Boundary Detection Using Open Spline Curve Based on Mumford-Shah Model

Transcription

1 Vol. 35, o. 2 ACTA AUTOMATICA SIICA February, 2009 Boundary Detecton Usng Open Splne Curve Based on Mumford-Shah Model LI ao-mao 1, 2 ZHU Ln-Ln 1, 2 TAG Yan-Dong 1 Abstract Inspred by Cremers s work, ths paper proposes a novel method for detectng open boundares, such as coastlne and skylne n an mage. Ths method s based on B-splne functon, curve evoluton, and the cartoon model of Mumford-Shah functonal (M-S model). Because the object to be detected s an open curve n the mage doman, two constrant equatons are ntroduced nto the M-S model. Thus, the problem of open boundary detecton becomes a mnmal partton problem. Wth the partal dfferental equatons (PDEs) of control ponts and constrant equatons, the curve wll stop on the desred boundary. The method can be used to detect automatcally a curve that separates an mage nto two dstnct regons and s not necessarly defned by gradent, even f the mage s very nosy. In adon, wth two open curves, our model can be extended to detect belt-lke objects, such as rvers and roads. Key words Open curve, dffuson snake, splne curve, boundary detecton, M-S model Automatc detecton of an open boundary, s of fundamental mportance n processng such mages as coastlne for cartography and shp postonng [1 4] and detectng skylnes for navgaton and automatc target recognton [5 6]. umerous approaches to ths partton problem have been proposed, ncludng contour detecton, edge detecton, texture feature analyss, etc. [1 6]. Then, the open boundary to be detected must be dentfed among the detected edges, whch usually nclude many pseudo-edges and much more other rrelevant edges. Moreover, due to the noses and complex texture, the edge usually cannot be detected as a contnuous one. Thus, post-processng, such as mathematcal morphology [3] or curve approxmaton [5], s also needed for the connectvty of edges to form a whole open boundary or a contnuous curve. Reference [7] proposed a method for lnear features detecton based on the snake model [8] usng closed parametrc curve. However, due to ts shortcomngs, the snake model s senstve to nose and the ntal curve poston. Snce Osher and Sethan proposed the level sets [9] and varatonal framework for mage segmentaton, numerous approaches known as actve contour models have been proposed for mage segmentaton usng level set theory [10 11]. One of the most famous approaches s the C-V (Chan-Vese) model [11] whch s based on M-S segmentaton technques [12] and the level set method. For C-V model, curve evoluton s based on mage regon nformaton and level set method. The segmentaton model s robust to nose and the ntal curve poston. In almost every actve contour model, contours are supposed to be closed curves, and wth level set method they are represented mplctly as the zero level set of a Lpschtz functon defned n a hgher dmenson. In some applcatons, our nterest les solely n open boundary (edge) detecton [13 14], such as the detecton of the coastlne and the edge n streak-camera mages. In these cases, f we drectly apply a general actve contour model for open boundary detecton, some other objects, such as solated sland, etc., are also detected and t s necessary to recognze whch segment of the contour s the desred open boundary. Recently, Leu and Mn [13 14] almost smultaneously proposed Receved ovember 23, 2007; n revsed form February 24, 2008 Supported by atonal atural Scence Foundaton of Chna ( , ) and the Open Foundaton of State Key Laboratory of Robotcs (07A ) 1. State Key Laboratory of Robotcs, Shenyang Insttute of Automaton, Chnese Academy of Scences, Shenyang , P. R. Chna 2. Graduate Unversty of Chnese Academy of Scences, Bejng , P. R. Chna DOI: /SP.J the same varatonal