INTERFEROMETRIC TECHNIQUES FOR TERRASAR-X DATA. Holger Nies, Otmar Loffeld, Baki Dönmez, Amina Ben Hammadi, Robert Wang, Ulrich Gebhardt

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1 INTERFEROMETRIC TECHNIQUES FOR TERRASAR-X DATA Holger Nies, Omr Loffeld, Bki Dönmez, Amin Ben Hmmdi, Rober Wng, Ulrich Gebhrd Cener for Sensorsysems (ZESS), Universiy of Siegen Pul-Bonz-Sr. 9-, D-5768 Siegen Tel , FAX nies@ze.uni-siegen.de ABSTRACT Genering Digil Elevion Models (DEMs) ou of inerferomeric imge pirs is one of he well-known SAR pplicions which re used ll over he world. The im of our projec is o find new/exended mehods for InSAR proceing o improve he quliy of he finl DEM resul. We invesige in he fields of imge regisrion, bseline esimion/inerpolion nd phse unwrpping. In ech discipline here is lo of poenil for improving he individul resuls. The pper gives summry of he mehods we hve pplied nd shows some resuls using TerrSAR-X es sies. Index Terms SAR inerferomery, imge regisrion, informion mesures, orbi inerpolion, phse unwrpping, Klmn filer. INTRODUCTION Addiionlly o he clic correlion bsed mehods, we will employ, compre nd combine heoreicl mehods, bsed on informion heory, such s muul informion, lignbiliy in conjuncion wih uomed bin size deerminion o chieve opimlly resuls in imge regisrion. Considering he co-regisrion of single look complex SAR d, where correlion pproches re sndrd, he exploiion of similriy mesures nd rnsformion, bsed on informion heory, is rher novel. Compring hese novel pproches wih sndrd references wih respec o quliy nd quniive mesures, will broden he scienific undersnding, nd, furhermore, provide new insighs concerning coregisrion echniques for inerferomeric pplicions. By chieving more ccure regisrion of inerferomeric pirs he phse error will be reduced becuse he coherence will be improved. Bseline esimion nd inerpolion for fl erh removl nd heigh generion: Firs we wn o improve he given posiion nd velociy d by Klmn filering nd smoohing pproch, which uses ccelerion informion he cul sellie s posiion for correcing nd inerpoling he orbi d. Besides he vilble orbi se vecor mesuremens, he Klmn filer incorpores relisic models for grviion nd ir drg, hus enbling very ccure orbi esimion, propgion nd inerpolion. Afer pplying his lgorihm o he firs nd o he repeed orbi, he geomeric bseline vecor cn be clculed s he difference vecor of wo refined orbi posiions; i cn be furher rnsformed ino he inerferomeric bseline nd hen be furher improved by using co-regisrion vecors, deermined during he coregisrion proce. Phse Unwrpping: Generlly Klmn filer is powerful ool o obin ccure model bsed esimes ou of differen sources of informion. All he given informion is fused in n efficien wy nd lso he noise is cncelled opimlly. Becuse of his Klmn filer bsed d fusion pproch o unwrp nd simulneously filer he phses of inerferomeric SAR imges is developed. The d fusion concep explois phse informion, exrced from he complex inerferogrm rher hn from he phse imge nd fuses h informion wih he coherence mrix nd he phse slope informion exrced from he power specrl densiy of he inerferogrm. I is no necery o genere phse error bsed on compliced sisics neiher doing phse noise reducion; phse unwrpping kes plce simulneously wih removing he phse errors. 2. THE METHOD OF IMAGE REGISTRATION Imge regisrion is known s n imporn pr of genering Digil Elevion Models (DEM) wih Inerferomeric SAR (InSAR) proceing nd is one of he criicl preproceing seps in remoe sensing. I is used in he formion of 3-D models bsed on 2-D imges ken differen poins of view s well s for mosicking pplicions. The pper gives n overview of he informion mesures which cn be used for uomic generion of reference poins und finding he correspondences in he second imge. The pproch is vlided wih simuled nd rel imge pirs origine from he TerrSAR-X sellie. The correlion coefficien is one of he populr similriy mesures used for inr-modliy imge regisrion. I mesures he similriy by clculing globl sisics such s men nd vrince. Anoher widely used similriy mesure is he esimed muul informion which does no ume ny funcionl relionship beween imges o be regisered. I mesures he redundncy beween wo imges by looking heir inensiy disribuions. Muul informion represens relive enropy beween wo ses. In mulimodl imge regisrion muul informion is sndrd reference. We show h in he regisrion of imges of he sme modliy, muul informion cn be more robus nd relible hn he correlion coefficien.

