. lnlern.tlanal Jouro.l of Computer Vi,ion. 321-331 (1988) @ 1987 Kluwer Academic Publishers. Boston. Manuf.crured in The Netherlands

Snakes: Active Contour Models

MICHAEL KASS, ANDREW WITKIN, and DEMETRI TERZOPOULOS Schlumberger Palo Alto Researrh, 3340 HilIvi!!:w Ave.. Pa/u AIro, CA 94304

Abstraet

A snake is an energy-minimizing spline guided by external constraint forces and infIuenced by. ¡mage forces tha! puIl it toward features such as lines /lnd edges, Snakes are active.contour models: they lock onto nearby edges, localizing them aecurately, Sea le-s pace continuation can be used to enlarge the cap­ ture region surrounding a feature. Snakes provide a unified account of a number ofvisual problems, in­ c1uding rletection of edges, linea, ami subjeetive contours; motion traeIcing; and stereo matching. We have used snakes successfully for interactive interpretation, in which user-imposed constraint forces guide the snake near features of interest.

1 Introduction organizations_ By adding suitable energy lerms lO the rninimization, it is possible for a user to push In recent computational VlSlon research, low­ the mode! out of a local minimum toward the level tasks such as edge or line' detection, stereo de.ired solution. Thc rcsult is aI! active model matching, aud motion tracking have been widely tha! falls into the desired solution when placed regarded as autonomous bottom-up processes. near it Marr and Nishihara [11), in a strong statement of Energy minimizing rnudels have a nch histo!)' thi, view, say Iha! up to the 2.5D sketch, no in vision golng back at least to Sperling's stereo "higher-IeveJ" information is yet brought to bear: model [16}. Such models have typically been the computations proceed by utilizing only what regarded as autunumous, but we have developed is available in the ¡mage itself. This rigidly se­ interactive techniques for guiding them. Interact­ quential approaeh propaga tes mistakes made at jng wilh such models allows us to explore the en­ a low level without opportunity for correction_ It ergy lautlscapeve!)' easily and develop effective Iherefore imposes stringent demands on the reH­ energy functions that have few local rninima lmd ability of low-Ievel Il1echanisms. As a weaker but little dependence on starting points. We hope more attainable goal for low-Ievel processing, we thereby to make the job of high-Ievel interpreta­ argue that it ought to provide sets of alternative tion manageable yet not' constrained un­ . organizations among which higher-Ievel pro­ necessarily by irreversible low-level decisions_ cesses may choose. rather Ihan shackling them The problem domajn we address is tha! of prematurely wilh a unique answer. findíng salient image contours-edges, lines, and In this paper we investigate the use of energy subjeclive contours-as well as tracking those minimization as a framework with!n which to contours during motion and matching them in realize this goal. We seek to design energy fune­ stereopsis. Our variational approach lO finding tions whose local minima comprise the set of image contours differs from the traditional ap­ alternative solutions available to high~['-Ievel proach of detecting edges and then !inIcing them. processes. The choice among these alternatives In our model, issues such as !he connectivity of could require sorne type of search or high-Ievel Ihe contours anrl the presencc of cOUlt:rs atIecl reasoning_ Tn the absencc of a weU-developed the energy functional and hence the detailed high-Ievel mechanism, however, we use lin in­ structu re of the locally optimal con tour. These teractive approach 10 explore the alternative , issues can. in principIe, be resolved by ve!)' high- 322 Kass, Witkin, and Tel'zopoulos

(a) (b)

(e) (d)

Fig. l. Lowcr-lc:ft: Orjginal wuutl photograph from Brodatz . . Others: Three difforent local minima for Ihe acrive cantanr modo!.

