New Algorithm for Level Set Evolution without Re-initialization and Its Application to Variational Image Segmentation

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1 03 ACADEMY PUBLISHER 305 New Algorth for Level Set Evoluto wthout Re-talzato ad Its Applcato to Varatoal Iage Segetato Culag Lu College of Iforato Egeerg, Qgdao Uversty, Qgdao, Cha Eal: Zheua Pa, ad Jg Dua College of Iforato Egeerg, Qgdao Uversty, Qgdao, Cha Eal: Abstract Tradtoally varatoal level set odel for age segetato s solved by usg gradet descet ethod, whch has low coputatoal effcecy ad eeds coplex re-talzato of level set fuctos as sged dstace fuctos. I ths paper, we frst reforulate the varatoal odel as a costraed optzato proble. The we preset a augeted Lagraga proecto ethod to preserve sged dstace fuctos ad accelerate the pleetato. By troducg auxlary varables, we covert dervatve costrats to algebrac equatos wth sple proecto. We apply the proposed algorth to the two-phase/ultphase Cha-Vese odels. Nuercal results are provded to copare our algorth wth soe others, whch deostrate effectveess ad effcecy of our approach. Idex Ters level set ethod, sged dstace fucto, augeted Lagraga ethod, proecto, segetato I. INTRODUCTION I the last twety years, ay of the ost geeral segetato odels have bee solved by the level set ethod (LSM) [-3]. The basc dea of the LSM s to plctly represet a cotour or terface as the zero level set of a hgher desoal fucto, called a level set fucto (LSF), ad forulate the oto of the cotour as the evoluto of the LSF. For closed cotours, sged dstace fuctos (SDFs) were orgally adopted to represet LSFs. Soe recet developets have proposed to use label fuctos [4], rather tha SDFs, to represet cotours. Ths chage allows us to use covex relaxato techques [5] ad fast algorths [6-8] to provde effectve alteratves to dstace preservg LSMs. Nevertheless, the LSM for age segetato uses zero level set of a cotuous SDF to express cotour, ad the geoetrc features such as oral, curvature ca be calculated aturally, whch s very coveet to post processg of curves ad surfaces [9]. For ths reaso, t s portat to desg fast ad accurate algorths for dstace preservg level set ethods. Correspodg author: Culag Lu I covetoal level set forulatos, the LSF s o loger preserved as a SDF durg cotour evoluto. To overcoe ths dffculty, two approaches have bee suggested to restore the regularty of the LSF ad ata stable terface evoluto. Re-talzato [0] s the ost coo ethod, whch s perfored by perodcally stoppg the evoluto ad reshapg the degraded LSF as a SDF. However, ths approach troduces the questos of whe ad how to re-talze the LSF. Also, t ay correctly ove the zero level set away fro the expected posto. I order to avod retalzato, the secod ethod as at costrag the LSF to preserve a SDF durg the cotour evoluto [9, -5]. I [], the authors troduce a ew forulato to restrct the LSF to a SDF. But ths forulato cossts of three PDEs, whch aes the uercal pleetato ore dffcult tha the stadard LSM. More recetly, L et al. [] has proposed to add a pealty ter to the orgal eergy fuctoal. The pealty ter elates the eed for re-talzato. However, the te step of ther ethod s restrcted by the Courat- Fredrchs-Lewy (CFL) codto [6] ad