method for the open boundary detecton based on C-V model. Ther method can automatcally detect a desred open curve, whch a regon boundary when t s a curve wth an explct functon y = f(x), whle t wll fal when the open boundary curve s a general curve and cannot be descrbed wth an explct functon y = f(x). The boundary curve wth explct functon and a general boundary curve are shown n Fg. 1. Fg. 1 (a) A boundary curve wth explct functon (b) A general boundary curve Illustraton of a boundary curve wth explct functon and a general boundary curve Based on M-S and varatonal partal dfferental equaton (PDE) models [15] and nspred by dffuson snake [16], we propose a novel method to automatcally detect open boundares. Unlke the method proposed n [13 14] that can only detect open boundary curves wth explct functon, n our method the evolvng open curve s represented by B-splne curve [17], so that a general boundary curve can be effectvely detected. Although t can be regarded as a specal case of dffuson snake, there are many dfferences between them. Frst, the evolvng contour s an open one n our method, whle t s a closed one n the dffuson snake. Second, dffuson snake uses much more control ponts than our method. Moreover, n many cases, dffuson snake stll needs a post-processng to recognze whch segment of the detected contour s the demanded boundary after contour detecton. Thrd, wth two open curves, our method can be extended to partton the gven mage nto three parts, whch can be used for belt-lke regons, such as rver and road. 1 Prevous work Let Ω be a bounded open subset of R 2, and u 0 : Ω R be a gven mage whose wh and heght are w + 1 and h + 1, respectvely. In order to automatcally detect an

2 o. 2 LI ao-mao et al.: Boundary Detecton Usng Open Splne Curve Based on Mumford-Shah Model 133 open boundary wth an explct functon n the gven mage u 0, the man dea n [13 14] s modelng t va an explct functon f : [0, w] [0, h]. Compared wth the method of modelng the detected boundary by the zero level set of a Lpschtz functon φ, the evolvng curve y = f(x) can be consdered as the zero level set of the functon φ = y f(x). In ths settng, the curve y = f(x) parttons the gven mage nto two dstnct regons. The formulaton of the energy functon, called 1-D modfed C-V model, s expressed as E(c 1, c 2, f) = λ 1 (u 0(x, y) c 1) 2 H(y f(x)) dxdy + Ω λ 2 (u 0(x, y) c 2) 2 (1 H(y f(x))) dxdy + ν C (1) Ω where H s the Heavsade functon. Analogous to the C-V model, the two phases of the segmentaton are regons above and below the curve y = f(x) represented by H(y f(x)) and 1 H(y f(x)) n [13 14] nstead of the nsde and outsde regons of a closed evolvng contour C represented by H(φ) and 1 H(φ) n C-V model [11]. In whatever cases, the essentalty of pecewse constant mage segmentaton s to frstly establsh the normalzed orthogonal bass functons characterzng phases of segmentaton, such as H(y f(x)) and 1 H(y f(x)) n 1-D modfed C-V model and H(φ) and 1 H(φ) n C-V model, and then to approxmate the orgnal mage u 0 optmally n a functonal framework. The approxmatng mage u can be wrtten as u = c ϕ (x, y) (2) =1 where ϕ s the characterzng functon for the -th phase and c s the correspondng average ntensty of the orgnal mage. The evoluton equaton deduced by varatonal method for (1) s f(t, x) t = λ 1(u 0(x, f(t, x)) c 1) 2 λ 2(u 0(x, f(t, x)) c 2) 2 + ( ) f(t, x) ν x x ( ) 2 (3) f(t, x) 1 + x 2 Model descrpton 2.1 Open curve representaton strategy Our method uses open, unform, quadratc, and perodc B-splne curve for open curves approxmaton (for more detals, please refer to [18]). Let P 1, P, and P +1 be three consecutve control ponts. The -th quadratc B-splne segment C can be defned by C (s) = P 1 0,3(s) + P 1,3(s) + P +1 2,3(s) (4) where 0,3(s), 1,3(s), and 