HYPERSPECTRAL IMAGE FEATURE EXTRACTION BASED ON GENERALIZED DISCRIMINANT ANALYSIS

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1 HYPERSPECRAL IMAGE FEAURE EXRACIO BASED O GEERALIZED DISCRIMIA AALYSIS Guopeng Yang a, *, Xuchu Yu, Xn Zhou c, a Zhengzhou Insue of Surveyng and Mappng,455, Henan, Chna - yangguopeng@homal.com Zhengzhou Insue of Surveyng and Mappng,455, Henan, Chna - c_yu@yahoo.com c Insue of Informaon Engneerng,455, Zhengzhou, Henan, Chna - Commsson VII, WG VII/3 KEY WORDS: Hyperspecral Image, Feaure Eracon, Generalzed Dscrmnan Analyss, Kernel Funcon ABSRAC: he hyperspecral mage enrches specrum nformaon, so compared h panchromac mage and mulspecral mage; can classfy he ground arge eer. he feaure eracon of hyperspecral mage s he necessary sep of he ground arge classfcaon, and he ernel mehod s a ne ay o erac he nonlnear feaure. In hs paper, Frs he mahemacal model of he generalzed dscrmnan analyss as descred, and hen he processng mehod of hs model as gven, fnally, e dd o epermens. hrough he ess, e can see ha, n he feaure space eraced y generalzed dscrmnan analyss, he samples of he same class are near h each oher; he samples of he dfferen classes are far aay. I can e concluded ha he mehod descred n hs paper s suale o hyperspecral mage classfcaon, and can do eer o han he mehod of lnear dscrmnan analyss.. IRODUCIO Hyperspecral remoe sensng echnology, hch frsly comes ou n he early 98s, organcally hangs he radaon nformaon hch relaes o he arges arue, and he space nformaon hch relaes o he arges poson and shape ogeher. he specrum nformaon, hch he hyperspecral mage enrches, compared h panchromac remoe sensng mage and mul-specral remoe sensng mage, can e used o classfy he ground arge classfcaon eer. Hyperspecral remoe sensng has very de elecromagnec ave range, from vsle lgh o shorave red, even o medum nfrared and hermal nfrared. I has hgh specral resoluon, and has los of ands, so can ge he ground arge s specral feaure curve, and recognze he arges y selecng and eracng he ands. We can ge he arge s specral radan parameers, and he quanave analyss of he earh's surface arge and eracon ecome possle. Because of he advanages of hyperspecral remoe sensng, a presen, los of counres n he orld have respec for hs ype of remoe sensng. Hyperspecral remoe sensng craf s form aeral o space aerospace. I ll ecome an mporan pah of map carography, vegeaon nvesgaon, ocean remoe sensng, agrculure remoe sensng, amosphere research, envronmen monorng, mlary nformaon acqurng (ong e al., 6). he hyperspecral mages have so hgh dmenson and he ground arges are so complcaed, ha s dffcul o oan enough ranng samples (Hoffec e al., 996). Hoever, he radonal mage classfcaon mehod, such as he sascal paern recognon and neural neors mehods, hch are ased on large numer samples hypohess, need o ge enough ranng samples o evaluae he pror classes nformaon hch ofen cause he Hughes phenomenon. So, he feaure eracon s one of he mos mporan seps hen e analyze he hyperspecral mages (Zhang, 3). In he md 99s, h he ernel mehod appled o suppor vecor machne successfully, people ry o eend he ordnary lnear mehods of feaure eracon and classfcaon o nonlnear suaon y usng ernel funcon. Kernel mehods for paern analyss are developng so fas ha here are so many achevemens n he appled felds. I s named as he hrd revoluon of paern analyss algorhms follong he lnear analyss algorhms, neural neors and decson rees learnng algorhms. Kernel mehods