Active Contour Model driven by Globally Signed Region Pressure Force

Transcription

1 Ative Contour Moel riven y Glolly Signe Region Pressure Fore Mohmme M. Aelsme n Sotirios A. Tsftris IMT Institute for Avne Stuies, Pizz S. Ponzino, 6, Lu, Itly mohmme.elsme@imtlu.it; s.tsftris@imtlu.it Astrt One of the most populr n wiely use glol tive ontour moels (ACM) is the region-se ACM, whih relies on the ssumption of homogeneous intensity in the regions of interest. As result, most often thn not, when imges violte this ssumption the performne of this metho is limite. Thus, hnling imges tht ontin foregroun ojets hrterize y multiple intensity lsses present hllenge. In this pper, we propose novel tive ontour moel se on new Signe Pressure Fore (SPF) funtion whih we term Glolly Signe Region Pressure Fore (GSRPF). It is esigne to inorporte, in glol fshion, the skewness of the intensity istriution of the region of interest (ROI). It n urtely moulte the signs of the pressure fore insie n outsie the ontour, it n hnle imges with multiple intensity lsses in the foregroun, it is roust to itive noise, n offers high effiieny n rpi onvergene. The propose GSRPF is roust to ontour initiliztion n hs the ility to stop the urve evolution lose to even ill-efine (wek) eges. Our moel provies prmeterfree environment to llow minimum user intervention, n offers oth lol n glol segmenttion properties. Experimentl results on severl syntheti n rel imges emonstrte the high ury of the segmenttion results in omprison to other methos opte from the literture. Inex Terms region-se segmenttion; vritionl level set metho; tive ontours; signe pressure fore I. INTRODUCTION One of the most known pprohes to ojet segmenttion re tive ontour methos. Overll, suh moels n e tegorize into three tegories: ege-se, region-se n hyri moels tht omine the vntges of oth ege n regionl informtion. One of the most populr ege-se moels is the geoesi tive ontours (GAC) moel, first propose in [1]. This moel, n its mny vrints, uses n ege-etetor, usully the grient of the imge, to stop the initil ontour on the ounry of the ojets of interest [1] [7]. As result, the moel hs the ility to hnle well imges with well-efine ege informtion; however, when imges hve high level of noise or the ojet is hrterize y wek eges, they nnot onverge t the right ounries. An lternte pproh, the Chn-Vese tive ontours (C V ) moel [8], is one of the most ommon region-se moels. The min ie ehin this kin of moel is to use region s sttistil intensity informtion to onstrut stopping funtion tht n stop the ontour evolution mong ifferent regions. Compre to ege-se moels, the region-se moel usully performs etter in imges with lurre eges n is less sensitive to the ontour initiliztion. However, y esign, this moel ssumes ertin hrteristi shpe for the intensity istriution for the foregroun n kgroun. A moel tht omines the vntges of the ege-se n region-se moels is the Geoesi-Aie Chn-Vese (GACV ) moel [9]. This hyri moel inlues region n ege informtion in its level set flow funtion. Thus, it n seletively just to lol or glol segmenttion. Zhng et l. [10] propose region-se tive ontour moel (ACM). It utilizes sttistil informtion insie n outsie the ontour to onstrut region-se signe pressure fore (SPF) funtion, whih is le to etter ontrol the iretion of evolution. However, oth moels still ssume Gussin intensity within the ROI. Severl uthors hve onsiere to introue terms tht relte to lol n glol intensity informtion in the SPF funtion to hnle itionl intensity lsses n imge inhomogeneity [11] [14]. However, these moels re sensitive to ontour initiliztion n itive noise of high strength. Furthermore, when the ontour is lose to the ojet ounries, the influene of the glol intensity fore my istrt the ontour from the rel ojet ounry, leing to ojet leking [15]. It is evient tht glol moels nnot ommote ojets tht re onstitute of more thn one intensity lsses n on the other hn lol moels lthough they my e le to hnle suh osions, they re sensitive to initiliztion n my le to leking. In this pper, motivte y these oservtions we propose new energy formultion tht inorportes higher orer sttistis for the intensity istriution insie the ontour. To eliminte the nee for re-initiliztion n elerte the urve evolution, we propose new SPF funtion whih we term Glol Signe Region Pressure Fore (GSRPF) funtion, whih n urtely moulte the signs of the pressure fores insie n outsie the ROIs. This pper is orgnize s follows. In Setion 2 we isuss previous glol tive ontour moels. Setion 3 presents the mthemtil formultion of the propose moel n its numeril implementtion. Setion 4 presents the experimentl results ompring the GSRPF with known moels in the literture on segmenttion ury se on numer of syntheti n rel imges. Finlly, Setion 5 offers onlusions.

