LAND COVER CLASSIFICATION USING E-SAR POLARIMETRIC DATA
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1 LAND COVR CLAIFICATION UING -AR POLARIMTRIC DATA V. Ktnssi *, M. Dbboo Lbotoy of Remote ensing, cool of Rul nd uveying ngineeing, Ntionl Tecnicl Univesity of Atens, Heoon Polytecniou 9, Zogpos, 15780, Geece Commission VII, WG VII/3 KY WORD: Lnd cove, Poliztion, Detection, AR, Accucy, Teoy, Test ABTRACT: Diffeent decomposition ppoces ve been poposed in ode to nlyse nd intepet AR polimetic imges. Tese e bsed eite on te complex voltge eflection mtix, like Puli, o on powe eflection mtix, like te covince o coeency mtix. Tey poduce polimetic pmetes wic e ppopite to etieve infomtion on te sctteing pocess of te tget. If te tget is distibuted, polimetic pmetes e ffected by speckle. Te obectives of tis wok e to sec nd point out te pmetes most ppopite fo te intepettion of diffeent lnd uses in AR imge; to evlute Mximum Likeliood ML clssifiction esults poduced by two diffeent polimetic input sets: te full polimetic, nd te Puli imges; to investigte te most ppopite size of te Lee filte window fo polimetic speckle eduction. Bsed on te full polimetic L-bnd, poliztion signtues wee extcted nd nlyzed fo fou lnd cove clsses: ubn, foest, vegettion nd smoot sufces. Te sctteing mecnism of tese lnd cove clsses ws lso nlysed bsed on te imges geneted by Puli decomposition nlysis. Te Mximum Likeliood clssifiction ws pefomed on te mgnitude content of te oiginl polimetic dt, b imges poduced by te Puli nlysis, nd c bot pevious cses dt. Te ccucy of ec clss confimed te contibution of polimetic dt nd Puli pmetes in te intepettion of te sctteing mecnism. To educe speckle effects nd impove clssifiction esults, te Lee filte ws pplied on te bove imges sevel times, ec time incesing te size of te moving window. Te ML clssifiction ws pefomed on te despeckled imges. Clssifiction ccucy pointed out te most ppopite size of te filte window fo speckle eduction. 1. INTRODUCTION Full polimetic dt cn define te sctteing beviou of lnd use/cove, tus giving bette lnd use/cove clssifiction esults tn single-cnnel AR mit, Boek, Dekke, evel pmetes ve been poposed to ssist te intepettion nd te clssifiction of polimetic AR dt. Tese pmetes e deduced fom te decomposition of eite te complex voltge eflection mtix o te powe eflection mtices. Te fist decomposition ctegoy dels wit te spee/deplne/elix decomposition nd te Puli decomposition. Te ltte ctegoy cn be divided into te Mulle/Kennug mtix nd covince o coeency mtix Hellmnn, Among tem te bsic Cloude-Pottie pmetes, entopy nd lp, Cloude, Pottie, 1997 deduced fom te Cloude decomposition teoem pplied on te coeency mtix e te most investigted Hellmnn, Ktzscm, 1998; Titin-cnide, 1999; ceucl, Cves, Cumming, tples, 001 fo lnd use/cove intepettion. Te numbe of tese pmetes ws fte incesed by te ddition of two polizing pmetes, te popgtion nd elicity pse ngles nd tee depolising pmetes, te nisotopy A nd two depolising eigenvecto ngles Cloude, Potie, Boene, 00. Pmetes deduced by te spn nomlistion of te Muelle mtix ve lso been investigted in ode to etieve sctteing electomgnetic mecnisms Titin-cnide, 1999 nd intepet polimetic dt. Bsed on te coeency mtix, te complex Wist clssifie, wic uses te complex Wist distibution of te coeency mtix nd mesues n ppopite distnce, d, ccoding to mximum likeliood clssifiction Lee, Gunes, Kwok, 1994, s been investigted fo supevised lnd use/cove clssifiction Lee, Liew, Kwo, Nkym, 001. An unsupevised clssifiction metod bsed on te Wist clssifie ws lso developed ceucl, Cves, Cumming, tples, 001, s well s, weiged Wisd clssifie ccoding to wic ec polimetic component is weiged eltive to te bsolute mplitude of te mesuements mit, Boek, Dekke, Focusing on te evlution of te polimetic infomtion deduced by te simplest pocessing metods, te obectives of tis wok e to: 1. Intepet lnd use/cove sctteing beviou by nlysing te poliztion signtues egding to te ellipticity ngle, oienttion nd intensity, nd b signtues extcted by te Puli decomposition metod. Clssify lnd use/cove of te test e bsed on te mgnitude content of te oiginl full polimetic dt, b te dt poduced by te Puli decomposition metod, nd c bot pevious cses dt. 3. Define te most ppopite size of te Lee filte window, pplied fo speckle eduction. Te full polimetic ibone dt sets wee cquied wit DLR s xpeimentl AR -AR. Te test egion is te e of Obepfffenofen, Gemny. Fo te study needs, te L bnd ws used wic s esolution of 3 metes.
