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158 IEEE TRANSACTIONS ON PATTERNANALYSIS ANDMACHINEINTELLIGENCE,VOL. 17, NO. 2, FEBRUARY 1995

Shape Modeling with Front Propagation:A Level Set Approach

Ravikanth Malladi, James A. Sethian, and Baba C. Vemuri

Abstract - hape modeling is an important constituent ofcomputer vision as wellas computer graphics research. Shapemodels aid the tasks of object representation and recognition.This paper presents a new approach to shape modeling which re-tains some of the a ttractive featur es of ex isting methods and over-comes some of th eir limitations. Our techniques can be appliedtomodel arbitrarily complex shapes, which include shapes withsignificant protrusions, and to situations where noa pr ior i as-sumption abo ut the objects topology is made.A single ins tance ofour model, when presented with an image having more than oneobject of interest,has the ability to split freely to represent eachobject. This m ethodis based on the ide as developed by O sher andSethian to model propagating solidhiquid interfaces with curva-ture-de pende nt speeds. The interfac e (front) is a closed, noninter-secting, hypersurface flowing along its gradient field with con-stant speedor a speed that depends on the curvature.It is movedby solving a H amilton-Jac ob? type equation written for a func-tion in which the interface is a particular level set.A speed termsynthesizpd from th e im age is usedto stop the interface in the vi-cinity of object boundaries. The resulting equation of motion issolved by employing entropy-satisfying upwind finite differenceschemes. We presen t a variety of ways of com puting evolvingfront, including narrow bands, reinitializations, and differentstopping criteria. T he efficacy of the scheme is demon strated withnume rical experimen ts on some synthesized images and some lowcontra st medical images.

Index Terms - hape modeling, shape recovery, interface mo-tion, level sets, hyperbolic conservation laws, Hamilton-Jacobiequation, entropy condition.

I. INTRODUCTION

N this paper, we describe a m odeling technique based on aI evel set approach for recovering shapes of objects in twoand three dimensions from various types of image data. Themodeling techniq ue may be viewed as a form of active model-ing such as snakes[151 and deformable surfaces [34] since,the model which consists of a moving front, may be moldedinto any desired shape by externally applied halting criteriasynthesized from the image data. The snakes or deformablesurfaces may be viewed as Lagrangian geometric formulations

Manuscript received May17, 1993; revised June 21, 1994.R. Malladis and J. Sethians research supported in part by the Applied

Mathem atical Sciences Subprogram of the Office of Energy Research,

USDept . of Energy under Co ntract DE-AC03-76SD 00098 and by theNSF ARPA under grant DMS -8919074 .B. C. V emuris work sponsored in part byNSF rant ECS-9210648.R. M alladi andJ. Sethianare with Lawrence Berkeley Laboratory andDe-

partment of Mathematics, University of Califomia, Berkeley, CA 94720USA. B. Vemuri is with the Departmentof Com puter and Information Sci-ences, University of Florida, Gainesville,FL 32611 USA.

IEEECS Log Number P95015.

wherein the boundary of the model is representedin aparametric form. These parameterized boundary representa-tions will encounter difficulties when the dynamic model em-bedded in a noisy data set is expandinghhrinking along itsnormal field[ 101 and sharp corners or cus ps develo p or piecesof the boundary intersect. By exploiting recent advances ininterface techniques, our m odeling technique avoids this La-grangian geome tric view and instead cap italizes on a relatedinitial value partial differential equation. In this setting, severaladvantages are apparent, including the ability to evolve themodel in the presence of sharp corners, cusps and changes in

topology, model shap es with significa nt protrusions and holesin a seam less fashion, and ex tension to three dimen sions inanextremely straightforward way.

A . Background

An im portant goal of com putational vision is to recove r theshapes of objects in 2D and 3D from various types of visualdata. One way to achieve this goal is via model-based tech-niques. Broadly speaking, these techniques involve the use of amodel whose boundary representa tion is matched to the imageto recover the object of interest. These models can either berigid, such as corre lation-base d temp late matching techniques,or nonrigid, as those used in dyna mic model fitting technique s.

Shape recovery from raw data typically precedes its sym-

bolic representation. Shape models are expected to aid the re-covery of detailed structure from noisy data using only theweakest of the possible assumptions about the observed shape .To this end, several variational shape reconstruction methodshave been proposed and there is abundant literature on thesame (see [4],[27], [35], [38], [17] and references therein).Generalized spline models with continuity constraints are wellsuited for fulfilling the goa ls of shape recovery (see[ 6 ] , 331).Generalized splines are the key ingredient of the dynamicshape modeling paradigm introduced to vision literature byKass et a1 [15]. Incorporating dynamics into shape m odelingenables the creation of realistic animation for computergraphics applications and for tracking moving objects in com -puter vision. Following the advent of the dynamic shapemodeling paradigm[ 151, [34], conside rable research follow ed,with num erous application specific modification sto the model-ing primitives, and external forces derived from data con-straints [391, [181,[ I 11, [241, [%I , [W.

The final recovered shape in these schemes can depend onan initial guess which is reasonably close to th e desired sh ape.One so lution to this problem in the one-dim ensiona l case has

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MALLADI, SETHIAN, ANDV E M U R I : SHAPE MODELING WITH FRONT PROPAGATION: A LEVEL SET APPROACH 159

(a) C T image (b) DSA image ( c )Shapeswith holesFig. I . Test bed for our topology-independentshape modelingscheme.

been presented by Amini et al[2].They use a disc rete form ofdynamic programming to optimize the univariate variationalproblem.

The framework of energy minimization (snakes) has beenused successfullyin the past for extracting salient image con-tours suchas edges and lines byKass et a1 [15].To make thefinal result relatively insensitive to the initial conditions, Co-hen [ l o ] suggested the use of an inflation force which makesthe snake behave like an e dge seekin g active model. Althoughthe inflation force p revents the curv e from ge tting trapped byisolated spurious edges, the active contour model cannot bemade to ex trude through any sign ificant protrusion s that ashape may possess (see Fig. l(b) ) without resorting to cumber-some resamp ling technique s. In this paper, we present a tech-nique which overcom es this problem and accurately models bi-furcations and protrusionsin complex shapes. Most existingshape modeling schem es require that the topology of the objectbe known before the shape recovery can commence. It is,however, not always possible to specifythe topology of anobject prior to its recov ery. For examp le, an important concernin object tracking and motion detection applications is topo-

logical change resulting from tracking the positions of objectboundariesin an image sequence through time. During theirevolution , these closed contours may change connectivity andsplit, thereby underg oing a topolog ical transform ation. Onesuch exam ple is the splitting of cell boundaryin a sequence ofimages depicting cell division. A heuristic criterion for split-ting and merging of curvesin 2D which is based on monitoringdeformation energiesof points on the elastic curve has beendiscussed in [26]. In the context of static problems, morere-cently, particle systems have been used to model surfaces ofarbitrary topology[ 3 2 ] .Here, particles can be added and de-leted dynamic ally to enlarge, and trim the surface respectively.