2 II. DISCUSSION OF PREVIOUS ACMS In this setion, to ppreite the ontriution of the propose moel we riefly review two wiely use glol region-se tive ontour moels. A. Chn-Vese (C V ) Moel The (C V ) moel [8], the lssil region-se moel, uses the region s sttistil informtion to onstrut region stopping funtion tht n stop the ontour evolution etween ifferent regions. The level set formultion of the C V moel, regring the time evolution of the level set funtion φ, n e esrie s = δ (φ)[µ ( φ/ φ ) v λ 1 (I (x) 1 ) 2 +λ 2 (I (x) 2 ) 2 ], (1) where, I(x) enotes n imge inexe y pixel lotion x, µ 0, v inreses the propgtion spee, λ 1 0, n λ 2 0 re prmeters tht ontrol the influene of eh term. The first term keeps the level set funtion smooth, while the seon n thir terms re the internl n externl fores respetively tht rive the ontour towrs the ojet ounries, n δ (φ) is the Dir funtion. 1 n 2 re efine s follows: Ω I (x) H (φ)x 1 = Ω H (φ)x, (2) 2 = Ω I (x) (1 H (φ))x Ω (1 H (φ))x, (3) where, H (φ) is the Hevisie funtion n Ω is the imge omin. Plinly esrie the C V moel rives the ontour (towrs smooth solution) to enlose regions tht mximize the ifferene in their verge intensity. Overll, ompre to GAC moels tht rely on ege grients the C V moel is less sensitive to initiliztion n n reognize the ojet s ounries effiiently. Furthermore, the implementtion of this moel requires the re-initilize of the evolution urve to e signe istne funtion, whih is omputtionlly expensive opertion. B. Seletive Binry n Gussin Filtering Regulrize (SBGFRLS) Moel The SBGFRLS moel [10] omines the vntges of the C V n GAC moels. It utilizes the sttistil informtion insie n outsie the ontour to onstrut region-se signe pressure fore (SPF) funtion, whih is use in ple of the ege stopping funtion (ie., the informtion relte to imge grients) in the GAC moel. Its level set formultion n e esrie s = sp f (I (x)) α φ, (4) where, α is the lloon fore prmeter (ontrolling the rte expnsion of the level set funtion) n the sp f is efine s I (x) sp f (I (x)) = mx ( I (x) 1 + ), 2 (5) 2 where, 1 n 2 re efine in Eqs. 2 n 3. Oserve tht ompre to the C V moel, in Eq. 1 here the Dir funtion δ(φ) hs een reple y φ whih oring to the uthors hs n effetive rnge of the whole imge, rther thn the smll rnge of the Dir funtionl. Also, the rket in Eq. 1 is reple y the sp f funtion efine in Eq. 5. To regulrize the urve the uthors in [10] (following the prtie of others, e.g., [3], [10], [16]), rther thn relying on the omputtionlly ostly µ ( φ/ φ ) term, they onvolve the level set urve with Gussin kernel (K σ ), ie., φ = K σ φ. (6) This σ ontrols the interfe of the urve s µ oes in Eq. 1 of the C V moel. If the vlue of σ is smll, then the level set funtion is sensitive to the noise n it oes not llow the level set funtion to flow into the nrrow regions of the ojet. Overll this moel is fster, omputtionlly effiient, n performs etter thn the onventionl C V moel s pointe in [10]. III. THE PROPOSED MODEL The mjority of glol intensity se tive ontour moels (s reviewe in the previous setion) ssume tht regions of interest re ompose y flt homogeneous (in intensity) regions. Consequently, when these ssumptions re violte the performne of these moels is fr from the esire. We propose new intensity riven moel tht n effiiently moel the foregroun (ie., the ojet(s)) when they re hrterize y non symmetri istriution. This non symmetry oul rise either from intensity vritions or from the ft tht the ojet oul e ompose y two or more intensity lsses. To provie omputtionlly effiient solution n reue the possiility of trpping into lol minim we provie n SPFlike formultion (whih we term GSRPF). A. Moel Desription It is ovious tht relying only on the glol men (insie n outsie the ontour s in C V moel is not suffiient when esriing intensity istriutions when imges hve foregrouns with more omplex intensity istriutions. To overome this prolem, we minimize the segmenttion energy y introuing the glol mein in ition to the glol men. Assuming ontour C, x pixel lotion in the imge I(x), the energy term is efine s E G ( C, +,m +, ) = + in(c) out(c) λ + e + (x)x 2λ e (x)x (7) e + (x) = I(x) I(x) m + 2, (8) e (x) = I(x) 2, (9) where, λ + n λ efine the weight of eh term (insie n outsie the ontour), + n m + re the slr pproximtions of the men n mein respetively for imge I insie the

