Mining Inexact Spatial Patterns

Transcription

1 Mnng Inexact patal attens engyu Hng Cdnated cence Labaty nvesty f Illns at bana-champagn bana, IL 680 Emal: hng@fp.uuc.edu Abstact Ths k ppses the methdlgy f autmatcally mdelng and mnng nexact spatal pattens. We fst chse attbuted elatnal gaph (AR as the nfmatn epesentatn f the data. The patten s mdelled as a cmpact paametc patten AR (AR. We fmulate the pblem as a paamete estmatn task and pesent the pcedue f estmatng the paametes f the AR mdel gven the data. Expemental esults f the ppsed appach ae shn.. Intductn A spatal patten epesents an bject/cncept that cnssts f many bject/cncept pmtves amng hch vaus spatal ( cntextual elatns ae defned. A cncept can be an bject a set f elated bjects. A spatal patten cnssts f patten pmtves and patten elatns. atten pmtves epesent bject/cncept pmtves. atten elatns epesent the elatn the cntextual elatns amng the bject/cncept pmtves. The patten pmtves encde nn-spatal ppetes (e.g., lcatn, aea, cl, etc.. The patten elatns encde cntextual ppetes amng the patten pmtves (e.g., the elatve pstn, sze, appeaance beteen t patten pmtves, etc.. In many spatal patten mnng lteatues, the nfmatn encded by the patten pmtves s called as nn-spatal ppetes gemetc ppetes. And the nfmatn encded by the patten elatns s called as spatal ppetes tplgcal ppetes. The gal f spatal data mnng s t extact mplct egula knledge encded n spatal pattens that ae nt explctly sted n spatal databases [3]. The extacted spatal pattens captue the chaactestcs f the data n cncse mannes and can be used f vsualzng spatal data, dscveng elatnshps amng data elements, cnstuctng spatal knledge-bases, eganzng data n spatal databases, etc. Reseaches have ppsed many technques f mnng the nn-spatal and/ spatal ppetes f spatal pattens [], [6], [], [2], [3], [4], [5], [7], [8]. Mst f the appaches deal th detemnstc spatal and nn-spatal ppetes. sually, the use shuld Thmas. Huang Beckman Insttute nvesty f Illns at bana-champagn bana, IL 680 Emal: huang@fp.uuc.edu pepcess (e.g. classfy the a data tems and epesent the data tems by the detemnstc spatal and nn-spatal ppetes. Theefe, the mnng esults depend n the esults f the pepcessng step. Heve, n sme applcatns, the use des nt have enugh p knledge abut the data t btan gd enugh fnal pepcessng esults that can be used by the abve appaches. The nse n the a data makes the pblem even hade. Theefe, n many cases, t s bette t use ethe the a data dectly the ntemedate esults f the pepcessng step hch ae elable. Theefe, the nstances f the patten embedded n the data ae nt exactly the same. In addtn, spatal ppetes and nn-spatal ppetes shuld nt be cnsdeed as ndependent nfmatn. A patten s called a spatal patten because the spatal ppetes f the patten pmtves and the nn-spatal ppetes amng the patten pmtves ae nt ndependent. In the est f the pape, e ll fst pesent the methdlgy f autmatcally mdelng and extactng nexact spatal pattens n ectn 2. We d nt eque classfyng the patten pmtves. Instead, e can use ethe the a data dectly sme featues extacted fm the a data. Hence, e ae nt tyng t extact pattens hse nstances appea exactly the same n the database. In the ds, e ae lkng f nexact spatal pattens n the database. Me mptantly, e cnsde bth the spatal ppetes and nn-spatal ppetes f a patten smultaneusly. We then pesent the algthms t mdel and extact the data n ectn 3. The mplemental ssues ae dscussed n ectn 4. ectn 5 pesents expemental esults. The pape s cncluded th summay n ectn The Methdlgy 2. blem fmulatn Fst, e need t defne u gal and fmulate the pblem. We select attbuted elatnal gaph (AR [9] t epesent the data befe e can pefm calculatn n the data. AR s a vey geneal epesentatn and can be used t epesent all knds f spatal pattens. An AR cnssts f a set f ndes and acs. The ndes f an AR epesent the pmtves. The attbutes f the ndes encde the nn-spatal ppetes f the pmtves. The acs f an AR epesent the ela-

