MEAN SHIFT ALGORITHM AND ITS APPLICATION IN TRACKING OF OBJECTS

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1 Proceens o e F Inernaonal Conerence on Macne Learnn an Cbernecs Dalan 3-6 Auus 006 MEAN SHIF ALGORIHM AND IS APPLICAION IN RACKING OF OBJECS ZHI-QIANG WEN ZI-XING CAI Collee o Inormaon Scence an Enneern Cenral Sou Unvers HuNan Cansa Cna zqwen000@63.com zxca@csu.eu.cn Absrac: Mean s alorm s recenl wel use n racn clusern ec owever converence o mean s alorm as no been rorousl prove. In s paper mean s alorm w Gaussan prole s sue an apple o racn o objecs. e mprecse proos abou converence o mean s are rsl pone ou. en a converence eorem an s rorous converence proo are prove. Lasl racn approac o objecs base on mean s s moe. e resuls o expermen sow e moe approac as oo perormance o objec racn apple o occluson. e conrbuons n s paper are expece o urer su an applcaon n mean s alorm. Kewors: Mean s alorm; Converence; Kernel uncon; racn o objec; Baacara coecen. Inroucon Mean s wc was propose n 975 b Fuunaa an Hoseler[] s a nonparamerc erave proceure a ss eac aa o local maxmum o ens uncon. In spe o s oo properes as been nore unl Cen s paper[] renews our neres n. Cen n [] revse mean s evelopn a more eneral ormulaon an emonsran s poenal uses n clusern an lobal opmzaon. Snce en mean s as been wel use n objec racn[3-7] mae semenaon[89] paern reconon an clusern[0] lern[] normaon uson[3] an ec. Cen[] scusse e mean s alorm n ree was an cel sue e blurrn process. Le aa be a ne se S embee n e -mensonal Euclean space X. Le X be a ne se prouce n erave proceure. Wen S e mean s proceure s a blurrn process namel e npu aa s recursvel moe aer eac mean s sep. Cen n [] prove e proos or e converence o blurrn process. However wen S s xe rou e process an s nalze o S s no loner a blurrn process an Cen s eores o no appl. Comancu n [4] propose e mean s proceure rom ens esmaon an prove convere a neares saonar pon o e unerln ens uncon. Comancu s wor n [4] ncel s wa Cen as no eal w. L n [5] oun e msae n Comancu s converence proo n [4] an ave some counerexamples. Comancu also mae e same msae n [46]. L prove e converence o mean s n a new wa. In ac e proo n [4456] s mprecse an e converence o mean s nee o be sue. Armn a mean s alorm w Gaussan prole s paper sues above problem. e man conrbuons o s paper are: e mprecse proos abou converence o mean s n [4456] are pone ou an a rorous converence proo s prove. Moreover e racn approac o objecs base on mean s s moe. e paper s oranze as ollows: mean s alorm s nrouce n secon. Secon 3 proves e proo or e converence o mean s. Moe racn approac o objecs an s expermen are presene n secon 4. Secon 5 s e concluson.. Mean s Gven n aa pons x n n e -mensonal space R erave ormula o mean s s as ollows. n x x n x We can el λ Were λ c > /06/$ IEEE 404

2 Proceens o e F Inernaonal Conerence on Macne Learnn an Cbernecs Dalan 3-6 Auus 006 n c x x s a mulvarae ernel ens n esmaor w prole wc s Gaussan prole. c n x x were c n c s e corresponn normalzaon consan. We ene ervave n x x x 3 m x n x x From we can el m 4 were x m c sows mean s alernaes owar e raen recon an leas e ornal pons s o a local maxmum pon o srbun ens uncon. e sep sze λ canes on w e wole erave process. 3. Converence o mean s Leraure [446] prove e converence eores. However ern rom [44] Leraure [6] sue e varable-banw mean s. L n [5] oun a e proos wc were prove or converence o mean s n leraure [446] are wron. Wen leraure [4] was provn e converence o sequence {...} L n[5] consere s wron o euce {...} convere rom - converes o zero an ave some counerexamples. L n [5] use a wen parameer c n Comancu s meo. L prove e converence o mean s n a new wa. e wor L n [5] s o n above msaes an correc bu e no rascall correc. L[5] an Comancu [446] use e proper o convex uncon o prove e sequences... convere an s monooncall ncreasn bu uncon x s rewren n -x wc s no alwas a convex uncon possbl s a concave uncon or neer o em. Wen x s a convex uncon sequences are also no alwas convere o a local maxmum pon. e enons an properes abou convex se convex uncon an concave uncon ma consul leraure[7] or oer leraures. eorem. an s proo are prove as ollows or e mprecse proo n [45]. eorem. e proo o eorem or example: I e ernel K as a convex an monooncall ecreasn prole e sequences { }... an { K }... convere an { K }... s monooncall ncreasn[4] n [45] s mprecse. Proo. e eals abou e proo can be oun n [45]. We suppose e proo n [45] s accurae so sequences convere an monooncall ncrease. From [45] we now convere a a pon assumn an en e expresson 0 s rue. Leraure [4] prove e proo or s eor accorn o proper o convex uncon: x x x x -x as ollows e expresson n [4] s use: c n x K j K j x j x n Here e auor consere x s a convex uncone proo n [5] s smlar o a n [4]. Accorn o e proper o convex uncon s convex an Hessan marx s a posve semene marx. Accorn o secon-orer alor seres expanson ormula aroun we ave: were η 0 <η <. Aer s replace b we el From > we ave < 0. Assumn e as connuous paral ervaves o secon orer ere exs a Neboroo Ω o ŷ sasn s a neave ene marx were Ω. Wen an. So ere a leas exs a τ τ... sasn τ Ω wc mples τ s a neave ene marx. s s paraoxcal w e proper a... s a posve semene marx. So e proos n [45] are mprecse. From e proo o eorem we suppose sequence... convere an monooncall ncrease e Hessan marx s a neave ene 405