2 To reduce compuionl cos, we use uomic subimge selecion s reducion in serch d sregy. We propose mesure, clled lignbiliy, which shows he biliy of subimge o provide relible regisrion. This feure cn be more relible hn oher subimge selecion mehods such s using grdien, enropy nd vrince, e.g. In firs sep will serch for pproprie reference poins wih high enropy. We compre rdiionl mehods (clculing he effecive specrl bndwidh, Shnnon s enropy ) wih oher (novel) informion mesures. Ler we wn o deec hese reference poins in he second imge pplying differen mehods nd sregies. For qunizing he errors which occur we genered d se of imge pirs where he differences (rnslion, roion, skewne, noise ) beween mser nd slve imge re excly known. Afer hving evlued he quliy of differen informion mesures for imge regisrion, we will lso presen regisrion resuls of rel d ses. Using repe p imges pirs from he recen Germn TerrSAR-X sellie we hve he poibiliy o regiser imges of high quliy nd high resoluion (up o m). The regisrion proce consi more or le of hese four individul seps: ) Auomicl generion of reference poins. This involves he exrcion of feures o be used for he mching proce. Some disincive feures include edges, conours nd lines of inersecion. 2) Feure mching. In his secion he correspondence beween he feures deeced in he inpu imge nd feures deeced in he reference imge. Similriy merics such s muul informion nd feure descripors re used for his purpose. 3) Trnsformion model esimion. The mpping funcions beween he inpu nd reference imges re esimed by mching he corresponding feures. 4) Imge resmpling nd rnsformion. The inpu imge is rnsformed ccording o he esimed mpping funcion. In his pper he firs wo poins re considered. These seps could be repeed for pplying course nd fine regisrion. The reference poins should be seleced regrding heir vlue of informion nd opiml geomericl disribuion specs. 2.. Informion mesures for he generion of reference poins 2... Shnnon Enropy The Shnnon Enropy is common informion mesure of he unceriny ocied wih rndom vrible. n n IShnnon = pk ln = pk ln pk () k= pk k= The enropy of rndom vrible is defined in erms of is probbiliy disribuion nd cn be shown o be good mesure of rndomne or unceriny. A subimge wih shrp probbiliy densiy disribuion correspond o low enropy, wheres dispersed disribuion yields high enropy vlue Alignbiliy Alignbiliy is he biliy of n imge o provide relible imge regisrion resuls by showing he correc rnsformions. Alignbiliy (A y ) is compued by muul informion vlues of subimge gins he rnsformed version of iself. A se of muul informion (2.2.2) vlues, MI S i re obined by compring he subimge S i () by roed version of iself, S i (θ j ), ( (, ) ( )) MI = MI S S θ (2) Si i i j where θj is he ngle of roion. In our work, θj is vried beween - o wih n incremen of. The scled difference of highes nd he second highes MI vlues is defined s lignbiliy. This cn be expreed s Specrl Bndwidh A = MI MI (3) y 2 For opimizing he regisrion resuls using he correlion coefficien (2.2.) we hve implemened mesure for he effecive specrl bndwidh. From he signl heory we know h he lrger he bndwidh, he shrper he disribuion of he correlion coefficien. For his reson we hve o find he cuou of he imge which hs he bigges vlue for he specrl bndwidh. This gives higher chnce for deecing his cuou in he second imge. Sring from he energy densiy specrum (EDS) φ ( f ) = S( f) (4) which hs o be normlized, φ( f ) φ ( f ) = φ ( f ) df (5) he ol power could be clculed. 2 f = 2 P f φ ( f) df (6) The normlized EDS is posiive definie, hs per uni re nd feures in his wy he min properies uf probbiliy densiy funcion (PDF). From his he men, he covrince μf = f φ ( f) df (7) f f f f σ = P μ = ( f μ ) φ ( f) df (8) nd he specrl bndwidh 2 2 f f f f σ = P μ = ( f μ ) φ ( f) df (9) cn be clculed.