3 ontour, n is the slr pproximtion of the men outsie the ontour. Following stnr level set formultions [8] we reple the ontour urve C with the level set funtion φ [17] E G ( φ, +,m +, ) = + φ 0 φ 0 λ + e + (x)x 2λ e (x)x. (10) The sttistil esriptors +, m +, n now n e efine in similr fshion to other intensity riven tive ontour moels s sttistil verges n meins + (φ) = verge(i φ(x) 0), m + (φ) = mein(i φ(x) 0), (11) (φ) = verge(i φ(x) 0), Using the level set funtion φ to represent the ontour C in the omin Ω, the energy funtionl n e written s follows: E G ( φ, +,m +, ) = λ + e + (x)h(φ(x))x Ω + 2λ e (x)(1 H(φ(x)))x, (12) Ω where H is the Hevisie funtion. By keeping +, m +, n fixe, we minimize the energy funtion E G (φ, +,m +, ) with respet to φ to otin the grient esent flow s = δ (φ) [ λ + e + (x) + 2λ e (x) ], (13) where δ is the Dir elt funtion. By onsiering the higher orer sttistis, our moel n overome the limittion of the C V moel s symmetri sttistil ssumption, whih is not urte most of the rellife imges. In the inry gry level imges, our moel s n energy minimiztion moel ehves extly the sme s C V moel where m + = +. However, y hving our moel s GSRPF, implementtion with SPF funtion, it is still more roust to the initiliztion thn C V in hnling inry gry imges. B. The GSRPF sign pressure funtion formultion Although we oul rely on Eq. 13 to upte our level set, otining n SPF like formultion woul reue the possiility of trpping into the lol minimum y well moulting the interior n exterior fores. In this setion we propose suh formultion whih we term Glolly Signe Region Pressure Fore (GSRPF) funtion. It is erive suh tht it n moulte the signs of the pressure fore insie n outsie the ojet of interest using the sttistil quntities efine in Eq. 11 n the minimiztion of the propose energy funtionl of Eq. 13. First, we ssume λ + = λ = 1, then we efine the SPF funtion s follows: sp f (I(x)) = sp f 1 sp f 2 (I(x)), (14) where, { sp f1 = sign( m + 4 ), sp f 2 (I(x)) = sign(i(x) +2 +m m + 4 ), (15) where, +,m +, n re efine in Eq. 11. Rther thn onstnt fore (the α in Eq. 4), we use fore tht is qurti funtion of I(x) to ontrol the propgtion of the evolving urve α(i(x)) = (I(x) +2 + m ) m + 4. (16) The signifine of the propose propgtion funtion α(i(x)) is to ynmilly inrese the interior n exterior fores of the urve when it is fr from the ounries (thus reuing the possiility of entrpment in lol miniml) n erese the fores when the urve is lose to the ounries (thus llowing the urve to stop very lose to the tul ounries). The (per-pixel) multiplition of the propose α(i(x)) n sp f (I(x)) results in new region-se signe pressure fore funtion, whih we term Glolly Signe Region Pressure Fores (GSRPF): gsrp f (I(x)) = α(i(x)) sp f (I(x)). (17) The propose GSRPF hs the pity to moulte the sign of the pressure fores n impliitly ontrol the propgtion of the evolving urve so tht the ontour shrinks when it is outsie the ojet of interest n expns when it is insie the ojet. Following the sp f formultion in setion II-B the finl level set formultion of our moel is: = gsrp f (I (x)) φ. (18) For omputtionl effiieny, s in setion II-B, we use Gussin kernel to regulrize the level set funtion to keep the interfe regulr. The σ of the smoothing kernel is the only ontrol prmeter of the moel. As we will emonstrte in the results setion the propose moel: is ple of ientifying ojets of omplex intensity istriution (y onsiering the skewness of the istriution); is roust to itive noise (e.g. higher orer sttistis is onsiere in our moel to ommote the non symmetri n noisy istriutions); is not sensitive to initiliztion (sine only glol informtion is onsiere for the urve evolution); is omputtionlly effiient (sine it oes not require reinitiliztion of the level set funtion n regulrizes the ontour effiiently); n requires few itertions to onverge.