2,3(s) are blendng functons defned n [19], and 0,3(s) = 1 2 (1 s)2 (5) 1,3(s) = s 2 + s (6) 2,3(s) = 1 2 s2 (7) where s [0, 1]. Gven a set of control ponts P, 1, an open curve C md, composed of all quadratc B-splne segments, can be defned as C md (s) = 1 =2 C (s) = P B (s) (8) where B (s) are the unform, quadratc, and perodc B- splne bass functons. As s obvous n Fg. 2 (a), the open splne curve defned above, together wth mage boundares, cannot drectly form closed regons that are necessary for M-S model. In order to form closed regons, we add another two control ponts P 0, P +1 to the open curve to ensure that the open boundary pases through two ponts (0, y 1) and (w, y ), where P = (x, y ). In adon, let C 1(0) = (0, y 1) and C (1) = (w, y ). Then, the whole open curve C can be defned as =1 C(s) = C 1(s) + C md (s) + C (s) (9) The closed regons are establshed for regon-based open boundary detecton (see Fg. 2 (b)). (a) Open boundary wthout boundary conons Fg. 2 (b) Open boundary wth boundary conons Illustraton of open boundares wth and wthout boundary conons 2.2 Boundary detecton model In 1989, Mumford and Shah proposed a varatonal approach to mage segmentaton known as M-S model. When the orgnal mage u 0 s approxmated by a pecewse constant functon on the mage plane, the M-S model s smplfed as E MS (c, C) = 1 1 u 0 c 2 dxdy + ν C s 2 ds 2 Ω 0 Ω = Ω C, C s = dc ds (10) By the open curve representaton strategy proposed n Subsecton 2.1, we ncorporate the evolvng contour and mage boundares nto a smplfed dffuson snake [16] for the open boundary detecton modeled as E MS (c, C) = 1 λ u 0 c 2 Ω 2 dxdy + ν 1 0 C 2 s ds (11) s. t. P 0 = 2C 1(0) P 1 P +1 = 2C (1) P (12) where λ s the weghtng parameter for every phase Ω. We use the constrant equaton (10) nstead of the closed curve conon,.e., C(0) = C(1), to form closed regons n M-S model for open boundary detecton. Mnmzng the energy functonal n (9) wth respect to the contour C md, we have the followng evoluton equaton for C md :

3 134 ACTA AUTOMATICA SIICA C md C C md )ss (s, t) = [λ1 e+ (s, t) λ2 e (s, t)] n (s, t) + ν(c t (13) where n s the unt nner normal vector on the contour wth respect to the regon above the contour C. The two terms e+ and e denote the energy densty[16] of regons above and below the contour C, respectvely, defned as e+ = (u0 u+ )2, e = (u0 u )2 (14) where u s gven by (2). Evolvng equatons for the mddle control ponts P, 1 are obtaned by nsertng (6) n (11). P dp B (s) = [λ1 e+ (s, t) λ2 e (s, t)] n(s, t) + =1 ν =1 P (t) d2 B (s) ds2 (15) Dscretzng (13) wth a set of nodes s along the contour C md, a set of lnear dfferental equatons for these evolvng control ponts P, 1 are deduced. At last, the evoluton equatons for the coordnates of each mddle control pont (x, y ) are deduced. Vol. 35 trx B s obtaned by evaluatng the splne bass functons at these nodes: Bj = B (sj ), where s corresponds to the maxmum of B. We can see that f there only exsts the second term, t enforces an equdstant spacng of control ponts. (10) are the Evoluton equatons for the end control ponts,.e., P 0 and P +1. The prncpal steps of our algorthm are as follows: ntalzng an open curve that satsfes boundary conons; startng teraton. For each teraton, we frst evolve these mddle control ponts P, 1 va (14) and (15), and then obtan the frst and last control ponts wth (10). The teraton step s terated untl the soluton s statonary. The statonary soluton s the desred boundary. We also extend our method belt-lke regons usng two open boundary curves that partton the nput mage nto three phases and gve a correspondng expermental result. The evolvng equatons for each curve are smlar to (14) and (15). 