3 2.2. Similriy merics Correlion Coefficien. The correlion coefficien (CC) is one of he mos widely used similriy merics. I is second order meric defined by E{ ( AμA) ( BμB) } ρ ( AB, ) = () σσ A B where A nd B re porions of he mser nd he slve imge. Figure : Digrm of incorrecly regisered sub imges (Muul Informion) for differen simuled scenes nd bin sizes Muul Informion Muul informion (MI) hs emerged in recen yers s populr similriy meric in he regisrion of imges (especilly in he field of medicl pplicions). If we erm he enropy of rndom vrible A H( A) = pa ( ) log A pa ( ) () = p log p A A ( ) ( ) nd he join enropy of A,B H ( AB, ) = pab ( b, ) log pab ( b, ) (2) A, b B he muul informion beween wo rndom vribles A nd B is given by pab (, b) MI = pab (, b) log (3) A, b B pa( ) pb( b) In erms of enropy nd join enropy we cn wrie MI = H( A, B) + H( A) + H( B ) (4) Ofen he normlized version of muul informion is used H( A) + H( B) MI( A, B) NMI = = + (5) H A, B H A, B ( ) A ( ) which gives only vlues beween nd nd simplifies he inerpreion he obined vlues of he muul informion. The mximizion of muul informion crierion posules h muul informion is mximized if imges re correcly regisered Chi 2 divergence χ 2 divergence (disnce o independence) exclusively uses he esimion of he join probbiliy densiy funcion nd does no use he rdiomeries of he pixels. Disnce o independence is normlized version of he χ 2 es [3]: ( AB, ) ( p p p ) 2 2 χ = b b (6) i, j pp b I mesures he degree of he sisicl dependence beween wo imges. Figure 2: Normlized vlue of MI using differen bin sizes Choosing more bins llows for more deiled represenion of probbiliy densiies. However, his deil my be nohing more hn noise, cused by smll smple size in ech bin. In he ps, invesigors hve used fixed bin size which ws deermined empiriclly. We chose o formlize he pproch by using vrible number of bins, clculed by Surges' formul, Sco's or Freedmn-Diconis' choice. hsurges = + log2n= log n (7) While he Surges formul only depends on he size of he dse, he rules of Sco or Freedmn/Diconis depends lso on he chrcerisics of he hisogrm: hsco = 3.49 /3 σ n (8) For exmple Sco s rule uses he sndrd deviion of he hisogrm; Freedmn-Diconis' choice is bsed on he inerqurile rnge. h = 3.49 Q Q n (9) FD ( ) /3 3 In figure 3 he influence of he bin size of he hisogrm is shown. The four brs for he differen bin sizes snd for using rnsled, noisy, roed nd roed + scled imge pirs The influence of he bin size The bin size of he hisogrm used in he esimion of muul informion is criicl iue. The relibiliy nd robusne is enhnced if he righ bin size is chosen. Figure 3: Mismching resuls for differen d ses nd bin size

4 2.4. Simulion Resuls Figure 6 shows he compuionl cos of he differen mehods. To number he quliy of our regisrion pproch we hve genered simuled InSAR imge pirs ou of rel TerrSAR-X d. Time [s] Chi2 KD MI(6) CC Figure 6: Needed ime for differen similriy merics Obviously he rdiionl correlion coefficien is he mos cos expensive wy he compre he cuous of he imges. The discrepncy rises he lrger hese cuous re. Figure 4: Auomic genered reference poins in n SAR imge The dvnge of simulion is h we cn excly mesure how ccure our mehod is. We cree imge pirs mued by rnslion, roion, scling or noise. Firs we compre he differen informion mesures for uomic reference poin generion lised up in 2.. Shnnon Enropy is he fses mehod, bu regrding he correc mching of he reference cuou in he second imge he specrl bndwidh mehod nd lignbiliy give he bes resuls. As similriy mesures we used he correlion coefficien (CC), Kolmogorov disnce (KD), muul informion (MI) nd chi 2 divergence. Mesure f(x) Kolmogorov Disnce Muul Informion ½ x- x log x χ 2 Divergence ½ (x-) 2 Tble : Comprison of he f Divergence Fmily For compring he similriy merics figure 5 is used. In his experimen he imge pirs re lwys roed (2 ), hen scling, noise nd higher roion ngle (6 ) re dded. Figure 5: Mismching resuls for differen d ses nd merics If here is no big difference beween he wo chosen cuous chi 2 divergence is good choice; regrding noise, scling nd roion i ofen leds o misregisrion. Figure 7-9: Imge shif experimen wih rnsled (2), noisy (3) nd roed+scled+noisy (4) d From figure 7-9 we cn see h MI nd CC lwys led o good regisrion resuls, even if here is big