the SDF property s oly ecouraged but ot eforced. I [4], Lu et al. proposes a augeted Lagraga ethod (ALM) to get rd of re-talzato. Ther ethod splfes the treatet of costrat greatly, but t does ot avod the coputato of curvature, whch s te-cosug. I ths paper, we propose a costraed optzato approach, splt augeted Lagraga proecto ethod (SALPM), to get rd of re-talzato ad prove the coputato effcecy. We corporate the varable splttg techque to update the Lagrage ultpler, ad costra level set fuctos to stay dstace fuctos va drect proecto. We apply our algorth to the Cha- Vese odels [7, 8]. Coparsos wth other ethods have show the hgh effcecy of our proposed approach. The rest of ths paper s orgazed as follows. I Secto II, we revew brefly the LSM appled to age segetato. I Secto III, we dscuss the fraewor of our ew odel detal. Nuercal results are gve Secto IV. Secto V draws the coclusos. 03 ACADEMY PUBLISHER do:0.4304/sw

2 ACADEMY PUBLISHER A. LSM ad VLSM II. RELATED WORKS We frst recall the tradtoal LSM. Let R be a ope bouded doa, f ( x ): R be a gve age, where x = ( x, y) s a pxel. The LSF φ s orally defed as a SDF φ ( x t) ( (), x) x sdec() t x C() t d( C() t, x) x outsdec() t d C t, = 0 where dc (,x) deotes the Eucldea dstace fro x to C. A costrat to () s the equato ( t) () φ x, = () To satsfy (), [0] used a re-talzato schee to solve the followg equato to steady state φt + sg ( φ0 )( φ ) = 0 R (3) φ( x,0) = φ0 where φ 0 s the fucto to be re-talzed. The varatoal LSM (VLSM) proposed [9] offers us a way to ebed the LSF drectly to the eergy fuctoal by utlzg the followg facts ( ) d, ( ) C = δ φ φ x S = H φ dx (4) I the above, C s the legth of C, S s the area of S (a ope set S,.e. C = S ). H ( z ) ad δ ( z) are, respectvely, Heavsde fucto ad Drac delta fucto. To avod sgularty uercal pleetato, H ( z ) ad δ ( z) are usually expressed regularzed versos wth paraeter ε > 0 to approxate the orgal oes as H ε ( z) z = + arcta π ε δ ε z = π ε, ( ) z + ε B. The Cha-Vese Model wthout Re-talzato We here adopt the Cha-Vese odel for age segetato, as t represets a large class of actve cotour odels publshed the lterature. The Cha- Vese odel proposed to use level set fuctos to represet = phases. If =, t s called the twophase Cha-Vese ode, otherwse t s the ultphase Cha-Vese odel [7, 8]. For =,,..., ad =,,...,, let ( b... b b ) be the bary represetato of, where b = 0. The characterstc fucto χ ( x ) ca be wrtte as the followg geeral expresso [0] ( ) b ( ) ( ) b H ε ( φ ) = (5) χ x = + (6) where φ s the level set fucto. The Cha-Vese odel becoes the followg zato proble γ δε ( φ) φ d Q( u, ) d φ, u x+ x χ x (7) = = where γ s a postve tug paraeter, u s the ea testy value, ad Q( u, x ) s defed as ( u f). Cosderg the costrat (), we ca forulate the proble (7) as a costraed zato proble γ δε ( φ) φd Q d φ, u x+ χ x s.t. φ = (8) = = I order to force the LSF to be a SDF durg evoluto, great efforts have bee ade to eforce () to satsfy (). The authors [] add a quadratc pealzato ter of () to the fuctoal ad obta the followg ucostraed zato proble γ δε ( φ) φ dx+ Q d χ x = = φ, u ( φ ) μ + - d x = The zato proble (9) s usually solved by usg a alteratg optzato schee ad u = f χ dx χ dx φ φ χ = δ ( φ) γdv Q t φ = φ φ + μ Δφ dv φ (9) (0) () Theoretcally, μ should be a large pealty paraeter, but t was poted out [] that the te step Δ t > 0 ad the paraeter μ > 0 ust satsfy μδ t < 0.5 for stablty. Therefore, there s a cotradcto betwee the accuracy of the costrat ad the choce of large te steps. To prove the accuracy ad stablty, the authors [4] troduce a ALM to solve the segetato proble (8). Defe D ( φ) = φ -, the augeted Lagraga fuctoal s ax γ δε ( φ) φ dx+ Q d χ x λ φ = = μ + λ Ddx+ Dd x = = where λ s called the Lagrage ultpler. () 03 ACADEMY PUBLISHER

3 03 ACADEMY PUBLISHER 307 The zato of () s solved by the followg teratve schee ad φ φ χ = γδε ( φ ) dv Q t φ = φ φ φ + dv λ + μ Δφ dv φ φ λ λ μ φ ( ) + + = + - (3) (4) Ths ethod splfes the treatet of costrat greatly, but t does ot avod the coputato of curvature evoluto equato (3) as (). III. THE PROPOSED METHOD A. SALPM The SALM ths secto s spred by the researches [6, 7]. Dfferet fro the costrat D = φ - (), we troduce a ew varable, say w, to serve as the arguet of the fuctoal D = w u, uder the costrat w = u. Ths leads to the followg costraed proble ax ( ) λ φ, w = = μ λddx Dd x = = + + Qχ dx+ γ δ φ w dx s.t. w = (5) Note here that the SALM reduces the possblty of llcodtog by troducg the Lagraga ultpler λ ad varable w at each step to the eergy fuctoal (5). Therefore, the covergece of ths algorth ca be guarateed wthout creasg μ to a very large value as the pealty ethod (9). Sce (5) volves ultple varables, we also use the alteratve zato ethod to fd the uercal soluto to (5). A saddle pot of the ax- proble (5) eeds the followg three equatos γ w ( ) δ φ ( φ ( w )) ( λ ) χ + Q + dv φ = φ μ Δ dv = 0 (6) w + + γ δ ( ) ( w ) 0 w ε φ + λ + μ φ = s.t. w = (7) λ λ μ φ ( w ) + = (8) We use the se-plct dfferece schee ad Gauss-Sedel teratve ethod to obta a steady-state soluto to the sub-proble (6). Mzato wth respect to w + ca be perfored by usg the followg shrage operator [7] λ γ w = ax φ δε ( φ ),0 μ μ 0, 0 = 0 (9) + λ + λ 0 φ φ μ μ At last, techque as w + s obtaed va a sple proecto w + w = (0) w + + B. Algorth Detals Now we preset the teratve augeted Lagraga proecto ethod Algorth I. Algorth I (SALPM) Italzato: φ, λ, w ad set = 0, for =,,...,, =,,...,.. Repeat 3. Update each u by (0); 4. Copute each φ + by (6); 5. Copute each w + by (9) ad (0); 6. Copute each λ by (8); 7. = + ; 8. Utl a stoppg crtero E E E η s satsfed, where η s a sall postve value. IV. EXPERIMENTAL RESULTS Ths secto shows uercal results of our SALPM for both sythetc ad real ages. All the experets are perfored by usg MATLAB v00b o a Wdows XP platfor wth a Itel Core Duo CPU at.80 GHz ad GB eory. To set up a relatvely eutral crtero for coparso, we use the sae tal cotour for all the ethods each experet. Moreover, soe paraeters are fxed for geeralty as follows: ε =, Δ t = 0.0. γ s requred to be tued for each exaple, ad s usually foratted by γ = α 55, α (0,). A. Coparso ad Aalyss of Two-phase Experets We frst copare our proposed algorth to retalzato ethod [0], L et al. s ethod [] ad ALM [4] desged to preserve the SDF the LSM. The test age s preseted Fg. (a). For the retalzato ethod, the tal LSF s a SDF. The tal LSF for the other three approaches s a pecewse 03 ACADEMY PUBLISHER