2 . POLARIZATION AND PAULI DCOMPOITION THORY.1 Wve Poliztion Fo plne tnsvese electomgnetic wve, te vecto of te electic field oscilltes in plne pependicul to te popgtion diection. Te vectoil ntue of tese wves is clled poliztion nd is independent of te cosen coodinte system. In te cse tt te tce of te tip of te field vecto witin plne pependicul to te popgtion diection is n ellipse, te wve is ellipticlly polized. pecil cses of te ellipliclly poliztion e line o cicul poliztion. Fo te desciption of poliztion, coodinte system nd efeence diection of popgtion e needed. Fo comptibility wit fully polimetic d systems tt use two otogonl line polized ntenns, Ctesin coodinte system is intoduced, wee + k is te popgtion diection of te plne tnsvese electomgnetic wve, nd nd v e te oizontl nd veticl diections of te plne of te electic field. Te eqution of tnsvese electomgnetic wve s function of its position is: ik = exp 1 Te vecto of te complex electic field consists of component nd v v component, wic e pependicul to te popgtion diection. = + v v = cos δ cos δ 1 + = sin wic is te eqution of n ellipse figue 1 wit n oienttion ngle ψ suc tt tn ψ = cos δ Consequently, poliztion my lso be descibed by te geometicl popeties of te ellipse wic e te oienttion ngle ψ nd te ellipticity ngle χ sin δ δ 6 7 tn χ = ± 8 b wee nd b e te mino nd mo xes of te ellipse, nd χ specifies te spe of te ellipse s well s te sense of ottion of te vecto. Te poliztion is left nded fo 0<χ<π/4 nd igt nded fo π/4<χ<0 fo n obseve looking in te diection of te popgtion. Te poliztion ngles ψ nd χ e elted to te wve pmetes v,, nd δ by sinχ = sinα sinδ tnψ = tnα cosδ 9 wee te ngle α is defined s Tese components cn be expessed on te bsis of tei el mplitude = nd pse expiδ : = = exp iδ wee = o v. Te component of te electic field cn lso be witten s function of time t nd position : tnα = 3 10 v { exp i k ω t } = Re{ exp i k ωt + δ } = cos k ωt + δ, t = Re = cos τ + δ 4 wee τ = k ω t. By defining ngle δ s te eltive pse diffeence between te two components, δ = δ v δ, it esults tt:, t = cos τ + δ + δ = cos τ + δ cos δ sin τ + δ sin δ = cos τ + δ cosδ 1 cos τ δ sinδ 5 + ubstituting cosτ +δ by eqution 4 we obtin: Figue 1. Wve ellipticlly polized in v, plne wit popgtion diection + k If bot mplitudes e equl v =, nd χ = ± π/4, te poliztion is cicul. Fo χ = +π/4 te poliztion is left nd cicul nd fo χ = -π/4 te poliztion is igt nd cicul.