level computations. Perhaps more importantly, Without detailed knowledge about the object high-Icvclmechanisllls can interact with the con­ in view, it is difficult to justify a choice among the topr model by pushing ir toward an appropriate three interpretations. Knowing that wood is a local minimum. Oprimization and relaxation layered structure, or perhaps inferring its laye red have been used previously in edge and line detec­ structure from elsewhere in the picture could tion, (3,5,13,24,25], but without theinreractive help to rule out interpretation (b). Beyond Ihat, guiding used here. . the 'corree!' interpretation conld be very task de­ In many image interpretation tasks. the correet penden!. In many dornains, su eh as analyzing interpretation of low-Ievel events can require seismíc data, ¡he choice of interpretation can de­ high-Ievel knowledge. Considero for example. the pend on expert knowledge. Different seismic in­ ¡hree perceptual organizations of two dark lines terpreters can derive significantly different per­ in figure 1. The ¡hree ditTeren! organizations cor­ ceptual organízations [rom the same seismic respond to three different local minima in our sections depending on thcir knowledge amI line-contour mode!. It is important to notice thar training. Because a single 'corree!' interpretation the shapes of the lines are materially different cannot always be defined, we suggest low-Ievel in the three examples. no! just becau~e of ~ dif­ mechanisms which seek apIJrupriarelocal min­ ferent linking of line segments. The segments ima instead of searching for global mínima. themselves are changed by the perceptual Unlike most other techniques for finding organization. . salient contours, QUl" mudel is active. It is always Snakes 323 minimizing its cnergy functiollal and therefore toward salient image features Iike Hnes, edges, exhibits dynamic behavior. Because of the way and subjective contours. The externa! constraint the contours slither while minimizing their en­ forces are responsible for putting the snake near crgy, we caU them snakes. Changes in high-Ievel the desired local minimum. These forces can, for interpretation can exert forces on a snake as it example, come from a user interface, automatic continues its minimization. Even in the absence attentional mechanisms. or high-lev~l ¡n.terpre­ oC suc:h forces, snakes exhibit hysteresis when ex­ tations. posed to moving stimuli. Representing the position of a snake paramet­ Snakes do nOI try to solve the entire problem of rically by ves) = (x(s),y(.<)), we c.an write its energy finding salient image contours. They rely on functional as olher mechanisms to place them somewhere near the desired contour. However, even in cases E~"" = f E,.,,,(v(s)) ds where no satisfactory automatic starting mech­ anism exisls, snakes can stilI be used for semi­ E¡n,(v(s)) Eim ••.(v(s)) automatic image interpretation. If an expert tI

siderations are relatively importan!. In a fullyex­ only 10 push a snake near the feature. Once clase plicit Euler method, it takes O(I/~) iteralions each enough, the energy minimization will pull Ihe of 0(11) tímc for an impulse IU lravel down lhe snake in the rest ofthe way. Accurale tracking of length of a snake. The resulting snakes are Ilac­ contour features can be specified in this way with cid. In order lO erecl more rigid snakes. it is vital tittle more effort than poinling. The snake energy to use a more staLJle melhod lhal can accom­ minimization provides a 'power assis!' for modale lhe large internal forces. Our semi­ image interpretation. implicil melhod allows forces to lrave! the entire Our interface allows the user to connect a length oC a snake in a single 0(11) ¡teration. spring to any poi nt on a snake. The olher end of the spnng can be anchored at a fixed position, connected to another point on a snake, or 2.2 Sl1ake Pit dragged around using Ihe mouse. Creating a spring between . XI and X2 sirnply adds 1n order lo experiment with di fferent energy -k(x, - X,)2 lo Ihe external constraint encrgy

functions for low-level visual tasks. we have E.:on o . developed a user-interface for snakes on a Sym­ In addition lo springs, the user interface pro­ holies Lisp Machine. The interface allows a user vicies a 1/; repulsion force controllable by the to select starting poinls and exert forces on mouse. The Ilr energy functional i5 clipped near snakes interactively as they minimize their en­ I "" O to prevent numencal instability. so the ergy. In addition to its value as a researeh 1001. resulting potential is depicted by a vol¡;ano icono the user-interface has proven very useful for The volcano is very use fui for pushing a snake semiautomatic image interpretation. In order to out of one local mínimum and ¡nto another. specify a particular image featul'f', Ihe user has Figure 2 shows the snake-pit interface beíng

N/f. 2. The Snake Pit user-interface. Snakes are shoWII in black, springs and Ihe volcano are in white. Snakes 325

used. The two dark lines are different snakes E ¡mage ¡.::;:;; W lineE line + W edge!:.' edge + W (crmE ICfm which the user has connected with two springs (3) shown in white. The olher springs attach points on ¡he snakes to fixed positions On the screen. In By adjuslíng the welghls, a wide range of snake the upper right, the volcano can be seen bending behavior can be created. a nearby snake. Each oC the snakes has a sharp comer which has becn spccificd IJ y the user. 3.1 Line Functional

