Two Fast Alternating Direction Optimization Methods for Multiphase Segmentation

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1 It'l Cof. IP Cop. Vso ad Patter Recogto IPCV'5 3 Two Fast Alteratg Drecto Optzato Methods for Multphase Segetato u Ta WeboWe ad Zheua Pa College of Iforato Egeerg Qgdao Uversty Qgdao Shadog Cha lua086@63.co ustwwb@63.co Abstract - I ths paper two ew ethods assocated wth alteratg drecto optzato fast alteratg drecto ethod of ultplers(fastadmm ad fast alteratg zato algorth(fastama are proposed for age segetato usg the ultphase Cha-Vese odel whch s o the bass of pecewse costat optal approxatos. For these ethods we corporate the varable splttg approach ad a reset codto order to update the agrage ultpler ad ae sure the value of eergy fuctoal s always postve. The Osher ad Stetha level set ethod bary level set fuctos thresholdg ethod ad proecto forula are appled the pleetato. Fally uercal results wth rapd covergece are obtaed by our ethods whch are also copared wth those of soe other fast varatoal ethods to deostrate better effectveess of our ethods. Keywords: Multphase segetato fast alteratg drecto ethod of ultplers fast alteratg zato algorth actve cotours level sets. Itroducto I age segetato aor advaces were ade two-phase age segetato[-3] the early days. Muford-Shah odel proposed by Muford D ad Shah J [] s regarded as the ost sgfcat rego-based odel. It has bee exteded to a great deal of applcatos. I 00 Toy F. Cha ad uta A. Vese proposed the Cha-Vese odel [5] for actve cotours to detect obects a gve age. It s oe of the splfed varats of Muford-Shah odel. Nevertheless as the coplexty of the ages creases - phase age segetato s ot able to eet the actual eeds. Therefore ultphase segetato s appled to satsfy the deads. O the bass of Potts odel [6][7] fro statstcal echacs Zhao et al. [8] started to study ultphase oto segetato by usg the level set ethod ad proposed a odel whch ca represet dfferet regos by level set fuctos. I order to reduce the uber of level set fuctos Cha et al. cotued ther wor ad proposed ultphase segetato odel [9] whch s a geeralzato of Cha- Vese odel. Ther schee ca aturally avod "overlap" ad "leaage" proble. But there are stll soe probles about solvg the global optzato accuracy stablty ad speed. Alteratg drecto ethod of ultplers(admm was frst descrbed by Glows ad Marocco [0] ad alteratg zato algorth(ama was preseted by Tseg []. These techques are cooly ow as the Splt Brega Method [] ad are ow to be a effcet solver for probles volvg the total-varato or [3]. These ethods ca be accelerated usg optal frst order ethods of the type frst proposed by Nesterov []. Ad accelerated varats of ADMM ad AMA ca be called FastADMM ad FastAMA. I our paper we desg ethods (FastADMM ad FastAMA that ca acheve coputatoal effcecy ad ow eve faster covergece to solve the fuctoal of ultphase Cha-Vese odel based o bary level sets fraewor [56]. The gradet descet ethod (GDM [7] Chabolle s dual ethod (DM [8] alteratg drecto ethod of ultplers(admm [9].e. the augeted agraga ethod (AM [0] ad alteratg zato algorth(ama [] are used to copare wth our ethods. But results of these ethods whch are used ultphase segetato odel wll be obtaed a slow covergece. To Goldste et al. [9] troduced FastADMM ad appled t to solve the TV odel as a exaple ad soe other strogly covex probles. A predctor-corrector type accelerato step s used ths ethod. The reag of ths paper s orgazed as follows. I Secto the bary level set forulato of the fuctoal of ultphase Cha-Vese odel used our paper s revewed alog wth ts four tradtoal soluto ethods. Our proposed ethods are dscussed Secto 3 ad ts teratve dscrete forulas for pleetato wll be preseted detal. I Secto soe uercal experets are gve to llustrate the effectveess of our ethod by coparg wth other ethods. Fally a cocluso s gve secto 5. The ultphase Cha-Vese odel ad ts four tradtoal ethods. The bary level set based forulato I order to separate a age doa to subdoas wth ad.vesead Cha defed up to phases ad level set fuctos. Ths way aes sure that each pxel! x y # wll belog to oe ad oly oe phase.

