

Letℋ119970119970119900119900119860119860 represent the basic amplitude PDF and let 119970119970119900119900

119860119860be the compared amplitude one e 120575120575 values of the previoussix areas in this study are given in Figure 3 (see the yellowtextbox) It can be clearly seen that the proposedℋ119970119970119900119900

119860119860 betteragrees with the amplitude histograms of all six terrains thanthe 119970119970119900119900

119860119860 as expected because the 120575120575 values are larger than1 for all six areas e same conclusion can be conrmedfrom a visual point of view and implies the higher ttingprecision using ℋ119970119970119900119900 than using 1199701199700 over homogeneousheterogeneous and extremely heterogeneous regions

5 Conclusion and Perspective

We have developed an empirical model ℋ1199701199700 to exploitthe knowledge of statistical characteristics of SAR ampli-tude or intensity images over the wide terrain classes withhomogeneous heterogeneous and extremely heterogeneousbackscattering properties e parameter estimators of this

model based on the MoLC are also provided Consequentlywe report the performances of different land-over typologieswith ℋ1199701199700 distribution ts e experimental results showthat the ℋ1199701199700 distribution is a more advanced model com-pared with the known 1199701199700 distribution to characterize themultilook processed SAR data

As we know a preliminary statistical analysis of SARclutter data is important for designing signal processing algo-rithms such as speckle ltering target detection buildingextraction image segmentation and classication In futureit is worth expecting to use the ℋ1199701199700 distribution in theseelds Herein we rstly attempt to give an analytical deriva-tion for constructing a constant false alarm rate (CFAR)detector to promote the upcoming studies of target detectionin SAR images

Given the propose amplitude modelℋ119970119970119900119900119860119860 shown in (1)

its cumulative distribution function (CDF) is written as (seeAppendix B)

119865119865ℋ119970119970119900119900119860119860(119909119909120572 120575 minus

120572120572120572120572120572120572120572120572119907(120572120572minus120572120572120572119907120572120572120572120572120572120572minus119894Γ ((120572120572 minus 120572120572120572 119907120572120572120572120572120572Γ (120572120572120572120572120572 Γ (120572120572120572

1199091199092120572120572

times 2119865119865119894 10076521007652120572120572 minus 120572120572120572120572

120572 minus120572120572120572120572 119894 minus

120572120572120572120572 minus120572120572120572120572119909119909minus212057212057210076681007668 120572

minus 120572120572120572 minus120572120572120572 120572120572120572 120572120572120572 120572120572120572 119909119909 120572 0120572

(19)

where 2119865119865119894(sdot120572 sdot sdot sdot120572 is the Gauss hypergeometric function For agiven value of the false alarm probability denoted by 119875119875119891119891119891119891 thecorresponding CFAR threshold 119879119879 for theℋ119970119970119900119900

119860119860 distributioncan be obtained from

119894 minus 119875119875119891119891119891119891 120575 119865119865ℋ119970119970119900119900119860119860(119879119879120572

120575 minus120572120572120572120572120572120572120572120572119907(120572120572minus120572120572120572119907120572120572120572120572120572120572minus119894Γ ((120572120572 minus 120572120572120572 119907120572120572120572

120572120572Γ (120572120572120572120572120572 Γ (1205721205721205721198791198792120572120572

times 2119865119865119894 10076521007652120572120572 minus 120572120572120572120572

120572 minus120572120572120572120572 119894 minus

120572120572120572120572 minus120572120572120572120572119879119879minus212057212057210076681007668

(20)

Considering 119865119865ℋ119970119970119900119900119860119860(119879119879120572 is strictly monotonously increas-

ing the threshold119879119879 can be accurately calculatedwith the helpof the numerical solution or a simple bisection method [16]

Our future work will focus on demonstrating the perfor-mances of the proposed CFAR detector using somemeasuredSAR data and investigating how the ℋ1199701199700 distribution per-forms when extending it to other application elds

International Journal of Antennas and Propagation 7

0 05 1 15 2 25 3 350

02

04

06

08

1

12

14

Amplitude

PD

FWater body

= 10042

(a)

0 05 1 15 2 25 3 350

05

1

15

Amplitude

PD

F

Drying riverbed

= 35351

(b)

0 05 1 15 2 25 3 350

02

04

06

08

1

12

14

Amplitude

PDF

Mountain

= 23306

(c)

0 05 1 15 2 25 3 350

02

04

06

08

1

12

14

Amplitude

PDF

Town

= 26756

(d)

0 05 1 15 2 25 3 350

02

04

06

08

1

12

14

Amplitude

PD

F

Urban1

= 1944

Histogram

The proposed model

(e)

0 05 1 15 2 25 3 350

02

04

06

08

1

12

14

Amplitude

PD

F

Urban2

= 13562

Histogram

The proposed model

(f)

F 3 Plots of aplitude istogras and of te estiated PFs (a)ndash(f) are corresponded to te tting results for te scenes son inFigures 2(a)ndash2(f) respectively

8 International Journal of Antennas and Propagation

Appendices

A The Derivation of119898119898th OrderMoments oftheℋ119970119970119900119900

119868119868 Distribution

Via (2) the119898119898th order moments of theℋ119970119970119900119900119868119868 are expressed as

119864119864 10076491007649119911119911119898119898119868119868 10076651007665 = 10045601004560infin

0119911119911119898119898119901119901119885119885119868119868

(119911119911) 119889119889119911119911

= minus120590120590120590120590119907119907119907119907119899119899119899119899Γ ((119907119907 minus 119899119899) 119907119907)

119907119907Γ (119907119907119907119907) Γ (119899119899)

times 120590120590(119899119899minus119907119907)119907119907 10045601004560infin

0119911119911119898119898119898119899119899minus11989810076521007652119898 119898

119899119899120590120590119911119911minus11990711990710076681007668

(119899119899minus119907119907)119907119907119889119889119911119911119889

(A1)

A variable replacement 119909119909 = 119911119911minus119907119907 leads to 119911119911 = 119909119909minus119898119907119907 and119889119889119911119911 = minus(119898119907119907)119909119909minus(119898119907119907)minus119898119889119889119909119909 thus (A1) can be rewritten as

119864119864 10076491007649119911119911119898119898119868119868 10076651007665 = minus120590120590120590120590119907119907119907119907119899119899119899119899Γ ((119907119907 minus 119899119899) 119907119907)

1199071199072Γ (119907119907119907119907) Γ (119899119899)

times 120590120590(119899119899minus119907119907)119907119907 10045601004560infin

0119909119909((119898119898119898119899119899)119907119907)minus11989810076521007652119898 119898

11989911989912059012059011990911990910076681007668

(119899119899minus119907119907)119907119907119889119889119909119909119889

(A2)

According to the integral formula intinfin0 119909119909120583120583minus119898119889119889119909119909(119898119898119889119889119909119909)119907119907 =119889119889minus120583120583Γ(120583120583)Γ(119907119907minus120583120583)Γ(119907119907) | arg 119889119889| 120573 120573120573Re 119907119907 119907 Re 120583120583 119907 0 [15 3194Equation 3] one can obtain (8)