3 In te cse tt te pse ngles e equl δ v =δ δ = 0 te tce of te tip of te electic field vecto is stigt line.. Te Puli decomposition ppoc If scttee is illuminted by n electomgnetic plne tnsmitted by n ntenn, te incident wve t te scttee is given by t = + v 11 t t v nd tt induces cuents in te scttee, wic in tun edite sctteed wve. In te f zone of te scttee, te sctteed wve cn be consideed s plne wve. Te sctteing pocess cn be modelled s line tnsfomtion, descibed by mtix. Te eceived field is ten given by: e t e t exp ik 1 = [ ] = = e t Te [] mtix is efeed to s Jones mtix Jones, 1941, nd is complex x mtix, contining infomtion on te scttee. nd vv e clled co-pol nd v nd v cosspol components. Accoding to te ecipocity teoem, te coss-pol components e equl. Insted of te mtix nottion, one my use fou element complex vecto k, wic contins complete infomtion on te [] mtix T [ ] = k = Tce[ ] Ψ = [ k 0, k1, k, k3 ] wee Tce [] is te sum of te digonl elements of [] nd Ψ is complete set of x complex bsis mtices unde emitin inne poduct. Any complete otonoml bsis set of fou x mtices cn be used. A bsis wic is moe elted to te pysics of wve sctteing, te Puli bsis, is fomed by te Puli spin mtices Cloude, Ψ P = 1 0 0, , , 0 0 i i 14 0 Puli mtix ctteing type Intepettion 1 0 odd-bounce ufce, 0 1 spee,coneeflectos 1 0 even-bounce diedl even-bounce titled 45 ο 45 ο titled diedl 0 i coss-polise Not existent fo i 0 bcksctteing Tble. Puli mtices nd tei intepettion in te poliztion bsis, e v 3. INTRPRTATION OF TH POLARIMTRIC DATA 3.1 Intepettion of poliztion signtues A poliztion signtue is given by its 3-D pesenttion. X,Ψ,Ζ xes e ssigned to te ellipticity ngle χ, te oienttion ngle ψ, nd te intensity of te co-pol o coss-pol components of te d signl Z. As mentioned peviously, te ellipticity ngle tkes vlues in te intevl [ 45 o, 45 o ] nd te oienttion ngle in te intevl [0, 180 o ]. Poliztion signtues wee extcted fo fou lnd uses: ubn, foest, vegettion, nd smoot sufces ods, unwys, etc. Te signtue of te ubn clss pesents ig vlues of intensity fo n ellipticity ngle close to 45 o nd n oienttion ngle close to 180 o figue 3. Tis mens tt poliztion is left nd cicul i.e. te mplitudes of te co-pol components nd vv e equl nd te eltive diffeence ngle close to 90 o, nd te oienttion of te wve tnsmitted cnges 180 o eltive to te oienttion of te wve eceived. Te bove desciption fits well to te even-bounce sctteing type of oizontl diedls, wic coespond to te buildinggound intection tt we encounte in ubn es. In figue 3 we lso obseve vey low vlues of intensity fo n ellipticity ngle close to 0 o nd n oienttion ngle close to 0 o. Tis mens tt in ubn es, poliztion oizontlly oiented, wic is te dipole like sctteing cse, is missed. Te low intensity vlues fo n ellipticity ngle close to -45 o nd n oienttion ngle close to 0 o denote te sufce sctteing type of smoot sufces e.g. ods found in te ubn es. Te coesponding vecto k p is ten 1 k P = [ +,, +, i ] T 15 Te Puli decomposition ppoc ssigns te ppopite deteministic sctteing mecnisms to ec one of te fou elementy sctteing mtices. Te bsic sctteing mecnisms e: isotopic sufce, igt wound elix nd left wound elix. Consequently, te Puli mtices cn be intepeted s sown in tble Hellmnn, b Figue 3. Te co-pol nd b te coss-pol signtue of te ubn e Ubn es lso ve lge coss-pol contibution figue 3. Te nge entie of vlues of te ellipticity ngle is encounteed fo ig intensity of te coss-pol components nd fo n oienttion ngle close to 9 o, nd b 16 o. Tis poves tt discontinuities of bot sctteing types, diedls nd sufces, esult in ndom poliztion ellipses. Tose wit n oienttion
