Change detection in remote sensing images with graph cuts

Transcription

1 Change deteton n remote sensng mages wth graph uts Fernando Pérez Nava a Aleandro Pérez Nava a José M. Gálvez Lamolda b Maros Fuentes Redondo a Dept. de Estadísta I. O. Computaón. Unversdad de La Laguna. Islas Canaras pan b Dept. de Físa Fundamental. Unversdad de La Laguna. Islas Canaras pan Es. é. up. de Ingenería Informáta. Unversdad de La Laguna. Islas Canaras pan ABRAC Change deteton s an mportant part of man remote sensng applatons. hs paper addresses the problem of unsupervsed pel lassfaton nto 'Change' and 'No Change' lasses based on Hdden Markov Random Feld HMRF models. HMRF models have long been reognzed as a method to enfore spatall oherent lass assgnment. he optmal lassfaton under these models s usuall obtaned under the Mamum a Posteror MAP rteron. However the MAP lassfaton n HMRF models leads n general to problems wth eponental omplet so appromate tehnues are needed. In ths paper we show that the smple struture of the hange deteton problem makes that MAP lassfaton an be eatl and effentl alulated usng graph ut tehnues. Another problem related to HMRF modellng and to an hange deteton tehnue s the determnaton of the parameters or thresholds for lassfaton. hs learnng problem s solved n our HMRF model b the Epetaton Mamzaton EM algorthm. Epermental results obtaned on four sets of multspetral remote sensng mages onfrm the valdt of the proposed approah. Kewords: Change deteton hdden Markov random models epetaton mamzaton graph uts multtemporal mages remote sensng. 1. INRODUCION Automat hange deteton from mages of the same sene aured at dfferent tmes s a problem of wdespread nterest due to a large number of applatons n dverse dsplnes [1] [2]. Important applatons of hange deteton nlude vdeo survellane medal dagnoss treatment and remote sensng [3]. Usuall hange deteton n remote sensng nvolves the analss of two regstered aeral or satellte multspetral mages from the same geographal area obtaned at two dfferent tmes. uh an analss ams at dentfng hanges that have ourred n the same geographal area between the two tmes onsdered. wo man approahes have been proposed to solve the hange deteton problem [4]: the supervsed approah and the unsupervsed approah. he former s based on supervsed lassfaton methods whh reure a learnng set wth multtemporal ground truth whle the latter perform hange deteton wthout relng on an addtonal nformaton. As the generaton of a learnng set s usuall a dffult and epensve task the use of unsupervsed methods s of great nterest n man applatons n whh a learnng set s not avalable. A varet of approahes have been proposed for unsupervsed lassfaton. tatstal approahes both parametr and non-parametr have been emploed. hs tpe of methods labels pels aordng to probablt values whh are determned based on some dstrbuton on the features for the pels n the mage. he smplest dstrbutons are based on fnte mture models FM both parametr and non-parametr [5]. However the FM model has an ntrns lmtaton: spatal nformaton s not taken nto aount beause all the data ponts are onsdered to be ndependent samples drawn from a populaton. An approah to obtan a spatall oherent lusterng s to use a hdden Markov random feld HMRF whh s an stohast proess generated b an Markov random feld MRF [6] [7] and whose lass labels annot be observed dretl but whh an be observed through a feld of observatons. tatstal approahes attempt to solve the problem of estmatng the assoated lass label gven onl the values for eah pel. he most popular tehnue to estmate an HMRF s mamum a posteror MAP estmaton [6]. he MAP framework was popularsed n the feld of mage analss b Geman and Geman [8]. MAP estmaton onssts of mamzng the posteror probablt of the labels gven the pel ntenstes. hs s an optmsaton problem that an be solved b a varet of tehnues. he man problem of ths approah s ts hgh omputatonal ost. For man