2 It'l Cof. IP Cop. Vso ad Patter Recogto IPCV'5 The level set ethod proposed by Osher ad Setha [] s a effectve represetato for evolvg curves ad surfaces because of autoatc chage of topology. The a dea of the level set forulato s to plctly represet a gve terface % t as the zero level set of a pschtz cotuous fucto! R & R. s defed as follows: (! x t 0 f x s sde %! t! x t 0 f x s at %! t! x t + 0 f x s outsde %! t It s oral to defe as a sged dstace fucto order to eep stablty uercal pleetato. The dstace fucto obeys the Eoal equato.! xt ( ( The varatoal level set ethod [3] gves a way to apply the level set fucto to the eergy fuctoal. For a gve ope rego wth sooth boudary %! t sple facts ca be got as follows:.. (3 legth % H H! x ad! x area. H ( where are Heavsde fucto ad Drac delta fucto respectvely. Accordg to ther wor for =... let! b... / b/ b/ be the bary represetato of - b / :. The characterstc fucto where 0 ca be wrtte as the followg geeral expresso: b! x b / / / H 0 ; = > x of 3. (5 The eergy fuctoal for phases s obtaed:!.!. (6 E H Q s postve paraeters. The where! ad!... fucto Q s defed as c / f c s a costat vector whch ca be obtaed by the ea testy value of f sde as follows:. f c. (7. O ths bass a ew approach called a bary level set fucto s troduced by Joha e et al. [5] ad Bresso et al. [6] whch has a spler defto about talzato of level set fucto. Frstly assue that the terface s eclosg?. A dscotuous level set fucto s used stead. It s defed as follows: ( f x # t! x (8 0 f x # ext If the level set fucto ca use the bass fucto! x satsfy! x! x boudary of ad the area sde. legth area! 5 thewe to calculate the legth of the. x (9. x (0 Wag Q et al. [] proposed a ultphase Cha-Vese odel based o a pluralty of bary level set fuctos ad alteratg covex optzato 00. They rewrte the ultphase Cha-Vese odel based o bary level set as: E! Q.. ( where s the bary level set fucto ad the characterstc fucto! x should be restated as follows: x 0 b ; = > 3. ( b / / /. GDM DM ADMM or AM ad AMA for zg ultphase Cha-Vese odel.. Gradet descet ethod (GDM The eergy fuctoal zato proble assocated wth Equato ( ca be solved by coputg the evoluto equato of va gradet descet flow as: ( 5 6 5! Β / Q t 7 Α Χ 5 0 o 5 5. (3 Where the secod forula of (3 s the boudary codto. But (3 cludes fourth order dervatves eed to be dscretzed usg coplex fte dfferece forulas ad the tegrato steps of te archg deped o rght had ters heavly... Dual ethod (DM I order to speed up the calculato Dual forula of TV or proposed by.. sup Β p p9 Chabolle [8] ca be appled the eergy fuctoal.