2 tns amng the pmtves. The featue vects f the elatns descbe the spatal ( cntextual nfmatn f the pmtves. The elatns f a nde unquely specfy that nde gven the denttes f the ndes. An example f AR epesented vsual patten s shn n Fgue. (a (b Fgue. A vsual patten s epesented as an AR. (a The vsual patten cnssts f a gup f mage segments n dffeent cls. (b The AR epesentatn f (a. The cl f a nde epesents the mean cl attbute f ts cespndng mage segment. The acs epesent the adjacent elatns amng the mage segments. Ou gal s t develp the they and algthms f buldng a cmpact patten AR (AR mdel t summazed a lage set f sample ARs (ARs hch epesent the data. The AR mdel encdes the chaactestcs f bth the spatal ppetes and nn-spatal ppetes f the nstances f the patten. The AR mdel can be used t detect the nstance f the patten late. We use t dente the AR mdel and use { } t dente the AR set. Each epesents a data tem (e.g., an mage. We assume that the s gvened by sme undelyng pbablstc dstbutn functn (DF f(, hee s a vaable epesents a gaph dan fm. Ou gal s t estmate f( gven. Ths s ften teated as a statstcal nfeence pblem that estmates a pbablstc mdel t appxmate f(g by mnmzng the Kullback- Leble dvegence KL(f ( [4]: * f ( ag mn f ( lg d ( ag mn E ag mn KL( f ( f ( [lg f ( ] E f ( [lg ] The AR mdel shuld encde the full nfmatn f. nce the fst tem n eq. ( s a cnstant egadless hat f( exactly s, the pblem f eq. ( s equal t * ag max E f ( [lg ] (2 In pactce, e nly have lmted samples. Theefe, e appxmate the pblem f eq. (2 by * p ( ag max f ( lg It s n geneal dffcult t dectly estmate thut any p knledge abut f(. In pactce, f( s usually assumed t be a paametc functn hse paamete set s dented by Θ. Cnsequentally, the AR mdel s a paametc mdel hse paamete set s Θ. We can ete eq. (3 as * Θ ag max f ( lg (4 Θ (3 nce the fm f a functn can be vey cmplcated, e futhe assume f( s a lnea cmbnatn f a set f mxtue densty functns t make the pblem tactable. Theefe, e desgn the AR t cnsst f a set f mdel cmpnents. Each mdel cmpnent f s a paametc AR and cespnds t a mxtue f f(. Each nde n the AR s asscated th an attbuted DF descbed by the paametes f that nde. Each elatn n the AR mdel s asscated th a elatnal DF descbed by the paametes f that elatn. Theefe, cnssts f a set f attbuted DFs and a set f elatnal DFs. We futhe assume that the fms f the attbuted DFs and the elatnal DFs ae aussan exp[ ( x u x 2 ξ / (2π T 2 C C ( x u] (5 hee x s ethe an attbute vect a elatnal vect, u s the mean f the aussan dstbutn, C s the cvaance matx f the aussan dstbutn, and ξ s the dmensn f x. 2.2 tatns We n defne the fllng mathematc ntatns that ll be used n the est f pape: (a Each AR < A, R >. The sample ndes f s dented by A k a, k, k s the numbe f the sample ndes, k s a sample nde n, a k s the attbute vect f k. We all j f j because the nstances f the patten may nt appea alne. In the the ds, each sample may cntan the nstance f the patten and smethng lese that ae nt elated t the patten. The elatns amng the ndes ae descbed by R { cd, b cd } c, d. Each sample elatn cd has a elatnal vect b cd t descbe the elatn be-