2 nterestng problems fndng the eat soluton s NP-hard and so there s no hope to fnd a tratable method to fnd an eat mamum. Reentl Mn-Cut algorthms on graphs [9] have emerged as an nreasngl useful tool for eat or appromate optmzaton n low-level vson. he bas tehnue s to onstrut a spealzed graph for the funton to be optmzed suh that the mnmum ut on the graph also provde the optmum of the funton ether globall or loall. he mnmum ut n turn an be omputed ver effentl b ma-flow algorthms. hese methods have been suessfull used for a wde varet of mage analss nludng mage restoraton [10] or stereo and moton [11]. he mnmum ut soluton omes wth some nterestng theoretal ualt guarantee. In some ases t s the global mnmum n other ases a loal mnmum n a strong sense that s wthn a known fator of the global mnmum [12]. 2. AN HMRF FORMULAION OF HE CHANGE DEECION PROBLEM Problems where part of the data s mssng or unobservable are ommon n mage analss. he observatons ma represent measurements n the form of multdmensonal varables for eah pel n the mage whle the hdden data ould onsst of an unknown label assgnment to be estmated from the observatons for eah pel. Dependng on the partular problem labels ma represent lasses segmentaton problem dspartes stereo problem or dsplaement flow problem. In ths paper we fous on hange deteton as a btemporal segmentaton problem. However unlke a smple segmentaton problem unknown lghtnng transforms between mages ma be present. In eton 2.1 we present the bas defntons onernng the Markov models for the unobservable data. In eton 2.2 we spef the omplete parametr models for the observed and unobserved data. In eton 2.3 we restrt ths general formulaton for the hange deteton problem and n eton 2.4 we present the MAP soluton to the hange deteton problem Markov Random Felds A Markov Random Feld MRF s omposed of three sets. A fnte set of stes. A neghborhood sstem N defned as N { N } where eah N s a subset of whh form the neghborhood of ste. In ths paper we wll use a frst order neghborhood sstem: for eah ste the neghbors are the four stes surroundng t. he thrd set s a olleton of random varables also alled the feld X { }. In order for X to be a MRF the ont probablt dstrbuton must satsf: P { } P N P > where M denotes a realzaton of the feld restrted to M. Propert 1 means that the nteratons between ste and the other stes atuall redue to nteratons wth ts neghbors. Propert 2 s mportant for the Hammersle Clfford theorem to hold. hs theorem states that the ont probablt dstrbuton of a Markov feld s a Gbbs dstrbuton gven b: P 1 Z ep H 3 where H s the energ funton: H V 4 C where C omprses the lues sets where all stes are neghbors n N. he V C are the lue potentals and tpall depend on parameters. Fnall Z X ep H 5

3 s the normalzng fator also alled the partton funton. he alulaton of Z nvolves all possble realzatons of the Markov feld. herefore ts eat omputaton s eponentall omple. hs s a problem when usng these models n stuatons where an epresson of the ont dstrbuton s reured Hdden Markov models Problems nvolvng nomplete data where part of the data s hdden or unobservable are ommon n mage analss. Observatons ma represent measurements for eah pel of an mage whle the hdden data ould onsst of an unknown lass assgnment to be estmated from the observatons for eah pel. In ths paper we fous on ths ase usuall referred to as mage segmentaton. he unobserved data s modeled as a dsrete Markov random feld X as defned n 3 wth energ funton H dependng on a parameter. In hdden Markov models the observatons are ondtonall ndependent gven X aordng to a lkelhood L Q where Q are the parameters that full desrbe the lkelhood. We wll asume the lkelhood to be: L Q L Q ep log Q ep Q ep L l D D l Q 6 assumng that all the l are postve. hs makes the model smlar to a fnte mture FM model. A FM model ould be seen as a hdden Markov model where the hdden feld X s one of ndependent varables. hs s a partular ase that makes FM models more tratable. In the general ase the omplete lkelhood s gven b: P Q L Q P Z L Q ep V Z ep H + D 7 C Note that both the normalzaton onstant Z the luster potental V and the energ H depend on. he ondtonal feld X gven s a Markov feld wth energ funton: C H V + D 8 In the mage segmentaton problem we have to reover the unknown lass labelng X gven the observatons. hs lassfaton problem usuall reures values for the parameters Q. If the are unknown the must be estmated HMRF formulaton of the hange deteton problem In ths seton we formulate the hange deteton problem as an mage segmentaton problem. In ths ase observatons ome from two multspetral mages u and v aured n the same geographal areas at two dfferent tmes t 1 and t 2. A ommon assumpton s to onsder that mages have been oregstered [13] and that the possble dfferenes n lght and atmospher ondtons have been orreted [14]. We wll see n eton 4 that our formulaton s nvarant to global llumnaton hanges. In the hange deteton problem we have two lasses: "Change" CH and "No hange" NC. o formulate the hange deteton problem as a segmentaton problem we wll defne an approprate lkelhood L X and Markov Random Feld he lkelhood model he lkelhood for both mages gven the lasses wll be based on a gaussan dstrbuton N µ wth ommon mean n eah lass:

4 D l u v -d / 2 log 2π ep 2 - µ - µ 9 { CH NC} where d s the sum of the number bands for mages u and v. upersrpt denotes matr transpose. We wll restrt CH to represent ndependent varables so the orrelaton part between u and v of matr CH s fed to 0. Our formulaton of the lkelhood depend on the set of parameters Q {m CH NC }. Another possblt s to use a low-dmensonal transform of the two mages uv. A ommon transform for hange deteton n monospetral mages s the dfferene transform u-v [4]. In ths ase we have: D l σ u v 2 log 2πσ { CH NC} 2-1/ 2 ep-1 2σ 2 - µ he Markov Random Feld model A wdel used model for the hdden lass labels s the sotrop Isng model. Here the set of stes s a latte and the neghbor sstem s omposed of the four nearest neghbors of eah pel. Formall: P Z δ 1 0 ep H Z ep C V Z ep δ C 11 where must be adaent stes to be n the lue set C. Note that the lue potental funton s: V δ 12 hs shows that the probablt assgned to an mage s a funton onl of the number of homogeneous lues or euvalentl the boundar length between the lasses. We onsder > 0 to enfore a spatall oherent lassfaton MAP estmaton of hanges n btemporal mages tatstal approahes attempt to solve the problem of estmatng the lass label gven onl the data reorded for eah pel. he most popular wa to estmate an HMRF s mamum a posteror MAP estmaton. MAP estmaton onssts of mamzng the posteror probablt of the hdden lass labels gven the observed pel values: ˆ arg ma P Q arg ma Z arg ma Z ep C V ep H Q D arg mn C V + D 13 Greg et al. [10] were frst to dsover that powerful mn-ut/ma-flow algorthms from ombnatoral optmsaton ould be used to mnmze the energes represented n 13. Greg et al. onstruted a two termnal graph suh that the mnmum ost ut of the graph gves a globall optmal bnar labellng n ase of the Isng model of nteraton 11. Prevousl eat mnmzaton of energes lke 13 were not possble and suh energes were approahed manl wth teratve algorthms lke smulated annealng. Unfortunatel the graph ut tehnue remaned unnoted for almost 10

5 ears. In the late 90 s a large number of new omputer vson tehnues appeared that fgured how to use mn-ut/maflow algorthms on graphs for solvng more omple problems [10] [11] [12]. o appl the mn-ut approah t s neessar buld a graph so that the global mnmum of the energ funton an be omputed b solvng a standard two termnal mnmum ut problem over t. Consder a graph G defned as follows: there are two termnals: the soure CH and the snk NC. For eah pel we reate a verte. hese verte are onneted to the termnals b t-lnks {CH } {NC} wth weghts D CH+K and D NC+K. Constant K must be greater than N where N s the number of neghbours four n our ase. For eah par of neghbourng pels and we reate an n-lnk {} wth weght. An eample for a 33 mage s shown below: CH tlnk graph ut slnk NC Fgure 1 Eample of graph onstruton for the mage hange problem adapted from [12]. he mnmum ut parttons the pels n two sets. Pels n the same ut as verte CH are lassfed as "Change" pels and the rest are the "No Change" pels. he mnmum ut gves the global mnmum of the energ funton and an be omputed n polnomal tme. Note that algorthms based n the mage dfferenes lke [15] that were prevousl solved usng appromate teratve tehnues ma n fat be solved eatl. 3. LEARNING HE PARAMEER OF HE ENERG FUNCION A potental problem wth the presented approah s to set the orret value for the parameters Q. Parameter estmaton for Markov HMRF models has alwas been onsdered a dffult problem. hs has lead to man ad ho shemes for estmatng the parameters of the model [7] [16] [17] nludng hoosng parameter values b hand. he ommon approah to the parameter estmaton problem s to obtan the mamum lkelhood estmate MLE of the parameters gven the observatons. he log-lkelhood of the model s: log P log P 14 It s mpossble to obtan an analtal soluton of the MLE so numeral methods are needed. he Epetaton Mamzaton EM [18] algorthm s an teratve algorthm that tres to mamze the log-lkelhood b optmzng at teraton : Q Elog P X 15 the epetaton of the omplete log-lkelhood gven the observaton and the urrent estmate of the parameters he EM algorthm proeeds as follows: Intalzaton tart from an ntal guess 0 of Epetaton tep Compute Q