B The Derivation of the CumulativeDistribution Function ofℋ119970119970119900119900

119860119860

Based on the denition of the CDF the CDF ofℋ119970119970119900119900119860119860 is

119865119865ℋ119970119970119900119900119860119860(119909119909) = 10045601004560

119909119909

minusinfin119901119901119885119885119860119860

(119911119911) 119889119889119911119911 = 10045601004560119909119909

0119901119901119885119885119860119860

(119911119911) 119889119889119911119911

= minus2120590120590120590120590119907119907119907119907119899119899119899119899Γ ((119907119907 minus 119899119899) 119907119907)

119907119907Γ (119907119907119907119907) Γ (119899119899)

times 120590120590(119899119899minus119907119907)119907119907 10045601004560119909119909

01199111199112119899119899minus11989810076521007652119898 119898

119899119899120590120590119911119911minus211990711990710076681007668

(119899119899minus119907119907)119907119907119889119889119911119911119889

(B1)

Similarly utilizing a symbol change 119905119905 = 119911119911minus2119907119907 (B1) turnsout to be

119865119865ℋ119970119970119900119900119860119860(119909119909) =

120590120590120590120590119907119907119907119907119899119899119899119899Γ ((119907119907 minus 119899119899) 119907119907)1199071199072Γ (119907119907119907119907) Γ (119899119899)

times 120590120590(119899119899minus119907119907)119907119907 10045601004560119909119909minus2119907119907

0119905119905(minus119899119899119907119907)minus11989810076521007652119898 119898

11989911989912059012059011990511990510076681007668

(119899119899minus119907119907)119907119907119889119889119905119905119889

(B2)

Likewise applying the integral formula int1199061199060 119909119909120583120583minus119898119889119889119909119909(119898 119898 119889119889119909119909)119907119907

= (119906119906120583120583120583120583)2119865119865119898(119907119907119907 120583120583119907 119898 119898 120583120583119907 minus119889119889119906119906) | arg(119898 119898 119889119889119906119906)| 120573 120573120573 Re 120583120583 119907 0[15 3194 Equation 1] we arrive at (19) by simplifying (B2)

References

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

[2] G Gao ldquoStatistical modeling of SAR images a surveyrdquo Sensorsvol 10 no 1 pp 775ndash795 2010

[3] A Achim E E Kuruolu and J Zerubia ldquoSAR image lteringbased on the heavy-tailed rayleigh modelrdquo IEEE Transactionson Image Processing vol 15 no 9 pp 2686ndash2693 2006

[4] T Esch M iel A Schenk A Roth A Muumlller and S DechldquoDelineation of Urban footprints from TerraSAR-X data byanalyzing speckle characteristics and intensity informationrdquoIEEE Transactions on Geoscience and Remote Sensing vol 48no 2 pp 905ndash916 2010

[5] 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

[6] C Tison J M Nicolas F Tupin and H Maicirctre ldquoA newstatistical model for Markovian classication of Urban areas inhigh-Resolution SAR imagesrdquo IEEE Transactions on Geoscienceand Remote Sensing vol 42 no 10 pp 2046ndash2057 2004

[7] A C Frery A H Correia and C D C Freitas ldquoClassify-ing multifrequency fully polarimetric imagery with multiplesources of statistical evidence and contextual informationrdquoIEEE Transactions on Geoscience and Remote Sensing vol 45no 10 pp 3098ndash3109 2007

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

[9] Y Delignon R Garello and A Hillion ldquoStatistical modeling ofocean SAR imagerdquo IEE Proceedings Radar Sonar and Naviga-tion vol 144 no 6 pp 348ndash354 1997

[10] Y Delignon and W Pieczynski ldquoModeling non-Rayleighspeckle distribution in SAR imagesrdquo IEEE Transactions onGeoscience and Remote Sensing vol 40 no 6 pp 1430ndash14352002

[11] A C Frery H-J Muumlller C D C F Yanasse and S J SSantrsquoAnna ldquoAmodel for extremely heterogeneous clutterrdquo IEEETransactions on Geoscience and Remote Sensing vol 35 no 3pp 648ndash659 1997

[12] K L P Vasconcellos A C Frery and L B Silva ldquoImprovingestimation in speckled imageryrdquo Computational Statistics vol20 no 3 pp 503ndash519 2005

[13] 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

[14] S N Annsen and T Elto ldquoApplication of the Matrix-variatemellin transform to analysis of polarimetric radar imagesrdquo IEEETransactions on Geoscience and Remote Sensing vol 49 no 6pp 2281ndash2295 2011

[15] S I Gradshteyn and I M Ryzhik Table of Integrals Series andProducts Academic Press San Diego Calif USA 7th edition2007

[16] G Moser J Zerubia and S B Serpico ldquoSAR amplitudeprobability density function estimation based on a generalizedGaussian modelrdquo IEEE Transactions on Image Processing vol15 no 6 pp 1429ndash1442 2006

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of Antennas and Propagation 5

(a) (b)

(c) (d)

(e) (f)

F 2 e Tested TerraSAR-X SAR image (a) ldquoWater bodyrdquo patch with a size of 332 px times 322 px (b) ldquodrying riverbedrdquo image (size 1597times 1734 pixels) (c) ldquomountainrdquo image (size 1996 times 2034 pixels) (d) ldquotownrdquo image (size 525 times 560 pixels) (e) ldquourban1rdquo image (size 1198 times1342 pixels) and (e) ldquourban2rdquo image (size 705 times 728 pixels)

6 International Journal of Antennas and Propagation

T 1 e main parameters of TerraSAR-X space-borne systems for the data in this study

Testing site Acquisitionmode Polarization Resolution

(az times rg) Date Size Azimuthlooks

Rangelooks

Beijing China SpotLight HH 192m times 226m 2008-01-31T1003 UTC 21333 px times 20000 px 119 102Sanchagang China StripMap VV 611m times 598m 2008-01-01T2220 UTC 11909 px times 22363 px 102 267

T 2 Parameter estimations of noted areas in Figure 2

Area ℋ119970119970∘119860119860 (120572120572120572 120572120572120572 120572120572120572 120572120572120572 120572120572120572

ldquoWater bodyrdquo (minus66029 minus11001 63048 10709 3683)ldquoDrying riverbedrdquo (minus60382 minus10555 56813 11897 3683)ldquoMountainrdquo (minus29788 minus05779 24694 01426 3683)ldquoTownrdquo (minus5062 minus07271 45872 04861 3683)ldquoUrban1rdquo (minus11774 minus04895 06905 111945 222)ldquoUrban2rdquo (minus08641 minus07065 03227 23517 222)

to reading their metadata les ext the estimates of allother parameters inℋ1199701199700 are accomplished by the estimatorsderived in Section 3 which are shown at Table 2

Furthermore in order to quantitatively assess the perfor-mances that different distributions t we dene an error ratiofactor (ERF) as

120575120575 120575sum119873119873119894119894120575119894 10092651009265119901119901119888119888 1007649100764911991111991111989411989410076651007665 minus 119902119902 100764910076491199111199111198941198941007665100766510092651009265