5 diimilriy beween he wo imges. All mehods seem o be robus o noise; chi 2 divergence fils in he cse roion nd scling. 3. ORBIT INTERPOLATION The firs sep of our mehod of orbi inerpolion is o genere n orbi model bsed on relisic grviion model. By clculing he grdien of he grviion field we obin he ccelerion h influences he sellie. The inegrion of his ccelerion is done by Klmn filer h conins se spce model where posiion, velociy, ccelerion nd is derivives up o he 3 rd order re included. The ccelerion, which is clculed by evluing he grviion model he prediced posiion of he sellie, serves s observion. Thus predicor correcor srucure is performed. Simuled noisy GPS posiion mesuremens re included by simply dding hem o he observion vecor. For he clculion we use wo differen coordine frmes: U = n n= P nm m= n e n+ μ r ( sin() δ ) ( c cos( mλ) + s sin( mλ) ) nm where: n - grde of he poenil model m - order of he poenil model r - disnce o he cener of he Erh δ - geocenric liude λ - geogrphic lengh P nm - fully normlized Legendre funcions C nm ; S nm - hrmonic coefficiens of he Erh s poenil e - semi-mjor xes of he Erh The geocoefficiens re ken from he CHAMP-only Erh Grviy Field Model EIGEN-2. nm (22) 3.. The used Frmes The erh cenered ineril frme is defined s follows: r r r r IN = xin ex + y IN IN e y + z IN IN ezin (2) where he m cener of erh is he poin of origin, he x- xis poins owrds he vernl equinox, he z-xis poins owrds he norh pole nd he y-xis complees he righ hnded Cresin frme. The vecors in his frme cn lso be convered ino polr coordines ϕ, ϑ, r, where ϕ is he ngle in he x-y-plne mesured from he x-xis owrds he y-xis (rnge=[..2π]), ϑ is he ngle beween he vecor nd he z-xis mesured from he z-xis owrds he vecor (rnge=[..π]) nd r is he norm of he vecor (rnge=[..]). For he clculions in his pper ϑ is convered ino rnge of [-π/2..π/2] o mch he geocenric liude in he erh cenered erh fixed frme. The erh cenered erh fixed frme is defined by: r r r r EF = xef ex + y EF EF e y + z EF EF ezef (2) where gin he m cener of erh is he poin of origin. The x-xis poins owrds he inersecion of he zero meridin nd he equor. The z-xis is he sme s in he erh cenered ineril frme nd he y-xis complees he righ hnded Cresin frme. Compred o he ineril frme he erh fixed frme roes round he z-xis wih n ngulr re of ω= rd/s. In some cses he Cresin coordines re expreed in geocenric longiude λ (rnge=[..2π]), geocenric liude δ (rnge=[-π/2..π/2], δ=π/2 he norh pole) nd disnce o he poin of origin r (rnge=[..]) The Grviion Model A good descripion of he erh s grviion cn be chieved by hrmonic nlysis of he poenil [6]. The equion is: Figure : Grviion model of he Erh The fully normlized Legendre funcions re clculed using recursive lgorihm [6]: P P P n,m n,n n,n = η sin = τ sin = ν cos wih he sr vlues ( δ) Pn,m σ Pn ( δ) Pn,n ;n > ( δ) P ;n > n,n 2,m ; n > m + ( δ), P = ( δ) P,,, 3 nd η = σ = =, P = 3 sin cos ( 2n + ) ( 2n ), τ = 2n + ( n + m) ( n m) ( 2n + ) ( n + m ) ( n m ) 2n +, ν = ( 2n 3) ( n + m) ( n m) 2n Clculing he ccelerion (23) Becuse he ccelerion depends on he posiion reled o he WGS84 ellipsoid he clculion cn no be done in he ineril coordine frme. To evlue he grviion model he righ posiion he vecor h is expreed by ineril coordines mus be rnsformed ino erh cenered erh fixed coordines. This is done by roion