have ecome focus of machne learnng, applcaon sasc, paern recognon, and daa mnng, successfully appled n face recognon, speech recognon, characer recognon, machne malfuncon classfcaon and so on (John e al., 5). We don need o no he concree form and parameers of he nonlnear mappng, he changes of form and parameers of ernel funcon can change he mappng from he npu space o feaure space, and change he performance of ernel mehods. We can avod dmenson dsasers phenomenon hch es n radonal mode analyss mehods y usng he ernel funcon, and also can smplfy compuaon, herefore, Kernel mehods can precede he npu h hgh dmensons. he ernel mehods can comne h he dfferen analyss algorhms, desgn he dfferen ernel algorhms, and he o pars can e desgned separaely, so e can selec dfferen ernel funcon and analyss algorhm n dfferen applcaon felds. In order o mprove classfcaon accuracy of hyperspecral remoe sensng mage, e can use he specal classfer, such as SVM and KFDA. If e erac suale feaure of he hyperspecral mage, he common classfer also can e used. One of he research rends n hyperspecral mage s he * Correspondng auhor. el.: ; E-mal address:.yangguopeng@homal.com. 85

2 he Inernaonal Archves of he Phoogrammery, Remoe Sensng and Spaal Informaon Scences. Vol. XXXVII. Par B7. Beng 8 nonlnear mehods, and he ernel mehods provde a ne approach o he feaure eracon. Some research scholars have suded he feaure eracon mehods of hyperspecral ased on ernel funcon, such as ernel prncpal componens analyss (KPCA) and ernel Bhaacharyya feaure eracon (KBFE) (Lu, 5). In, he Generalzed Dscrmnan Analyss (GDA) as rough forard y Bauda (Bauda e al., ), hch s he nonlnear eracon of Lnear Dscrmnan Analyss, has een successfully used n face recognon (Gao e al., 4) and mechancal falure classfcaon (L, 3). In hs paper, e frs nroduced he mahemacal model and he soluon of he GDA, appled hs mehod o erac feaures from he hyperspecral mage. hen e made epermens h o groups of he hyperspecral mages hch ere oaned y dfferen nds of hyperspecral magng sysem. A las he resul as analyzed. he man conens ere descred n deal as follo.. GEERALIZED DISCRIMIA AALYSIS hrough mappng samples from he npu space o he feaure space h hgh dmensons, e carry on he lner mehods of feaure eracon n hs feaure space. Because of he dmenson n he feaure space s very large, and may e nfnude, n order o avod deal h he samples perceply,e use he ernel funcons o compue he nner produc n he feaure space.. heory of Feaure Eracon Based on GDA Suppose here are C classes of samples, hch are elong oω, ω, L, ωm, and he orgnal sample has n dmensons, so R n. If e map he sample o feaure space H h hgher dmensons y he mappng, n he feaure space, ll e( ) H.If all he samples are mapped o he fuure space H, he nraclasses scaer mar S, he nerclasses scaer mar S and he oal scaer mar S of he ranng samples, ll e descred as follos: c S = ( ( ) m )( ( ) m ) () = = c S = ( m m )( m m ) () = S = ( ( ) m )( ( ) m ) (3) = s he amoun of ranng samples elongng o he class ω, s he amoun of all he ranng samples. In he feaure space H, ( ) s he sample ( =, L ) of class ( =, L, C ), ) s he sample ( =, L, ) of ( all he samples, m = E{ ( ) ω } s he mean of samples n he class, m = P( ω ) m C = S, S and S are all nonnegave mares. s he mean of all he samples. In he feaure space H, he Fsher dscrmnan funcon can e defned as J( ) = S (4) S s a nonzero vecor. In he feaure space H, Generalzed Dscrmnan Analyss (GDA) s o fnd a group