4 ACADEMY PUBLISHER costat fucto. Nuercally, we ca chec that the four ethods gve the sae soluto as show Fg. (b). Fgs. (c), (e), (g), ad () show the evoluto of the level set fucto for the four dfferet ethods. I Fgs. (d), (f), (h), ad (), we plot the correspodg ea devato of φ, whch easures the dstace betwee the coputed LSF at the th terato. (a) (b) (c) (d) (e) (f) (g) (h) () () Fgure. Segetato of crcle age wth two phases. (a) Test age of sze 00 00; (b) Sae segetato result usg re-talzato ethod, L et al. s ethod, ALM ad SALPM, respectvely; (c)-(d) Results wth re-talzato for γ = ; (e)-(f) Results wth L et al. s ethod for γ = ; (g)-(h) Results wth ALM for γ = ; ()-() Results wth SALPM for γ = ACADEMY PUBLISHER

5 03 ACADEMY PUBLISHER 309 Although the fal LSF provdes the desred results Fg. (c), the perodc re-talzato process produces a o-sooth zato Fg. (d). Besdes, we do ot ow geeral whe to re-talze the LSF as a SDF. I ths experet, we apply the re-talzato for every 5 terato. We the cosder the pealty ethod []. Fg. (e) shows ther ethod does ot costra exactly the LSF to be a SDF. Moreover, ther approach slows dow the zato process as the uber of teratos to reach the covergece state creases cosderably as show Fg. (f). We observe fro Fg. (g) that the ALM [4] coverges faster tha L et al. s ethod, but the LSF dffers fro a SDF. Our algorth s preseted Fgs. () ad (). Our forulato costras the LSF to be a SDF due to proectos, ad the proposed SALPM s fast because t avods the calculato of the curvature ter. (a) (b) (c) (d) (e) (f) Fgure. Segetato of Europe ght-lghts age wth two phases. (a) Test age of sze 38 88; (b) Result wth o re-talzato for γ = ; (c) Result wth re-talzato for γ = ; (d) Result wth L et al. s ethod for γ = ; (e) Result wth ALM for γ = ; (f) Result wth SALPM for γ = Next, we show the segetato results o a atural Europe ght-lghts age Fg. (a). We ca see fro Fgs. (b) ad (c) that the segetato results vsually have dfferet topologes for the two-phase Cha-Vese odel wth o or wth re-talzato process usg the sae paraeters, whch deostrates that whether or ot the re-talzato s doe affects segetato accuracy. The the segetato results by L et al. s ethod, ALM ad our SALPM are gve Fg. (d)-(f). Fro the segetato results, we see that all the three ethods wor for ths age ad get the satsfactory results. However, whe better ad detaled segetato results are eeded, our proposed ethod deed perfors better Fg. (f). TABLE I COMPARISON OF ITERATIONS AND COMPUTATION TIME USING DIFFERENT SEGMENTATION METHODS Methods Iteratos CPU te (s) Fg. Fg. Fg. 3 Fg. 5 Fg. Fg. Fg. 3 Fg. 5 Re-talzato ethod [0] L et al. s ethod [] ALM [4] SALPM See Table I for the correspodg teratos ad coputato te for segetato of ths exaple. For the re-talzato ethod, a lot of te s spet o retalzato. L et al. s ethod s faster tha the retalzato ethod, although t requres a cosderable uber of teratos. We ca see that the ALM ad our ethod are uch faster tha both of the re-talzato ethod ad L et al. s ethod. Moreover, our ethod s uch faster tha the ALM due to sple coputato of Laplaca, geeralzed soft thresholdg forula ad proecto. 03 ACADEMY PUBLISHER