6 Mamzaton tep Update the urrent estmate to arg ma 1 Q + or ust obtan +1 nreasng Q Generalzed EM Repeat the Epetaton and Mamzaton steps untl onvergene. he EM algorthm guarantees an nrease n the log-lkelhood untl a loal optma s found. o proeed wth the Epetaton step note that usng 8 Q an be wrtten as: log C P V Z l P Q Q 16 he frst term does not depend on whle the last two do not nvolve Q. We rearrange those terms and wrte: log C P V Z Q l P Q Q Q here are several problems to ompute both terms. In order to evaluate QQ we need the ondtonal probablt P that annot be omputed eatl. o ompute Q we need both P and the partton funton Z that also annot be omputed eatl. here are several approahes to appromatel solve these problems: mean feld-lke appromatons Classfaton EM or tohast EM [16]. Our approah s based on Montearlo EM MCEM [19] n whh epetatons are estmated from samples of the dstrbutons. 3.1 Estmaton of parameter b o estmate parameter we dfferentate n 18 and make the resultng epreson eual to zero obtanng the euaton: C C V V E E 19 In our ase Isng model we have: E E C C δ δ 20 he nterpretaton of the above euaton s lear. he mean number of homogeneous lues n both dstrbutons must be the same. Note that the frst term s a funton of whle the seond term s a onstant. Both terms are appromatel omputed b samplng from the orrespondent dstrbutons usng the wendsen-wang algorthm [20] [21]. o solve ths undmensonal euaton we use the bseton method. hs method has the advantage of reusng funton evaluatons done n prevous teratons of the EM algorthm. Note also that the estmaton of the funton does not depend on the mages to proess onl of ts sze and ould be omputed before learnng. 3.2 Estmaton of parameter Q In our ase 9 the set of parameters Q omprses a ommon mean for the "Change" and "No Change" lass and two separate ovarane matres CH NC that s Q {m CH NC }. Instead of optmzng QQ dretl we wll onl

7 seek to nrease t. hs orresponds to the Generalsed EM algorthm GEM. We nrease the log-lkelhood ths proeedng n two steps. Frst we mamze m wth CH NC fed to ther prevous values CH NC obtanng: µ + 1 A P π π P / A π π 21 where s the number of stes the number of pels n the mage. hen we obtan the optmal values for CH NC wth the value for m obtaned before as: P µ + 1 π µ Fnall ovarane terms between the two mages n CH are set to zero to ensure modelng ndependene between the two mages for the "Change" lass. hese update formulas n 21 and 22 guarantee an nrease n the log-lkelhood untl a loal mamum s found. 4. RADIOMERIC INVARIANCE Before remote sensng mages an be used for a hange deteton stud the mages must frst be standardzed for ondtons outsde of real surfae hange. Dfferenes n the sensor solar llumnaton or atmospher ondtons make t dffult f not mpossble to auratel ompare satellte or aeral mages aured on dfferent dates and/or dfferent platforms. everal methods [14] have been proposed for the relatve radometr normalzaton of multspetral mages taken under dfferent ondtons at dfferent tmes. All proeed under the assumpton that the relatonshp between the radanes reorded at two dfferent tmes from regons of onstant refletane s spatall homogeneous and an be appromated b lnear funtons. In ths seton we show that our approah s nvarant to global affne transforms of both mages Let us wrte agan our orgnal observatons for pel as u v. If eah mage s affnel transformed we obtan for eah pel a new value: u u Mu 0 nu M + n M n 23 v v 0 Mv nv he update euatons n eah teraton of the EM are now assumng that M s nvertble: µ Mµ M + n + 1 M 24 hen we obtan as optmal parameters Q { m CH NC } and the log-lkelhood for the transformed mages l Q s eual to the orgnal log-lkelhood l Q makng the whole hange deteton proess nvarant.