2

sum119873119873119894119894120575119894 10092651009265119901119901119887119887 1007649100764911991111991111989411989410076651007665 minus 119902119902 100764910076491199111199111198941198941007665100766510092651009265

2 120572 (18)

where 119911119911119894119894 ∣ 119894119894 120575 119894119894119873119873119894 is a sample set 119901119901119888119888(sdot120572 represents thecompared theoretical PDF 119901119901119887119887(sdot120572 is the basic theoretical PDFand 119902119902(sdot120572 indicates the actual PDF from the observed dataesymbol sdot denotes 2-norm It is obvious that the numeratorand denominator in (18) separately imply the total ttingerrors with the 119901119901119888119888 and 119901119901119887119887 to approximate 119902119902 e better theperformances of 119901119901119888119888 related to 119901119901119887119887 t the smaller 120575120575 is and viceverse Specically 120575120575 120575 119894 if both 119901119901119888119888 and 119901119901119887119887 have the identicalcapability for tting the measured data

Letℋ119970119970119900119900119860119860 represent the basic amplitude PDF and let 119970119970119900119900

119860119860be the compared amplitude one e 120575120575 values of the previoussix areas in this study are given in Figure 3 (see the yellowtextbox) It can be clearly seen that the proposedℋ119970119970119900119900

119860119860 betteragrees with the amplitude histograms of all six terrains thanthe 119970119970119900119900

119860119860 as expected because the 120575120575 values are larger than1 for all six areas e same conclusion can be conrmedfrom a visual point of view and implies the higher ttingprecision using ℋ119970119970119900119900 than using 1199701199700 over homogeneousheterogeneous and extremely heterogeneous regions

5 Conclusion and Perspective

We have developed an empirical model ℋ1199701199700 to exploitthe knowledge of statistical characteristics of SAR ampli-tude or intensity images over the wide terrain classes withhomogeneous heterogeneous and extremely heterogeneousbackscattering properties e parameter estimators of this

model based on the MoLC are also provided Consequentlywe report the performances of different land-over typologieswith ℋ1199701199700 distribution ts e experimental results showthat the ℋ1199701199700 distribution is a more advanced model com-pared with the known 1199701199700 distribution to characterize themultilook processed SAR data

As we know a preliminary statistical analysis of SARclutter data is important for designing signal processing algo-rithms such as speckle ltering target detection buildingextraction image segmentation and classication In futureit is worth expecting to use the ℋ1199701199700 distribution in theseelds Herein we rstly attempt to give an analytical deriva-tion for constructing a constant false alarm rate (CFAR)detector to promote the upcoming studies of target detectionin SAR images

Given the propose amplitude modelℋ119970119970119900119900119860119860 shown in (1)

its cumulative distribution function (CDF) is written as (seeAppendix B)

119865119865ℋ119970119970119900119900119860119860(119909119909120572 120575 minus

120572120572120572120572120572120572120572120572119907(120572120572minus120572120572120572119907120572120572120572120572120572120572minus119894Γ ((120572120572 minus 120572120572120572 119907120572120572120572120572120572Γ (120572120572120572120572120572 Γ (120572120572120572

1199091199092120572120572

times 2119865119865119894 10076521007652120572120572 minus 120572120572120572120572

120572 minus120572120572120572120572 119894 minus

120572120572120572120572 minus120572120572120572120572119909119909minus212057212057210076681007668 120572

minus 120572120572120572 minus120572120572120572 120572120572120572 120572120572120572 120572120572120572 119909119909 120572 0120572

(19)

where 2119865119865119894(sdot120572 sdot sdot sdot120572 is the Gauss hypergeometric function For agiven value of the false alarm probability denoted by 119875119875119891119891119891119891 thecorresponding CFAR threshold 119879119879 for theℋ119970119970119900119900

119860119860 distributioncan be obtained from

119894 minus 119875119875119891119891119891119891 120575 119865119865ℋ119970119970119900119900119860119860(119879119879120572

120575 minus120572120572120572120572120572120572120572120572119907(120572120572minus120572120572120572119907120572120572120572120572120572120572minus119894Γ ((120572120572 minus 120572120572120572 119907120572120572120572

120572120572Γ (120572120572120572120572120572 Γ (1205721205721205721198791198792120572120572

times 2119865119865119894 10076521007652120572120572 minus 120572120572120572120572

120572 minus120572120572120572120572 119894 minus

120572120572120572120572 minus120572120572120572120572119879119879minus212057212057210076681007668

(20)

Considering 119865119865ℋ119970119970119900119900119860119860(119879119879120572 is strictly monotonously increas-

ing the threshold119879119879 can be accurately calculatedwith the helpof the numerical solution or a simple bisection method [16]

Our future work will focus on demonstrating the perfor-mances of the proposed CFAR detector using somemeasuredSAR data and investigating how the ℋ1199701199700 distribution per-forms when extending it to other application elds

International Journal of Antennas and Propagation 7

0 05 1 15 2 25 3 350

02

04

06

08

1

12

14

Amplitude

PD

FWater body

= 10042

(a)

0 05 1 15 2 25 3 350

05

1

15

Amplitude

PD

F

Drying riverbed

= 35351

(b)

0 05 1 15 2 25 3 350

02

04

06

08

1

12

14

Amplitude

PDF

Mountain

= 23306

(c)

0 05 1 15 2 25 3 350

02

04

06

08

1

12

14

Amplitude

PDF

Town

= 26756

(d)

0 05 1 15 2 25 3 350

02

04

06

08

1

12

14

Amplitude

PD

F

Urban1

= 1944

Histogram

The proposed model

(e)

0 05 1 15 2 25 3 350

02

04

06

08

1

12

14

Amplitude

PD

F

Urban2

= 13562

Histogram

The proposed model

(f)

F 3 Plots of aplitude istogras and of te estiated PFs (a)ndash(f) are corresponded to te tting results for te scenes son inFigures 2(a)ndash2(f) respectively

8 International Journal of Antennas and Propagation

Appendices

A The Derivation of119898119898th OrderMoments oftheℋ119970119970119900119900

119868119868 Distribution

Via (2) the119898119898th order moments of theℋ119970119970119900119900119868119868 are expressed as

119864119864 10076491007649119911119911119898119898119868119868 10076651007665 = 10045601004560infin

0119911119911119898119898119901119901119885119885119868119868

(119911119911) 119889119889119911119911

= minus120590120590120590120590119907119907119907119907119899119899119899119899Γ ((119907119907 minus 119899119899) 119907119907)

119907119907Γ (119907119907119907119907) Γ (119899119899)

times 120590120590(119899119899minus119907119907)119907119907 10045601004560infin

0119911119911119898119898119898119899119899minus11989810076521007652119898 119898

119899119899120590120590119911119911minus11990711990710076681007668

(119899119899minus119907119907)119907119907119889119889119911119911119889

(A1)

A variable replacement 119909119909 = 119911119911minus119907119907 leads to 119911119911 = 119909119909minus119898119907119907 and119889119889119911119911 = minus(119898119907119907)119909119909minus(119898119907119907)minus119898119889119889119909119909 thus (A1) can be rewritten as

