Stable Model for Active Contour based Region Tracking using Level Set PDE

Transcription

1 666 Suk-Ho Lee : STABLE MODEL FOR ACTVE CONTOUR BASED REGON TRACKNG USNG LEVEL SET DE hp://dx.doi.org/.69/iice Sable Model for Acive Coour based Regio Trackig usig Level Se DE Suk-Ho Lee, Membe KCE Absrac his pape we propose a sable acive coour based rackig mehod which uilizes he bimodal segmeaio echique o obai a backgroud color dimiished image The proposed mehod overcomes he drawback of he Masouri model which is liable o fall io a local miimum sae whe colors appear i he backgroud ha are similar o he arge colors. The Masouri model has bee a foudaio for acive coour based rackig mehods, sice i is derived from a probabiliy based ierpreaio. By sabilizig he model wih he proposed speed fucio, he proposed model opes he way o exed probabiliy based acive coour rackig for pracical applicaios. dex Terms Trackig, Acive Coou Segmeaio, Level Se.. NTRODUCTON N his pape we propose a sable rackig model which is a exesio of he Masouri model [] ha has bee proposed for regio rackig wihou moio compuaio. Trackig regios i a image sequece is a challegig ad difficul problem i image processig ad compuer visio. Howeve regio rackig has also may impora applicaios such as iellige video surveillace, auomaed video ediig, video daabase search ad rerieval, ec. Recely, acive coour based approaches have bee sudied for he aim of simulaeous rackig ad segmeaio of he obec []-[7]. The approach aims for obaiig he obec boudary for furher high level compuer visio asks, while rackig he obec. f he boudary of he obec is kow, he iellige aalysis ca be doe of he siuaio i which he obec is ivolved (See Figure ). The Masouri model [8] has bee proposed for regio rackig ha derives from a Bayesia formulaio which uses o moio field or moio parameers compuaio. This model has become a foudaio for may acive coour based rackig models sice he problem of regio rackig is formulaed as a Bayesia esimaio problem, ad he resulig rackig algorihm is expressed as a level se parial differeial equaio. Mauscrip received Ocober, ; revised November 7, ; acceped November 3,. Suk-Ho Lee is wih he Deparme of Sofware Egieerig, Dogseo Uiversiy, Busa, 67-76, Korea ( pera@gdsu.dogseo.ac.kr) Uil his pape oher acive coour based algorihms us used moio iformaio (opical flow), or backgroud subracio o defie he speed fucio of he acive coour. Aoher ovely of he Masouri model is ha very lile a priori iformaio abou he regio beig racked is used i he algorihm. This kid of expressio has opeed a wide possibiliy for ierpreig he rackig process as parial differeial equaio solvig problem. Howeve he Masouri model is raher sesiive o he color variaio i he backgroud. f a cerai color appear i he backgroud such which is similar o he color of he obec, he he acive coour ries o iclude he backgroud regio also i he arge regio, ad resul i a failure of he rackig. his pape we propose a ew speed fucio which is a exesio of he speed fucio of ha used i he Masouri model o sabilize he rackig process. We also use a bimodal segmeaio echique o obai a ew image i which he arge regio is beer discrimiaed from he backgroud. We uify boh echiques i a sigle framework. Experimeal resuls verifyig he sabiliy of he proposed model are give a he ed of his paper. Fig.. Examples whe he boudary of he obec ca give some clue o he siuaio of he scee.. MATHEMATCAL FORMULATON FOR ROBABLTY BASED ACTVE CONTOUR MOTON TRACKNG : THE MANSOUR MODEL his secio, we iroduce how he Masouri model resuls from a probabiliy based formulaio of he acive coour based moio rackig. We follow he mai process described i [7]. The, we refer o he problems wih he Masouri model. he probabilisic approach,

2 NTERNATONAL JOURNAL OF NFORMATON AND COMMUNCATON ENGNEERNG, VOL. 9, NO. 6, DECEMBER 667 he shape ad he moio parameers of he acive coour are updaed such ha he coour's a poseriori probabiliy is maximized. To formulae he poseriori probabiliy fucio, i is assumed ha he image is composed of he obec regio ad he backgroud regio. R, R ob R where R correspods o he full image regio ad Rob ad R correspod o he obec ad he backgroud regios, respecively. The likelihood of observig he boudary is equal o he likelihood of pariioig he image io he obec ad he backgroud regios: ) ( ϕ ( R)) { R ob, R }, () ( where ϕ is he pariioig operaor. The obec rackig problem is formulaed i erms of he boudary probabiliy,, defied by he probabiliies of he regios, give he curre image,, ad he previous i collecio of obec boudaries, : i,,..., as E( ) Ψob