3 It'l Cof. IP Cop. Vso ad Patter Recogto IPCV'5 5 E p! Β p Q!.. A way of alteratg optzato s used to copute dual varable p : 5 5 /! Β p / Q 5t 5 5p / p / ( ad (5 (6 5t Though dual ethod ca speed up the processg to soe degree t ca ot obta the deal covergece rate...3 Alteratg drecto ethod of ultplers (ADMM The basc dea of [3] s to use low-order varables stead of hgh-order varables ad obta a approxate result. The costrat w s added eergy fuctoal. E w! w Q.. #! w! w (7.. / / s called the agrage ultpler ad # s a where pealzato paraeter. The the varables are optzed as: ( w 5 Q Β # Β w / 0 5!#! / w / Β 0 o 5 6 / ax / / 0 7 # # Α 8 Χ /! # # (8 (9.. Alteratg zato algorth (AMA The coputg fraewor of AMA s slar to ADMM. Both of the adopt the alteratg drecto ethod whle AMA s spler. The oly sgfcat dfferece s about the calculato of. Here Gradet descet ethod s used to esure ts covergece of teratve ethod. The we ca get through the followg terato: ( t o 5 5 / Β / Q. (0 3 Our proposed ethods Though the GDM DM ADMM or AM ad AMA have bee successfully exteded to the ultphase Cha-Vese odel they caot get results wth exact values as well as rapd speed. Through troducg a restart rule -.e. the accelerato paraeters are reset whe certa codtos are et. So both expesve coputg process ad coplex te appearace the evoluto equatos are able to be avoded by our proposed fast ethod. 3. Fast alteratg drecto ethod of ultplers (FastADMM Frst the basc dea of usg FastADMM s proposed. to replace the so the hgh-order varables ca be splfed by low-order varables. The the varables wll be optzed respectvely. Followg covetos the sybol ' & ' s gogtobeusedtodeotevectorfuctos.the eergy fuctoal s rewrtte as follows: Δ Ec! w;.! w Q. ( Δ #! w/! w. /. We troduce auxlary varables v!... Where varable Δ If the value of should be carefully calculated by the teredate. Please otce that w should be updated by v. s obtaed by solvg the Euler-agrage v wll be updated by w. Detaled equato pleetato of FastADMM s show Algorth. Algorth : FastADMM for ultphase Cha-Vese odel Δ0 0.Italzato:... # 0. w v. For Ε.. Subproble about solve the followg probles alteratvely: c :... c arg % c E c w ; Γ.. Subproble about : arg % E c v ; Γ!. Δ. ( (3.3. Subproble 3 about 3 ( w w arg % 3 w E c w ; : Γ Δ.. Update agrage ultpler : (5

4 6 It'l Cof. IP Cop. Vso ad Patter Recogto IPCV'5.5. f E 0.6. #! w Δ /. (6 the a restart rule ad a relaxato factor are used to update. else v ad Δ. (7 v / w w / w. (8 Δ / /. (9 Δ v w eed to be processed by threshold: ( a 0 otherwse. (30 3. The overall loop wll be terated f the stoppg crteros (descrbed secto are satsfed. The proecto 3! Β equato ( s a sple to the terval Η Ι trucato of 0.For 0... the zers of varables c w subprobles -3 ca be obtaed by zg the followg eergy fuctoals: % c Q. (3 % Q v 3 Δ!.! /. #! v. / Δ!.!. / #.! w / % w! w w (3 (33 Thus the eergy fuctoal s splfed whch ca be coputed by easer teratve algorth ad avod the subproble whch occurs assocatg wth o covergece successfully. Next (3 to (33 wll be solved respectvely by dfferet teratve ethods. 3.. Estatos of pecewse costat paraeters We ca obta c... as Equato ( Coputg of the bary level set fucto Whe s beg coputed se-plct Gauss-Sedel teratve schee ca be used because t ca esure ts fast covergece. The! th value of c ad the th v auxlary varable should be fxed. Ths cocept s also appled the followg paragraphs. The correspodg Euler- agrage equato of (3 s: 5 # Β! v / Β 0 Δ. Q 5. (3 I experets we fd that two or three teratve steps are eough to acheve a good zer of. It s a powerful guaratee of the eergy fuctoal zato. Now becoes the ostadard bary level set fucto. It ust be proected to [0] as Equato (:!! Max M 0. ( Calculato of the auxlary varable The soft thresholdg forula [5-7] s used ths part to calculate the varable w. It s oe of the ost classcal algorths whch has bee wdely used to obta zers of the varables ths d of equato. The varables c ad are fxed. The calculato result s show as : Δ Δ 6! / # w Max7 / / 0Α.