6 30 03 ACADEMY PUBLISHER B. Coparso ad Aalyss of ultphase Experets To provde soe ore sghts, we copare our SALPM wth the L et al. s ethod ad ALM o ultphase age segetato. Fg. 3(a) s a sythetc age wth three regos. Fg. 3(b) s the degraded age wth Gaussa ose. We observe fro Fgs. 3(c)-3(e) that three algorths wor for ths osy age ad get the desrable results, but our ethod gves better segetato results tha the other two ethods. I Fg. 4, we preset quattatve coparsos aog the three ethods by gvg the plots of the error rato (deoted as ER []) vs. the terato uber. Here, the results are cosstet wth the cocluso Fg. 3. (a) (b) (c) (d) (e) Fgure 3. Segetato of sythetc age wth three phases. (a) Orgal sythetc age of sze 56 8; (b) Degraded age ad the sae tal cotour for all the ethods; (c) Result wth L et al. s ethod for γ = ; (d) Result wth ALM for γ = ; (e) Result wth SALPM for γ = L's ethod SALPM ALM ER(%) 迭代次数 Fgure 4. Evoluto of error rato for segetato of the sythetc age usg dfferet ethods. 03 ACADEMY PUBLISHER

7 03 ACADEMY PUBLISHER 3 I Fg. 5, we preset the experetal results for fourphase atural age segetato of the L et al. s ethod, ALM ad SALPM. Aga, we observe fro Fgs. 4(b)-(d) that our algorth exhbts a better perforace tha the other two the aspect of segetato accuracy. Meawhle, we coclude fro Table I that the SALPM coverges faster ad cosues less te tha ay of the other ethods. (a) (b) (c) (d) Fgure 5. Segetato of real age wth four phases. (a) Test age of sze 39 5; (b) Result wth L et al. s ethod for (c) Result wth ALM for γ = ; (d) Result wth SALPM for γ = V. CONCLUSIONS I ths paper, we have troduced a ew varatoal level set forulato that copletely elates the eed of the re-talzato ad overcoes the speed ltato. By troducg soe auxlary varables, we desg ts fast splt augeted Lagraga proecto ethod, whch does ot volve dfferece of curvatures, ad ca preserve SDFs autoatcally va a sple proecto. I addto, eve f the tal LSF s ot a SDF, t ca be corrected autoatcally. The dea of ths paper ca be easly exteded to other odels uder the varatoal level set fraewor, such as oto segetato, 3D recostructo, ad geoetrc surface processg etc. ACKNOWLEDGMENT Ths wor was supported by Natoal Natural Scece Fuds of Cha (No ). REFERENCES [] S. Osher, ad J. A. Setha, Frots propagatg wth curvature-depedet speed: Algorths based o Halto-Jacob forulato, Joural of Coputatoal Physcs, vol. 79, o., pp. -49, 988. [] T. Adersso, G. Läthé, R. Lez, ad M. Borga, Modfed gradet search for level set based age γ = ; segetato, IEEE Trasactos o Iage Processg, vol., o., pp , 03. [3] H. Zhag, Z. Dua, Z. Zhu, ad Y. Wag, Fast ovg obect segetato based o actve cotours, Joural of Coputers, vol. 7, o. 4, pp , 0. [4] J. Le, M. Lysaer, ad X.-C. Ta, A bary level set odel ad soe applcatos to Muford-Shah age segetato, IEEE Trasactos o Iage Processg, vol. 5, o. 5, pp. 7-8, 006. [5] T. F. Cha, S. Esedoglu, ad M. Nolova, Algorths for fdg global zers of deosg ad segetato odels, SIAM Joural o Appled Matheatcs, vol. 66, o. 5, pp , 006. [6] X. Bresso, S. Esedoglu, P. Vadergheyst, J.