The schemes describedin this paper offer a new approachto some of the abov e problem s. To b egin, the convergence tothe final result is re latively independ ent of the shape initializa-tion. The algorithm allows branchesto sprout automatically asthe front moves.The scheme describedin this paper can beapplied where noa priori assumption about the objects topol-ogy is made. A single instance of our model, when presentedwith an image having more than one shape of interest (seeFig. I(c)),has the ability to split freely to repre sent each shap e[19], [20].W e show that by using our a pproach , it is also pos-

sible to extract the bounding contours of shap es with holes in aseamless fashion (see F ig.13).

Our method is inspired by ideas first introducedin Osherand Sethian [23], [29],which grew out of workin Sethian[28], to model propagating fronts with curvature-dependentspeeds. Two such examples are flame propagation and crystalgrowth, in which the speed of the moving interfac e normal toitself depen ds on transport terms modified by the local curva-ture. The challengein these problems is to devise numericalschemes for the equ ations of the propagating fro nt which willaccurately app roxima te these highly u nstable physical phe-nomena. Osher and Sethian[23] achieve this by viewing thepropagating surface as a specific level set of a higher-dimensional function. The equ ation of motion fo r this functionis reminiscent of an initial value H amilton -Jacob? equationwith a parabo lic right-hand side and is closely related to a vis-cous hyperbolic conservation law .

In our work, we ad opt these level set techniqu es to theproblem of shape recovery. To isolate a shape from its back-ground, we first consider a c losed, non intersecting, initial hy-persurface placed inside(or outside) it. This hypersurface is

then made to flow along its gradient field with a speedF(K),where K is the curvature of the hyper surface. Unknown shapesare recovered by making the front adhere to the objectboundaries. This is done by synthesizing a speed term fromimage data which acts as a halting criterion. Finally, we notethat a separate study also applying a level set approach hasbeen performed independen tly by Case lles et a1[7].

The o utline of this paper isas follows. In Sec tion 11, webriefly explain the level set approach to front propagationproblems and the accompanying numerical algorithm s. In Sec-tions111 and IV, we discuss the applicationof this technique toshape recovery prob lems, and co nsider various speed fun ctionsand appr oaches tothe problem, such as the effect of globalspeed law s, narrow band formu lations, reinitialization andstopping criteria. In Section V, we present some experim entalresults of applying ou r method to some synt hetic and low con-trast medical images. We c oncludein Section VI.

11. F RON T ROPAG ATIONROBLEM

In this section we present the level set technique due toOsher and Sethian[23]. For details and an expository review,see Sethian[29].

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As a starting point and motivation for the level set ap-proach, consider a closed curve moving in the plane, that is, let

0) be a smooth, closed initial curve in Euclidean planeS2and let t ) be the one-pa rameter family of cu rves generated bymoving 0) along its normal vector field with speedF(K), agiven scalar function of the curvatureK . Let x(s, f , be the

position vector which parameter izest )

bys, 0

5S

S.One num erical approac h to this problem is to take the aboveLagrangian description of the problem, produce equations ofmotion for the position vectorx(s, t), and then discretize theparame terization with a setof discrete marker particles lyingon the moving front. These discrete markers are updated intime by approximating the spatial derivatives in the equationsof motion, and advancing their positions. However, there areseveral problems with this approach, as discussedin Sethian[28]. First, small errors in the computed particle positions aretremendously a mplified by the curv ature term, and calculation sare prone to instability unless an extremely small time step isemployed. Second, in the absence of a smoothing curvature(viscous) term, singularities develop in the propagating front,and an entropy condition must be observed to extract the cor-rect weak solution. Third, topological changes are difficult tomanage as the evolving interface breaks and merges. Andfourth, significant bookkeeping problems occur in the exten-sion of this technique to three dimension s.

As an alternative, the central idea in the level set approachof Osher and Sethian [23] isto represent the front f) as thelevel set {W = 0) of a functionW. Thus, given a moving closedhypersurfacefi t) , that is, t = 0) : [0 ) + S N , e wish toproduce an Eulerian formulation for the motion of the hyper-surface propagating along its normal direction with speedF,where F can be a functio n of various argum ents, including thecurvature, normal direction, etc. The main idea is to embedthis propagating interface as the zero level set of a higher di-mensional functionW. Let y x, t = 0), wherex E 91N s defined

by

where d is the distance fromx to it = 0), and the plus (m inus)sign is chosen if the poin tx is outside (inside) the initial hyper-surface t=O). Thus, we have an initial functiony ( x , t = 0 ) 91N+ with the property that

y ( t = O ) = ( x I W(X, t = O ) = O ) (2)

As illustration, consider the exam ple of an expanding circle.Suppose the initial fronty at t = 0 is a circle in the xy-plane(Fig. 2(a)). We imagine that the circle is the level set{ I// = 0 )of an initial surfacez = H x , y, t = 0) in 913(see Fig. 2(b)). Wecan then match the one-p aramete r family of moving curv esr t)with a one-p arame ter family of m oving surfaces in such a waythat the level set{ tu = 0) always yields the moving front (seeFig. 2(c) and Fig. 2(d)).

Our goal is to now produce an equation forthe evolvingfunctionw(x ,t) which contain s the embedded motion oft) asthe level set{ v/= 0) . Here, we follow the derivation presented

Fig. 2. Level se t formulation of equations of motion- a) and (b) show thecurve y and the surface H x ,y )at t = 0, and c) and (d) show the curve y andthe corresponding surfacew x, y ) at time t .

in [22]. Letx(t) ,f E [0 ,=) be the path of a point on the propa-gating front. Thatis, x ( t=O) is a point on the initial frontt = 0), and x, = F(x ( t ) )with the vectorx , normal to the front

at x(t) . Since the evolving functionw is always zero on thepropagating hypersurface, we must have

By the chain rule,

wherexi is the ith c ompo nent ofx . Let

Since

4)

5 )

we then have the evolutio n equation forW, namely

with a given value ofy ( x , t = 0). We refer to this as a Hamil-ton-Jacobi type equation because, for certain forms of thespeed functionF, we obtain the standard Hamilton-Jacobiequation.

There are four major advantages to this Eulerian Hamilton-Jacobi form ulation. The first is that the evolv ing functionI t ) always remains a functionas long as F is smooth.However, the level surface( W = 0}, and henc e the propagatinghypersurface t ) may change topology, break, merge, and

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MALLADI, SETHIAN, A N D VEMURI: SHAPE MODELINGWITH FRONT PROPAGATION:A LEVEL SET APPROACH 161

form sharp corners as the function yevo lves, see [231.The second advantage of thisEt~kian ormulation concerns

numerical approximation. Becauseyfx, t ) remains a functionas it evolves, we may use a discrete gridin the domain of xand substitute finite difference approximations for the spatialand temporal derivatives.For example, using a uniform mesh

of spacingh , with grid nodesi j and employing the standardnotation that I,Y~ is the approximation to the solution

W ih,jh, dt), where At is the time step, we might write

y ' - ys'I + ( F ) ( V u y ; )= 0.