(36 Δ 7 # # Α 8 Χ / # After the zers of subprobles -3 are foud agrage ultplers should be updated accordg to (6. Now let us troduce the restart rule whch s able to guaratee covergece for wealy-covex probles. If the udgg crtera E s greater tha 0 the paraeter sequece Γ s used to over-relax the sequece of terato ad help update v ad Δ. The defto of E s descrbed [9]. At the ed of the pleetato we ca wor out the threshold a for barzato of o the bass of the hstogra of ts result (as descrbed (30. I secto the stoppg crteros whch ca be used to terate the overall loop are preseted. 3. Fast alteratg zato algorth (FastAMA The basc dea of usg FastAMA s slar to FastADMM. I ths algorth oly oe teredate varable

5 It'l Cof. IP Cop. Vso ad Patter Recogto IPCV'5 7 Δ s used to accelerate the covergece. Detaled pleetato of FastAMA s show Algorth. Algorth : FastAMA for ultphase Cha-Vese odel Δ0 0. Italzato:... # 0.. For Ε.. Subproble about wv solve the followg probles alteratvely: c :... c arg % c E c w ; Γ.. Subproble about : Γ!. Δ ; arg % E c w. (37 (38.3. Subproble 3 about w : 3 (39 Γ w arg % w E c w 3 ; Δ.. Update agrage ultpler : # w / Δ.5. Update the relaxato paraeter:. (0. (. (.6. Update the teredate varable Δ : Δ / /. (3.7. eed to be processed by threshold: ( a. ( 0 otherwse 3. The overall loop wll be terated f the stoppg crteros (descrbed secto are satsfed. The zers of varables c w subprobles -3 ca be obtaed by zg the followg fuctoals: % c Q. (5 Δ!..! / % Q v Δ % 3! w.! w! w. / #! w /. (6. (7 Where equato (5 ca be zed as preseted (7. of (6 ca be obtaed as equato (0 please otce that should be replaced by the teredate varable.ad of (7 ca be obtaed as (36. Δ w After the zers of subprobles -3 are foud agrage ultplers should be updated accordg to (. The paraeter sequece Γ s used to over-relax the sequece of terato ad help update pleetato barzato of Δ. At the ed of the s requred as well. We use the sae stoppg crteros to terate the overall loop as preseted secto. Nuercal experets I ths secto the uercal results of our proposed ethods are appled o soe real cases ad they wll be copared wth dfferet ethods (GDM DM ADMM or AM AMA to deostrate the effectveess ad effcecy of our ethods. All the experets are operated o the sae platfor (Matlab7.8 o a PC (Itel (R CPU.60GHz. The sae tal cotours ad tatos of varables for all the ethods each experet are used order to have a relatvely eutral crtero for coparso. To clarfy ths the tal values of varables are show as follows: 0 0 GDM: c 0 # 0Γ c 0 # 0 p 0 DM: Γ ADMM or AM: c Γ AMA: c Γ w FastADMM: Γ FastAMA: c Γ 0 # 0 w # c 0 # 0 w0 0 v 0 Δ = # 0 w 0 0. As descrbed [8] the teratos eed to be terated whe the followg crteros are satsfed. I ths paper the sae stoppg crteros ca be used the proposed two ethods. We eed to otor the costrats errors teratos: wth Β where R + % w R w R w 0 R w deotes the!... R w / w (8 (9 or o age doa.if ( % s a sall eough paraeter terato of outer repeat wll be stopped. These good uercal dcators are also used to deterate the values of # whch ca be the bass of pealty paraeter adustet.