3 Image Forees The simples! use fuI image functional is the image intensity itself. If we set In order to make snakes useful for early vision we E , ~ [(x, y) (4) nccd energy fUllctionals Iha! attract them to un salient [eatures in images. In this section, wepre­ then depending on the sign oC WUn" the snake wilI sent !hree different energy functionals which at­ be attracted either ta light linl's oc dark lines. traet ¡¡ snake to lines, edges, and terminations. Subject to its olher constraints, the snake will try The total image energy can be expressed as to align itself with the lightest or darkest nearby a weighted combinatian af the three energy contour. This energy functional was used with functionals the snakes shown in figure 1. By pushing with the

r/.R.3. Two edee snAk~~ ('In a pe:u.and potCLtO. Uppcr.. lt::fI.: Tbe user has puJled one ofthe sna.kes away from the edge ofthe pear. Other.: Nlerthe user let' go. (he sn.ke .nap' back 10 the edge of lhe pear. 326 Kass, Witkin, and Terzopoulos

volcano, a user can rapidly move a snake from Hildreth Iheory. Adding Ihis energy lerm lo a one ofthese positions to another. The coarse con­ snake means that the snake is attracled lo zero­ trol necessary to no so suggesls Ihal symbolic at­ crossings, bUl slill constrained by ilS own tenlional meehanisms mighl be able to guide a smoothness. Figure 4 shows scale-space con­ snake effectively. tinuation applied to this energy functional. The upper len shows the snake in equilibrium at a very coarse scale. Since ¡he edge-energy function 3.2 Edge Functional is very blurred, lhe snake does a poor job of Jocalizing lhe edge, but is attracted lo this local Finding edges in an image can also be done with minimum from very far away. Slowly reducing a very simple energy funclional. Ir we sel E'd¡¡e '" the blurring leads the snake 10 the po

In figure the snake was attracted lO the pear 3, 3.4 Terminaríon Functional boundary [rom a fairly large distan ce away be­ cause of the spline energy termo This type of con­ In order to find terminations of line segments vergence is rather common for snakes.lfpart of a and comers, we use the curvature nflevellines in snake finds a low-energy image feature, the ' spline term wil! pul! neighboring parts of the snake loward a possible continuation of lhe fea­ ture. This effeclively pl~ces a large energy well around a good local minimum. A similar effecl can be achieved by spalially smoolhing the edge­ or line-energy functional. One can allow the snake lo come to equilibrium on a very blurry en­ ergy functional and lhen slowly reduce lhe bluf­ ringo The result is minimization by scale-con­ linuation [20,21 J. In orderto show the relationship of scale-space continuation to the Marr-Hildreth theory of edge-detection (10], we have experimenled with a slightly different edge funclional. The edge­ energy func:tional is