4 C. Implementtion To illustrte the ese of implementtion of our moel, the min steps of the lgorithm n e summrize s: 1) Initilize the level set funtion φ to e inry s follows: ρ x Ω 0 Ω 0 φ(x,t = 0) = 0 x Ω (19) 0 ρ x Ω Ω 0 where ρ 0 is onstnt, Ω 0 is suset in the imge omin Ω n Ω 0 is the ounry of Ω 0; 2) Clulte the GSRPF with Eq. 17; 3) Evolve the level set oring to Eq. 18; 4) Regulrize the level set oring to Eq. 6; 5) If the urve evolution hs onverge, stop n return the result. Otherwise return to Step 2. IV. EXPERIMENTAL RESULTS In this setion we emonstrte the superiority of the propose metho, ompre to referene implementtions of the methos propose in Setion II, when presente with hllenging syntheti n rel imges. We implemente the propose lgorithm in Mtl R2009 on PC (2.5-GHz Intel(R) Core(TM) 2 Duo, 2.00 GB RAM). For fir omprison we use referene Mtl implementtions of the C V n SBGFRLS. To emonstrte the effetiveness of our pproh in hnling imges where the kgroun hs multiple intensity lsses we rete syntheti imge for this purpose shown in Fig. 1, without itive noise n with noise. We ompre the performne of the propose moel with the C V n SBGFRLS moels, n vry the prmeters. As Fig. 1() illustrtes, y inresing the vlue of σ, the propose GSRPF is less sensitive to the noise n fins ll the regions of the ojet for lrge spn of σ. On the other hn, the SBGFRLS moel (Fig. 1()) is not le to evolve properly through the noisy regions even when ltering the vlues of α n σ vlues. Similrly, s Fig. 1(e) shows, the C V moel is unle to segment the imge with ifferent µ vlues. To emonstrte the ury of the propose metho quntittively we opt the preision n rell metris, n ompre the lgorithms result with the groun truth. Fig. 2 shows the effet of σ on the ury of the segmenttion result using the syntheti imge with noise shown in Fig. 1() using the groun truth. Bse on this experiment, σ = 1.4 is reommene to hnle noisy imges with multiple lsses in the foregroun. Tle I shows the roustness of our moel when ifferent levels of noise is e to the syntheti imge of Fig. 1. The high preision t most noise levels onfirms the ility of the propose GSRPF to fin ll the regions of the ojet irrespetive of noise strength. Fig. 3() illustrtes the ility of the GSRPF moel to fin urtely the ounries of ojets with vrious onvexities, shpes, n noisy kgroun. SBGFRLS n ientify the ojets, however, it is unle to segment the hole insie the ojet, s shown in Fig. 3(). The C V moel is unle to e Fig. 1: A syntheti imge with multiple lsses in the foregroun n the performne of the propose, SBGFRLS, n C V, moels s funtion of their prmeters. () the originl 123 x 80 imge with three ifferent intensities 100, 150 n 200, n its histogrm; () the sme imge with Gussin noise e of stnr evition (SD) 30, n its histogrm. Overli lso is the initil ontour (in re) use in ll susequent tests. From left to right the segmenttion results in () of our moel with ifferent σ vlues (1.4, 1.6, 1.8, n 2); () of SBGFRLS with ifferent σ n α vlues ((2,10), (2,50), (2.5, 10), n (2.5,50) respetively); n (e) of the C V moel with ifferent µ vlues (1.4, 1.6, 1.8, n 2). TABLE I: The roustness of GSRPF moel (σ = 1.4) to noise level: the preision n rell with ifferent Gussin noise levels ontrolle y stnr evition (SD). SD Preision(%) Rell (%) segment this imge (s shown in Fig. 3()) euse C V moel is trppe into the lol minim. To emonstrte the spee n ptility of the propose funtion, in Fig. 4 we show the urve evolution for few itertions. It is reily evient tht our moel onverges fst to n urte elinetion of the foregroun ojet.