dx + Ψ ( x3) dx3 dx, (7) x x x3 Where Ψ ob log ( ), Ψ log ( ) Rob R The speed fucio of he Masouri model is compued by miimizig he eergy fucioal i (7). By reformulaig (7), he speed fucio F is compued based o he relaioship bewee he iesiy values of he backgroud pixels ad he foregroud pixels iside a local widow. where Ψ ob wih F ( Ψob χob Ψ χ ) dx, (8) x Ω log ( ), Ψ log ( ) Rob ( σ α max e z: z m ( x+ z)). R ( ϕ ( R ),,,... ). () The usig he Bayes rule, his ca be reformulaed as ( ϕ ( R ),,,... ) ( ϕ( R ),,... ) (3) (,,... ) Here, he firs erm i he umeraor ca be spli io wo erms such as ad ob ( R,,,... ) (4) ( R,,,... ) (5) Usig he assumpio ha each pixel i he image is idepede of each oher such ha: Rob ob x x ob ( ) R C C ( ( x )) 3 x ( ) ( ( x3)) Here, x, x, x are pixels ha lie i a small pach 3 which ceer is o he acive coou each correspodig o he iside, he boudary ad he ouside of he acive coour (Fig. ). The by maximizig his poseriori probabiliy, we obai he Eergy fucioal for rackig: (6) Fig.. ixels iside he local pach alog he acive coour. The speed fucio decides he speed of he followig curve evoluio equaio: dc d FN (9) which evolves he curve C accordig o he speed field F i he ormal direcio N. he classical level-se mehod his evoluio is achieved by umerically solvig he followig DE φ φ i F, () o he regular grid where is he level-se fucio. As he fucio evolves coiuously, so does he implicily represeed curve. The level se fucio i he foregroud regio is assiged a posiive value while he backgroud regio is

3 668 Suk-Ho Lee : STABLE MODEL FOR ACTVE CONTOUR BASED REGON TRACKNG USNG LEVEL SET DE assiged a egaive value. The he L differece values of he iesiy values iside he local widows which are se a he boudary of he obec are compued o decide he speed fucio F. The speed fucio F is compued by he followig hree seps: i. Se V, o he miimum value of i over all ieger pairs i+ l, + l ( l, l ) i he widow if ( i + l, + l ) Obec( φ ) i he previous ou. Se V, o he miimum value of i over all ieger pairs i+ l, + l ( l, ) l i he widow if ( i + l, + l) Backgroud ( φ ) i he previous i ou 3. F V V. Here, he superscrip i deoes he curre frame, while he superscrip i deoes he previous i+ l, + l Figure shows how he acive coour is moved by he above hree seps. The lef upper image i Fig. shows he previous frame, while he righ upper image shows he curre The lower row shows elarged local regios of he upper row frames. from (). The color a poi (b) i Fig. 3 is closer o he backgroud color i he previous frame, i.e., is close o he color a poi (a). Therefore, a his po i becomes i ou ha V > V, ad he sig of F a his poi becomes posiive. This makes he value of φ o decrease a his po ad whe φ becomes egaive, he his poi becomes a member of he backgroud regio. Therefore, he poi (b) i Fig. 3 has chaged from he foregroud regio o he backgroud regio. Howeve a classificaio ca happe if a ew color appears i he backgroud regio i he curre frame which is closer o he foregroud regio i he previous Figure 4 shows such a case. Here, compared wih colors iside he widow (show as a blue box i Fig. 4) i he previous frame, a ew color has appeared iside he widow i he curre frame (idicaed by he leer (c)). This color had bee occluded by he rackig obec, ad herefore had o bee observed i he previous The problem is ha he ew color is more close o he obec color ha he backgroud color i he L differece orm sese. Therefore, he Masouri algorihm regards his color as he obec color raher ha he backgroud colo ad he speed fucio of he acive coour is compued such ha i icludes his color. As a resul he regio wih his color is icluded i he obec regio as ca be see i he righ image of Fig. 4. Fig. 3. Showig how he Masouri model acs: Lef : previous frame Righ: curre frame, (a) backgroud color i he previous frame, (b) backgroud color i he curre ca be see ha he sig of he speed fucio F is i ou i ou posiive if V > V ad egaive if V < V. This decides he direcio of he moio of he coou sice he sig of F decides wheher he value of he level se fucio a ( i, ) decreases or icreases as ca be see Fig. 4. Showig he case whe he Masouri model acs abormal Lef : previous frame Righ: curre frame, (a),(b) backgroud ad foregroud colors i he previous frame, (c) ew backgroud color i he curre Oce he false regio is icluded i he obec regio, i remais as he obec regio, ad herefore, he error of he acive coour regio ad he obec regio icreases wih ime ad he rackig fails as ca be see i Fig. 4. Therefore, a ew speed fucio has o be proposed which ca deal wih his problem.