119864119864 10076491007649119911119911119898119898119868119868 10076651007665 = minus120590120590120590120590119907119907119907119907119899119899119899119899Γ ((119907119907 minus 119899119899) 119907119907)

1199071199072Γ (119907119907119907119907) Γ (119899119899)

times 120590120590(119899119899minus119907119907)119907119907 10045601004560infin

0119909119909((119898119898119898119899119899)119907119907)minus11989810076521007652119898 119898

11989911989912059012059011990911990910076681007668

(119899119899minus119907119907)119907119907119889119889119909119909119889

(A2)

According to the integral formula intinfin0 119909119909120583120583minus119898119889119889119909119909(119898119898119889119889119909119909)119907119907 =119889119889minus120583120583Γ(120583120583)Γ(119907119907minus120583120583)Γ(119907119907) | arg 119889119889| 120573 120573120573Re 119907119907 119907 Re 120583120583 119907 0 [15 3194Equation 3] one can obtain (8)

B The Derivation of the CumulativeDistribution Function ofℋ119970119970119900119900

119860119860

Based on the denition of the CDF the CDF ofℋ119970119970119900119900119860119860 is

119865119865ℋ119970119970119900119900119860119860(119909119909) = 10045601004560

119909119909

minusinfin119901119901119885119885119860119860

(119911119911) 119889119889119911119911 = 10045601004560119909119909

0119901119901119885119885119860119860

(119911119911) 119889119889119911119911

= minus2120590120590120590120590119907119907119907119907119899119899119899119899Γ ((119907119907 minus 119899119899) 119907119907)

119907119907Γ (119907119907119907119907) Γ (119899119899)

times 120590120590(119899119899minus119907119907)119907119907 10045601004560119909119909

01199111199112119899119899minus11989810076521007652119898 119898

119899119899120590120590119911119911minus211990711990710076681007668

(119899119899minus119907119907)119907119907119889119889119911119911119889

(B1)

Similarly utilizing a symbol change 119905119905 = 119911119911minus2119907119907 (B1) turnsout to be

119865119865ℋ119970119970119900119900119860119860(119909119909) =

120590120590120590120590119907119907119907119907119899119899119899119899Γ ((119907119907 minus 119899119899) 119907119907)1199071199072Γ (119907119907119907119907) Γ (119899119899)

times 120590120590(119899119899minus119907119907)119907119907 10045601004560119909119909minus2119907119907

0119905119905(minus119899119899119907119907)minus11989810076521007652119898 119898

11989911989912059012059011990511990510076681007668

(119899119899minus119907119907)119907119907119889119889119905119905119889

(B2)

Likewise applying the integral formula int1199061199060 119909119909120583120583minus119898119889119889119909119909(119898 119898 119889119889119909119909)119907119907

= (119906119906120583120583120583120583)2119865119865119898(119907119907119907 120583120583119907 119898 119898 120583120583119907 minus119889119889119906119906) | arg(119898 119898 119889119889119906119906)| 120573 120573120573 Re 120583120583 119907 0[15 3194 Equation 1] we arrive at (19) by simplifying (B2)

References

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

[2] G Gao ldquoStatistical modeling of SAR images a surveyrdquo Sensorsvol 10 no 1 pp 775ndash795 2010

[3] A Achim E E Kuruolu and J Zerubia ldquoSAR image lteringbased on the heavy-tailed rayleigh modelrdquo IEEE Transactionson Image Processing vol 15 no 9 pp 2686ndash2693 2006

[4] T Esch M iel A Schenk A Roth A Muumlller and S DechldquoDelineation of Urban footprints from TerraSAR-X data byanalyzing speckle characteristics and intensity informationrdquoIEEE Transactions on Geoscience and Remote Sensing vol 48no 2 pp 905ndash916 2010

[5] 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

[6] C Tison J M Nicolas F Tupin and H Maicirctre ldquoA newstatistical model for Markovian classication of Urban areas inhigh-Resolution SAR imagesrdquo IEEE Transactions on Geoscienceand Remote Sensing vol 42 no 10 pp 2046ndash2057 2004

[7] A C Frery A H Correia and C D C Freitas ldquoClassify-ing multifrequency fully polarimetric imagery with multiplesources of statistical evidence and contextual informationrdquoIEEE Transactions on Geoscience and Remote Sensing vol 45no 10 pp 3098ndash3109 2007

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

[9] Y Delignon R Garello and A Hillion ldquoStatistical modeling ofocean SAR imagerdquo IEE Proceedings Radar Sonar and Naviga-tion vol 144 no 6 pp 348ndash354 1997

[10] Y Delignon and W Pieczynski ldquoModeling non-Rayleighspeckle distribution in SAR imagesrdquo IEEE Transactions onGeoscience and Remote Sensing vol 40 no 6 pp 1430ndash14352002

[11] A C Frery H-J Muumlller C D C F Yanasse and S J SSantrsquoAnna ldquoAmodel for extremely heterogeneous clutterrdquo IEEETransactions on Geoscience and Remote Sensing vol 35 no 3pp 648ndash659 1997

[12] K L P Vasconcellos A C Frery and L B Silva ldquoImprovingestimation in speckled imageryrdquo Computational Statistics vol20 no 3 pp 503ndash519 2005

[13] 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

[14] S N Annsen and T Elto ldquoApplication of the Matrix-variatemellin transform to analysis of polarimetric radar imagesrdquo IEEETransactions on Geoscience and Remote Sensing vol 49 no 6pp 2281ndash2295 2011

[15] S I Gradshteyn and I M Ryzhik Table of Integrals Series andProducts Academic Press San Diego Calif USA 7th edition2007

[16] G Moser J Zerubia and S B Serpico ldquoSAR amplitudeprobability density function estimation based on a generalizedGaussian modelrdquo IEEE Transactions on Image Processing vol15 no 6 pp 1429ndash1442 2006

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

6 International Journal of Antennas and Propagation

T 1 e main parameters of TerraSAR-X space-borne systems for the data in this study

Testing site Acquisitionmode Polarization Resolution

(az times rg) Date Size Azimuthlooks

Rangelooks

Beijing China SpotLight HH 192m times 226m 2008-01-31T1003 UTC 21333 px times 20000 px 119 102Sanchagang China StripMap VV 611m times 598m 2008-01-01T2220 UTC 11909 px times 22363 px 102 267

T 2 Parameter estimations of noted areas in Figure 2

Area ℋ119970119970∘119860119860 (120572120572120572 120572120572120572 120572120572120572 120572120572120572 120572120572120572

ldquoWater bodyrdquo (minus66029 minus11001 63048 10709 3683)ldquoDrying riverbedrdquo (minus60382 minus10555 56813 11897 3683)ldquoMountainrdquo (minus29788 minus05779 24694 01426 3683)ldquoTownrdquo (minus5062 minus07271 45872 04861 3683)ldquoUrban1rdquo (minus11774 minus04895 06905 111945 222)ldquoUrban2rdquo (minus08641 minus07065 03227 23517 222)

to reading their metadata les ext the estimates of allother parameters inℋ1199701199700 are accomplished by the estimatorsderived in Section 3 which are shown at Table 2