8 5. EXPERIMENAL REUL In order to asses the effetveness of the proposed tehnues for hange deteton we onsder four dfferent data sets: the three frst ones are aeral mages and the last one s a satellte mage. he frst dataset s a multspetral RGB par of mages orrespondng to the sland of Gran Canara pan. he three last data sets are monospetral blak and whte mages two orrespondng to the sland of enerfe pan and the last one orrespondng to Madrd pan. In the followng both the data sets and the eperments are detaled Data sets he frst of the four data sets onssted on two multspetral mages aured from an arborne platform n the sland of Gran Canara pan n 1996 and he are olor mages and have 3 hannels for the Red Green and Blue omponents. he area seleted for the eperments was a seton pels of the full mages. he seond data set onssted on two aeral monospetral mages n an area of La Laguna n the sland of enerfe pan aured n 1996 and he area seleted for the eperments was a seton pels of the full mages. he thrd data set onssted on two aeral monospetral mages n an area of the t of anta Cruz de enerfe n the sland of enerfe pan aured n 1996 and he area seleted for the eperments was a seton pels of the full mages. he last data set onssted on two satellte monospetral mages aured b the IR-1C satellte n an area of the t of Madrd pan n 1999 and he area seleted for the eperments was a seton pels of the full mages. All mages are orthophotos so no geometr regstraton was emploed. Hstograms for eah band n the frst mage were transformed to math the orrespondng band n the seond mage to partall remove an non lnear transform n the lght ondtons at the tme of the two austons. he nose affetng the data values was redued b applng a smoothng flter 33 wndow sze n all mages Desrpton of the eperments For all the datasets we assumed that the model parameters were unknown so we emploed the EM algorthm. he ntalzaton of parameter n the EM algorthm was done as follows: Parameter was set to 1.5. An ntal estmaton of the posteror feld was done omputng the dfferene mage. We set a probablt of hange of 1 to all pels whose Euldean dstane between the multspetral data n the two mages was above a threshold 40% of the mamum dstane. he ntal value for m was taken from the mean of both mages. From the ntal posteror feld and mean we alulated the startng values for CH NC usng euaton 22. he ma-flow/mn-ut ode used for MAP omputaton s a freel avalable C++ mplementaton from [12]. he evaluaton of the MAP soluton n all the eperments took no more than two seonds. he EM algorthm onverged tpall n less that ten teratons but the learnng tme s hgh due to the omplet of the samplng proess Epermental results for the frst data set In ths eample we show the results of the hange deteton proess for the two olour orthophotos wth 1:10000 sale 1 meter per pel of the frst data set. In Fgure 2 we an see the two olour mages prnted n blak and whte that were taken n 1996 and 1998 over an urban development area n Gran Canara Canar Islands. In ths ase we have a trul multspetral mage wth 3 omponents orrespondng to the Red Green and Blue RGB omponents. he man hanges are new buldngs on the rght of the mages and the ompleton of part of the roads. he parameters found b the EM algorthm were: 1.07

9 µ CH 10 NC and the hange deteton results are: Fgure 2 Up: Aeral mages from Gran Canara used n the eperments. Down: MAP soluton blak pels are lassfed as 'No Change' and whte pels as 'Change'. he MAP lassfaton mage shows that man hanges are deteted. here are two man soures of errors. he frst soure s due to geometr regstraton errors n the ortophotos and an be deteted n the borders of the roads and buldngs. he seond soure s due to non unform lghtnng hanges n the mages. We also ompared the hange deteton results for a monohrome verson of the two mages. As epeted the use of 3 omponents dereases the hange deteton errors.

10 Epermental results for the seond data set In ths eample we show the results of the hange deteton proess for two blak and whte orthophotos wth 1:5000 sale 0.5 meters per pel. In Fgure 3 we an see two mages that were taken n 1996 and 1998 over a regon n La Laguna enerfe. he man soures of hange are a new parkng floor on the left and new buldngs on the rght. he parameters found b the EM algorthm were: µ CH NC and the hange deteton results are: Fgure 3 Up: Aeral mages from La Laguna used n the eperments. Down: MAP soluton blak pels are lassfed as 'No Change' and whte pels as 'Change'. he MAP lassfaton mage shows that man hanges are deteted. We an see agan two man soures of errors. he frst soure s due to geometr regstraton errors n the ortophotos and an be deteted n the borders of the streets and buldngs. he seond soure s due to non unform lghtnng hanges n the mages manl n the roofs of the buldngs present n the two mages.