6 round he common z-xis of boh he ineril nd he erh fixed frme. The roion ngle is Θ+ω* where Θ is he de dependen ngle beween he ineril frme nd he erh fixed frme he sring ime of he lgorihm. This ngle is clculed using he MJD (Modified Julin De). To go on wih he clculion he so obined ccelerion vecor mus be rnsformed bck ino he ineril frme by x r IN = y = z sin cos ( ϕ) sin( δ) cos( ϕ) cos( δ) cos( ϕ) ϕ ( ϕ) sin( δ) sin( ϕ) cos( δ) sin( ϕ) δ ( ) ( ) cos δ sin δ r 3.3. Inegrion A gre dvnge of using Klmn filer is he fc h mesuremens cn be esily included by dding hem o he observion. In he presen cse he se rnsiion mrix corresponds o Tylor series of he equion of moion. For he one-dimensionl cse he equion is: s( k + ) v( k ) + ( k + ) x( k + ) = = ( & k + ) ( && k + ) &&& ( k + ) T T T T T 2! 3! 4! 5! (24) s( k ) T T T T v( k )!!! T T ( k ) = T = A x( k ) 2! 3! ( & k ) 2 T ( k ) T && ( k ) T! 2 &&& where s is he posiion, v is he velociy, is he ccelerion nd T is fixed discree ime inervl. The observion mrix is dped in cse of rriving posiion mesuremens by seing he elemens which correspond o he observed posiion in he se vecor o. Oherwise ll elemens re excep hose h correspond o he observed ccelerion Mesuremen Noise To obin usble resuls i is necery o model he noise of he mesuremens. This is done by dding digonl mrix which conins he squres of he sndrd deviion (σ 2 ) when clculing he Klmn gin. The elemens corresponding o he posiion re se o σ 2 of he simuled posiion mesuremens. The elemens corresponding o he ccelerion ken from he grviion model depend on he weigh h shll be given eiher o he orbi model or o he mesuremens. Bsiclly when using Klmn filer he noise is umed o be uncorreled, Guin disribued whie noise by pproximion. Anoher imporn feure is he driving noise, which indices he quliy of he model on which he Klmn filer is bsed. I is dded ech predicion sep nd cn lso be used o give more or le weigh o he underlying model. 4. PHASE UNWRAPPING Nerly ll known phse unwrpping echniques ry o unwrp he mpped phses by sequence of differeniing, king he principl vlue of he discree derivive nd inegring i gin. A serious drwbck of ny differeniion of funcions which re modulo mpped nd noise conmined is bis resuling from he discree derivive of noisy modulo-2pi mpped phses. As consequence of his bis, phse slopes re lwys underesimed. The resuling bis depends on he phse slope iself, s well s on he coherence. Our phse unwrpping lgorihm is bsed on n Exended Klmn filer. The Klmn filer explois so clled "Bsic Slope Model" enbling he filer o incorpore ddiionl locl slope informion obined from he smple frequency specrum of he inerferogrm by locl slope esimor. The locl slope informion is hen opimlly fused wih he informion direcly obined from rel nd imginry pr of he inerferogrm. For his reson i is no necery o genere phse error bsed on compliced sisics, phse unwrpping kes plce simulneously wih removing he phse errors. Wheres some echniques ry o reduce he phse noise by filering before unwrpping he phse, he Klmn filering pproch simulneously unwrps he phses nd elimines he phse noise, so h no pre-filering is necery. Bsed on our experience in his re [8], we hve done some refinemens for improving he unwrpping resuls. There some mehods for reducing he compuionl lod of our lgorihm re given s well s new wy o use weighed 2D Klmn filer. Therefore he filer cn yield excellen resuls of he unwrpped phse even in regions wih seep nd rough opogrphy. 4.. The Inerferogrm The phses of n InSAR imge re ll mpped ino he sme 'bsebnd' inervl (e.g. -π, π), while ny bsolue phse offse (n ineger muliple of 2π) is los. Furhermore hey re subjec o phse noise cused by he superimposed mpliude noise in he rel nd imginry prs of he InSAR imge. For he complex SAR inerferogrm we hve in polr noion poin (n,m): znm (, ) = nm (, ) exp [ j% ϕ ( nm, )] (25) wih (n,m) being he observed inerferomeric mpliude nd being he inerferomeric phse modulo mpped, where he modulo mpping is generlly expreed by: α = [ α] π = α ± π ( π π ] 2 n 2, nd : α π (26)