3 Proceens o e F Inernaonal Conerence on Macne Learnn an Cbernecs Dalan 3-6 Auus 006 marx so ere s eorem. In orer o prove proo or e eorem a lemma s ven as ollows Lemma I x s Gaussan uncon e ollown expresson s rue. > Proo. From [4] we now a x s Gaussan uncon e ollown expresson s rue m m > 0 m m 6 From e 6 we can el c c > 0 So > 0 eorem. Converence eorem Assumn S R s nonemp open convex se : S R as connuous paral ervaves o secon orer n S. I S s Gaussan uncon an... e Hessan Marx o sequences convere an monooncall ncrease moreover sequences { } convere. Proo. Accorn o e properes o prole x s boune so s also boune. I wll explan e converence o monooncall ncreases. As accorn o rs-orer alor seres expanson ormula aroun an aer s replace b λ S we ave: λ λ θλ were 0 <θ <. Assumn ϕ θ λ λ θλ e ollown expresson s rue. lm θ 0 ϕ θ λ θλ λ 0 > lm ϕ θ θ λ λ λ > 0 Lemma Funcon ϕ θ s connuous an erenable mpln ϕ θλ ' θ λ θλ < 0 Easl eln θλ S. So ϕ θ monooncall ecrease an ϕ θ > 0 θ 0 an expresson > can be conclue. Consequenl sequence converes an monooncall ncreases. Assumn sequence converes o e pon S we can rve e equal lm ϕ θ 0. In aonor 0 < ϕ < ϕ θ we ave wo equales 7a an 7b lm 0 7a lm 0 7b e le erm o 7a an 7b are mulple b eac oer eln lm 0 wc mples lm 0 a s lm 0 so s converen. For e connu o uncon s easl nown o convere o pon an 0 e aapve sep szes s aope n mean s o uaranee converence wc also elmnaes e nee or aonal proceures o coose e aequae sep szes or example nn e opmum sep szes n seepes escen meo[78]. s s a major avanae over e raonal raen-base meos. e banw onl eces e number o observe pea values[0] an e sze o reon were s concave. Generall e number o peas ecreases w e ncrease o e x 0 banw. So or Hessan marx H s mean 0 s can convere o a sea pon. 4. Expermen on racn base on mean s e racn alorm base mean s n [9] s aope n s paper an more eals abou s alorm can be oun n [9]. Bu n s paper ere are some erences rom [9] as ollows. 0 x Hessan marx H s were x s e 0 w an e o objec respecvel. 406

4 Proceens o e F Inernaonal Conerence on Macne Learnn an Cbernecs Dalan 3-6 Auus 006 e prole s Gaussan uncon. x e x λ x 0 x > λ 3 Opmum x s searce sasn a s maxmum. e veo n our expermens s acqure n ouoor a srbuon o Baacara coecen Fure. Analss n process o expermen b erave mes a 0 rame b 30 rame c 70 rame 0 rame Fure. e resul o expermen wenλ Fure 3. Alerave Baacara coecen envronmen rou vson ssem nsalle n robo o our Lab[0]. A objec an s nal locaon are poecall nown. rou e mean s alorm e locaon o objec s oun aer some mes. Furea sows e srbuon o Baacara coecenρaroun e nal locaon o objec. Aer some erave mes e nex a 0 rame b 30 rame Fure 4. e resul o expermen wenλ locaon s oun. Fure b sows e erave mes n process o nn e objec rom e rs rame o 0 rame. Fure sows e resul o expermen. Aer 0 rames e objec can be oun; even ere are some occlusons. Fure 3 sows e alerave Baacara coecen n eac rame so e curve n Fure3 as wo mnmum pons A an B a sow ere are wo 407