-P. Thra, ad S. Osher, Fast global zato of the actve cotour/sae odel, Joural of Matheatcal Iagg ad Vso, vol. 8, o., pp. 5-67, 007. [7] T. Goldste, X. Bresso, ad S. Osher, Geoetrc applcatos of the Splt Brega ethod: Segetato ad surface recostructo, Joural of Scetfc Coputg, vol. 45, o., pp. 7-93, 00. [8] C. Lu, Y. Zheg, Z. Pa, J. Dua, ad G. Wag, SAR age segetato based o fuzzy rego copetto ethod ad Gaa odel, Joural of Software, vol. 8, o., 8-35, 03. [9] V. Estellers, D. Zosso, R. La, S. Osher, J.-P. Thra, ad Xaver Bresso, Effcet algorth for level set ethod preservg dstace fucto, IEEE Trasactos o Iage Processg, vol., o., pp , 0. [0] M. Sussa, P. Serea, ad S. Osher, A level set approach for coputg solutos to copressble twophase flow, Joural of Coputatoal Physcs, vol. 4, pp , ACADEMY PUBLISHER

8 3 03 ACADEMY PUBLISHER [] J. Goes, ad O. Faugeras, Recoclg dstace fuctos ad level sets, J. Vs. Cou. Iage Represet., vol., o., pp. 09-3, 000. [] C. L, C. Xu, C. Gu, ad M. D. Fox, Level set evoluto wthout re-talzato: A ew varatoal forulato, Proceedgs of IEEE Coputer Socety Coferece o Coputer Vso ad Patter Recogto (CVPR), vol., pp , 005. [3] C. L, C. Xu, C. Gu, ad M. D. Fox, Dstace regularzed level set evoluto ad ts applcato to age segetato, IEEE Trasactos o Iage Processg, vol. 9, o., pp , 00. [4] C. Lu, F. Dog, S. Zhu, D. Kog, ad K. Lu, New varatoal forulatos for level set evoluto wthout retalzato wth applcatos to age segetato, Joural of Matheatcal Iagg ad Vso, vol. 4, pp , 0. [5] K. Zhag, L. Zhag, H. Sog, ad D. Zhag, Retalzato-free level set evoluto va reacto dffuso, IEEE Trasactos o Iage Processg, vol., o., pp. 58-7, 03. [6] R. Courat, K. Fredrchs, ad H. Lewy, O the partal dfferece equatos of atheatcal physcs, IBM J., vol., o., pp. 5-34, 967. [7] T. F. Cha, ad L. A. Vese, Actve cotours wthout edges, IEEE Trasactos o Iage Processg, vol. 0, o., pp , 00. [8] L. A. Vese, ad T. F. Cha, A ultphase level set fraewor for age segetato usg the Muford ad Shah odel, Iteratoal Joural of Coputer Vso, vol. 50, o. 3, pp. 7-93, 00. [9] H.-K. Zhao, T. F. Cha, B. Merra, ad S. Osher, A varatoal level set approach to ultphase oto, Joural of Coputatoal Physcs, vol. 7, o., pp , 996. [0] Q. Wag, Z. Pa, ad W. We, Splt-Brega ethod ad dual ethod for ultphase age segetato, Joural of Coputer-Aded Desg & Coputer Graphcs, vol., o. 9, pp , 00. [] S. Ha, W. Tao, ad X. Wu, Texture segetato usg depedet-scale copoet-wse Reaacovarace Gaussa xture odel KL easure based ult-scale olear structure tesor space, Patter Recogto, vol. 44, o. 3, pp , 0. Culag Lu was bor 977. He receved Ph.D. degree the college of forato scece & egeerg, Shadog Uversty of Scece & Techology, Qgdao, Cha, 0. He s curretly a lecturer the College of Iforato Egeerg at Qgdao Uversty. Hs research terests clude age segetato, the varatoal ethods ad PDEs age processg. Zheua Pa was bor 966. He receved hs Ph.D. degree the egeerg echacs fro Shagha Jao Tog Uversty, Cha 99. Curretly he s a professor of the College of Iforato Egeerg at Qgdao Uversty. Hs research terests clude dyacs ad cotrol of ultbody systes, coputer sulato ad varatoal age processg. He has publshed uerous papers ad coferece papers the area of age processg ad obect recogto. Jg Dua was bor 989. Curretly he s a postgraduate studet of the College of Iforato Egeerg at Qgdao Uversty. Hs research terests clude varatoal age processg, ad three-desoal recostructo. 03 ACADEMY PUBLISHER