7 Simplifying he noion o one-dimensionl posiion dependence we wrie: ϕ ( k) = ϕ( k) + e ( k) = [ ϕ ] 2π ϕ( k) [ eϕ ( k) ] 2 [ ϕ( k) + e ϕ ( k) ] 2 π = + π 2π (27) where is he rue unmbiguous phse ime or poin k, is he rue phse error is he mpped phse error poin k, he sochsic prmeers, such s probbiliy densiy. 4.. The Principle of he lgorihm Bsed on he specific model of he inerferomeric phse n exended Klmn filer is implemened, which ddiionlly fuses phse slope esimion, clculed from he inerferogrm s power specrl densiy. Complex SAR Inerferogrm Observionmodel Locl - Slope Esimor Phse Vrince Phse Slopes 2-D Klmn-Filer Errror Vrince Se Spce Model Unmbiguous Phse Figure : Scheme of he unwrpping mehod Sring poin is complex SAR inerferogrm from which he vlues of he individul pixels re used s observions of he Klmn filer (3). I is lso poible o exrc ddiionl informion from his inerferogrm, e.g. o clcule he vrince of he phse [8] for unsure opiml filering resuls. Imporn o menion re he esimes of he phse slopes which be used in he predicion sep. To complee he se spce model, we mus deermine he error vrince of he esime of he men phse slope The Slope Esimor The inerferomeric phse ime is decomposed ino he sum of hree erms: ϕ() = ϕ( ) + & ϕ( τ) dτ = ϕ( ) + 2 π (f + f( % τ)) dτ = ϕ( ) + 2π f ( ) + 2π f( % τ) dτ men phse vriion dynmic phse vriion (28) Subsiuing his decomposiion ino () wih normlized mpliude nd using complex noion we hve: z() = exp jϕ( ) + 2π f ( ) + 2π f( % ) () τ dτ + n = c exp j2π f( % τ) dτ exp{ j2 π f } + n( ) 4243 { (29) specrl shif compl. noise frequency modulion compl. frequ. mod. signl s() = s () exp{ j2 π f} + n () The men fringe frequency f, corresponding o he men phse slope wih respec o given observion window cn be observed s specrl shif in he inerferogrm s power specrl densiy. Hence we migh use ny locl frequency esimor nd pply i o he complex inerferogrm, o esime he specrl shif from he complex correlion kernel of he inerferogrm. Likewise we could lso pply ny oher locl frequency esimion, bsed on he fc h he insnneous frequency cn be direcly clculed from he complex d by: d z& ( ) dz( ) f () = ϕ() = where: z& () = i (3) 2π d 2 π z( ) d Rher hn h, we will esime he specrl shif in he frequency domin from he inerferogrm s power specrl densiy, obined in loclly shifed window. The echnique is rher convenionl, clculing he power specrl densiy in locl window nd pplying some subpixel resoluion echnique o idenify he mximum wih subpixel resoluion. Good resuls for finding he posiion of mximum of he inerferogrm s power specrl densiy wih subpixel ccurcy gives mehod where we fi he originl curve wih qudric curve. Afer minimizing he qudric error of originl nd fied curve we cn deermine he needed frequency which corresponds o he phse slope. A drwbck of he slope esimion echnique is he high compuionl cos. For beer performnce of he complee lgorihm we found wys o sve loops or mke individul procedures fser. Becuse mos of he informion of he new (sliding) window is lso conined in he previous window we cn subrc he no required informion of one row/column in he Fourier domin nd dd he new row/column informion. This does no ffec he quliy of he resul, bu increses he proceing speed The Klmn filer For solving he non-liner phse unwrpping problem 2D Klmn filer ws implemened, which cn be derived from he snd equions of D exended Klmn filer. The lgorihm hs recursive predicor-correcor srucure; he high-rnking - nd + -signs describe he poin in ime. While - mens shor before he income of he mesuremens, + snds for h poin in ime when he new mesuremens re vilble for he cul ime index (k). + xˆ ˆ k+ = A xk + uk (3) + T P ( k+ ) = A P ( k) A + Q( k)