3 teen c and d. elf-elatn, hch epesents the elatn beteen a nde and tself, s alled. F example, the dstance elatn beteen a nde and tself s 0. We have b cd bdc f the elatns ae dectnless. (b The AR mdel has M cmpnents: M { } Φ { Ω, Ψ } M, Θ. Ω s the mdel nde set. Ψ s the mdel elatn set. Θ s the paamete set. Ω { ω k } k 0. ω 0 s null mdel nde. ω k (k>0 s a nn-null mdel nde. ω 0 des nt have physcal exstence. It s used t pvde a mdelng destnatn f thse sample ndes that ae nt nstances f the mdel ndes. s the numbe f the mdel ndes. Actually, dffeent mdel cmpnents may have dffeent numbe f vald mdel ndes. shuld be the lagest numbe f the nnnull mdel ndes n all the cmpnents. If a cmpnent has less than vald ndes, null ndes that d nt have physcal nstances can be added. The easn e assume that all cmpnents have same numbe f ndes s t make pblem descptn ease. Heve, t n t affect the mdelng capablty f the AR mdel. d Ψ { ψ cd } c,. Each sample elatn ψ cd mdel the elatn beteen ω c and ω d. Θ α, { ek } k,{ β k } k,{ f cd } c, s the paamete set f Φ. The hle paamete set f M s Θ { Θ} The value f α dentes the pecentage f the patten nstances n s epesented by Φ. The value f α mples the numbe f the patten nstances n. e k µ k, Σ k dentes the paametes f the attbuted DF epesented by the mdel nde ω k. µ k and Σ k ae the mean and cvaance matx f the attbuted DF espectvely. β k dentes the pecentage f samples ndes that ae epesented by ω k and belng t the nstances epesented by Φ. f cd γ cd, Λ cd dentes the paametes f the elatnal DF epesented by the mdel elatn ψ cd. γ cd and Λ cd ae the mean and cvaance matx f the elatnal DF espectvely. 2.3 Illustatn f the pblem fmulatn ven the pblem fmulated n ectn 2. and the ntatns defned n ectn 2.2, e can an example t llustate h e fmulate the pblem (see Fgue 2. d The value f M shuld be much smalle s that f s that e have a cmpact epesentatn f the patten. Fgue 2. blem fmulatn 3. Infe the aametes f the AR mdel ven the abve pblem fmulatn, e ppsed t estmate bth the stuctue and paametes f the AR mdel teatvely (see Fgue 3. The pcedue s explaned as fllng. (a Repesent the samples as ARs (b Intalze the AR mdel (c Infe the paametes f the AR mdel usng the EM algthm. (d Mdfy the stuctue f? tp (e Delete spuus mdel cmpnents & mdel ndes fm. Yes Fgue 3. The leanng pcedue f the AR mdel. 3. tep (a Repesent each data sample as an AR. Take mages as an example. We can epesent each pxel f an mage as a nde n an AR and the elatns (e.g., dstance amng the pxels as the acs n that AR. O, e can apply mage segmentatn edge detectn t the

4 mages. The mage segments detected edges ae epesented as ndes n the ARs. The elatns f the ARs epesent the spatal elatns amng thse mage segments edges. 3.2 tep (b We ntalze the AR mdel by the fllng algthm s that the cmpnents f ae as dffeent as pssble. Algthm I: Intalze the AR Mdel Randmly select ne AR fm and use t ntalze Φ. Fst, e assgn the stuctue f t Φ. The attbute vects and elatn vects f ae used t ntalze the cespndng attbute means f the attbuted DFs and the elatn means f the elatnal DFs. The attbuted cvaance matxes and the elatnal cvaance matxes ae ntalzed as dentcal matxes. f k 2 t M Calculate the matchng sce Q( j,φ m beteen each AR j and each cmpnent Φ m ( m < k (see ectn 4. k elect an AR x ag mn Q( j m se the ppetes f x t ntalze Θ k. end-f j, Φ m. 3.3 tep (c Ths step estmates the paametes f va the EM algthm [5]. The EM algthm uns teatvely beteen t steps, the Expectatn step (E-step and the Maxmzatn step (M-step, untl t cnveges eaches the maxmum numbe f teatns. Algthm II: Estmate the paametes f E-step: Calculate the matchng pbabltes beteen each AR and the AR mdel. M-step: se the calculated matchng pbabltes t update the paametes f. H t calculate the matchng pbabltes n the E- step ll be dscussed n ectn 4. In [9], e pved that the paametes f the AR mdel shuld be updated usng the fllng expessn n the M-step. α ( t + β µ Σ γ Λ k k k cd cd hee M a j j a j j j j j j j j T j j x x b j k j (ω k k k σ τ σ τ σ τ y y ( ψ σ τ k ( ψ cd cd T ( ψ ( ψ cd cd ( t The supescpt t dentes the numbe f teatns. (t dentes the pbablty f matchng th Φ gven Θ Θ. (t dentes the pbablty f j match- j k ng th ω k gven x µ. j a j k (t Θ (t Θ. ( ψ cd ( ω c Θ σ y γ. b cd 3.4 tep (d & (e (t τ d (t. nce e select sme ARs t ntalze the AR mdel, the stuctue f the AR mdel may nt be apppate afte ntalzatn. Theefe, afte each stp f the EM teatn, e need t examne f the leaned AR mdel s a ell epesentatn f the