6 8 It'l Cof. IP Cop. Vso ad Patter Recogto IPCV'5 I teratos the relatve errors of agrage ultplers ad the soluto should be otced. They should reduce to a suffcetly sall level: / / / / / /!... The covergece of eergy fuctoal E! / E! be guarateed. E! (50. (5 E eed to 9% should be satsfed. Experet. Sythetc age of sze s used as the test age. I ths experet two bary level set fuctos are used to detect three dfferet subdoas (=. I Fg. soe results of GDM DM ADMM or AM AMA ad proposed two ethods are frstly preseted respectvely so that we ca ae vsual coparsos wth the segeted ages. Fg. (a shows the orgal age. The tal cotours are show Fg.(b. Fg. (c-(f shows the segetato results of GDM DM ADMM or AM ad AMA respectvely. The segeted ages show Fg. (g ad (h are fro our proposed two ethods. The paraeters used FastADMM for Fg. (g are gve as follows:! 0.5 # Ad the paraeters used FastAMA for Fg. (h are : t 0.! 5 # Fg.. The effects of GDM DM ADMM or AM AMA ad proposed two ethods. The frst row: orgal age ad the tal cotours. The secod ad thrd row: segetato results of GDM DM ADMM or AM AMA. The last row : results of our proposed two ethods. Fro left to rght we llustrate relatve resduals (8 relatve errors of agrage ultplers (50 relatve error of (5 ad eergy curve alog the outer repeat Fg.. The graphs coe fro Fg. (g ad (h respectvely. It ca be observed that the algorth has coverged log before 00 teratos. They also gve portat forato about how to choose pealty paraeter #. I order to guaratee covergece as well as the speed of covergece the costrat errors should coverge to zero wth early the sae speed. If R w R w R w goes to zero qucer tha others the decrease #. wll coverge to zero wth the sae speed as the terato proceeds ad the eergy wll decrease to a steady costat value whe # are chose properly. Ths experet pots out that the selecto of paraeter! has o obvous effect o the results. ( Relatve resduals ( Relatve errors of agrage ultplers (a Orgal age (b Ital cotours ( Relatve error of (l The eergy (c By GDM (d By DM (e By ADMM or AM (f By AMA ( Relatve resduals ( Relatve errors of agrage ultplers (g By FastADMM (h By FastAMA

7 It'l Cof. IP Cop. Vso ad Patter Recogto IPCV'5 9 (o Relatve error of (p The eergy Fg.. The plots of paraetrc errors ad the eergy curve. (-(l are obtaed by FastADMM fro Fg. (g. (-(p are obtaed by FastAMA fro Fg. (h. I the aspect of algorth effcecy teratos ad coputatoal te of ethods preseted ths experet are gve. It s easy to see that FastADMM ad FastAMA have the faster covergece rate. TABE. Coparsos of teratos ad coputatoal te Approaches Iteratos Te (sec Fg. -(c: GDM Fg. - (d: DM Fg. - (e: ADMM Fg. - (f: AMA 8 0. Fg. - (g: FastADMM Fg. - (h: FastAMA 0.08 Experet. I ths experet our ethods wll be copared wth GDM DM ADMM ad AMA by usg the o a age of sze The orgal age s preseted Fg. 3(a (= ad the sae tal cotours are used Fg. (b. Three parts cotaed by the ethods etoed above are gve Fg. 3(c-(h. We ca see all these ethods ca obta alost the sae segetao effects. Paraeters used FastADMM ad FastAMA are gve:! 0.5 # 0.. t 0.! 