At(7)

Here, we have used forw ard differences in time, and letV , y ibe some appropriate finite difference operator for the spatialderivative.

Th e correct technique for approximating the spatial deriva-tive in the above comes from respecting the appropriate en-tropy condition for propagating fronts, discussedin detail in

[29]. As brief motivation for these schemes, consider a peri-odic cosine curve propagating in its normal direction withspeed F = 1 - EK,where K is the curvature. This problem hasbeen discussed extensively in [28].For E > 0, the front stayssmooth for all time.For E = 0, the parameterized analytic solu-tion corresponds to a front which passes through itself and de-velops a swallowtail solution. In order for the propagatingfront to correspond to the boundary of an expanding region,we invoke the entropy condition, namely thatif the boundaryis viewed as a propagating flam e, thenonce a particle is burnt,it stays burnt. This entropy condition yields the front whichcorresponds to the limiting solutionas E + 0 of the smoothcase.

In order to build a correct entropy-satisfying approximation

to the difference operator, we ex ploit the technology of hyper-bolic conservation laws. Following [23], we use a modificationof an Engquist-Osher scheme [12]. That is, given a speedfunction F(K), we update the front by the following scheme.First, separateF ( K ) into a constant advection termFo and theremainderF I ( K ) , hat is,

The advection componentFo of the speed function is then ap-proximated using upwind schemes, while the remainder is ap-proximated using central differences. In one space dimension,we have

Extensionto higher dime nsions are straightforw ard; we use theversion givenin [30].

The third advantage of the above formulation is that intrin-sic geometric properties ofthe front may be easily determined

from the level functiony. For example, at any point of thefront, the norm al vector, is.given by

and the curvature is easily obtained from the divergen ce of the

gradient of the unit normal vector to front, that is,

Finally, the fourth advantage of the above level set approachis that there are no significant differences in f ollowing fronts inthree space dimensions. By simply extending the array struc-tures and gradients operators, propagating surfaces are easilyhandled.

Since its introduction in [23], the above level set approach

has been used in a wide collection of problems involvingmoving interfaces. Someof these application s include the gen-eration of m inimal surfaces [8], singularities and geodesic s inmoving curves and surfaces in [9], flame propagation [25],[40], and fluid interfaces [31], [22]. Extensions of the basictechnique include fast methodsin [ l ] and extensions to triplepoints in [3]. The fundamental Eulerian perspective presentedby this approach has since been adopted in many theoreticalanalyses of mean curvature flow,in particular, see [13]. Incomputer vision, a model for sh ape theory based on this workhas been presentedin [161.

111. SHAPERECOVERYWITH FRONT PROPAGATION

In this section, we describe how the level set formulation for

the front propagation problem discussed in the previous sec-tion can be used for shape recovery. First, note that the frontrepresents the boundary of an evolving shap e. Since the idea isto'extract objects, shapes from a given image, the front shouldbe forced to stop in the vicinity of the desired objects'boundaries. This is analogous to the force criterion used topush the active contour model towards desired shapes [15].We define the final shape to be the configuration when all thepoints on the front come to a sto p, thereby bringing the compu-tation to an end.

Our goal now is to define a speed function from the imagedata that can be applied on the propagating frontas a haltingcriterion. As before, we split the speed functionF into twocomponents:F = FA + FG. The term FA, referred to as the ad-

vection term, is independent of the moving front's geometry.The front uniformly expands or contractswith speed FA de-pending on its sign and is analogous to the inflation force de-fined in [lo]. The second termFG, is the part which dependson the geometry of the front, such as its local curvature. This(diffusion) term sm oothes out the high cu rvature regions of thefront and has the same regularizing effect on the front as theinternal deformation energy term in thin-plate-membrane

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splines [15] (see the Fig.9). W e rewrite Equation6 by split-ting the influence ofF as

First consider the case when the front moves with a constant

speed, that is,FG = 0 F = FA. Define a negative speedF, tobe

where M I and M2 are the maximum and minimum values of themagnitude of image gradient IVG,* I(x, j)I, x, y) E R . Theexpression Go I denotes the image convolved with a Gaussiansmoothing filter whose characteristic width is6 Alternately,we could use a sm oothed zero-crossing image to synthesize thenegative speed function. The zero-crossing image is producedby detecting zero-crossings in the functionVG, * I which isthe original image convolved with a Laplacian-of-Gaussian

filter whose characteristic width isQ. The value of F, lies inthe range [-FA, 01 as the value of image gradient varies be-tween M I and MZ From this argument it is clear that, ifIVG,* I (x ,y)I approaches the maximumM I at the objectboundaries, then the front gradually attains zero speed as itgets closer to the object boundaries and eventually comes to astop.

If FG 0, then it is not possible to find an additive speedterm from the image that will cause the net speed of the frontto approach zero in the neighborhood of a desired shape. In-stead, we multiply the speed functionF = FA + FG with aquantityk,. The termk,, which is definedas

1k f x , y )= 1+(VGU*I (x ,y ) l

has values that are closer to zero in regions of high imagegradient and values that are closer to unity in regions withrelatively constant intensity. If one desiresa speed functionthat falls to zero faster than the reciprocal function, the follow-ing definition canbe employed:

More sophisticated stopping criteria can be synthesized byusing the orientation dependent steerable filters[141.

Iv. EXTENDINGHE SPEEDFUNCTION

The image-based speed terms have meaning only on theboundary y( t ) , that is, on the level set{v = 0 ) . This followsfrom the fact that they were designed to force the propagatinglevel set ( ty = 0) to a com plete stop in the neighborhood of anobject boundary. However, the level set equation of motion iswritten for the function yd efi ne d over the entire domain. Con-sequently, we require that the evolution equation has a consis-

tent physical meaning for all the level sets, that is, at everypoint x . y ) E R. The speed functionFI derives its me aning notfrom the geometry ofy ut from the co nfiguration of the levelset w= 0 ) in the image plane. Thus, our goalis to constructan image-based speed function that is globally defined. W e

call it an extension of F, off the level set {y = 0) because itextends the meaning ofF, to other level sets[30]. Note that thelevel set ( 0 ) lies in the image plane and therefore@,ustequal F, on ( = 0 ) . The same argument applies to the coeffi-cient k,. With the extensionsso defined, the equation of motionfor the caseF = FA is given by

and

when F = FA+ FG.If the level curves are moving w ith a constant spe ed, that is,FG= 0, then at any timet , a typical level set{y = C) C E R, is adistance C away from the level set( y = ) (see Fig.3). Ob-serve that the above statement is a rephrased version ofH u y -gen s principle which, from a geometrical standpoint, stipu-lates that the position of a front propagating w ith unit speed ata given timet should consist of only the set of points located adistance t away from the initial front. On th e other hand, forFG

0, the level sets will not remain a constant distan ce apart.With this in mind, there are several Gays to extend the

speed function to the neighboring level sets.