5 ARs. If the numbe f the mdel cmpnents M has been ntalzed t lage, the ARs ll be veepesented by the leaned AR mdel. Ths can be fund ut by examnng the cespndng α f the mdel Φ. If the value f α s t small, say smalle than a use-defned theshld, t means that the patten epesented by Φ nly ccu a fe tmes n the ARs. We can then delete Φ. On the the hand, t s vey lkely t ntalze the cmpnents f the AR mdel th spuus ndes and elatns because the ARs may nt nly cntan the nstances f the pattens. T dentfy thse spuus ndes, e can examne the cespndng β k f a mdel nde ω k. If β k s t small, t means nly a fe sample ndes ae smla t ω k. Hence, ω k shuld be a spuus mdel nde and be deleted cnsequently. A easnable theshld f β k culd be (-εmα /, hee ε <. 4. Implementatn Issues T update Θ usng the expessns n ectn 3.3, e (t (t need t calculate j k and. In the ds, e need t egste each AR th each cmpnent f the AR mdel, hch tuns ut t be a t-gaph matchng pblem. T-gaph matchng as a fundamental pblem has been dely nvestgated. It s an pblem [8]. Many appaches have been ppsed t fnd a lcal ptmum slutn [2], [3], [6], [20], [2]. We use an mplementatn f the pbablstc elaxatn gaph matchng algthm [3] hch can pvde an appxmatn t (t. j k As e mentned befe, the samples ae usually nsy and cntan nn-patten nfmatn. Ths ll nt nly affect the featue extacted f the bject pmtves but als ceate spuus ndes n the ARs. Theefe, a null mdel nde s geneally used n the gaph matchng algthms. Ths has been taken cae hen e desgn the AR mdel (see ectn 2.. Each mdel cmpnent has a null nde hch pvdes a matchng destnatn f the spuus ndes. The matchng algthm uses aussan attbuted DFs and aussan elatnal DFs. It decdes hch mdel nde f the mdel cmpnent Φ t match th a sample nde j f by Γ ( j ag max ω k j (t We then defne as k (t hee j M j m j ( Γ( j j ( Γ( j ℵ( Γ m ℵ( Γ ( m j ( f Γ ( j ω 0 ℵ( Γ ( j 0 these We als defne the matchng sce Q(,Φ beteen each AR and each cmpnent Φ as Q(, Φ j j ( Γ( j ℵ( Γ ( 5. Expemental Results 5. nsupevsed vsual patten mnng fm ndependent mages We fst apply the ppsed appach t autmatc vsual patten mnng. Ou appach can be dectly appled t mage pxels because each pxel can be teated as a nde n an AR. Heve, n ths expement, the eslutn f the mages s 307,200 pxels. Wkng dectly n mage pxels esults n hgh cmputatnal cmplexty because thee ll be t many ndes and elatns. L-level mage pcessng, f example, segmentatn, can be used t educe the dmensn f the data and cnsequentally educe the cmputatnal cmplexty. We fst segment the sample mages usng a segmentatn pgam develped by Felzenszalb and Huttenlche [7]. The segmented mages ae epesented as ARs. Each nde f the ARs epesents an mage segment. The attbute f the nde epesents the mean and vaance f the cl featue (RB f the segment. The selected featues f the segments tun ut t be enugh n ths expement. The adjacent elatns amng the segments ae cnsdeed. Dung the leanng pcedue, the adjacent elatns ae updated as cntnuus vaables n the ange f [0, ]. A theshld f 0.5 s used at the end f the leanng pcedue t decde the fnal values (0/ f the elatns. An example s shn hee. The sample mages nclude 20 mages f ZI TM and 20 mages f McDnald TM n vaus backgunds. me f them ae shn n Fgue 4. Thse mages ae segmented (see Fgue 5 and epesented as ARs (see Fgue 6. We ntalze the AR mdel th fu mdel cmpnents (M 4 and set the cnstants as α > 0.5 & β k > 0.8Mα j j