5 # (f By AMA (g By FastADMM (h By FastAMA Fg.3. The effects of GDM DM ADMM AMA ad proposed two ethods. The frst row: orgal age ad the tal cotours. The secod ad thrd row: segetato results of these sx ethods. Here a threshold ethod should be used to realze the baryzato of.itsaportatwaytohelpfdthe accurate results. No-threshold solutos of the proposed ethods are show as follows. It ca be observed that othreshold ofte results fuzzy edges (red rectagles. ( By FastADMM (a Orgal age (b Ital cotours ( By FastADMM Fg.. No-threshold solutos of the proposed ethods. ( coes fro FastADMM. ( coes fro FastAMA. Next the hstogras of o-threshold solutos fro FastADMM are gve Fg. 5. It gves us a good way to choose the threshold of. I ths experet we fd the threshold a 0.5 could be applcable. (c By GDM (d By DM (e By ADMM

8 0 It'l Cof. IP Cop. Vso ad Patter Recogto IPCV'5 ( hstogra of (l hstogra of (o-threshold (o-threshold Fg.5. Hstogras of (o-threshold. Fro Table t s obvous that the total coputatoal cost requred by our ethods s uch less tha other four ethods fro the coparso. TABE. Coparsos of teratos ad coputatoal te Approaches Iteratos Te (sec Fg. 3-(c: GDM Fg. 3- (d: DM 8 0. Fg. 3- (e: ADMM Fg. 3- (f: AMA Fg. 3- (g: FastADMM 0.6 Fg. 3- (h: FastAMA Experet 3. The results of all these ethods are show o a bra agetc resoace age (MRI. Fro the orgal age of sze Fg. 6(a there are four parts eed to be segeted. Fg. 6(b shows the tal cotours. The segetato results fro dfferet ethods are gve Fgs. 6(c-(h ad local elarged results of (c-(h are show Fgs. 6(-(. Those subdoas separated fro Fg. 6(g ad (h wth proposed ethods are respectvely preseted Fgs. 6(o-(p. The paraeters used FastADMM for Fg. 6 (g are:! 0.5 # Ad the paraeters used FastAMA for Fg. (h are : t 0.! 0.5 # (a Orgal age 3 (b Ital cotours (c By GDM (d By DM (e By ADMM (f By AMA (g By FastADMM (h By FastAMA ( By GDM ( By DM ( By ADMM (l By AMA ( By FastADMM ( By FastAMA (o FastADMM segetato results (p FastAMA segetato results Fg.6. The coparso betwee other ethods ad our ethods o a MRI. The frst row: orgal age ad the tal cotours. The secod ad Thrd row: results of other ethods ad our ethods. The fourth ad ffth row: zooed sall subregos (purple rectagles. The last two row: four dfferet phases of (g ad (h obtaed by proposed ethods. I Table 3 coparsos of teratos ad coputatoal te usg dfferet ethods are gve. TABE 3. Coparsos of teratos ad coputatoal te Methods Iteratos Te (sec Fg. 6-(c: GDM 5.58 Fg. 6-(d: DM.8 Fg. 6-(e: ADMM Fg. 6-(f:AMA Fg. 6-(g ad (o: FastADMM 5 0. Fg. 6-(h ad (p: FastAMA Coclusos I ths paper by usg the relevat cocepts of Nesterov s accelerated algorth covex optzato ad ultphase Cha-Vese odel we propose FastADMM ad FastAMA for ultphase age segetato. Our proposed accelerated ethods have bee valdated by several uercal experets. The coparso of results obtaed by soe

9 It'l Cof. IP Cop. Vso ad Patter Recogto IPCV'5 other approaches ad our proposed approach dcate that our approach ows good eough effects ad t s a good way to effcetly ze the dffcult fuctoal. Our ethod ca also be appled to surface segetato 3D recostructo ad age deosg odels etc. the future wor. It s supposed to yeld shorter rute tha the tradtoal ethods whle the qualty of results s detcal. Acowledgeets The wor has bee partally supported by the Natoal Natural Scece Foudato of Cha (os ad Refereces [] Kass M Wt A Terzopoulos D Saes: actve cotour