4 NTERNATONAL JOURNAL OF NFORMATON AND COMMUNCATON ENGNEERNG, VOL. 9, NO. 6, DECEMBER 669. ROOSED METHOD We propose a ew speed fucio which ca deal wih he above meioed problem. We le he proposed speed fucio be F( mi{ ϕ,} + max{ ϕ,} e ε + ( r, +Δ Ba ( r ) σ () ) ( ) [ χ ]dr where B a ( r ) deoes a small ball regio wih radius a aroud r, ad χ R (r) is he characerisic fucio of he regio { r ϕ > }, ad σ ad ε are small posiive values. To see how he proposed speed fucio works, we firs observe how he erm afer he max{} fucio i (3) works. We le his erm be F e ε + ( r, +Δ ) Ba ( r ) σ ( ) R [ χ ]dr. () We see ha χ R is posiive, which is agai posiive if he poi r has bee he obec regio i he previous frame, i.e., φ has bee posiive i he previous Meawhile F is egaive if χ is egaive, which is egaive if he poi [ ] R R F is posiive if [ ] r has bee he backgroud regio i he previous The value F is compued iside he ball regio B a r ), ( σ ε + ( r, +Δ ( ) ad has a meaigful value oly if e is large eough, i.e., if he color value a r i he previous frame ad he color value a r i he curre frame are o oo differe. The max fucio muliplied o F has o be posiive if he fucio F is o have a effec o he speed fucio F (. Tha is, if ϕ is posiive he we le F o decide wheher F ( is posiive or egaive. f ϕ is egaive he F ( is egaive, regardless of he sig of F. Here, he fucio is he resul of bimodal ϕ segmeaio [8], which we have proposed o compue he Cha-Vese model i a speedy ad sable way. The fucio ϕ is compued as he miimizer of he followig fucioal: E( ϕ ) ˆ( ave ˆ( ave { ϕ < } { ϕ } ϕ H ( δ + ϕ ) dr ϕ H ( δ ϕ ) dr (3) Here, H is he Heaviside fucio, ad δ is a small posiive value. The miimizaio above leads o a segmeaio of Ω Ωi Ωou { ϕ > } { ϕ < }. Here, we ry o force he speed fucio F o be F < i he regio Ω ou. For his aim, we defied he ew differece image ˆ( as 3 β ˆ( : β exp (4) q ( λ ( where β ad β are scalig parameers, ad ( is he origial image, he icomig he ew image ˆ(, he colors correspodig o λ ( dimiish, sice he deomior becomes small. We choose a domia color λ ( ) i such a way ha λ ( σ arg sup φ ( s, ds (5) ad λ φ ( s, ds > α Ω, (6) λ σ where φ ( s, deoes he hisogram of a he level s ad a ime. The supremum of (5) correspods o he domia colo while he codiio i (6) correspods o he color ha covers a regio of reasoable size. Here, we carefully choose δ ad σ ha represe he variace ad he porio of he image, respecively. We usually choose < δ <. 9 Similarly, we ca choose he secod domia color (if i exiss) i such a way ha σ λ ( arg sup φ ( s, ds (7) ad λ φ ( s, ds > α Ω. (8) λ σ This makes he colors i he ew image o dimiish, if hey are close o λ (. Similarly, we choose he hird domia color if i exiss, ad coiue i he same way wih he fourh colo ec. Usig his ew image we obai a fucio ϕ as a miimizer of he eergy fucioal i (3). This miimizer is he used o compue he proposed speed fucio usig equaio (). Due o he bimodal segmeaio, he maor backgroud colors are elimiaed from he backgroud regio. The mior backgroud colors which are close o he arge colors become ow segmeed o he backgroud, ad hus he acive coour will o fall io a local miimum. This makes he rackig more sable ha he Masouri model.