Furthermore in order to quantitatively assess the perfor-mances that different distributions t we dene an error ratiofactor (ERF) as

120575120575 120575sum119873119873119894119894120575119894 10092651009265119901119901119888119888 1007649100764911991111991111989411989410076651007665 minus 119902119902 100764910076491199111199111198941198941007665100766510092651009265

2

sum119873119873119894119894120575119894 10092651009265119901119901119887119887 1007649100764911991111991111989411989410076651007665 minus 119902119902 100764910076491199111199111198941198941007665100766510092651009265

2 120572 (18)

where 119911119911119894119894 ∣ 119894119894 120575 119894119894119873119873119894 is a sample set 119901119901119888119888(sdot120572 represents thecompared theoretical PDF 119901119901119887119887(sdot120572 is the basic theoretical PDFand 119902119902(sdot120572 indicates the actual PDF from the observed dataesymbol sdot denotes 2-norm It is obvious that the numeratorand denominator in (18) separately imply the total ttingerrors with the 119901119901119888119888 and 119901119901119887119887 to approximate 119902119902 e better theperformances of 119901119901119888119888 related to 119901119901119887119887 t the smaller 120575120575 is and viceverse Specically 120575120575 120575 119894 if both 119901119901119888119888 and 119901119901119887119887 have the identicalcapability for tting the measured data

Letℋ119970119970119900119900119860119860 represent the basic amplitude PDF and let 119970119970119900119900

119860119860be the compared amplitude one e 120575120575 values of the previoussix areas in this study are given in Figure 3 (see the yellowtextbox) It can be clearly seen that the proposedℋ119970119970119900119900

119860119860 betteragrees with the amplitude histograms of all six terrains thanthe 119970119970119900119900

119860119860 as expected because the 120575120575 values are larger than1 for all six areas e same conclusion can be conrmedfrom a visual point of view and implies the higher ttingprecision using ℋ119970119970119900119900 than using 1199701199700 over homogeneousheterogeneous and extremely heterogeneous regions

5 Conclusion and Perspective

We have developed an empirical model ℋ1199701199700 to exploitthe knowledge of statistical characteristics of SAR ampli-tude or intensity images over the wide terrain classes withhomogeneous heterogeneous and extremely heterogeneousbackscattering properties e parameter estimators of this

model based on the MoLC are also provided Consequentlywe report the performances of different land-over typologieswith ℋ1199701199700 distribution ts e experimental results showthat the ℋ1199701199700 distribution is a more advanced model com-pared with the known 1199701199700 distribution to characterize themultilook processed SAR data

As we know a preliminary statistical analysis of SARclutter data is important for designing signal processing algo-rithms such as speckle ltering target detection buildingextraction image segmentation and classication In futureit is worth expecting to use the ℋ1199701199700 distribution in theseelds Herein we rstly attempt to give an analytical deriva-tion for constructing a constant false alarm rate (CFAR)detector to promote the upcoming studies of target detectionin SAR images

Given the propose amplitude modelℋ119970119970119900119900119860119860 shown in (1)

its cumulative distribution function (CDF) is written as (seeAppendix B)

119865119865ℋ119970119970119900119900119860119860(119909119909120572 120575 minus

120572120572120572120572120572120572120572120572119907(120572120572minus120572120572120572119907120572120572120572120572120572120572minus119894Γ ((120572120572 minus 120572120572120572 119907120572120572120572120572120572Γ (120572120572120572120572120572 Γ (120572120572120572

1199091199092120572120572

times 2119865119865119894 10076521007652120572120572 minus 120572120572120572120572

120572 minus120572120572120572120572 119894 minus

120572120572120572120572 minus120572120572120572120572119909119909minus212057212057210076681007668 120572

minus 120572120572120572 minus120572120572120572 120572120572120572 120572120572120572 120572120572120572 119909119909 120572 0120572

(19)

where 2119865119865119894(sdot120572 sdot sdot sdot120572 is the Gauss hypergeometric function For agiven value of the false alarm probability denoted by 119875119875119891119891119891119891 thecorresponding CFAR threshold 119879119879 for theℋ119970119970119900119900

119860119860 distributioncan be obtained from

119894 minus 119875119875119891119891119891119891 120575 119865119865ℋ119970119970119900119900119860119860(119879119879120572

120575 minus120572120572120572120572120572120572120572120572119907(120572120572minus120572120572120572119907120572120572120572120572120572120572minus119894Γ ((120572120572 minus 120572120572120572 119907120572120572120572

120572120572Γ (120572120572120572120572120572 Γ (1205721205721205721198791198792120572120572

times 2119865119865119894 10076521007652120572120572 minus 120572120572120572120572

120572 minus120572120572120572120572 119894 minus

120572120572120572120572 minus120572120572120572120572119879119879minus212057212057210076681007668

(20)

Considering 119865119865ℋ119970119970119900119900119860119860(119879119879120572 is strictly monotonously increas-

ing the threshold119879119879 can be accurately calculatedwith the helpof the numerical solution or a simple bisection method [16]

Our future work will focus on demonstrating the perfor-mances of the proposed CFAR detector using somemeasuredSAR data and investigating how the ℋ1199701199700 distribution per-forms when extending it to other application elds

International Journal of Antennas and Propagation 7

0 05 1 15 2 25 3 350

02

04

06

08

1

12

14

Amplitude

PD

FWater body

= 10042

(a)

0 05 1 15 2 25 3 350

05

1

15

Amplitude

PD

F

Drying riverbed

= 35351

(b)

0 05 1 15 2 25 3 350

02

04

06

08

1

12

14

Amplitude

PDF

Mountain

= 23306

(c)

0 05 1 15 2 25 3 350

02

04

06

08

1

12

14

Amplitude

PDF

Town

= 26756

(d)

0 05 1 15 2 25 3 350

02

04

06

08

1

12

14

Amplitude

PD

F

Urban1

= 1944

Histogram

The proposed model

(e)

0 05 1 15 2 25 3 350

02

04

06

08

1

12

14

Amplitude

PD

F

Urban2

= 13562

Histogram

The proposed model

(f)

F 3 Plots of aplitude istogras and of te estiated PFs (a)ndash(f) are corresponded to te tting results for te scenes son inFigures 2(a)ndash2(f) respectively

8 International Journal of Antennas and Propagation

Appendices

A The Derivation of119898119898th OrderMoments oftheℋ119970119970119900119900

119868119868 Distribution

Via (2) the119898119898th order moments of theℋ119970119970119900119900119868119868 are expressed as

119864119864 10076491007649119911119911119898119898119868119868 10076651007665 = 10045601004560infin

0119911119911119898119898119901119901119885119885119868119868

(119911119911) 119889119889119911119911

= minus120590120590120590120590119907119907119907119907119899119899119899119899Γ ((119907119907 minus 119899119899) 119907119907)

119907119907Γ (119907119907119907119907) Γ (119899119899)

times 120590120590(119899119899minus119907119907)119907119907 10045601004560infin

0119911119911119898119898119898119899119899minus11989810076521007652119898 119898

119899119899120590120590119911119911minus11990711990710076681007668

(119899119899minus119907119907)119907119907119889119889119911119911119889

(A1)