of dscrmnan vecors (, L d ), hch can mamze he Fsher dscrmnan funcon (4), and all he vecors are orhogonal. =, ;, =, L, d he frs dscrmnan vecor of GDA s also he fsher dscrmnan vecor, hch s he egenvecor correspondng o mamal egenvalue of egenfuncon S = λs.if e no he frs r dscrmnan vecors,, L r, he r + dscrmnan vecor r + can e goen hrough resolvng he follo opmzaon prolem. ma( J( )) ModelⅠ: =, =, L, r (5) H Accordng o he heory of he reproducng ernel Hler space, he egenvecors are lnear comnaons of H elemens, so can e epressed as = = α( ) = α (6) = ( ( ), L, ( )), α = ( α, L, α ), α s opmal ernel dscrmnan vecor, hch can map he sample ( ) n he feaure space o he drecon ( ) = ( ) = αξ (7) ξ ( (, ), (, ),, (, )) = L.For he n sample R, ξ s he ernel sample vecor hch relaes o,, L,, so he ernel mar s K = ( ξ, ξ, L, ξ ) In he feaure space H, he mean of each classes and he mean of all he samples can also e mapped o he drecon Τ m = α ( ) = α μ (8) = Τ = ( ) = = m α α μ (9) μ = ( ( ) ( ) ), L, ( ( ) ( )) () = = μ = = = ( ( ) ( ) ), L, ( ( ) ( )) () Accordng o he Equaon (8),()and (),here are f S = α Kα () S α K α (3) f = 86

3 he Inernaonal Archves of he Phoogrammery, Remoe Sensng and Spaal Informaon Scences. Vol. XXXVII. Par B7. Beng 8 S f = c α K α (4) K = ( μ μ)( μ μ ) = ( )( ) c K = ξ μ ξ = = K = ( ξ μ )( ξ ) = K s he ernel nerclasses scaer mar, (5) μ (6) μ (7) K s he ernel nraclasses scaer mar, and K s he oal scaer mar. All of hree mares are nonnegave mares, and her szes are. From Equaon () and (3), Fsher dscrmnan funcon (4) can e epressed as ( ) J α = α K α (8) α K α α s a nonzero vecor. he orhogonal consran condon can e epressed as = α α = α Kα =, ;, =, L, d So, he Model Ⅰcan e epressed y ernel mares as ma( J ( α)) ModelⅡ: α Kα =, =, L, r (9) α R ha s o say ha, f e no he frs r dscrmnan vecors α, L, αr, he r + dscrmnan vecor α r + can e go hrough resolvng he aove opmzaon prolem. α s he egenvecor correspondng o he mamal egenvalue of egenfuncon Kα = λk α. If { α, α,l, α d } s from he ModelⅡand {,, L, } s from Model Ⅰ, he relaonshp d eeen hem s α = = ( ) = α, =, L, d () ( ) = ( ), L, ( ). In Bauda s leraure (Bauda e al., ), nsead of J ( ), hey used J ( ) J ( ) = S S Correspondngly, he Model I of GDA can e reren as ma( J ( )) ModelⅠ: =, =, L, r () H and he Model Ⅱof GDA can e reren as ma( J ( α)) ModelⅡ: α Kα =, =, L, r () α R For Model I h J ( α ), f e have non he frs α r+ r ( r ) dscrmnan vecors, he can e goen y resolvng he follong egenfuncon. ΓK α = λk α (3) r+ r+ Γ = I KΛ ( ΛKK KΛ ) ΛKK mar. Λ= ( α, α, L, α ).Because s an deny vecor n r, I s an deny Model I, = α Kα =.If α has een non, α should e sandardzed y dvdng α Kα. In he feaure space H, f a group of dscrmnan vecors {,, L, d} have een non, for he sample ( ), s dscrmnan feaure s = α = α = = ( ) ( ) ( ) (, ) α ξ (4) ξ s ernel vecor of he npu sample. = he ransformaon funcon of GDA s y = W ( ) = [,,, L d ] ( ) (5) = [ α, α,, L αd ] ξ y s he feaure eraced y GDA hch has d dmensons.. Kernel Funcon Basng on he heory of ernel funcon, once a ernel funcon (, y ) accords h Mercer heorem, hen corresponds o a nner produc ernel funcon, mappng funcon and feaure space n a ceran space. In fac, o change ernel parameer s o mplcly change mappng funcon n order o change he compley of dsruon n sample su-space. here are hree nds of ernel ha are usually used. () Dmensonal polynomal ernel of degree d (, y) = [( y ) + p] d p and d are cusom parameers. If p = and