5 67 Suk-Ho Lee : STABLE MODEL FOR ACTVE CONTOUR BASED REGON TRACKNG USNG LEVEL SET DE V. EXERMENTAL RESULTS Experimes have bee performed o a whole image sequece o compare he rackig resuls of he Masouri model ad he proposed model. Figure 5 shows he rackig resuls wih he Masouri model. ca be observed ha he coour becomes rapped a he regio where he backgroud has a ew color. As a resul, he coour eludes from he rue obec boudary, ad he whole rackig fails. Figure 6 shows he rackig resul wih he proposed model. As ca be observed, he ew speed fucio ca hadle he ew colors i he backgroud, ad as a resul, he coour sicks o he obec boudary. The rackig becomes more sable ha he Masouri model, as he ceer of he obec regio is more accuraely compued, ad a more accurae boudary of he obec is obaied. V. CONCLUSON his pape we correced he speed erm used i he coveioal Masouri model such ha he acive coour becomes o rapped by backgroud colors similar o hose i he arge regio. The correcio is doe by usig he bimodal level se segmeaio algorihm o deermie he domia backgroud colors. Experimeal resuls verify he performace of he proposed mehod. ACKNOWLEDGMENT This work was suppored by he Naioal Research Foudaio of Korea(NRF) gra fuded by he Korea goverme(mest) (No. -564). REFERENCES Fig. 5. Trackig resul wih he coveioal Masouri model. The coour is rapped by he backgroud colors as ca be see i he red circles i he boom row. [] S. Osher ad J. Sehia, Fros propagaio wih curvauredepede speed: Algorihms based o Hamilo Jacobi formulaios, J. Compu. hys., vol. 79, pp. 49, 988. [] N. aragios ad R. Deriche, Geodesic acive coours ad level ses for he deecio ad rackig of movig obecs, EEE Tras. aer Aal. Mach. ell., vol., o. 3, pp. 66 8, Mar.. [3] T. Cha ad L. Vese, Acive coours wihou edges, EEE Tras. mage rocess., vol., o., pp , Feb.. [4] Alper Yilmaz, Xi L ad Mubarak Shah, Coour-Based Obec Trackig wih Occlusio Hadlig i Video Acquired Usig Mobile Cameras, EEE Tras. AM, vol. 6, o., pp , Nov., 4. [5] G. Sudaramoorh A. Yezz A.C. Meucc Coarse-o-Fie Segmeaio ad Trackig Usig Sobolev Acive Coours, EEE Tras. AM, vol. 3, o. 5, pp , May, 8. [6] Wei Yu, F. Frache Yao-Je Chag, Tsuha Che, Fas ad robus acive coours for image segmeaio, roceedigs of EEE C, pp , Sep.. [7] A. Masour Regio Trackig via Level Se DEs wihou Moio Compuaio, EEE Tras. AM, vol. 4, o. 7, pp , July, [8] Suk-Ho Lee ad Ji Keu Seo, "Level se-based bimodal segmeaio wih saioary global miimum", EEE Tras. mage rocess., vol. 5, o. 9, pp , Sep. 6. Suk-Ho Lee received his BS, MS, ad hd i elecroics egieerig from Yosei Uiversiy, Korea, i 993, 998, ad 3, respecively. He worked as a researcher i.5cm x 3cm he mpedace magig Research Ceer from 3 o 6 ad was a assisa professor a Yosei Uiversiy from 6 o 8. He has bee i he Deparme of Mulimedia Egieerig a Dogseo Uiversiy, Busa, Korea, sice 8, where he is ow a professor. His curre research ieress iclude image ad video filerig based o DEs, medical imagig, ad compuer visio. Fig. 6. Trackig resul wih he proposed model. The coour is o rapped by he backgroud colors as ca be see i he boom row.