5 % e f g h i j k l m n o p 40 Rell Preision Σ Fig. 2: The sensitivity of our moel to the prmeter σ, in segmenting the imge in Fig. 1 with Gussin noise, SD = 30, in terms of Rell n Preision. Fig. 3: The segmenttion results on 101 x 99 syntheti imge ontining ifferent ojets of vrile onvexity n shpe, n noisy kgroun. Left to right: the originl imge (with the initil ontour), propose (σ = 1.4), SBGFRLS, n C V moels. Fig. 4: Demonstrting the rpi evolution of the propose moel (σ = 3.5) on 481 x 321 rel imge (ownloe from [18]). Left to right: initil ontour, ontour t 6 n 9 itertions, n finl ontour (15 itertions). Fig. 5 shows the roustness of the propose GSRPF moel ut the sensitivity of the SBGFRLS n C V moels to ifferent ontour initiliztions. The interior n exterior fores re urtely efine inepenent of the initil ontour s lotion. The initil position of the ontour oes not ffet the finl segmenttion, s Fig. 5(), (f), (j), n (n) show, n the presene of the plne s show oes not le to oversegmenttion. On the other hn, the SBGFRLS moel is unle to urtely segment the ojet when the ontour is initilize outsie the ojet, s shown in Fig. 5(g), (k), n (o). On the other hn, the C V moel is more roust to the initiliztion ompre to SBGFRLS, with the exeption of Fig. 5(p). Fig. 6 emonstrtes the ility of our metho in hnling imges rising in the life n nturl sienes. In Fig. 6() ll moels urtely elinete the ounries of rin mlignny. Fig. 6() shows the ility of our moel to extrt urtely n Ariopsis rosette from omplite kgroun (e.g. soil, pot, try); however, the other two Fig. 5: Testing roustness to initiliztion when segmenting 135 x 125 plne imge otine from [19]. Arrnge s olumns re the originl imge with ifferent ontour initiliztions, n then from left to right the results of the propose GSRPF (σ = 1.4), SBGFRLS (with σ = 1 n α = 25), n C V (with µ = 0.2) moels respetively, when using the sme initil ontour. moels re not le to extrt ll the plnt prts, s seen in Fig. 6() n 6(). Similrly, Fig. 6() n () show the ility of GSRPF to segment multiple ojets in the sene, suh s ells n hromosomes. On the other hn the segmenttion results of the SBGFRLS n C V moels re not stisftory. This is minly ttriute to the ft tht oth moels impose ertin onitions on the foregroun intensity istriution, n s suh they nnot minimize the overlp etween the ojet n kgroun istriutions. To emonstrte the omputtionl effiieny of the propose metho when ompre to other glol methos, Tle II shows the CPU time in seons n finl numer of itertions (to onvergene) for ll the imges use here. Overll it is le to segment the imges in roughly hlf the numer of itertions when ompre to SBGFRLS, nother sp f -like moel. V. C ONCLUSION In this pper, we propose novel energy se-tive ontour moel se on new Glolly Signe Region Pressure Fore (GSRPF) funtion. GSRPF onsiers the glol informtion extrte from n ROI n ommotes lso foregroun intensity istriutions tht re not neessrily symmetri. It utomtilly n effiiently moultes the signs of the pressure fores insie n outsie the ontour. The resulting lgorithm is less sensitive to noise, ontour initiliztion, n n hnle imges with omplexity in the foregroun