d =, ll e called lnear ernel funcon. () Radal ass funcon (RBF) ernel y (, y ) = ep σ σ >. (3) eural eor ernel funcon (, y) = anh( μ( y ) + v) μ and v are parameers. Dfferen from polynomal ernel and RBF ernel, he neural neor ernel accords h he Mercer heorem only hen ( μ, v) are ceran values..3 Flo of Feaure Eracon ased on GDA Accordng o Bauda s leraure (Bauda e al., ), e selec J ( ) as he Fsher dscrmnan funcon, hrough he analyss aove, he seps of feaure eracon ased on generalzed dscrmnan are descred as follos., and s parameers, and () Selec he ernel funcon ( ) he amoun d of he feaure ll e eraced. 87

4 he Inernaonal Archves of he Phoogrammery, Remoe Sensng and Spaal Informaon Scences. Vol. XXXVII. Par B7. Beng 8 () Calculae he ernel mar K, and calculae K and K accordng o he Equaon (5) and (7). (3) Resolve he Equaon Kα = λk α n order o ge he egenvecor α correspondng o he mamum egenvalue. (4) Ge oher dscrmnan vecors α, α3, L, αd y Equaon (), and sandardze hem y dvdng α Kα. (5) Erac he feaure usng Equaon (5) for any npu sample. 3. EXPERIME In order o no heher he feaure eracon ased on GDA could mprove he classfcaon precson of hyperspecral mage, e dd o epermens. he epermens daa are oaned y dfferen remoe sensors (AVIRIS and PHI). We also compared he GDA h oher feaure eracon mehods, ncludng Prncpal Componen Analyss (PCA), Kernel PCA (KPA), and Lnear Dscrmnan Analyss (LDA). 3. Epermen Flo he seps of he epermens e have done are gven elo: () Collec he samples of dfferen ground ypes accordng he specral lrary or he non ground cover nformaon. And hen, dvde he samples no ranng samples and es samples. () Usng he ranng samples, calculae he ransform mares of dfferen feaure eracon mehods separaely, ncludng PCA, KPCA, LDA and GDA. (3) From he ransform mares hch e go n Sep e eraced he feaure of he hyperspecral mages. (4) ran he Mnmum Dsance Classfer (MDC) hrough ranng samples h feaure eraced y Sep 3. And hen, evaluae he classfcaon resul of he esng samples. Class ame Samples umer Alune 64 Buddngone 89 Dce 395 Kaolne 9 Le 76 Quarz 85 Sal 38 uff 33 ale. Samples of hs hyperspecral mage Amospherc radaon correcon ased on AREM has een appled o he AVIRIS mage. Afer elmnang he ands hch have oo much nose and hch are asored y he vapour, e used 9 ands n he epermen. We seleced 5 samples each class randomly as he ranng samples, and aled he ohers as esng samples. In he es, e seleced he Poly ernel and RBF ernel for KPCA and GDA. he feaure mages eraced ased on s shon n Fgure. () Image of he frs feaure 3. Epermen Epermen Daa: he ASA AVIRIS (Arorne Vsle/ Infrared Imagng Specromeer) nsrumen acqured daa over he Cupres mne feld, evada, USA. AVIRIS acqured daa n 4 ands of nm dh h cenre avelenghs from 4-5 nm. he mage of hs daa s shon n Fgure. here are egh nds of ores n hs area; he samples of hem are descred n ale. () Image of he second feaure Fgure. Hyperspecral mage from AVIRIS (R:78,G:,B:33) (3) Image of he hrd feaure 88