3 Expermental results where 1, nx and ny are x and y components of unt normal drecton n. The cyclc trdagonal ma- In ths secton, we present some expermental results on synthetc and real mages, as well as some comparsons wth some relatve methods, to valdate our method. In these numercal experments, the ntensty of nput mage s [0 255] and the weghtng parameter ν s chosen as Frst, we mplement our method to automatcally detect a coastlne n a nosy mage wthout denosng (see Fg. 3). As shown n Fg. 4, we do experments on a synthetc mage to show the advantage of our method over the one proposed n [13 14]. From ths example, we can see that the method proposed n [13 14] fals to detect the desred open boundary when t s a general curve that cannot be descrbed wth explct functon, whle our method can do t. (a) Intal curve (c) Fnal result dx (t) = (B 1 )j [(λ1 e+ (sj, t) λ2 e (sj, t)) j=1 nx (sj, t) + ν(xj 1 2xj + xj+1 )] (16) dy (t) (B 1 )j [(λ1 e+ (sj, t) λ2 e (sj, t)) = j=1 ny (sj, t) + ν(yj 1 2yj + yj+1 )] Fg. 3 (a) Intal curve Fg. 4 (17) (b) Mddle curve A coastlne detecton experment usng our method (b) Our result (c) 1-D C-V result A comparson between our method and 1-D modfed C-V model

4 o. 2 LI ao-mao et al.: Boundary Detecton Usng Open Splne Curve Based on Mumford-Shah Model 135 In Fg. 5, we gve an expermental result on skylne detecton to show the dfferences between our method and dffuson snake. From the detecton results n Fg. 5, we can learn that the dffuson snake method requres a post-processng to determne whch segment of the detected closed curve s the desred open boundary, whle our method has no such need. In adon, the dffuson snake needs more control ponts than our method for curve evoluton. As we have mentoned above, our method can also be extended to automatcally detect belt-lke regons, such as rvers and roads, by usng two open curves to partton the nput mage nto three dstnct regons. We also gve an expermental result n Fg. 6. All results n ths paper are obtaned under the envronment of Matlab 7.0. The CPU tme (n seconds) and teratons of our method for all experments are lsted n Table 1. (a) Intal curves (b) Mddle curves (c) Fnal result Fg. 6 An experment on a rver detecton (The rver s treated as a belt-lke regon usng our method.) Table 1 Segmentaton tmes of our method Image Sze (pxvals) Iteratons Tme (s) Fg Fg Fg Fg (a) Orgnal mage Fg. 5 (d) Result of our method (b) Intal curve of our method (c) Intal curve of dffuson snake (e) Result of dffuson snake (f) Comparson of control ponts between our method and dffuson snake Comparson between our method and dffuson snake 4 Concluson and dscusson We proposed a method for automatcally detectng general open boundares such as coastlne or skylne, represented by B-splne curve based on M-S model. Then, we extended the model to deal wth automatc belt-lke regons through usng two open curves to partton the gven mage nto three dstnct regons, such as rvers and roads. The essentalty of our method s that the mddle control ponts are evolvng va M-S model, whle the end ponts are evolvng va reasonable boundary conons n (10). In our method, the evolvng contour s the only open boundary wth less control ponts than dffuson snake method, whch, besdes the ntal open boundary, contans the mage boundares formng the closed curve. Wth our method, the open boundary curve and ts B-splne expresson can be drectly obtaned wthout any preprocessng and postprocessng, even f the mage s nosy. Although the method n ths paper s more robust to nose and the ntal curve poston than the classcal snake method, t cannot accurately detect boundares when they vary very sharply [13]. Acknowledgements We thank Prof. Dr. Cremers for dscussng some ssues about dffuson snake and SU Jng for her assstance. References 1 Tello M, Lopez-Martnez C, Mallorqu J, Bonastre R. Automatc detecton of spots and