Fig. 3 . Huygens principle construction.

A . Global Extension

As a first attempt, we require that the external (image-based)speed function be such that level sets moving under this speed

function cannot collide.W e can construct one such extension to the image-based

speed function by (see Fig.4) letting the value of kI i , ) t a

point P lying on a level set{y = ) be the value of4 i I )t apoint Q , such that pointQ is closest toP and lies on the level

set {v= ) . Thus, k, L/) educes toFI k,) on {y = ) .

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~

163

By updating the level set function ona grid, we are moving thelevel sets without constructing them explicitly. Therefore astraightforward algorithm consists of advancing from one time stepto the nextas follows:

I X

Fig. 4. Extensionof image-based speed terms toother level sets.

Algorithm 1

1) At each grid point(iAx, Ay) ,whereAx and Ay are step sizesin either coordinate directions, theextension of image-basedspeed term is computed.This is done in accordance with theconstruction described in the previous section; that is, bysearching for a pointq which lies on the level set{ ty= O ,

and is closest to the point(iAx,jAy) . The value of image-based speed term at the current point is simply its value at thepointq.

2) With the value of extended speed term( ')i,j and Y:,~, calculateI,u~,;'sing the upwind, finitedifference schemes given in[30]

3) Construct an approximation for the level set( ty = 0) fromyts . This is required to visualize the current position of thefront in the image plane. A piecewise linear approximationfor the front~ t )s constructedas follows. Given a cellC(i,j ) , f

or

then C(i, j ) @ t) and is ignored, else,the entry and exitpoints wherey = 0 are found by linear interpolation. Thisprovides two nodes onr t) and thus, one of the line segmentswhich form the approximation to t). The collection of all

such line segments constitutes the approximation to the levelset [ ty= 0}, which is used for future evaluation of the image-based speed term in the update equation.

4) Replacen by n + 1 and return to step1.

B. Global Extension with Reinitialization

The above construction can create a discontinuous velocity

extension away from the zero level set, since the distance func-tion is not differentiable. One solution t o this is t o reinitializationthe level set function every fixed number of time steps to keepthe level sets evenly spaced around the front. A straightforwardway to do this is to recompute the distance from each point ofthe grid to the zero level set. However, this is anO(N3)opera-

tion, if we assume that thereare N points in each coordinate di-rection, plus approximatelyO(N) points on the interfaces.An alternative to this reconstruction is provided by[31],

based on an idea of M orel. The idea is simply to iterate on thelevel set function at a given time accordingto the followingequation:

In the limit ask + m this convergences to the distance func-tion, with some error in relocating the original ze ro level set. Fordetails,see [ 5 ] .

The most expensive step in either of these algorithms is thecomputation of theextension for image-based speed term. Thisis because at each grid point, we must search for the closestpoint lying on the level set{ ty = 0). Moreover, ifFG = 0, thenthe stability requirement for thi explicit method for solving ourlevel set equation isAt = O(Ax). For the full Equation12), thestability requirement isAt = O Axz). This could potentially forcea very small time step for fine grids. These two effects, indi-vidually and compounded, make the computation exceedinglyslow. In the case of reinitializing using the above iteration for-mula, additional labor is involved.

C.Narrow-Band Extension with Reinitialization

As a efficient alternative, we observe that the front can bemoved by updating the level set function at a small set of pointsin the neighborhood of the zero set instead of updating it at all

the points on the grid. InFig. 5 the bold curve depicts the levelset { t y = 0) and the shaded region around it is the narrow band.The narrow band is bounded on either side by two curves whichare a distance6 apart, that is, the two curves are the level sets( ty= f 2). The value of 6 determines the number of grid pointsthat fall within the narrow band. Since, during a given time stepthe value ofwjis not updated at points lying outside the narrowband, the level sets[lM> 6'2) remain stationary. The zero setwhich lies inside moves until it collides with the boundary of thenarrow band. Which boundary the front collides with depends onwhether it is moving inwardor outward; either way, it cannotmove past the narrow band. A complete discussion of the narrowband techniques for interface propagation m ay be found in[13.

As a consequence of our update strategy, the front can bemoved through a maximum distance of8 2 , either inward oroutward, at which point we must rebuild an appropriate (anew) narrow band. We reinitialize the tyfunction by treatingthe current zero set configuration, that is,{ ty = 01, as the ini-tial curve 0). Chopp [8] observed that the reinitialization stepcan be made cheaper by treating the interior and exterior meshpoints as sign holders. Note that the reinitialization proceduremust account for the case when( ty = 0 ) changes topology.This procedure will restore the meaning ofty function by cor-

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recting the inaccur acies introduce d as a result of o ur update al-gorithm. Once a newwfunction is defined on th e grid, we cancreate a new narrow band around the zero set, and go throughanother set of, sayI, iterations in time to move the front ahead

Y

Fig. 5. A narrow band of width 6 round the leve l set( y = 0).

by a distance equal to&2. The value of I is set to the numberof time steps required to m ove the front by a distance roughlyequal to &2. This choice dependson some experimentation.Thus, a faster algorithm for shape recovery consists of thefollowing steps:

the narrow band, the issue of spec ifying boundary cond itionsfor points lyingon the edge of the band becomes pertinent.With our relatively simple speed motion, the free-end bound-ary condition is a dequate, however,in more complex applica-

tions such as crystal growth, and flame pro pagation , accuratespecification of boundary cond itions is necessary[13.

We now show that this new faster approach provides acorrect approximation to the propagating front problem. InFig. 6, we show the result of app lying narrow-band algorithmto a star shaped front propagating with speedF = -K, whereK is the cu rvature as in Equatio n (1 1). The calculation wasdone on a unit box with 64 pointsin either direction, and atime step of At = 0.00003 was employ ed. The width of thenarrow band has been set to6 = 0.075, and the wfunctionwas recomputed once every(I =) 40 time steps. In Fig. 6(a),

we show the initial curve along with the level sets(I /le 0.2).After 40 narrow-band updates (Fig. 6(b)), only the level sets{lv/le 0.0375) move and the rest remain statio nary. We notethe inconsistency between the leve l sets lying on eithe r side ofthe narrow band, making the reinitialization step necessary inorder to restore the meaning of thety function. Following thereinitialization step, another 40 update steps are applied(Fig. 6(c)), which diffuses the high curva ture regions of thefront even further. In subse quent figures, the results of repeat-edly applying the same strategy are shown. Finally, inFig. 6(f), the peaks and troug hs on the front get comp letely dif-fused, and it attains a smooth circular config uration after four

Algorithm 21) Set the iteration numberm = 0 and go to step 2.2)At each grid point i , j ) lying inside the narrow band,

3) With the above value of extended speed term@), and yTj, alculatey: using the upwind, finite

the extension k , Of image-based peed term. reinitialization steps and a total of 2 00 time steps.