8 Correcion Predicion Kk ( + ) = P( k+ ) C ( k+ ) T C ( + ) ( + ) ( + ) + ( + ) F k P k CF k R k r = y h( x ˆ ) T F k + k + k + + k+ k+ x ˆ = x ˆ + K ( k+ ) r k + + P ( k+ ) = P ( k+ ) K( k+ ) CF ( k+ ) P ( k + ) A is he se rnsiion mrix, describing he dynmicl chnges of he vrible, which is o be esimed wih respec o ime or spce, u is ny deerminisic or known influence chnging he se x(k) from one pixel o he nex x(k+). R(k+) is he mesuremen noise covrince nd Q(k) he driving noise covrince describing he unceriny of he se rnsiion. P is he covrince mrix which exprees mesure for he discrepncy of he prediced/upded esime o cul se. The linerized observion mrix is given by: d C F ( k + ) = h[] x xˆ d x k + (32) Due o he nonliner observion mpping, he Klmn filer will uomiclly unwrp he inerferomeric phses. In doing so, he Klmn filer fuses he informion gined from he complex inerferogrm wih he slope informion exrced from he inerferomeric power specrum. 3. WEIGHTED 2D KALMAN FILTERING In he cse of phse unwrpping we hve, sclr se nd vecoril observion, conining he complex inerferogrm pixel. Due o he nonliner observion mpping, he Klmn filer will uomiclly unwrp he inerferomeric phses. The Klmn filer fuses he informion gined from he complex inerferogrm wih he slope informion exrced from he inerferomeric power specrum. For he wo-dimensionl unwrpping he Klmn filer lgorihm works long he skeched inegrion phs. Any predicion esime is clculed, depending on wo neighbors (figure 2), so h only he predicion equions in he Klmn filer lgorihm hve o be modified. The sme pplies for he error covrince mrix, which is weighed sum of he wo involved neighbored covrince mrices. ( ) (, ) + xˆ r, u r D Klmn filer Figure 2: Principle of he 2D predicion srep ( ) r ( ) ( ) ( ) xˆ r r,,p r, xˆ r,,p r, D Klmn filer ( r ) (, ) + xˆ, u r In our pproch he conrol vecor u(k) corresponds o he esimed slopes in rnge nd zimuh direcion nd includes lso dpive clculed vrince (w(k)) o describe he unceriny of he esimes. u (, ) ˆ r r =Δ ϕ r(, r) + wr (, ) (33) u (, ) ˆ r =Δ ϕ (, r) + wr (, ) For improving he lgorihm presened in [6] we dded some weighing fcors o obin he bes poible esime of he unmbiguous phse. ( ) ( ) xˆ r,,p r, r ϕ m ( r, ) (, ) = ( m, ) v r f ϕ γ r ( ) ( ) xˆ r,,p r, (, ) = (, ) + (, ) xˆ r W xˆ r W xˆ r r r (, ) = (, ) + (, ) P r W P r W P r r r Figure 3: The weighed 2D Klmn filering ( ) ( ) + + xˆ r,,p r, The used weighing fcors re resuling from he vlues of he involved covrince mrices of he esimes which weigh he predicion wih respec o he cul ccurcy of he esimed phses. The observion is done in sndrd wy; he phse vlue of he cul pixel is sepred ino rel nd imginry pr nd is compred wih he predicions. zr (, ) Re r (, ) cos ( ϕm ( r, )) v (, r ) yk ( ) = = + (, ) sin ( (, )) (, ) zr ϕ m r v r 2 (34) Im k ( ) = hxr ( (, )) + vr (, ) We obin he linerized observion mrix by he vecoril derivion of he given nonliner observion h(x). Afer compring he cul observion wih he prediced se, we cn compue he bes poible upded se esime for he phse for ech individul pixel (figure 3). 4. OPERATION IN AREAS OF LOW COHERENCE The resul of he unwrpped produc will be drmiclly reduced in quliy if he coherence (e.g. cused by rdr show, lyover, vegeion ) is regionlly low. Becuse of error propgion he filer cn never reurn bck o he correc phse. For geing rid of his problem we hve implemened severl mehods. The coherence

9 γ (, r) = E * E{ z( r, ) z2( r, ) } { z( r, ) } E z2( r, ) { } 2 2 (35) is lso n imporn prmeer o qunize he mesuremen noise (see figure 3) or o serve s n ddiionl source of informion for our se spce model. Becuse of our nlysis of he regrded inerferogrm we hve lo of informion (slopes, vrince of he slope, noise predicion) we cn use. Togeher wih oher informion (like coherence) we re ble o uomiclly msk ou res in which our lgorihm will probbly give wrong phse esimes. According o his only smll pr of he inerferogrm is negleced by he proceor while he mos res re proceed correcly. If here is some priori knowledge vilble (SRTM or oher vilble DEMs) we cn ke his informion nd complee our phse slope mrix, which oherwise will be filled wih zeros. Driven by his model he Klmn filer cn fill he gps of he msked ou res. Anoher soluion o void phse errors cused by res of low coherence