6 /. Afte leanng, the AR mdel nly has t mdel cmpnents. One mdel cmpnent has sx mdel ndes. The the ne has eght mdel ndes. The mages ae als classfed nt t classes, hch f them s epesented by ne f the mdel cmpnents. T vsualze the esults, e used each leaned mdel cmpnent t detect the nstances f pattens n the ARs f ts class. The detected nstances f the AR mdel cmpnents ae shn n Fgue 7. The gnal mage segments cespndng t the detected patten nstances ae shn n Fgue 8. Besdes the shape dffeences due dffeent vepnts, the appeaances ( nn-spatal nfmatn f nstances f a same patten ae dffeent. Take an example f the m n the mddle f the McDnald TM sgn. The mean RB cl featues f the m n the mages shn n Fgue 4 ae (240.8, 92.3, 20.3, (240.3, 80., 09.4, and (205.7, 44.3, 7.2 espectvely. Theefe, e ll nt be able t extact the pattens f e ae lkng f pattens hse appeaance n tems f extacted featues des nt change. Fgue 6. The ARs f the mages n Fgue 5. Fgue 7. The nstances f the ARs detected fm the ARs n Fgue 6. Fgue 4. A subset f the sample mages. Fgue 8. The gnal mage segments cespndng t the nstances f the ARs. Fgue 5. The segmentatn esults f the mages n Fgue nsupevsed vsual patten mnng fm mage sequence In ths expement, e sh h t use the ppsed appach t extact pattens fm mage sequences. The vde s fst dgtzed nt mage fames. Agan, t educe the cmputatnal cmplexty, the mage fames ae segmented ndependently and epesented by ARs n the same ay used n ectn 5.. The tempal nfmatn f the mage sequence mpses the mtn smthness cnstant. The mtn smthness cn-

7 stant eques an mage segment epesented by a nde f the AR f the cuent mage fame can nly mve t cetan ange n the next mage fame. The ange s decded by the mvng speed hch s set by the use. The AR s set t have nly ne cmpnent that s ntalzed as the AR f the fst mage fame f the sequence. We nly extact the spatal patten hse ndes appea n all the mage fames f the vde sequence. An example s shn as the by-un sequence n Fgue 9. As shn, e lean a spatal patten cespndng t a by n the vde sequence. It s easy t fnd ut that the by s mvng t the ght dectn elatvely t the scene f e cmpae the mvement f the by th the dmnate mvement f the the pat f the scene. In ths ay, u methd autmatcally summaze the vde clp as a defmable spatal patten mvng aganst the backgund fm left t ght. The appeaance chaactestcs f the patten ae captued by the leaned AR mdel. Fgue 9. Typcal esults f the by-un sequence. Thse fu clumns cespnd t the esults f the st, 6th, th, and 5th fames n the sequence. The fst shs the gnal mages n the sequence. The secnd shs the segmentatn esults. The thd shs the cespndng ARs f the segmented mages. The futh shs the detected nstances f the AR mdel. The ffth shs gnal the mage segments cespndng t the detected patten nstances. 6. ummay Ths pape pesent a methdlgy f evdence cmbnng that fuses the nn-spatal nfmatn and the spatal nfmatn f the multple samples. Based n t, e develped a cmputatnal mdel and asscated algthms f autmatcally mdelng the emegng spatal patten fm data. We demnstate h t use the ppsed appach f unsupevsed mage patten extactn fm multple ndependent mages and mage sequences. Althugh the expements ae manly caed ut usng t-dmensnal vsual patten (mages n ths pape, the ppsed appach s sutable f geneal spatal patten leanng and dscvey. F example, AR can be used t mdel mlecules n 3D space. AR can als be used t heachcally epesent a cmplex patten. The ntenal stuctue f a nde f an AR can be an AR. Futue k ll be devted t extensvely applyng the ppsed appach t the data n hghe dmensnal space. Refeence [] D. A. Bell,.. Anand, and C. M. hapctt, Database mnng n spatal databases, Intenatnal Wkshp n pat-tempal Databases, 994. [2] B. Bhanu, and O. D. Faugeas, hape matchng f t-dmensnal bjects, IEEE Tans. atten Analyss and Machne Intellgence, vl. 6, pp , 984. [3] W. J. Chstmas, J. Kttle, and M. etu, tuctual matchng n cmpute vsn usng pbablstc elaxatn, IEEE Tans. atten Analyss and Machne Intellgence, vl. 7, n. 8, pp , 995. [4] T. M. Cve, and J. A. Thmas, Elements f Infmatn They, e Yk: Wley, 99. [5] A.. Dempste,. M. Lad, and D. B. Rubn, Maxmum lkelhd fm ncmplete data va the EM algthm, J. Ryal tat. c. e. B, vl. 39, n., pp 38, 977. [6] M. Este, et al., A densty-based algthm f dscveng clustes n lage spatal databases, In c. f the ecnd Intenatnal Cnfeence n Data Mnng KDD-96, pp , tland, Oegn, August 996. [7]. F. Felzenszalb and D. O. Huttenlche, Image segmentatn usng lcal vaatn, n c. IEEE Cnfeence n Cmpute Vsn and atten Recgntn, 998, pp [8] M. R. aey and D.. Jhnsn, Cmputes and Intactablty: A ude t the They f -

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