D traighqorward Narrow-Band Extension1.1

difference schem e given in[30].4)Construct a polygonal approxim ation for the level set

[ VI= ) from wr . A contour tracing procedureis usedto obtain a polygon al approxim ation. Given a cell( i , J )

which contains t ) , this procedure traces the contour byscanning the neighboring cells in order to find the nextcell which contains t ) . Once such a cell is found, theprocess is repeated until th e contour closes on itself. Theset of nodes visited during this tracing process constitutesthe p olygonal approximation tost). n gen eral, to collectall the closed contours, the above tracing procedure isstarted at a new, as yet unvisited, cell which contains thelevel set {w= 0). A polygonal approximation is requiredin step2 for the evaluation of image-based speed termand more importantly,in step 6 for reinitializing thefunction.

5) Incrementm by one. If the value ofm equals I, go tostep 6, else, go to step2.

6)Compute the value of signed distance functionw bytreating the polygonal approximation of{w= 0) as theinitial contour 0). As m entioned earlier, a more generalmethod of reinitialization is required when{ ly = 0)changes topology.Go to step 1.

at points lying inn this approach, since we only update

The narrow-band approach, in addition to being computationallyefficient, allows us to return to the original constructio n of thespeed function extension and replace it with a more mathe-matically appealing version. Since the narrow-band mecha-nism periodically rec alibrates the front, we can in fact sim-ply move each level set with the spe ed determin ed by the im-age gradient as givenin Equations (14) and 15). In otherwords, for points inside the narrow band, the external speedvalues are picked directly from their corresponding image lo-cations. Thus, we can ignore the previous extension velocityand provide a purely geometric one based on the local imagegradient. Although this may cause many other level sets totemporarily stop, the narrow-band reinitialization resets themall around the zero level set. This will ensure that the zerolevel set is drawn clo se to the objec t boundary as well as retainother desirable properties of the level set approach, suchastopological m erge and split. Also, sin ce the extensio n compu-tation doe s not involve any search, the time complexity of thisapproach is identical to that of a basic narrow-band front

propagation alg orithm . We currently use this compu tationallyefficient algorithm and suggestit for others interested in thiswork.

V . SHAPERECOVERY ESULTS

In this section we present several shape recovery results thatwere obtained by applying the narrow-band level set algorithmto image data. Given an imag e, our method requires the user to

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MALLADI. SETHIAN, AND VEMURI:SHAPE MODELING WITH FRONT PROPAGATION:A LEVEL ET APPROACH I65

a) t = 0.0000 (b) t = 0.0012

(d) t = 0.0036c ) t = 0.0024

e) t = 0.0048 (f) t = 0.0060Fig. 6 .Narrow-band algorithm applied to a star-shaped front propagating with speed F = -K. alculation s were done witha 64 x 64 grid with a time stepAf = 0.00003. Ywas recomputed after every 40 time iterations.

provide an initial contoury(0). The initial contour can be

placed anywhere in the image plane. However, it must beplaced inside a desired shape or enclose all the constituentshape s. Our front seeks the object boundaries by either propa-gating inward or outward in the normal direction. This choiceis made at the timeof initialization. Note that after the specifi-cation of initial shapeof y(O), our algorithmdoes not requireany further user interaction. On the other hand, the user mayinteract with the model by varying the smoothness control pa-

rameter E until a desired amountof smoothness is achieved in

a given shape.The initial value of the functionIU that is, H x ,0) is com-

puted from$0). We first discretize the level set functionw o nthe image planeand denote w jas the value ofI Uat a grid pointidx, Ay ) ,where Ax and Ay are step sizes in either coordinate

directions. We define the distance from a pointi, j ) o the ini-tial curve to be the shortest distance from(i, j ) o y(0). Themagnitude ofw s set to this value. We use the plus sign if

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( a ) Original image (b ) Image-based speed t e rm

, with (T = 3.25 , synthesized from theCT mage.1

Fig. 7 . Image-based speed termkdx, y) =I+IVG, * I (X , Y ) l

is outsidey(0) and m inus sign if i, j ) s inside. Once the value

of yjj is computed at timet =

0 by follo wing the abov e proce-dure, we use algorithms from the previous section to move thefront.

We now presentour shape recov ery results in 2D. First, weconsider a 256x 256 C T (computed tomography) image ofanabdom inal section shown in Fig. 7(a), with the goal of recover-ing the shape of the stoma ch in this particular slice. The func-tion yh a s been discretized on a 128x 128 mesh, that is, calcu-lations are performed at every second pixel. In Fig. 8(a), weshow the closed contour that the user places inside the desiredshape at timet = 0. The function y i s then made to propagatein the normal direction with speed

F = k,(-1.0 - 0.025K).

W e emp loyed the narrow-band update algorithm to move thefront with a time step size set toAt = 0.0005, and the yfunc-tion was recomputed after every50 time steps. Fig. 7(b) showsthe image-based spe ed term which is synth esized accord ing toEquation (14). N ote thatin Fig. 7(b), kdx, y) values lying inthe interval [0..1] have been mapped into the interval [0..255].In Fig. 8(b) through F ig. 8(e) we depict the co nfigura tion ofthe level set{y = 0) at four interm ediate time instants. The fi-nal result is achieved after 575 time iterations and is shown inFig. 8(f). We em phasize that our m ethod does not require thatthe initial contour be placed close to the object boundary. Inaddition, observe how the front overshoots all the isolatedspurious edges present inside the shape (see Fig. 7(b)) and set-

tles in the neigh borho od of edges which correspo nd to the trueshape. This feature is a consequence ofEK component in thespeed which diffuses regions of high c urvature on the front andforces itto attain a smooth shape.As mention ed in Sec tion 111, smoo thn ess of the fr ont can becontrolled by choosing an appropriate curvature component inthe speed functionF = 1 - EK. The objective of our next ex-periment is to demonstrate smoothness control in the context

of shape recovery. In Fig. 9(a) through Fig. 9(c), we show the

results of applying our narrow-band sha pe recovery algorithmto an image consisting of three synthetic shapes. Initializationwas performed by drawing a curve enclosing each one of thethree shapes. We com pute the signed distance functionytx, y)from these curves. The level sets of are then made to propa-gate with speedF = k, (1 O - EK).First, as shown in F ig. 9(a),we perform shape recovery with the value ofE = 0.05. Theprocess is repe ated with diffe rent values ofE; 0.25 in Fig. 9(b)and 0.75 in Fig. 9(c). Clearly , with every increm ent in thevalue of E, the level set ( l y = 0} attains a con figuration that isrelatively smoother. This is analogous to the smoothness pro-vided by the second order term in the internal energy of a thinflexible rod [151.