extracton of fronters n SAR mages by means of the wavelet transform: applcaton to shp and coastlne detecton. In: Proceedngs of IEEE Internatonal Conference on Geoscence and Remote Sensng Symposum. Denver, USA: IEEE, ao L-Png, Cao Ju, Gao ao-yng. Detecton for shp targets n complcated background of sea and land. Opto- Electronc Engneerng, 2007, 34(6): 6 10 (n Chnese) 3 Ln Jn-Teng. The Study on Water Lne Extracton from Tme Seres of SAR Image [Master dssertaton], atonal Central Unversty, Chna, 2006 (n Chnese) 4 Kahraman M, Gumustekn S. Detectng coastlnes from aeral mages. In: Proceedngs of the 14th Sgnal Processng and Communcatons Applcatons. Antalya, Turkey: IEEE, Le W, Ln T C I, Ln T C, Hung K S. A robust dynamc programmng algorthm to extract skylne n mages for navgaton. Pattern Recognton Letters, 2005, 26(2):

5 136 ACTA AUTOMATICA SIICA Vol Zhang Chun-Hua, Zhou ao-dong, Lu Song-Tao. A small target auto-orentaton method of nfrared mage wth seasky background. Laser and Infrared, 2007, 37(1): (n Chnese) 7 Rocca M R D, Fan M, Fortunato A, Pstllo P. Actve contour model to detect lnear features n satellte mages. Internatonal Archves of Photogrammetry, Remote Sensng and Spatal Informaton Scences, 2004, 35(3): Kass M, Wtkn A, Terzopoulos D. Snakes: actve contour models. Internatonal Journal of Computer Vson, 1988, 1(4): Osher S, Sethan J A. Fronts propagatng wth curvature dependent speed: algorthms based on Hamlton-Jacob formulatons. Journal of Computatonal Physcs, 1988, 79(1): Caselles V, Kmmel R, Sapro G. Geodesc actve contours. Internatonal Journal of Computer Vson, 1997, 22(1): Chan T F, Vese L A. Actve contours wthout edges. IEEE Transactons on Image Processng, 2001, 10(2): Mumford D, Shah J. Optmal approxmatons by pecewse smooth functons and assocated varatonal problems. Communcatons on Pure and Appled Mathematcs, 1989, 42(5): Leu L H. Contrbuton to Problems n Image Restoraton, Decomposton, and Segmentaton by Varatonal Methods and Partal Dfferental Equatons [Ph. D. dssertaton], Unversty of Calforna, Los Angeles, USA, L Mn. Actve Contour Models for Image Segmentaton and Target Trackng [Ph. D. dssertaton], Shenyang Insttute of Automaton, Chnese Academy of Scences, Chna, 2006 (n Chnese) 15 Aubert G, Kornprobst P. Mathematcal Problems n Image Processng: Partal Dfferental Equatons and the Calculus of Varatons. ew York: Sprnger-Verlag, Cremers D, Tschhäuser F, Weckert J, Schnörr C. Dffuson snakes: ntroducng statstcal shape knowledge nto the Mumford-Shah functonal. Internatonal Journal of Computer Vson, 2002, 50(3): Blake A, Isard M. Actve Contours. Berln: Sprnger-Verlag, Farn G. Curves and Surfaces for Computer-Aded Geometrc Desgn. San Dego: Academc Press, Yamaguch F. Curves and Surfaces n Computer Aded Geometrc Desgn. ew York: Sprnger-Verlag, LI ao-mao Ph. D. canddate at the Shenyang Insttute of Automaton, Chnese Academy of Scences. He receved hs bachelor degree from Shenyang Insttute of Technology n Hs research nterest covers mage segmentaton and denosng based on varatonal methods. E-mal: lxaomaosa@gmal.com ZHU Ln-Ln Ph. D. canddate at Shenyang Insttute of Automaton, Chnese Academy of Scences. She receved her bachelor and master degrees from the Department of Automaton, ortheast Unversty n 2004 and 2007, respectvely. Her research nterest covers pattern recognton, level set methods, and trackng object n sequence frames. E-mal: zhulnln@sa.cn TAG Yan-Dong He receved hs bachelor and master degrees from the Department of Mathematcs, Shandong Unversty n 1984 and 1987, respectvely. In 2002, he receved hs Ph. D. degree n appled mathematcs from Unversty of Bremen, Germany. Hs research nterest covers numercal computaton, mage processng, and computer vson. Correspondng author of ths paper. E-mal: ytang@sa.cn