11 Epermental results for the thrd data set he thrd eample we show the results of the hange deteton proess for two blak and whte orthophotos wth 1:5000 sale 0.5 meters per pel. In Fgure 4 we an see two mages that were taken n 1996 and 1998 over a regon n the t of anta Cruz de enerfe enerfe. he man hanges are the dsappearng of the petrol tanks on the rght and the onstruton of a new ongress enter n the bottom of the mage. he parameters found b the EM algorthm were: µ CH NC and the hange deteton results are: Fgure 4 Up: Aeral mages from anta Cruz de enerfe used n the eperments. Down: MAP soluton blak pels are lassfed as 'No Change' and whte pels as 'Change'. he MAP lassfaton mage shows that man hanges are deteted. here are agan two man soures of errors. he frst soure s due to geometr regstraton errors n the ortophotos and an be deteted n the borders of the buldngs. he seond soure s due to the deteton of shadows as hanges.

12 Epermental results for the fourth data set he fourth eample we show the results of the hange deteton proess for two blak and whte satellte mages wth 5 meters per pel. In Fgure 5 we an see two mages over a regon n the t of Madrd pan that were taken n 1999 and he man hanges are land movements n several zones of the mage. he parameters found b the EM algorthm were: µ CH NC and the hange deteton results are: Fgure 5 Up: atellte mages from Madrd used n the eperments. Down: MAP soluton blak pels are lassfed as 'No Change' and whte pels as 'Change'.

13 Note the nrease the value of parameter wth respet to the prevous mages. hs means that gross hanges are deteted. he learned value of 1.43 for parameter beta mples that n mean eah pel shares hs lass assgnment wth 3.96 of hs 4 neghbours. Agan man hanges are deteted n ths ase. Geometr regstraton errors and lghtnng varatons have less mportane. 6. CONCLUION In ths paper we have presented an unsupervsed approah to the hange deteton problem on remote sensng mages. From a methodologal vewpont the man nnovaton of ths paper les n the formulaton of the hange deteton problem n the HMRF framework and ts eat and effent resoluton usng graph uts. hs makes the use of Markov Random Felds for hange deteton a vable alternatve to estng approahes. Another problem addressed n ths paper s how to determne the parameters n these models. he EM algorthm s used to solve the learnng problem. hs approah shares two errors soures wth other hange deteton tehnues: geometr regstraton unertant and non unform lghtnng varatons. Future etensons of ths work wll tr to solve these problems ntegratng them nto the HMRF model. ACKNOWELEDGEMEN hs work has been supported b "Programa para la Realzaón de Proetos de Investgaón Proet PI 2002/186 of the Conseería de Eduaón Cultura Deportes del Goberno Autónomo de Canaras and b "European Regonal Development Fund" ERDF. REFERENCE 1. Deer P.J. Dgtal Change Deteton ehnues: Cvlan And Mltar Applatons Internatonal mposum on petral ensng Researh IR ngh A. Dgtal Change Deteton ehnues Usng Remotel-ensed Data Internatonal Journal Of Remote ensng 10 6: Coppn P. Jonkheere I. Nakaerts K. Mus I. and Lambn E. Dgtal hange deteton methods n eosstem montorng: a revew. Internatonal Journal of Remote ensng 25: Radke R.J. Andra. Al-Kofan O. and Rosan B. Image hange deteton algorthms: A sstemat surve IEEE rans. Image Proessng 14 3: Blak M. J. Fleet D. J. and aob. Robustl estmatng hanges n mage appearane Computer Vson and Image Understandng peal Issue on Robust tatstal ehnues n Image Understandng 78 1: L. Z. Markov Random Feld Modelng n Image Analss prnger 2nd ed Pérez P. Markov Random Felds and Images CWI Quaterl 11 4: Geman and Geman D tohast relaaton Gbbs dstrbutons and the Baesan restoraton of mages IEEE ransatons on Pattern Analss and Mahne Intellgene 6 : Bokov. Veler O. and Zabh R. Markov random felds wth effent appromatons. Proeedngs of IEEE Conferene on Computer Vson and Pattern Reognton Greg D. Porteous B. and eheult A. Eat mamum a posteror estmaton for bnar mages. Journal of the Roal tatstal oet eres B 512: Kolmogorov V. and Zabh R Vsual orrespondene wth olusons usng graph uts. Internatonal Conferene on Computer Vson Bokov. Veler O. and Zabh R. Fast appromate energ mnmzaton va graph uts IEEE ransatons on Pattern Analss and Mahne Intellgene 2311: ownshend J. Juste C. Gurne and MManus J. he mpat of msregstraton on hange deteton IEEE rans. Geos. Remote ensng 30: Chavez P.. Radometr albraton of Landsat hemat Mapper multspetral mages.photogrammetr Engneerng and Remote ensng 559:

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