6 TABLE II: The CPU time n numer of itertions require y the propose GSRPF, SBGFRLS, n C V moels to segment the foregroun in some of the imges use here. Figure GSRPF SBGFRLS C-V CPU Time(s) Itertions CPU Time(s) Itertions CPU Time(s) Itertions Fig. 3() Fig. 5() Fig. 6() Fig. 6() Fig. 6() Fig. 6() Fig. 6: Segmenttion results when ifferent rel imges enountere in the life or nturl sienes re use. Arrnge s rows re: () 109 x 119 rin MRI imge, from [20]; () 436 x 422 Ariopsis optil imge with omplex kgroun; () 256 x 256 ellulose mirosopy imge, from [21]; n () 256 x 256 hromosome mirosopy imge, from [21]. Arrnge s olumns re the originl imge (with the initil ontour), n then from left to right the results of the propose GSRPF, SBGFRLS, n C V moels respetively, when using the sme initil ontour. (Prmeters s in Fig. 5, exept () of GSRPF (σ = 1)). n/or kgroun. Our moel is Gussin regulrizing level set moel tht relies only on single prmeter. It is esigne to hve qurti ehvior to onverge in few itertions without penlizing segmenttion ury. Results on syntheti n rel imges from vriety of senrios emonstrte the superiority of our moel in segmenttion ury when ompre with well regre glol level set methos, suh s the SBGFRLS [10] n C V [8] moels. REFERENCES [1] V. Cselles, R. Kimmel, n G. Spiro, Geoesi Ative Contours, Interntionl Journl of Computer Vision, vol. 22, no. 1, pp , Fe [2] S. Kihenssmy, A. Kumr, n P. Olver, Conforml Curvture Flows: From Phse Trnsitions to Ative Vision, [3] G. Zhu, Bounry-se imge segmenttion using inry level set metho, Optil Engineering, vol. 46, no. 5, pp , My [4] N. Prgios n R. Derihe, Geoesi tive ontours n level sets for the etetion n trking of moving ojets, Pttern Anlysis n Mhine Intelligene, IEEE Trnstions on, vol. 22, no. 3, pp , Mr [5] C. Xu n J. L. Prine, Snkes, Shpes, n Grient Vetor Flow, in IEEE Trnstions on Imge Proessing, 1998, pp [6] A. Vsilevskiy n K. Siiqi, Flux Mximizing Geometri Flows, in IEEE Trnstions on Pttern Anlysis n Mhine Intelligene, 2001, pp [7] C. Li, C. Xu, C. Gui, n M. D. Fox, Level set evolution without re-initiliztion: A new vritionl formultion, in in Pro. of IEEE Conferene on Computer Vision n Pttern Reognition, 2005, pp [8] T. F. Chn n L. A. Vese, Ative ontours without eges, Imge Proessing, IEEE Trnstions on, vol. 10, no. 2, pp , Fe [9] L. Chen, Y. Zhou, Y. Wng, n J. Yng, GACV: Geoesi-Aie CV metho, Pttern Reognition, vol. 39, no. 7, pp , Jul [10] K. Zhng, L. Zhng, H. Song, n W. Zhou, Ative ontours with seletive lol or glol segmenttion: A new formultion n level set metho, Imge n Vision Computing, vol. 28, no. 4, pp , Apr [11] L. Wng, L. He, A. Mishr, n C. Li, Ative ontours riven y lol Gussin istriution fitting energy, Signl Proessing, vol. 89, no. 12, pp , De [12] P. Wng, K. Sun, n Z. Chen, Lol n Glol Intensity Informtion Integrte Geoesi Moel for Imge Segmenttion, in Computer Siene n Eletronis Engineering (ICCSEE), 2012 Interntionl Conferene on, vol. 2. IEEE, Mr. 2012, pp [13] T.-T. Trn, V.-T. Phm, Y.-J. Chiu, n K.-K. Shyu, Ative ontour with seletive lol or glol segmenttion for intensity inhomogeneous imge, in Computer Siene n Informtion Tehnology (ICCSIT), r IEEE Interntionl Conferene on, vol. 1. IEEE, Jul. 2010, pp [14] U. Vovk, F. Pernus, n B. Likr, A Review of Methos for Corretion of Intensity Inhomogeneity in MRI, Meil Imging, IEEE Trnstions on, vol. 26, no. 3, pp , Mr [15] S. Liu n Y. Peng, A lol region-se ChnVese moel for imge segmenttion, Pttern Reognition, vol. 45, no. 7, pp , Jul [16] Y. Shi n W. C. Krl, Rel-time trking using level sets, in Computer Vision n Pttern Reognition, CVPR IEEE Computer Soiety Conferene on, vol. 2. IEEE, Jun. 2005, pp vol. 2. [17] H.-K. Zho, T. Chn, B. Merrimn, n S. Osher, A Vritionl Level Set Approh to Multiphse Motion, Journl of Computtionl Physis, vol. 127, no. 1, pp , [18] [19] skhzhng/. [20] liw/. [21]