A variable replacement 119909119909 = 119911119911minus119907119907 leads to 119911119911 = 119909119909minus119898119907119907 and119889119889119911119911 = minus(119898119907119907)119909119909minus(119898119907119907)minus119898119889119889119909119909 thus (A1) can be rewritten as

119864119864 10076491007649119911119911119898119898119868119868 10076651007665 = minus120590120590120590120590119907119907119907119907119899119899119899119899Γ ((119907119907 minus 119899119899) 119907119907)

1199071199072Γ (119907119907119907119907) Γ (119899119899)

times 120590120590(119899119899minus119907119907)119907119907 10045601004560infin

0119909119909((119898119898119898119899119899)119907119907)minus11989810076521007652119898 119898

11989911989912059012059011990911990910076681007668

(119899119899minus119907119907)119907119907119889119889119909119909119889

(A2)

According to the integral formula intinfin0 119909119909120583120583minus119898119889119889119909119909(119898119898119889119889119909119909)119907119907 =119889119889minus120583120583Γ(120583120583)Γ(119907119907minus120583120583)Γ(119907119907) | arg 119889119889| 120573 120573120573Re 119907119907 119907 Re 120583120583 119907 0 [15 3194Equation 3] one can obtain (8)

B The Derivation of the CumulativeDistribution Function ofℋ119970119970119900119900

119860119860

Based on the denition of the CDF the CDF ofℋ119970119970119900119900119860119860 is

119865119865ℋ119970119970119900119900119860119860(119909119909) = 10045601004560

119909119909

minusinfin119901119901119885119885119860119860

(119911119911) 119889119889119911119911 = 10045601004560119909119909

0119901119901119885119885119860119860

(119911119911) 119889119889119911119911

= minus2120590120590120590120590119907119907119907119907119899119899119899119899Γ ((119907119907 minus 119899119899) 119907119907)

119907119907Γ (119907119907119907119907) Γ (119899119899)

times 120590120590(119899119899minus119907119907)119907119907 10045601004560119909119909

01199111199112119899119899minus11989810076521007652119898 119898

119899119899120590120590119911119911minus211990711990710076681007668

(119899119899minus119907119907)119907119907119889119889119911119911119889

(B1)

Similarly utilizing a symbol change 119905119905 = 119911119911minus2119907119907 (B1) turnsout to be

119865119865ℋ119970119970119900119900119860119860(119909119909) =

120590120590120590120590119907119907119907119907119899119899119899119899Γ ((119907119907 minus 119899119899) 119907119907)1199071199072Γ (119907119907119907119907) Γ (119899119899)

times 120590120590(119899119899minus119907119907)119907119907 10045601004560119909119909minus2119907119907

0119905119905(minus119899119899119907119907)minus11989810076521007652119898 119898

11989911989912059012059011990511990510076681007668

(119899119899minus119907119907)119907119907119889119889119905119905119889

(B2)

Likewise applying the integral formula int1199061199060 119909119909120583120583minus119898119889119889119909119909(119898 119898 119889119889119909119909)119907119907

= (119906119906120583120583120583120583)2119865119865119898(119907119907119907 120583120583119907 119898 119898 120583120583119907 minus119889119889119906119906) | arg(119898 119898 119889119889119906119906)| 120573 120573120573 Re 120583120583 119907 0[15 3194 Equation 1] we arrive at (19) by simplifying (B2)

References

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

[2] G Gao ldquoStatistical modeling of SAR images a surveyrdquo Sensorsvol 10 no 1 pp 775ndash795 2010

[3] A Achim E E Kuruolu and J Zerubia ldquoSAR image lteringbased on the heavy-tailed rayleigh modelrdquo IEEE Transactionson Image Processing vol 15 no 9 pp 2686ndash2693 2006

[4] T Esch M iel A Schenk A Roth A Muumlller and S DechldquoDelineation of Urban footprints from TerraSAR-X data byanalyzing speckle characteristics and intensity informationrdquoIEEE Transactions on Geoscience and Remote Sensing vol 48no 2 pp 905ndash916 2010

[5] 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

[6] C Tison J M Nicolas F Tupin and H Maicirctre ldquoA newstatistical model for Markovian classication of Urban areas inhigh-Resolution SAR imagesrdquo IEEE Transactions on Geoscienceand Remote Sensing vol 42 no 10 pp 2046ndash2057 2004

[7] A C Frery A H Correia and C D C Freitas ldquoClassify-ing multifrequency fully polarimetric imagery with multiplesources of statistical evidence and contextual informationrdquoIEEE Transactions on Geoscience and Remote Sensing vol 45no 10 pp 3098ndash3109 2007

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

[9] Y Delignon R Garello and A Hillion ldquoStatistical modeling ofocean SAR imagerdquo IEE Proceedings Radar Sonar and Naviga-tion vol 144 no 6 pp 348ndash354 1997

[10] Y Delignon and W Pieczynski ldquoModeling non-Rayleighspeckle distribution in SAR imagesrdquo IEEE Transactions onGeoscience and Remote Sensing vol 40 no 6 pp 1430ndash14352002

[11] A C Frery H-J Muumlller C D C F Yanasse and S J SSantrsquoAnna ldquoAmodel for extremely heterogeneous clutterrdquo IEEETransactions on Geoscience and Remote Sensing vol 35 no 3pp 648ndash659 1997

[12] K L P Vasconcellos A C Frery and L B Silva ldquoImprovingestimation in speckled imageryrdquo Computational Statistics vol20 no 3 pp 503ndash519 2005

[13] 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

[14] S N Annsen and T Elto ldquoApplication of the Matrix-variatemellin transform to analysis of polarimetric radar imagesrdquo IEEETransactions on Geoscience and Remote Sensing vol 49 no 6pp 2281ndash2295 2011

[15] S I Gradshteyn and I M Ryzhik Table of Integrals Series andProducts Academic Press San Diego Calif USA 7th edition2007

[16] G Moser J Zerubia and S B Serpico ldquoSAR amplitudeprobability density function estimation based on a generalizedGaussian modelrdquo IEEE Transactions on Image Processing vol15 no 6 pp 1429ndash1442 2006

International Journal of

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Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of Antennas and Propagation 7

0 05 1 15 2 25 3 350

02

04

06

08

1

12

14

Amplitude

PD

FWater body

= 10042

(a)

0 05 1 15 2 25 3 350

05

1

15

Amplitude

PD

F

Drying riverbed

= 35351

(b)

0 05 1 15 2 25 3 350

02

04

06

08

1

12

14

Amplitude

PDF

Mountain

= 23306

(c)

0 05 1 15 2 25 3 350

02

04

06

08

1

12

14

Amplitude

PDF

Town

= 26756

(d)

0 05 1 15 2 25 3 350

02

04

06

08

1

12

14

Amplitude

PD

F

Urban1

= 1944

Histogram

The proposed model

(e)

0 05 1 15 2 25 3 350

02

04

06

08

1

12

14

Amplitude

PD

F

Urban2

= 13562

Histogram

The proposed model

(f)