5 he Inernaonal Archves of he Phoogrammery, Remoe Sensng and Spaal Informaon Scences. Vol. XXXVII. Par B7. Beng 8 (4) Image of he forh feaure Fgure. he feaure mages eraced y 7 ( σ = ) We evaluaed he classfcaon precson h he esng samples, usng he mnmum dsance classfer, and he resul as shon n ale. he classfcaon resul h he feaure eraced y as shon n Fgure 3. Fgure 4. Samples dsruon n hs PHI mage We assgned he samples each class randomly as he ranng samples and esng samples equally. he feaure as eraced y dfferen feaure eracon mehods. In he feaure space, he dsruon of samples as shon n Fgure 5. () PCA () Ploy-KPCA ( d =, p = ) Fgure 3. he classfcaon resul h feaure eraced y 7 ( σ = ) Feaure eraced Mehods Mss classfcaon (%) All ands 3.87 PCA 5.7 LDA 3.84 Ploy-KPCA d =, p = 4. RBF-KPCA 7 σ = 9. Ploy-GDA d =, p = σ = σ = (3) LDA (4) ( σ = ) Fgure 5. Samples dsruon n dfferen feaure space We assgned he samples each class randomly as he ranng samples and esng samples equally. he feaure as eraced y dfferen feaure eracon mehods. In he feaure space, he dsruon of samples as shon n Fgure 5. ale. he precson of classfcaon h feaures eraced h dfferen mehods. 3.3 Epermen Epermen Daa: he PHI nsrumen, creaed n Shangha Insue of echnology and Physcs, acqured daa over Changzhou, Jangsu, Chna, (E9, ). PHI acqures daa n 8 ands dh h cenre avelenghs from.4.85μm, and he sze of he mage s 346Х5. S nds of oecs es n he mage: (Colour-Class of he arge-amoun of sample): -house-, -aer-, 3-sol- 5, 4-ree-8, 5-vegeaon-66, 6-road-38, he resuls are vsualzed n fgure 4. Fgure 6. he classfcaon resul h feaure eraced y 3 ( σ = ) 89

6 he Inernaonal Archves of he Phoogrammery, Remoe Sensng and Spaal Informaon Scences. Vol. XXXVII. Par B7. Beng 8 Feaure eraced Mehods Mss classfcaon (%) All ands 8.77 PCA 9.4 LDA 8.5 Ploy-KPCA d =, p =.84 RBF-KPCA 7 σ = 6.8 Ploy-GDA d =, p = σ =.46 8 σ = 3.5 Bauda, G., Anouar, F..Generalzed dscrmnan analyss usng a ernel approach. eural Compuaon, (), pp Gao X.M., Yang, J.Y., Jn, Z., 4, Kernel-Based Foley- Sammon Dscrmnan Analyss and Face Recognon. Journal of Compuer-Aded Desgn & Compuer Graphcs, pp L W.H., 3, Mechancal Faul Feaure Eracon and Classfcaon Based on Kernel Mehods. Huazhong Unversy of Scence & echnology, pp.3-4. ale 3. he precson of classfcaon h feaures eraced h dfferen mehods. We evaluaed he classfcaon precson h he esng samples, usng he mnmum dsance classfer. he classfcaon resul h he feaure eraced y as shon n Fgure 6. he classfcaon resul of dfferen feaure eracon mehods s shon n ale COCLUSIO hrough he epermens of feaure eracon h AVIRIS and PHI mages e made some conclusons. he PCA s o fnd proec drecons, hch can mae he samples varance mamzed. he KPCA, usng he ernel funcon, can realze he nformaon compresson o a grea een, u s no good for classfcaon. When he ernel funcon and s parameers are correcly seleced, n he feaure space eraced y GDA, he samples of he same class are near h each oher; he samples of he dfferen classes are far aay. he GDA s a feaure eracng mehod hch s more suale o classfcaon han he LDA. When he ernel funcon and s parameers are correcly seleced, he classfcaon precson s much eer h he feaures eraced y GDA, han he feaures eraced y oher mehods. Ho o selec he ernel funcon and fnd suale parameer s our furher research. REFERECES ong, Q.X., Zhang, B., Zheng, L.F., 6. Hyperspecral Remoe Sensng. Hgher Educaon Press, Beng, pp Hoffec J.P., Landgree D.A., 996, Classfcaon of Remoe Sensng Images havng Hgh Specral Resoluon. Remoe Sensng of Envronmen, 57(3), pp.9-6. Zhang, L.P., 3. Sudy of Feaure Eracon and Classfcaon of Hyperspecral Remoe Sensng Image Based on Proecon Pursu and onlnear Prncple Curves. Shandong Unversy of Scence and echnology, pp.-8. John S.., ello C., 4. Kernel Mehods for Paern Analyss. Camrdge Unversy Press. Camrdge, pp.-4. Lu, W., 5. A Research of Feaure Eracon and Classfcaon echnques for arge Deecon n Hyperspecral mage. Informaon engneerng Unversy. Zhengzhou, pp