4 comptible wit te d look ngle pesent te igest vlues in te coss-pol signtue. Te cse of π/4 titled diedls is little pesented in tis figue, fo ig intensity of te cosspol components, n ellipticity ngle equl to 45 o nd n oienttion ngle close to 180 o. In te foest co-pol signtue, te igest vlues of intensity e obseved fo n ellipticity ngle close to 0 o, nd n oienttion ngle close to 160 o figue 4. In tis cse, poliztion is lmost oizontlly oiented i.e., pesents ig vlue nd vv vlue close to 0. Consequently, sevel es in te foest envionment denote dipole like sctteing, esulting fom te volume sctteing, i.e. wve sctteed by bnces nd/o leves, b contibution of two sctteing types, diedl nd sufces. Te ltte occus becuse esolution cell cn epesent eite tunk to gound intection wic denotes te double sctteing mecnism, o unifom smoot e wic denotes sufce sctteing mecnism. Altoug foest leves e ndomly oiented, tey fit moe to oizontl dipoles. Tus, te foest signtue pesents low vlues of copol intensity fo n ellipticity ngle close to 0 o nd n oienttion ngle close to 70 o figue 4, i.e. fo wves tt e veticlly polized vv pesents ig vlue nd vlue close to 0. In te foest coss-pol signtue figue 4, like in tt of ubn es, te d look ngle defines te oienttion of ig coss-pol intensities. Te cse of π/4 titled diedls is stongly pesented in tis figue fo ig intensity of te cosspol components, n ellipticity ngle equl to 45 o nd n oienttion ngle close to 180 o. In te unwy co-pol signtue figue 6, te lck of etuned wve is obseved fo lmost te entie nge of vlues of te ellipticity ngle, i.e. fo ndom ellipses. Indeed, low intensity vlues e obseved fo n oienttion ngle close to 0 o. b Figue 6. Te co-pol nd b te coss-pol signtue of te unwy Futemoe, tis signtue is simil to tt of te vegettion - dipole like sctteing pesenting, oweve, lesse intensity vlues. Dipoles e fomed long te unwy by te contibution of two sctteing types, diedl nd sufces. Tis occus becuse esolution cell cn epesent eite gss to gound intection wic denotes te double sctteing mecnism, o unifom smoot e wic denotes te unwy sufce. In te unwy coss-pol signtue figue 6, te cse of π/4 titled diedls is sligtly pesented. 3. Intepettion of te Puli signtues Te Puli components e computed s following: Puli 1 = Puli = Puli 3 = v + vv + v vv 16 b Figue 4. Te co-pol nd b te coss-pol signtue of te foest e In te vegettion co-pol signtue, we obseve ig vlues of intensity fo n ellipticity ngle close to 0 o, nd oienttion ngles close to 160 o nd 80 o espectively figue 5. Tis mens ig etun of te wve sctteed by oizontlly nd/o veticlly oiented dipoles. Te low intensity vlues, wic we obseve fo n ellipticity ngle close to -45 o nd n oienttion ngle close to 0 o, denotes te sufce sctteing type of smoot sufces like be soil o sowed fields. In contst to te foest signtue, vegettion signtue denotes n ogniztion of te wee Puli1 denotes te even bounce component, Puli te 45 o titled even bounce component, nd Puli3 te odd bounce component. Fo ec of te 3 components we clculte te bsolute vlue, we scle it wit n exponent of 0.7 nd ssign colo. Red colo is ssigned to Puli1, geen to Puli, nd blue to Puli 3. Puli signtues e extcted fom te colo composition of te tee components. Fo te ubn e te ed colo, i.e. diedls, domintes figue 7. Te oienttion of te steets is consequently extcted. ufce sctteing type geen nd 45 o titled diedls blue e less pesented. Tee e lso es in wic ll te sctteing mecnisms detected by te Puli metod e simultneously pesented. Tese es e sown wit wite b Figue 5. Te co-pol nd b te coss-pol signtue of te vegettion scttees wic llows tem to ct eite s oizontl leves o veticl tunks dipoles. In te vegettion coss-pol signtue figue 5, te cse of π/4 titled diedls is stongly pesented. c Figue 7. Colo composition of te Puli nlysis fo te ubn, b foest, c vegettion, nd d unwy clss b d
5 colo. In te foest signtue, colos e eqully pesented, i.e. none of te sctteing mecnisms wic e detected by te Puli metod, domintes. In te vegettion signtue, te blue colou domintes. Tis mens tt te sufce sctteing type is te most pesented. Te unwy signtue is pesented by dk tones. Tis mens tt none of te sctteing type domintes. %Totl ccucy CLAIFICATION Te window size 4.1 Clssifiction bsed on oiginl polimetic dt Bsed on te bsolute vlue of te nd vv nd v + v / of te polimetic components, we pplied te supevised Mximum Likeliood clssifie in ode to clssify te study e into fou clsses: ubn, foest, vegettion, nd unwys. Clssifiction ws pplied on w dt nd b on dt witout speckle, geneted by te ppliction of te Lee filte. In ode to investigte te most ppopite window size, we fist pplied te Lee filte 9 times, ec time incesing te size of te widow by step equl to. Te windows used wee 3x3, 5x5, 17x17. Ten clssifictions wee