In our third experiment we recover the complicated struc-ture of an arterial tree. The real image has been obtained byclipping a portion of a digital subtrac tion angiog ram. This isan example of a shape with extended branches or significantprotrusions. In this experiment we com pare the performance ofour scheme with the active contour mo del. First, an attemp t ismade to reconstruct the arterial structure using a snake modelwith inflation forces [lo]. In Fig. 10(a) through Fig. lO(i), weshow a sequence of pictures depicting the snake configurationin the image. We present the final equilibrium state of thesnake in Fig . lO(c), Fig. lO (f), and F ig. 1O(i) correspond ing tothree distinct initializations, each better than the preceding one- in terms of the closeness to the desired final shape. In allthree cases the active contour model, even after 1000 time it-erations, barely recovers the main stem of the arte ry and com-pletely fails to account for the branches. Due to the existence

of multiple local minima in the (noncon vex) energy fun ctionalwhich the numerical procedure explicitly minimizes, the finalresult depends on the initial guess. Observe how, in the thirdcase, despite a good initialization (Fig. lO(g)), the sna ke snapsback into a relatively bumpless conf iguration in Fig. 10(h).This is due to the snakes arc-length length (elasticity) and cur-vature (rigidity) minimizing property. Snakes prefer regularshapes because shapes with protrusions have very high defor

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a) = 0.0000 (b) t = 0.0500

c ) = 0.0875 (d) t = 0.1500

e ) = 0 .2250 ( f ) t = 0.2875Fig. 8. Recovery of the stomach shape from aCT image of an abdominal section. Narrow-band computation was done on a128 x 128 grid - he front was made

to propagate with speedF = k,(-1.0 0.025 K ) and the time stepAt was setto 0.OOOS. lwas recomputed once everySOtime steps.,.

mation energies. Note that it is important to maintain ab a l

ance between the image-based force and the inflation force.Therefore, we cannot increase the latter arbitrarily. One pos-sible way to account for significant protrusions in a shape isvia an adaptive resampling of the first order balloon-snakemodel. This however is a cumbersome solution to the problem.Now, we apply ou r level set algorithm to reconstruct the sameshape. After the initialization in Fig.1 (a), the front is made to

propagate in the normal direction. We employ the narrow-bandalgorithm with a band width of6 = 0.045 o move the front. Itcan be seen that in subsequent frames the front evolves into thebranches and finally in Fig.1 l(h) it completely reconstructsthe complex tree structure. Thus, a single instance of our shapemodel sprouts branches and recovers all the connected compo-nents of a given shape. Calculations were carried out on a128 x 128 grid and a time stepAt = 0.00025 was used. The

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( a ) E = 0 .05 (b) E

Fig. 9.Smoothnesscontrol in shape recoverycan be achievedby varying the CUI

plots of w x, t = 0) and w x, t = 0.375) are shown inFig. 1 (b) and Fig. 1 (i), respectively.

In the next exp eriment, we depict a situation when the frontundergoes a topological transformation to reconstruct theconstitue nt shapes in an imag e. The image shown in Fig. 12(a)consists of three distinct shapes. Initial curve is placed in sucha way that it envelopes all the objects. The front is then ad-vanced in the directionof the negativ e normal. Alternately, wecould perform the initialization by placing a curve in each oneof the individual shapes and propagating them in the normaldirection. We ch oose the form er option. The level set{ ty= 0 )first wraps itself tightly aro und the objects (see F ig. 12(d)through Fig. 12(f)). Subsequently it changes connectivity andsplits twice- in Fig. 12(g) and Fig. 12(h), thereby recoveringthree shape s. Fig. 12(i) shows the final result. Again it shouldbe noted that a sing le instance of ou r shape model dynamicallysplits into three instances to represent each object. The func-tion v/ was discretized on a64 x 64 grid and At was set to0.00025.

Next, we show that our approach can also be used to re-cover shapes with holes. The shapes in the Fig. 13 are exam-ples of shap es with ho les. The outer and in ner boundaries of agiven shape are recovered without requiring separate initiali-zations. In Fig. 13(a), we show the initial contour which en-closes both the shapes. This contour is then made to p ropagateinward with a constant speed . Fig. 13(b ) through Fig. 13(d) areintermediate stages in the front evolution. In Fig. 13(e), itsplits into two separate contours. The calculation comes to ahalt when, in Fig.13(f), the level set {v/ = 0) recovers theouter boundariesof two disconnected shapes. In the secondstage of our compu tation, we treat the zero set configuration nFig. 13(f) as an initial state, and propa gate the front inward bymomentarily relaxing the image-based speed term. This causesthe zero set to move into the shapes as shown in Fig. 13(g),

and recover the holes, thereby achieving a complete shape re-covery (see 13(h)). The calculations for this experiment weredone on a 128x 128 grid, and the time stepAt was set to0.00025.

In our last experiment, we recover the shapeof a flat super-quadric using the level set front propagation scheme in 3D. Vol-

= 0 .25

mature component in the speed F = k , (1.0 - E K )

c ) E = 0.75

ume data for this experiment consists of 32 slices each with aparticular cross section of the superquadric. The image-basedspeed term kI s computed from these images according toanequation in 3D which is analogousto Equation (14). A sphere,

which is the level surface{w =O }of a functionN x ,y, z) =2 +yz+ zz-O.Ol, formsour initialization (see Fig. 14 (a)). This initialsurface is moved with speedF = i , by updating the value ofwon a discrete 3D grid. The initial surface expand s smoothly inalldirections until a portion of it collide s with the superq uadric

boundary. At points with high gradient, thek, values are closeto zero and cause the zero set to locally come to stop near theboundary of the s uperqua dric shape. This situation is depicted inFig. 14(b) through Fig. 14(e), wherein the initial spherical shapetransforms into a flat superquadric. Finally, in Fig. 14(f), all thepoints on our shape model are stopped, thereby recovering theentire shape of the flat superquadric. Calculations were d one ona 32 x 32 x 32 grid with a time stepAf =0,0025.

e

VI. CONCLUDINGEMARKSIn this paper we have presented a new shape modeling

scheme.Our approach retains some of the desirable features ofexisting methods for shape modeling and overcomes someoftheir deficiencies. We adopt the level set techniques first intro-duced in Osher and Sethian [23] to the problemof shape recov-ery. With this approach, complex shapes can be recovered fromimages. The final result in our method is relatively independentof the initial guess. This is a very desirable featu re to have, es-pecially in applications suchas automatic shape recovery fromimage data. Mo reover,our scheme makes noa priori assump-tion about the objects topology. Other salient features ofourshape modeling scheme include its ability to split and mergefreely without any additional bookkeeping during the evolution-

ary process, and its easy extensibility to higher dimensions. Webelieve that this shape modeling algorithm will have numerousapplications in the areasof computer vision and computergraphics. Foran extensionof this work to a level set based shapedescription and recognition scheme, the reader is referred toMalladi and Sethian [21].