(5) Fig.4. Upper-left: Edge snake in equilibrimn.l co.",. '.'lIe. Upper-right: Snake in equilibrium at intermediate seale. where G. is a Gaussian of standard deviation G. Lower-Iefi: Final snake equilibrium atier scale-space con­ Minima of Ihis functional lie on zero-crossings tinuation. Zero-crossings overlayed 00 final 2 Lower-right: of G. * V¡ which define edges in the Marr- snakf!! position. Snakes 327 a slight1y smoothed image. Let qx,y) = G.(x,y) * using piecewise circular ares would probably ¡(x, y) be a slightly smoothed version oC the abo produce a very similar interpolation. An ap­ ¡mage. Let e = tan-1 (C/C,) be tbe gradient angle pealing aspect of the snake model is that tbe and let n = (cos 9, sin 9) and OJ. = (-sin 9, cos 9) same snake that finds subjective contours can be unit vectors along and perpendicular to the very effectively find more traditional edges in gradient dlrection. Tben the curvature ofthe level natural imagery. It may, moreover, provide sorne· contours in C(x,y) can be written insight into why the ability to see subjective con­ tours i5 important. (6) A Curther unusual aspeet of the snake model that bears on the psychophysics of subjective contonTS is hystheresis. Since snakes are constnn­ (7) tly minimizing their energy, they can exhibit hys­ teresis when shown moving stimuli. Figure 6 CyyC; - 2C"C,Cy + CxxC~ (8) shows a snake tracking a moving subjective con­ = (C; + C;)3I2 .- tour. As the horizontalline segment on the right By combining Eedll< and E"rm, we can create a moves over, the snake bends more and more until thc interna! spline forees overpuwer tbe snake tlla! 15 altla~tt:t1 lu edges or terminations. Figure 5 shows an example of such a snake ex­ image forces. Tben the snake falls offthe Hne and posed to a standard subjective contour illusion reverts to a smoother shape. Bringing the Hne [71. Tbe shape of the snake contour between the segment close enough to the' snake makes the edges and ¡ines in the illusion is entirely deter­ snake reattach. While it is difficult to show the mined by the spline smobthness termo The varia­ hysteresis in a still picture, the reader can easily lional problem solved bythe snake ls very closely verifY ¡he corresponding hysteresis in human vi­ related to a variational formulation proposed by sion by recreating the moving stimulus. Tbis type Brady et al. [21 for the interpolation of subjective ofhysteresis is uncharaeteristic of purely bottom­ contours. Ullman's [221 proposal of interpolating up processes and global optimizations.

Fig. 5. Righ': Standard subjective contour illusion. Left Edge/tennination sn.k. in equilibrium On the subjeclive c.ootoU¡;, 328 Kass. Wickin. al1d Ter:opoulos

",\1// ,,\1// ~\I/~ "'- /~ ~\I/~ "'- ~ ------l '- - - /"" "'- ~ "'- ~ "'- ~ "'- /1/\" /1/\", /1 r\'" /IT\" ",\11/ ,,\11/ '"\JI/ ",\11/ "'- ~ "'- ~ ~ ./ "'- ./ - - - - >--- - - ~ "'- ~/ ,"'- ./, ~ /1 I \" /1/\' / Ir\ ~/I \;:;- ,,\ 11/ ',\ I 1/ ,,\LI/ ",\1// "'- ~ "'- ~ '-¡. ./ "'- ~ ~ '- - - - - , ' - '- ./ "'- ~ "'- / "'- ~ "'- // I \'" //1 \" //1\" //1 \'" ",\11/ . ",\11/ '"\LI/ ",\l.!/ ~ ~ "'- ~ ~ ~ "'- ~ - - - - -; 1- ~. ~ ~ ~ /"//1\;;:: ./'./1 I \"-- /IT\'" //r\'" . -

Fig. 6, Above ¡eft: Oyo.mie subjective eoolour illusion. Se· line slides lo Ihe righ~ the snake bond. uotil il falls off Ihe quenc. is ¡eft 10 righl, 10p lO bonom. AboVe Righl: Soake at· lineo Bringing Ihe line close enough m.k•• Ihe snak. reatlach, tracted to edges and tenninations. As the movine horizontal

4 Stereo and Motion where r(s) and 0(8) are left and right snake eontours. Since the disparity smoothness constraint is 4.1 Scereo applied along contours, it shares a strong simi­ larity wíth Hildreth's (8) smóothness constraint Snakes can also be applied to the problem of for computing optic flow. This canstraint means stereo rnatching. In stereo, iftwo contours corre­ that during the process oflocalizing a contour in spond, then the disparity should vary slowly one eye, information about the corresponding along the contour unless the conlour rapidly con tour in the o/her eye is used. 1n stereo snakes, recedes in depth. Psychophysical evidence [4) of the stereo match actually affects the detection a disparity gradient tirnit in human stereopsis in­ and localization of the features on which the dicates that Ihe human visual systcm al least lo mMch is based. This diffcrs imponantIy, for ex' sorne degree assumes that disparities do not ample, from lhe Marr-Poggio stereo theory (12) change too rapidly wíth space. This constraint in which the basic stereo matching primitive can be expressed in an additional euergy fune­ zero-crossings 1l1ways Tema in unchanged by the tional ror a stereo snake: matching process. Figure 7 shows an example of a 3D surface (9) reconstructed from di~parities measured along a Snakes 329

single stereo snake on the outIíne oC a piece oC papero The surface is rendered from a very dif­ f~r~nt viewpoint than the original to emphasize Ihat a 3D model of lhe piece of paper has been computed rather Ihan merely a 2.5D model.