odels. It. J. Coput. Vs. ( [] Aubert G Barlaud M Faugeras O Jeha-Besso S Iage segetato usg actve cotours: calculus of varatos or shape gradet. SIAM J. Appl. Math. 63( [3] Jg D Zheua P Xagfeg Y Webo We Guodog Wag Soe fast proecto ethods based o Cha-Vese odel for age segetato. EURASIP Joural o Iage ad Vdeo Processg. (0: [] Muford D Shah J. Optal approxatos by pecewse sooth fuctos ad assocated varatoal probles[j]. Coucatos o pure ad appled atheatcs (5: [5] Cha T F Vese A. Actve cotours wthout edges[j]. Iage processg IEEE trasactos o 0(: [6] Potts R B. Soe geeralzed order-dsorder trasforatos[c]//proceedgs of the Cabrdge Phlosophcal Socety. 8(: [7] Glles Celeux Florece Forbes Nathale Peyrard. EMbased age segetato usg Potts odels wth exteral feld. [Research Report] RR [8] Zhao H K Cha T F Merra B Osher S. A varatoal level set approach to ultphase oto [J]. Joural of Coputatoal Physcs 7: [9] uta A. Vese Toy F. Cha A ultphase level set fraewor for age segetato usg the uford ad shah odel Iteratoal Joural of Coputer Vso vol.50 o.3 pp [0] R. Glows ad A. Marrocco If. Rech. Oper. vol. R- pp [] Paul T. Applcatos of splttg algorth to decoposto covex prograg ad varatoal equaltes. SIAM J. Cotrol Opt 9: [] T. Goldste ad S. Osher The splt brega ethod for regularzed probles UCA CAM Report [3] T. Goldste X. Bresso ad S. Osher Geoetrc applcatos of the splt brega ethod: Segetato ad surface recostructo J. Sc. Coput. vol. 5 pp October 00. [] Y. Nesterov A ethod of solvg a covex prograg proble wth covergece rate o(/^. Sovet Math. Dol. vol. 7 pp [5]e J ysaer M Ta X C. A bary level set odel ad soe applcatos to Muford-Shah age segetato [J]. IEEE Trasactos o Iage Processg 5(5: [6]X. Bresso S. Esedoglu P. Vadergheyst et al. Fast global zato of the actve cotour/sae odel. Joural of Matheatcal Iagg ad Vso 8(: [7]. I. Rud S. Osher E. Fate. Nolear total varato based ose reoval algorths. Physca D: Nolear Pheoea 60(: [8]A. Chabolle. A algorth for total varato zato ad applcatos. Joural of Matheatcal Iagg ad Vso 0(-: [9]T. Goldste B. O Dooghue S. Setzer ad R. Barau Fast alteratg drecto optzato ethods SIAM Joural o Iagg Sceces 7(3: [0] W. Zhu X.-C. Ta ad T. F. Cha Iage Segetato Usg Euler's Elastca as the Regularzato Joural of Scetfc Coputg 57(: [] Staley Osher Jaes A. Setha Frots propagatg wth curvature-depedet speed: algorths based o Halto-Jacob forulato Joural of Coputatoal Physcs vol.79 o. pp [] Q Wag Zheua Pa Webo We Splt- Brega ethod ad dual ethod for ultphase age segetato Joural of Coputer-Aded Desg & Coputer Graphcs vol. o.9pp [3]e J ysaer M Ta X C. A varat of the level set ethod ad applcatos to age segetato[j]. Matheatcs of coputato 75(55: []Wu C Zhag J Ta X C. Augeted agraga ethod for total varato restorato wth o-quadratc fdelty [J]. Iverse Probles ad Iagg 5: [5]P. K. Sahoo S. Solta AND A. K. C. Wog A survey of thresholdg techques Coputer vso graphcs ad age processg (: [6] A. Bec M. Teboulle A fast teratve shragethresholdg algorth for lear verse probles SIAM Joural o Iagg Sceces (: [7]J. Yag W. Y Y. Zhag Y. Wag A Fast Algorth for Edge-Preservg Varatoal Multchael Iage Restorato SIAM Joural o Iagg Sceces (: [8]X.-C. Ta Fast uercal schees related to curvature zato: a bref ad eleetary revew UCA CAM Report -0 May 0.