is o use Klmn smooher: We cn sr he 2D filering proce from ech edge of he inerferogrm in four differen direcions. The filer esime of he individul filers is hen combined wih he predicion esime of he oher filers where he weighing of he individul esimes is inversely proporionl o he corresponding error covrinces. + + xˆ ( ) = ( ) ( ) ˆ ( ) + ( ) ˆ ( ) s k Ps k Pf k x b f k P k xb k (36) ( ) ( ) ( ) ( ) ( ) ( ) + + s = f f + b b P k P k P k P k P k (37) Index f mens forwrd nd index b mens bckwrd direcion of he filer. Though he proceing burden resuls lmos compleely from he phse slopes esimion sep, using smooher will proporionl no rise much he compuion ime. Our pproch works very well for obining he unmbiguous phse of complex inerferogrm, which ws shown in [8] by using simuled d. The high compuion ime resuls only from exrcing he necery informion ou of noisy inerferogrm. For his reson we hve exended his pproch o mke i more robus nd mke he phse slope esimion fser. A furher improvemen of reducing he compuionl cos is reched by prllel compuing. A new muli-bseline pproch for he Klmn filer for elimining he mbiguiies cused by lyover is pr of our reserch. In some cses he long rck bseline migh be lrge in h wy here is lso lrge vrince of he shifing vecors in he scene (figure 4); h mens here is no consn or nerly consn offse in he shif. Especilly in his cse our uomic lgorihm leds o perfec mching of he wo imges. As n exmple we chose es sie of Ayers Rock in Ausrli ( lo of lyover). In figure 4 he shifing vecors re shown. The individul rrows show he direcion of he slve pixel o be found in he mser imge. Figure 4: The shifing vecors Figure 5: The inerferomeric phse 5. RESULTS USING TERRASAR-X DATA Becuse of he smll bseline of rel TerrSAR-X InSAR imge pirs he regisrion of hese imges ws succeful in ll cses/mehods. Since he MI mehod is fs nd robus his mehod is preferred. This lgorihm is fser hn he correlion coefficien lgorihm nd more relible hn he ohers. The le misregisered poins re presen he esier is he eliminion of he wrong vecors. Figure 6: The flened phse Figures 5 und 6 show he inerferomeric phse of he inerferogrm. The fl erh effec is removed by use of he inerpoled bseline.

10 6. ACKNOWLEDGEMENT The work repored herein hs been funded by he Germn Minisry of Educion nd Science (BMBF, Grn number 5EE68) which is grefully cknowledged. 7. REFERENCES [] R. Werninghus, W. Blzer, S. Buckreu, J. Miermyer, P. Mühlbuer, W. Piz, "The TerrSAR-X Miion," Proceedings of EUSAR 24, Ulm, Germny, My 25-27, 24 [2] L.G. Brown, A survey of imge regisrion echniques, ACM Compuing Surveys 24, pp [3] J. Ingld. nd A.J. Giros, On he Poibiliy of Auomic Mulisensor Imge Regisrion, IEEE Trnscions on Geoscience nd Remoe Sensing, vol. 42, iue, pp [4] A. Amnkwh nd O. Loffeld, "Imge Regisrion by Subimge Selecion nd Mximizion of Muul Informion," IGARSS 25 Inernionl Geoscience nd Remoe Sensing Symposium 25, Seoul, Souh Kore, 25 [5] H. Nies, O. Loffeld, B. Dömnez, A.B. Hmmdi, nd R. Wng Imge Regisrion of TerrSAR-X D Using Differen Informion Mesures, Proc. IGARSS 8 (IEEE 28 Geoscience nd Remoe Sensing Symposium), Boson, U.S.A., 28 [6] H. Klinkrd, Anlyische Berechnung erdnher Sellienbhnen uner Verwendung eines relisischen Orbimodells, Dierion, Universiä Crolo-Wilhelmin zu Brunschweig, November 983 [7] U. Gebhrd, O. Loffeld, M. Klkuhl, H. Nies, nd S. Knedlik, Orbi Trcking nd Inerpolion Using Relisic Grviion Model. Proc. IGARSS 4 (IEEE 24 Geoscience nd Remoe Sensing Symposium), Anchorge, Alsk, USA, 24 [8] O. Loffeld, H. Nies, S. Knedlik, nd R. Wng Phse Unwrpping for SAR Inerferomery;A D Fusion Approch by Klmn Filering, Geoscience nd Remoe Sensing, IEEE Trnscions on, 28, 46, (), pp [9] H. Nies, O. Loffeld,, nd R. Wng, Phse Unwrpping Using 2-D Klmn Filer - Poenil nd Limiions, Proc. IGARSS 8 (IEEE 28 Geoscience nd Remoe Sensing Symposium), Boson, U.S.A., 28

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