4.2 Motían

Once a snake finds a salient visual feature, it 'Iocks on.' If the feature then begins to move slowly, th~ snake will simply track the same local minimum. Movementthat is too rapid can cause a snake to flip ¡nto a different local mínimum, uut for ordinary speeds and video-rate sampling, Fíg. 7. Boltom: Slereogram of a. benl piece of papero Below: snakes do a goodjob oftracking motion. Figure 8 Surrace reconslruction from lhe outline ofthe paper matched shows eight selected frames out of a two-second using stereo snakes. The suñact" múdel is rendered from a video sequence. cdge-attracted snakes were in­ very different viewpoint lhan lhe original 10 emphasíze that ít itialized by hand on the speaker's Iips in lhe firsl is a fuIl 3D modeL rather lhan a 2.5D mode!. frame. After that, lhe snakes tracked lhe lip movements automatically. Tbe motíon tracking was done in this case wilhout any interframe constraints. Introducine 5uch constraints will doubtless make the tracking

Fig.8. Solocled frames from a 2·second video sequenc. show· the speaker's lips in the first frame. [he snakes automatically ing snakes used for mmion tracking. After being initialized to track t.he!: lip movements ..... ith high 'Ü;-l..:untcy. 330 Kass, ¡nIki/!, and Terzopou/os

more robust. A simple way 10 do so is ro give the Appendix: Numerical Methods snake mass. Then the snake will predict ils nex! position based on its previon. velocity, Let E,,, = E¡mag, + E,oo' When a(s) = a, and I)(s) '" 1) are constants, minímizing lhe energy fune­ lional of equalion (1) gives rise 10 the following 5 Conclusion two independent Euler equations:

¡:¡ vE", - O Snakes have proven useful for interactive specifi­ ox# + J-'XSS.~$ + a.\.- - (10) cation of image contours. We have bcgun to use Ihem as a basis for interactively matching 3D ¡:¡ vE,,, - O (11) models to images. As we develop better energy ay" + ,..y"" + --¡¡y - functionals the 'power nssist' of snakes becomes increasingly effective, Scale-space continuatíon When a(s) and I)(s) are not constant, it is sim­ can greatly enlarge the capture region around pler to go directly lo a discrete formulation ofthe feMores of interest. enclgy functional In equation (2), Then we can The sna ke model provides a uní fied ¡reatmenl write to a colleclion ol' visual problems that have been a treated differently in the pas!. Edges, lines, and E;;'a" = L E,.,U) + E,,,(i) (12) j "" I subjective contours can all be found by essen­ tially the same mechanisms. Tracking these fea­ Approxímating the derivatives with finite dif­ tutes through mulion and matching them in ferences and converting to vector notation with Vi stereo is easily handled ín the same framework, = (Xi,y,) = (x(ih),y(ih», we expand E¡o.(i) Snakes, perhaps, embody Marr's nOlion of E'n,(i) ~ a'¡v¡ - v¡_d 1!2hl 'leas! cOJ1lmitmenf (9] more than his bottom-up l 4 2.5D sketch. The snake provides a number of + 1),IVi - 1 - 2v, + vi +d /2h (13) widelyseparated local mínima 10 furtherlevelsof whcrc we define v(O) = ven). Let fx(i) = iJE ../iJx, prucessing. Instead of committing irrevocably to andf/i) = iJE.,,/Vy¡ where !he derivatives are ap­ a single interpretation,snakes can change their proximated by a finite dilTerence ir they cannot interpretatíon based on additional evidence from be computed analyticalIy. Now the correspond- higher levels of processing. They can, for exam­ . ing Euler equations a-re pIe, adjust monocular edge-finding based on binocular matches. We believe Ihat the ability to have alllevels' of + ~i-l[Vi-2 - 2v¡_J + Vi] visual processing influence the lowest-Ievel vi­ sual interpretations will turn out to be very im­ - 2~{[Vi_J - 2v{ + VI+I) portan!. Local energy-minimizing systems like + ~1+1[Vi - "2Vi+1 + V1+1) snakes olTer an attractive method for doing Ihis. The energy minimizaríon ¡caves a mueh simpler + (fx(i),f,(i)) = O (14) problem for higher level processing. The above Euler equations can be written in ma­ trix form as Acknowledgments Ax + r,(x, y) '" O (15) Ay f,(x. y) =0 Kurt Fleischer helped greatly with me snake pit + (16) user-interface and created the all-important vol­ where A is a pentadiagonal banded matrix. cano icono John Platt helped us develop snake To solve equ!ltions (15) Qnd (16), we set the theory and guided us llroulId ¡he Infiníte abyss oC right-hand sides of the equations equal to lhe numerical methods. Snake-Iíps Atkinson pro­ product of a step size and the negative time vided the visual motion stimulus, derivatives of the left-hnnd sides. Taking into ac- Snakes 331 count derivatives ofthe external forces we use re­ TIAL EQUATIONS. Clarendon: Oxfard. 1979. quires changing A at each iteration, so we achieve 7. Kanisza. 'Subjective contour.: SCIENTIF1C AMERI­ CAN. vol. 234, pp. 48-52. i976 faster iteration by simply assumin¡¡; tha! f, ~nd f, 8. E. Hildreth. "The compuratían of the velocity tield: are constan! during a time step. This yields an ex­ PROC. ROY. SOCo (LONDON). vol. B221. pp. 189-220. plicit Euler method with respect to the external 9. O. Marr. VISION. Freeman: San Francisco. 1982. rorces. The internal forces. howev"f, are com­ 10. D. Marr and E. Hildrerh. "A theory of edge delection; pletely specified by the banded matrlx, so we can PROC. ROY. SOCo (LONOON), vol. B207. pp. 187-217. 1980. t evaluate the time derlvative al time rather than 11. D. Marr and H.K Nishihara. "V¡ .. ual infoml3tion pro­ time t - I and therefore arrive at sn implicit cessing: Artificial Inlelligence and the sensorium of Euler step for the internal forces. The resulting sight: TECHNOLOGY REVIEW. vol. 81(1). October equations are 1978. 12. D. Man and T. Poggio. -A computational rheory of Ax, + f,(x,_" Y,-rJ = -y(x, - X,_I) (17) human slereo vision: PROC. ROY. SOCO (LONDON). vol. B204. pp. 301-328. 1979. Ay, + fy(X'_hY,_d = -:y(y,- Yt-I) (18) 13. A Martelli. <-IAn applicatinn of heunstio seorch mcthods where y is a step size. At equilibrium, the time to edge and contour delection: CACM. vol. 19. p. 73. 1976. derivátive vanishes and we end up with a solu­ 14. T. Poggio and V. Toree. "III-posed problems and regulari­ tion of equations (15) and (16). zation .n.ly,i, in early "sion: PROC. MRPA Image Equations (17) and (18) can be solved by ma- Understanding Workshop. New Orleans. LA, Baumann trix inversion: (ed.). pp. 257-263.1984. IS. T. Poggio. V. Toree. anfl C. Koch. "Campu~tional vision x, = (A + yl)-I(x'_1 - f,(x,_"y,_.) (19) and regularization theory; NATURE. vol. 317(6035), pp. 314-319. 1985. y, = (A + y1r'(Y'_1 - fy(X'_hY'_I» (20) 16. G. Sperling, "Binocular visiono a physical and a neuml theory; AM J. f'SYCHOLOGY. vol. 83, pp. 461-534. TIte matrix A + yl is a pentadiagonal banded 1970. matrix, so its inverse can be calculated by LV 17. D. Terzopoulos. A Witkin. 'ud M. Kass, "Symmetl)'­ decompositions in O(n) time [6,1], Hence equa­ seeking model1l: (úr 3D object reconstructiuIl," INT. J. tions (19) ami (20) provide a rapid solution to COMPUTER VISION. vol. 3. 1987. equations (15) and (16). The method is implicit 18. O. Terzopoulos. "Regularizarion of inverse visual prob­ lems involving discontinuities: IEEE TRANS. PAMI-R, with respect to the internal rorces, so it can solve p. 413. 1986. very rigid snakes with large step sizes. Ifthe exter­ 19. AN. 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