pefomed, nd tei ccucy ws clculted bsed on set of test es. Te totl ccucy of te clssifiction pplied on te w dt is 83.60%, nd te esults obtined fte te Lee filte ppliction e given in figue 8. % Totl ccucy Te window size Figue 9. % Totl ccucy of te polimetic dt bsed clssifiction s function of te size of te Lee filte We obseve tt te igest ccucy, 91%, is obtined fo window size 15x15. In tis clssifiction, te ccucy obtined fo te clss of te: ubn e is 45,84%, foest 94.7%, vegettion 95.43%, nd unwy 99.3%. Te ubn clss is most confused wit te foest. Altoug te poliztion signtues of te two ctegoies e quite diffeent figue 3 nd 4, due to te fct tt Byes clssifie is bsed on te mgnitudes only, te ubn nd foest clsses e confused. Fo te ote clsses, te mgnitude bsed clssifiction poduces ig ccucies. 4. Clssifiction bsed on Puli decomposition nlysis Byes clssifie ws lso pplied on te bsolute vlues of te tee Puli decomposition components, by using te sme tining set s in te pevious clssifiction. Clssifiction ws pplied on te initil Puli components, nd b on te Puli components fte te ppliction of te Lee filte. Te most ppopite window size ws investigted by te metod descibed in te pevious section. Accucy ws tested by te sme test set s in te pevious clssifiction. Te totl ccucy of te clssifiction pplied on te initil Puli components is 80.00%, nd te esults obtined fte te Lee filte ppliction e given in figue 9. Figue 9. % Totl ccucy of te Puli bsed clssifiction s function of te size of te Lee filte We obseve tt te igest ccucy, 91.41%, is obtined fo window size 3x3. In tis clssifiction, te ccucy obtined fo te clss of te: ubn e is 81,08%, foest 78.06%, vegettion 97.35%, nd unwy 99.01%. Te ubn nd foest clsses e confused. Altoug te ccucy fo te ubn clss is significntly incesed in compison to te clssifiction bsed on oiginl polimetic dt, te ccucy of te foest clss is educed. Te sctteing mecnisms tt Puli decomposition nlyzes, nd especilly te even bounce nd 45 o titled even bounce mecnisms, einfoce te discimintion of te ubn clss. On te ote nd, te wekness of te Puli decomposition in nlyzing te volume sctteing mecnism bsed on dipoles ffects te ccucy fo te foest clss, wic is nlyzed on te bsis of diedls. Fo te ote clsses, te Puli bsed clssifiction poduces ig ccucies. 4.3 Clssifiction bsed on oiginl nd Puli decomposition nlysis To educe confusion between ubn nd foest es, Byes clssifie ws pplied on te bsolute vlues of te polimetic dt, nd b dt geneted by Puli decomposition nlysis. In tis wy, odd nd even bounce sctteing mecnisms, s well s, oiginl full polimetic infomtion pticipte in mgnitude bsed clssifiction. Clssifiction ws pplied on te initil dt, nd b on dt fte te ppliction of te Lee filte. Te most ppopite window size ws investigted by te metod descibed in section 4.1. Accucy ws tested by te sme test set s in te pevious clssifiction. Te totl ccucy of te clssifiction pplied on te initil dt is 8.14%, nd te esults obtined fte te Lee filte ppliction e given in figue 10. % Totl ccucy Te window size Figue 10. % Totl ccucy of te clssifiction s function of te size of te Lee filte We obseve tt te igest ccucy, 91.53%, is obtined fo window size 3x3. In tis clssifiction te ccucy obtined fo te clss of te: ubn e is 81,65%, foest 78.31%, vegettion 97.0%, nd unwy 99.5%. Te ccucy of te ubn clss is sligtly incesed ltoug ubn nd foest clss
6 e still confused. Ubn pixels clssified s foest ccount fo 18.19% of te ubn smple. Foest pixels clssified s ubn ccount fo 1.69% of te foest smple. 5. CONCLUION Te potentil of te polimetic dt to disciminte mong diffeent lnd uses ws investigted in tis study. imple pocessing metods, suc s extction of polimetic signtues nd Puli decomposition nlysis wee used to intepete te sctteing mecnisms of ec lnd use. Polimetic signtues wee poved to be poweful tool fo tis pupose. evel sctteing mecnisms ve been ecognized on ec signtue. Te lge numbe of sctteing mecnisms poduced by non-deteministic tgets, suc s smples of lnd use, ws tei min disdvntge. Altoug given polimetic signtue we cn ecognize te sctteing beviou of te specific lnd use, te opposite is sometimes difficult to cieve. Due to te lge numbe of sctteing centes tt lnd use