F 3 Plots of aplitude istogras and of te estiated PFs (a)ndash(f) are corresponded to te tting results for te scenes son inFigures 2(a)ndash2(f) respectively

8 International Journal of Antennas and Propagation

Appendices

A The Derivation of119898119898th OrderMoments oftheℋ119970119970119900119900

119868119868 Distribution

Via (2) the119898119898th order moments of theℋ119970119970119900119900119868119868 are expressed as

119864119864 10076491007649119911119911119898119898119868119868 10076651007665 = 10045601004560infin

0119911119911119898119898119901119901119885119885119868119868

(119911119911) 119889119889119911119911

= minus120590120590120590120590119907119907119907119907119899119899119899119899Γ ((119907119907 minus 119899119899) 119907119907)

119907119907Γ (119907119907119907119907) Γ (119899119899)

times 120590120590(119899119899minus119907119907)119907119907 10045601004560infin

0119911119911119898119898119898119899119899minus11989810076521007652119898 119898

119899119899120590120590119911119911minus11990711990710076681007668

(119899119899minus119907119907)119907119907119889119889119911119911119889

(A1)

A variable replacement 119909119909 = 119911119911minus119907119907 leads to 119911119911 = 119909119909minus119898119907119907 and119889119889119911119911 = minus(119898119907119907)119909119909minus(119898119907119907)minus119898119889119889119909119909 thus (A1) can be rewritten as

119864119864 10076491007649119911119911119898119898119868119868 10076651007665 = minus120590120590120590120590119907119907119907119907119899119899119899119899Γ ((119907119907 minus 119899119899) 119907119907)

1199071199072Γ (119907119907119907119907) Γ (119899119899)

times 120590120590(119899119899minus119907119907)119907119907 10045601004560infin

0119909119909((119898119898119898119899119899)119907119907)minus11989810076521007652119898 119898

11989911989912059012059011990911990910076681007668

(119899119899minus119907119907)119907119907119889119889119909119909119889

(A2)

According to the integral formula intinfin0 119909119909120583120583minus119898119889119889119909119909(119898119898119889119889119909119909)119907119907 =119889119889minus120583120583Γ(120583120583)Γ(119907119907minus120583120583)Γ(119907119907) | arg 119889119889| 120573 120573120573Re 119907119907 119907 Re 120583120583 119907 0 [15 3194Equation 3] one can obtain (8)

B The Derivation of the CumulativeDistribution Function ofℋ119970119970119900119900

119860119860

Based on the denition of the CDF the CDF ofℋ119970119970119900119900119860119860 is

119865119865ℋ119970119970119900119900119860119860(119909119909) = 10045601004560

119909119909

minusinfin119901119901119885119885119860119860

(119911119911) 119889119889119911119911 = 10045601004560119909119909

0119901119901119885119885119860119860

(119911119911) 119889119889119911119911

= minus2120590120590120590120590119907119907119907119907119899119899119899119899Γ ((119907119907 minus 119899119899) 119907119907)

119907119907Γ (119907119907119907119907) Γ (119899119899)

times 120590120590(119899119899minus119907119907)119907119907 10045601004560119909119909

01199111199112119899119899minus11989810076521007652119898 119898

119899119899120590120590119911119911minus211990711990710076681007668

(119899119899minus119907119907)119907119907119889119889119911119911119889

(B1)

Similarly utilizing a symbol change 119905119905 = 119911119911minus2119907119907 (B1) turnsout to be

119865119865ℋ119970119970119900119900119860119860(119909119909) =

120590120590120590120590119907119907119907119907119899119899119899119899Γ ((119907119907 minus 119899119899) 119907119907)1199071199072Γ (119907119907119907119907) Γ (119899119899)

times 120590120590(119899119899minus119907119907)119907119907 10045601004560119909119909minus2119907119907

0119905119905(minus119899119899119907119907)minus11989810076521007652119898 119898

11989911989912059012059011990511990510076681007668

(119899119899minus119907119907)119907119907119889119889119905119905119889

(B2)

Likewise applying the integral formula int1199061199060 119909119909120583120583minus119898119889119889119909119909(119898 119898 119889119889119909119909)119907119907

= (119906119906120583120583120583120583)2119865119865119898(119907119907119907 120583120583119907 119898 119898 120583120583119907 minus119889119889119906119906) | arg(119898 119898 119889119889119906119906)| 120573 120573120573 Re 120583120583 119907 0[15 3194 Equation 1] we arrive at (19) by simplifying (B2)

References

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

[2] G Gao ldquoStatistical modeling of SAR images a surveyrdquo Sensorsvol 10 no 1 pp 775ndash795 2010

[3] A Achim E E Kuruolu and J Zerubia ldquoSAR image lteringbased on the heavy-tailed rayleigh modelrdquo IEEE Transactionson Image Processing vol 15 no 9 pp 2686ndash2693 2006

[4] T Esch M iel A Schenk A Roth A Muumlller and S DechldquoDelineation of Urban footprints from TerraSAR-X data byanalyzing speckle characteristics and intensity informationrdquoIEEE Transactions on Geoscience and Remote Sensing vol 48no 2 pp 905ndash916 2010

[5] 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

[6] C Tison J M Nicolas F Tupin and H Maicirctre ldquoA newstatistical model for Markovian classication of Urban areas inhigh-Resolution SAR imagesrdquo IEEE Transactions on Geoscienceand Remote Sensing vol 42 no 10 pp 2046ndash2057 2004

[7] A C Frery A H Correia and C D C Freitas ldquoClassify-ing multifrequency fully polarimetric imagery with multiplesources of statistical evidence and contextual informationrdquoIEEE Transactions on Geoscience and Remote Sensing vol 45no 10 pp 3098ndash3109 2007

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

[9] Y Delignon R Garello and A Hillion ldquoStatistical modeling ofocean SAR imagerdquo IEE Proceedings Radar Sonar and Naviga-tion vol 144 no 6 pp 348ndash354 1997

[10] Y Delignon and W Pieczynski ldquoModeling non-Rayleighspeckle distribution in SAR imagesrdquo IEEE Transactions onGeoscience and Remote Sensing vol 40 no 6 pp 1430ndash14352002

[11] A C Frery H-J Muumlller C D C F Yanasse and S J SSantrsquoAnna ldquoAmodel for extremely heterogeneous clutterrdquo IEEETransactions on Geoscience and Remote Sensing vol 35 no 3pp 648ndash659 1997

[12] K L P Vasconcellos A C Frery and L B Silva ldquoImprovingestimation in speckled imageryrdquo Computational Statistics vol20 no 3 pp 503ndash519 2005

[13] 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

[14] S N Annsen and T Elto ldquoApplication of the Matrix-variatemellin transform to analysis of polarimetric radar imagesrdquo IEEETransactions on Geoscience and Remote Sensing vol 49 no 6pp 2281ndash2295 2011

[15] S I Gradshteyn and I M Ryzhik Table of Integrals Series andProducts Academic Press San Diego Calif USA 7th edition2007

[16] G Moser J Zerubia and S B Serpico ldquoSAR amplitudeprobability density function estimation based on a generalizedGaussian modelrdquo IEEE Transactions on Image Processing vol15 no 6 pp 1429ndash1442 2006