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MALLADI. SETHIAN, AND VEMURI: SHAPE MODELINGWITH FRONT PROPAGATION: A LEVEL SET APPROACH 169

(a) I n i t i a l i z a t i o n1 (b) 500 i t e r a t i o n s (c) 1000 i t e r a t i o n s

(d) I n i t i a l i z a t i o n2 (e) 500 i t e r a t i o n s ( f ) 1000 i t e r a t i o n s

(g) I n i t i a l i z a t i o n3 (h) 500 i t e r a t i o n s (i) 1000 i t e r a t i o n s

Fig. 10. An un successful attempt to reconstruct a complex shape with significantprotrusions usingan active contour model. Three different results are shown inparts (c), 0, nd (i) correspondin g o three distinct initialization s n parts (a), (d), and (g), respect ively.The following parameter values were em ploye d in thisexperiment:y (damping)= 1.0, At = 0.50, W I (elasticity)= 0.035, w2 rigidity)= 0.015, coefficientof inflation force= 0.50, and coefficient of imageforce= 2.50.

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(a) t = 0 0000 ( c ) = 0 . 0 6 2 5

(d) t = 0 .1250 ( e ) = 0 .1875 (f) t = 0 . 2 5 0 0

g) t = 0 .3050 (h) t = 0 .3750 i) (x, 0 .375)

Fig. 11. Reconstruction of a shape with significan t protrusions: anarterial tree structure. Computation was done on a 128x 128 grid with a time stepAt = 0.000 25. The narrow-bandalgorithm was us ed with a band widthof S= .045.

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MALLADI, SETHIAN, AN D VEMURI: SHAPE MODELING WITH FRONT PROPAGATION: A LEVEL SET APPROACH 171

a) = 0.0000

(d) t = 0.1250

( 9 ) = 0.1875

(b) t = 0 .0250

e ) = 0.1625

( c ) = 0 . 0 8 7 5

(f) 2 = 0.1750

(h) t = 0 .2000 (i) t = 0 .2500

Fig. 12.Topological split:A single instanceof the shape model splits into three instances to reconstructthe individual shapes. Computation was doneon a64 x 64 mesh with a time step A t = 0.00025.

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a) = 0.0000 (b) t = 0 .0500 c ) t = 0 .1000

(d) t = 0 .1750 ( e ) t = 0 .2137 ( f ) t = 0 .2400

g) t = 0 .2500 (h) t = 0 .2700 (i) t = 0 .2950

Fig. 13.Shapes with holes:A two-stage sch eme is used to arrive at a complete shape descriptionof both simple shapes and shapes with h oles . Computation wasperformed on 128x 128 grid and the time stepA t was setto 0.00025.

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MALLADI, SETHIAN , AND V EMURI: SHAPE MODELING WITH FRONT PROPAGATION: A LEVEL SET APPROACH 173

a) = 0.0000 (b) t = 0.0500

c ) = 0.1000 (d) t = 0.1750

(e) 1 = 0.2250 ( f ) t = 0.3000

Fig. 14. Shape recovery in 3D :a flat superquadricshape.Calculationswere done ona 32 x 32 x 32 grid with a time stepAt = 0.0025.

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D. Adalsteinsson andJ. A. Sethian , A fast levelset method for propagatinginterfaces, submittedfor publication,J. of Compututionul Physics,1994.A. A. Amini, T. E. Weymouth,and R. C.Jain Using dynamic program-ming for solving variationa l problem s in vision,IEEE Tn . on PunemAnalysis and Machine Intelligence,vol. 12, No. 9, pp. 855-867, 1990J. Bence,B. Meniman, andS . Osher, Motion of multiple triple unc tions: Alevel set approach,to a p p r nJ. Compututionul Physics,1994.

R. M. B o k and B. C. Vemuri, On three-dimensional surfa ce reconstructionmethods, IEEE Truns. on Panem Analysis and Muchine Intelligence,vol. PAMI 13,N o. 1, pp . 1-13.1991.A. Bourliouxand 1. A. Sethian, Projectionmthods coupled to level set in-terfacemethods, o be submitted,J. qfComputuhonal Phy sics,1994.A. Blake and A. Zisserman,Vi.wul Reconstruction, MIT Press, Cam-bridge, MA.V. Caselles, F. Catte, T.Coll, and F. Dibos, A geometric modelforactive contours in image processing, Internal report no.9210,CEREMADE, Universiti de Paris-Dauphine, France.

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L. D. Cohen, On active contour models and balloons,Computer Vi-sion, Graphics, and Imuge Processing,vol. 53, No. 2, pp. 211-218,March 1991.H. D elingette, M. Hebert, andK. Ikeuchi, Shape representation andimage se gmen tation using deformable models, inProc. o IEEE Conon Computer Vision and Pattern Recognition,pp. 467472, Maui,Hawaii, June 1991.B. Engquist andS. Osher, Stable and entropy satisfying approxima-tions for transonic flow calculations,Math. Comp.,vol. 34,45, 1980.L. C. Evans and J. Spruc k, Motion of levelsets by mean curvature.I,J. ofDifJerentia1 Geo metry,vol. 33, pp. 635-681, 1991.W. T. Freeman and E.H. Adelson, Ste erable filters for early vision,image analysis, and wavelet decomposition,Proc. of ICCV, pp. 406-415, Osaka, Japan,1990.M. Kass, A. Witkin, and D. Terzopoulos, Snakes: Active contourmodels,Intl. J. o Computer Vision,pp. 321-331, 1988.B. B. Kim ia, A. R. Tannenbaum, andS. W. Zucker, Toward a compu-tational theory of shape: An overview, inProc. qf ECCV, Antibes,France, 1990.D. Lee and T. Pavlidis, One-dimensional regularization with discon-tinuities,IEEE Trans. on Pattern Analysisand Machine Intelligence,

R. Malladi, Deformable models: Canonical parameters for surfacerepresen tation and m ultiple view integra tion, Masters thesis, Dept.of CIS, Univ. of Florida, Gaine sville,FL, May 1991.