smple includes, sevel secondy sctteing mecnisms e simultneously poduced. Tis cn mke it difficult fo te intepete to distinguis te most ccteistic mecnism fo te smple unde intepettion. Te non deteministic ntue of lnd use tgets ws lso indicted by te Puli decomposition nlysis. Diedls nd sufce e te scttees tt Puli decomposition cn detemine. Te limited numbe of sctteing mecnisms ecognized by te Puli nlysis is its min disdvntge. Its bility to indicte diedls is its min dvntge. Ubn es wic pesent complexity by including sevel diffeent sctteing centes in te sme tget cn esily be intepeted fte tei nlysis in diedls. Regding te definition of te size of te window of te Lee filte wic is used fo speckle suppession, clssifiction bsed metod ws developed nd pplied. Due to te non deteministic ppoc of te scttees, te optimum size of te window s been found to be quite lge, 17x17 pixels fo te oiginl dt nd 3x3 pixels fo te dt sets wic include imges geneted by te Puli decomposition metod. Tis documents te non deteministic ntue of lnd uses, s lge filte window is necessy fo ig clssifiction ccucies to be cieved. To evlute clssifictions bsed on te mgnitude of te polimetic dt, te mximum Likeliood clssifie ws pplied on te full polimetic dt, b te dt poduced by te Puli decomposition metod, nd c bot pevious cses dt. In ll cses speckle suppession peceded. Te totl ccucies obtined in ll cses wee stisfctoy %. Te vey ig esolution of te -AR dt lso contibuted to tis. Howeve, te ccucy obtined fo te ubn clss ws medioce 45.84% in te fist clssifiction esults, ponouncing te wekness of te mgnitude of te polimetic dt to disciminte ubn es fom foest. Te dt geneted by Puli decomposition contibuted to te impovement of clssifiction esults egding te ubn clss, poducing n ccucy equl to 81.08% nd 81.65% fo clssifictions b nd c, espectively. On te ote nd, tey educed te ccucy of te foest clss by ppoximtely 15%, 78.31%. Diedls e well disciminted by te Puli nlysis nd ccteize te ubn clss moe tn ote clsses. Wen te imges geneted by te Puli decomposition e intoduced in te Mximium Likeliood clssifiction contibute significntly to te sttisticl definition of tis ctegoy but tey poduce misclssifictions to te ote ctegoies wic lso contin diedls, suc s te foest. Te key subect fo fute wok will be te intoduction of te pse of te full polimetic dt in te clssifiction lgoitm. RFRNC Cloude,.R., Pottie,., Boene W., Unsupevised Imge Clssifiction using te ntopy/alp/anisotopy Metod in Rd Polimety. In AIRAR t cience nd pplictions Woksop, Mc Cloude,.R., Pottie,., An entopy bsed clssifiction sceme fo lnd pplictions of polimetic AR. I Tnsctions on Geoscience nd Remote ensing, Vol 35, n o 1, pp Hellmnn, M., Clssifiction of Full Polimetic AR- Dt fo Ctogpic Appliction. Tesis, Tecnicl Univesity of Desden, August Hellmnn, M., Ktzscm,., Intepettion of AR- Dt using Polimetic Tecniques. In: Te etievl of Biond Geo-pysicl pmetes fom AR Dt fo lnd ppliction Woksop, TC, 1-3 Octobe Lee, K.Y., Liew,.C., Kwo, L.K., nd Nkym, M., 001. Lnd Cove Clssifiction Using NAA/JPL Polimetic yntetic Apetue Rd POLAR Dt. In te nd Asin Confeence on Remote ensing, ingpoe, 5-9 Novembe 001. Lee, J.., Gunes, M., Kwok, R., Clssifiction of multilook polimetic AR imgey bsed on complex Wist distibution. Intentionl Jounl of Remote ensing, Vol. 15, no 11, pp ceucl, B., Cves, R., Cumming, I., nd tples, G., 001. Automted Ice Clssifiction Using pcebone Polimetic AR Dt. In: IGAR Poceedings, Austli. mit, A. J., vn den Boek, A. C., nd Dekke R. J., Lnduse Clssifiction Using PHARU Polimetic Rd. In Poceedings of te nd Intentionl Woksop on Retievl of Bio- & Geo-pysicl Pmetes fom AR Dt fo Lnd Applictions, 1-3 Octobe 1998, Netelnds. Titin-cnide, C., Rd polimety fo vegettion obsevtion. In: CO AR Woksop, Netelnds, 6-9 Octobe ACKNOWLDGMNT Autos e gteful to D J. Moei DLR wo kindly povided te -ARpolimetic dt.
The Disk Method. 456 CHAPTER 7 Applications of Integration
56 CHAPTE 7 Applictions of Integtion Section 7 Volume: Te Disk Meto Fin te volume of soli of evolution using te isk meto Fin te volume of soli of evolution using te wse meto Fin te volume of soli wit known
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