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

8 International Journal of Antennas and Propagation

Appendices

A The Derivation of119898119898th OrderMoments oftheℋ119970119970119900119900

119868119868 Distribution

Via (2) the119898119898th order moments of theℋ119970119970119900119900119868119868 are expressed as

119864119864 10076491007649119911119911119898119898119868119868 10076651007665 = 10045601004560infin

0119911119911119898119898119901119901119885119885119868119868

(119911119911) 119889119889119911119911

= minus120590120590120590120590119907119907119907119907119899119899119899119899Γ ((119907119907 minus 119899119899) 119907119907)

119907119907Γ (119907119907119907119907) Γ (119899119899)

times 120590120590(119899119899minus119907119907)119907119907 10045601004560infin

0119911119911119898119898119898119899119899minus11989810076521007652119898 119898

119899119899120590120590119911119911minus11990711990710076681007668

(119899119899minus119907119907)119907119907119889119889119911119911119889

(A1)

A variable replacement 119909119909 = 119911119911minus119907119907 leads to 119911119911 = 119909119909minus119898119907119907 and119889119889119911119911 = minus(119898119907119907)119909119909minus(119898119907119907)minus119898119889119889119909119909 thus (A1) can be rewritten as

119864119864 10076491007649119911119911119898119898119868119868 10076651007665 = minus120590120590120590120590119907119907119907119907119899119899119899119899Γ ((119907119907 minus 119899119899) 119907119907)

1199071199072Γ (119907119907119907119907) Γ (119899119899)

times 120590120590(119899119899minus119907119907)119907119907 10045601004560infin

0119909119909((119898119898119898119899119899)119907119907)minus11989810076521007652119898 119898

11989911989912059012059011990911990910076681007668

(119899119899minus119907119907)119907119907119889119889119909119909119889

(A2)

According to the integral formula intinfin0 119909119909120583120583minus119898119889119889119909119909(119898119898119889119889119909119909)119907119907 =119889119889minus120583120583Γ(120583120583)Γ(119907119907minus120583120583)Γ(119907119907) | arg 119889119889| 120573 120573120573Re 119907119907 119907 Re 120583120583 119907 0 [15 3194Equation 3] one can obtain (8)

B The Derivation of the CumulativeDistribution Function ofℋ119970119970119900119900

119860119860

Based on the denition of the CDF the CDF ofℋ119970119970119900119900119860119860 is

119865119865ℋ119970119970119900119900119860119860(119909119909) = 10045601004560

119909119909

minusinfin119901119901119885119885119860119860

(119911119911) 119889119889119911119911 = 10045601004560119909119909

0119901119901119885119885119860119860

(119911119911) 119889119889119911119911

= minus2120590120590120590120590119907119907119907119907119899119899119899119899Γ ((119907119907 minus 119899119899) 119907119907)

119907119907Γ (119907119907119907119907) Γ (119899119899)

times 120590120590(119899119899minus119907119907)119907119907 10045601004560119909119909

01199111199112119899119899minus11989810076521007652119898 119898

119899119899120590120590119911119911minus211990711990710076681007668

(119899119899minus119907119907)119907119907119889119889119911119911119889

(B1)

Similarly utilizing a symbol change 119905119905 = 119911119911minus2119907119907 (B1) turnsout to be

119865119865ℋ119970119970119900119900119860119860(119909119909) =

120590120590120590120590119907119907119907119907119899119899119899119899Γ ((119907119907 minus 119899119899) 119907119907)1199071199072Γ (119907119907119907119907) Γ (119899119899)

times 120590120590(119899119899minus119907119907)119907119907 10045601004560119909119909minus2119907119907

0119905119905(minus119899119899119907119907)minus11989810076521007652119898 119898

11989911989912059012059011990511990510076681007668

(119899119899minus119907119907)119907119907119889119889119905119905119889

(B2)

Likewise applying the integral formula int1199061199060 119909119909120583120583minus119898119889119889119909119909(119898 119898 119889119889119909119909)119907119907

= (119906119906120583120583120583120583)2119865119865119898(119907119907119907 120583120583119907 119898 119898 120583120583119907 minus119889119889119906119906) | arg(119898 119898 119889119889119906119906)| 120573 120573120573 Re 120583120583 119907 0[15 3194 Equation 1] we arrive at (19) by simplifying (B2)

References

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

[2] G Gao ldquoStatistical modeling of SAR images a surveyrdquo Sensorsvol 10 no 1 pp 775ndash795 2010

[3] A Achim E E Kuruolu and J Zerubia ldquoSAR image lteringbased on the heavy-tailed rayleigh modelrdquo IEEE Transactionson Image Processing vol 15 no 9 pp 2686ndash2693 2006

[4] T Esch M iel A Schenk A Roth A Muumlller and S DechldquoDelineation of Urban footprints from TerraSAR-X data byanalyzing speckle characteristics and intensity informationrdquoIEEE Transactions on Geoscience and Remote Sensing vol 48no 2 pp 905ndash916 2010

[5] 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

[6] C Tison J M Nicolas F Tupin and H Maicirctre ldquoA newstatistical model for Markovian classication of Urban areas inhigh-Resolution SAR imagesrdquo IEEE Transactions on Geoscienceand Remote Sensing vol 42 no 10 pp 2046ndash2057 2004

[7] A C Frery A H Correia and C D C Freitas ldquoClassify-ing multifrequency fully polarimetric imagery with multiplesources of statistical evidence and contextual informationrdquoIEEE Transactions on Geoscience and Remote Sensing vol 45no 10 pp 3098ndash3109 2007

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

[9] Y Delignon R Garello and A Hillion ldquoStatistical modeling ofocean SAR imagerdquo IEE Proceedings Radar Sonar and Naviga-tion vol 144 no 6 pp 348ndash354 1997

[10] Y Delignon and W Pieczynski ldquoModeling non-Rayleighspeckle distribution in SAR imagesrdquo IEEE Transactions onGeoscience and Remote Sensing vol 40 no 6 pp 1430ndash14352002

[11] A C Frery H-J Muumlller C D C F Yanasse and S J SSantrsquoAnna ldquoAmodel for extremely heterogeneous clutterrdquo IEEETransactions on Geoscience and Remote Sensing vol 35 no 3pp 648ndash659 1997

[12] K L P Vasconcellos A C Frery and L B Silva ldquoImprovingestimation in speckled imageryrdquo Computational Statistics vol20 no 3 pp 503ndash519 2005

[13] 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

[14] S N Annsen and T Elto ldquoApplication of the Matrix-variatemellin transform to analysis of polarimetric radar imagesrdquo IEEETransactions on Geoscience and Remote Sensing vol 49 no 6pp 2281ndash2295 2011

[15] S I Gradshteyn and I M Ryzhik Table of Integrals Series andProducts Academic Press San Diego Calif USA 7th edition2007

[16] G Moser J Zerubia and S B Serpico ldquoSAR amplitudeprobability density function estimation based on a generalizedGaussian modelrdquo IEEE Transactions on Image Processing vol15 no 6 pp 1429ndash1442 2006

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of