5 Proceens o e F Inernaonal Conerence on Macne Learnn an Cbernecs Dalan 3-6 Auus 006 occlusons n process o objec movn. For occlusons e Baacara coecen value s qucl ecreasn. Aer a s qucl ncreasn. e me o occluson can no be oo lon; oerwse e objec wll be los. λ s mporan or e perormance o objec racn apple o occluson. I λ s oo small e objec s easl los. I λ s oo lare e compue me s oo lon an real me perormance s ba. Wen λ e resuls o expermen n Fure4 sow e objec s los. Generall λ wen λ n Fure sows e oo resuls. 5. Conclusons s paper prmarl sues e converence on mean s alorm w Gaussan prole an presens a moe racn approac o objec base on mean s. A rs s paper revew e mean s alorm en e mprecse proo n [45] s pone ou. r a new converence eorem an s proo are prove; lasl a moe racn approac o objec s presene. s approac can be apple o occluson ssue. However e parameer λ as an mporan eec on perormance o racn o objec n occluson envronmen. Generall λ.5-.5 can ensure e oo resuls. Acnowleemens s wor was suppore b e Naonal Naural Scence Founaon o Cna uner Gran No e auors an e wole member n nsue o nellen robo o Cenral Sou Unvers. Beses we an e anonmous revewers or er oo avce. Reerences [] Fuunaa K an Hoseler LD e esmaon o e raen o a ens uncon w applcaons n paern reconon IEEE rans. Inormaon eor vol. pp [] Cen Y Mean s moe seen an clusern IEEE rans. on Paern Analss an Macne Inellence vol.7 no.8 pp [3] Comancu D an Rames V Mean s an opmal precon or ecen objec racn In: Mojslovc A Hu J es. Proc. o e IEEE In l Con. on Imae Processn ICIP pp [4] Comancu D Rames V an Meer P Real-me racn o non-r objecs usn mean s In: Proc. o e IEEE Con. on Compuer Vson an Paern Reconon CVPR pp [5] Collns R Mean s blob racn rou scale space In: Proc. o e Con. on Compuer Vson an Paern Reconon CVPR pp [6] San C We Y an an e al Real me Han racn b Combnn Parcle Flern an Mean S In: Proc. o e 6 IEEE Inernaonal Con. on Auomac Face an Gesure Reconon 7-9 Ma pp [7] Mao E an Cavallaro A Hbr Parcle Fler an Mean S racer w aapve ranson moel In: Proc. o IEEE Snal Proc. Soce In. Con. on Acouscs Speec an Snal Processn ICASSP Plaelpa PA USA Marc pp [8] Comancu D Imae semenaon usn clusern w sale pon eecon In: Proc. o e IEEE In l Con. on Imae Processn ICIP pp [9] Wan J esson B an Y. Xu e al Imae an Veo Semenaon b Ansoropc Kernel Mean S In: Proc. European Con. on Compuer Vson ECCV 004. [0] Comancu D Rames V an A. D. Bue Mulvarae sale pon eecon or sascal clusern In: Proc. o e European Con. Compuer Vson ECCV. Pp [] Georescu B Smson I an Meer P Mean S Base Clusern n H Dmensons: A exure Classcaon Example In: Proc. ICCV Oc. pp [] Comancu D an Meer P Mean s analss an applcaons In: Proc. o e IEEE In l Con. on Compuer Vson ICCV pp [3] Comancu D Nonparamerc normaon uson or moon esmaon In: Proc. o e IEEE Con. on Compuer Vson an Paern Reconon CVPR pp [4] Comancu D an Meer P Mean s: A robus approac owar eaure space analss IEEE rans. on Paern Analss an Macne Inellence vol.4 no.5 pp [5] L X an Wu F e al Converence o a mean s alorm Journal o Soware vol.6 no.3 pp n Cnese. [6] Comancu D Rames V an Meer P e varable banw mean s an aa-rven scale selecon In: Proc. o e IEEE In l Con. on Compuer Vson ICCV pp [7] Xe Z L J an an Z non-lnear opmzaon Cansa: Naonal Unvers o Deence ecnolo Publs House pp n Cnese. [8] S Y Globall converen alorms or unconsrane opmzaon Compuaonal Opmzaon an Applcaons vol.6 pp [9] Comancu D Rames V Meer P Kernel-Base Objec racn IEEE rans. on Paern Analss an Macne Inellence Vol. 5 no pp [0] Ca ZX Zou XB e al Desn o Dsrbue Conrol Ssem or Moble Robo J. Cen. Sou Unv. scence an ecnolo vol.36 no.5 005pp: n Cnese 408