R. Malladi, A topology-independent hape modeling scheme, Doc-toral dissertation, Dept. of CI S, Univ. of Florida, Gainesville,FL, De-cember 1993.R. Malladi, J. A. Sethian, and B. C. Vemuri, Evolutionary fronts fortopology-independent shape modeling and recovery, inProc. oThird European Con on Computer Vision,LNCS vol.800, p. 3-13,Stockholm,Sweden, May 1994.R. Ma lladi and J. A. Sethian , A unified framework for shape segmen-tation, represen tation, and recognition, Report LBL -36069, LawrenceBerkeley L aboratory, Univ. of C alifornia,Berkeley, Aug. 1994.W. Mulder, S . Osher, and J. A . Sethian, Computing nterface motionin com pressible gas dynamics,J. qf Computational Physics,vol. 100,no. 2, pp. 209-228, 1992.S. Osher and J. A. Sethian, Fronts propagating with curvature de-pendent speed: Algorithms based on Hamilton-Jacobi formulation,J .o Computational Physics,vol. 79, pp. 12-49, 1988.A. Pentland and S . Sclaroff, Closed-form solutions for physicallybased shape modeling and recognition,IEEE Trans. on PatternAnulysisund Machine Intelligence,vol. 13, No. 7, July 1991.C. Rhee, L. Talbot, and J. A. Sethian, Dynamical behavior of a pre-mixed turbulent open V-Flame, submitted for publication,J. of FluidMech., 1994.R. Samadani, Changes in connectivity in active contour models,Proc. of the Workshop on Visual Motion, pp. 337-343, Imine , CA,March 1989.L. L. Schum aker, Fitting surfaces o scattereddata, n ApproximationTheory I I G. G .Lorentz, C.K. Chui, and L . L. Schum aker, (eds.). NewYork: Academ ic Press, 1 976, pp. 203-267.J. A. Sethian, Curvature and the evolution of fronts,Comm. in Math.Physics,vol. 101, pp. 487499 , 1985.J. A. Sethian, Numerical algorithms for propagating interfaces:Hamilton-Jacobi equations and conservation laws,J . o DifJerentialGeometry,vol. 31, pp. 131-161,1990.J. A. Sethian and J. Strain, Crystal growth and dendritic solidifica-tion, J . of Computational Physics,vol. 98, pp. 231-253, 1 992.

M. Sussman, P. Smereka, andS. Osher, A level set approach forcomputing solutions to incompressible wo-phase flow, UCLA CAMReport 93-18, 1993.R. Szel iski and D. Tonne sen, Surface modeling with oriented particlesystems, Computer Graphics SIGGRAPH,vol. 26, No. 2, pp.185-194, July 1992.D. Terzo poulos, Regulariza tion of inverse visual problems involvingdiscontinuities,IEEE Trans. on Pattern Analysis and Machine Intel-ligence,vol. 8 , no. 2, pp. 41 342 4, 1986.

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D. Terropoulos,lie omputationof visiblesurface epresentations.IEEETrans. on Panern Anulysis and Machine Intelligence,vol. 4, no.10,pp. 417438,1988.B. C. Vemuriand R. Malladi,Surface griding with intrinsic parameters,PatlernRecognifionLeners,vol. 13, No. 11, pp. 805-8 12, Nov. 1992.B. C. Vemuriand R. Malladi, C hW t in g intrinsic parameters with activemodels for invaiant surface nxonstruction,IEEE Trans. on PanernAnalysis ndMachine Intelligence,vol. 15, No. 7,pp. 668-681, July 1993.B. C. Vemuri, A. Mitiche, nd J. K. Aggarwal, Curvature-based e p m t a -tion of objects fromrange W ntl.J. o Image and VisionComputing, 4,

Y. F. Wang and J.F. Wang,Surface econstruction using deformablemod-els with interior and boundary constraints, inProc. qf ICCV, pp. 3W303 ,Osaka, Japan,1990.J. Zhu and J. A. Sethian, Pmjection methods coupled to level set interfacetechniques,J. qf Comp~ta tioml hysics,vol. 102(1), pp. 128-138.1992.

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[40]

Ravikanth Malladj received his B.Eng. degreewith honors in electrical engineering and an MS c.degree in physics from Birla Institute of Technologyand Science, India in 1988. He subsequently re-ceived the MS degree in 19 91 and the Ph.D. degreein 1 993; both in com puter vision from the Univer-sity of Florida, Gainesville, Florida, USA. He iscurrently working atthe Lawrence Berkeley Labora-tory, University of California, Berkeley, California,USA.

In the summer of 1 985 he worked at the NationalGeophysical Research Institute, India. During the

spring of 1988, he held a research assistant position with the semiconductorphysics group at the Ce ntral Electrica l and Ele ctronic s Research Institute,In-dia, From summer 1990,till fall 1993, he was a graduate research assistant atthe University of Florida, and was affiliated with the Cen ter for ComputerVision and Visualization.

His research interestsace focused on comp utationa l vision, shape modelingand recognition, computer graphic s, and medical ima ge interpretatio n.Dr. Malladi was awarded the NSF Postdoctoral Fellowship in Com putatio nal

Science and Engineering in June 1994, and his work received best peer re-views at the Third European Conferenceon Computer Vision, Stockholm,Sweden, in May 1994.

James A. Sethian is professor of mathematics atthe University of C alifornia , Berkeley, California,USA, and senior scientist in the physics divisionat the Lawrence Berkeley Laboratory. Beforecoming to Berkeley, he was at the Courant Insti-tute of Mathematics.

He received a Ph.D. in mathematics from theUniversity of C alifornia , Berkeley, and has been arecipient of the Presidential Young InvestigatorAward and a Sloa n Foundation Fellowship.

His research interests include computationalphysics, parallel processing, image processing,

and material science.

Baba C. Vemuri received the PhD degree in elec-trical and compute r engine enng from the Universityof Texas at Austin in 1987, and joined the Depart-ment of Computer and Information Sciences at theUniversity of Florida, Gainesville, where he is cur-

rently an associate professor of computer and in-formation sciences and electrical engineering.In the summer of 1 988 and 1992, Prof. Vemuri

was a Visiting Faculty Member at the IBM ThomasJ. Watson Research Cente r, Yorktown Heig hts, NY,associate d with the Exploratory Computer Vision

Grou p there. During the academic year 198 9-1990 , he was a V isiting Re-search Scien tist with the G erman Aerospace Research Institute , DLROberpfaffenhofen, Germany.

His research interests include computational vision, modeling for vision and

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graphics, medical imaging, and applied mathematics. In the past severalyears, his research has focused primarily on efficient algorithm s for3D hapemodeling and recovery from image data. He h as published numerous refereedjournal and conference articles in the field of computer vision and its appli-cations.

Dr. Vemuri received theNSF Research Initiation andthe Whitaker Founda-tion Awards in 1988 and 1994, respectively. He wasa recipient of the firstpriu: for best intemational paper in Computer Vision from the Norwegian

branch of the IAPR in1992.He subsequ ently received thebest review scoresfor his co-authored paper in ECCV94.Dr. Vemuri servedas a guest editorfor the April 1992special issue on Range Image Understandingof the Joumalof Image and Vision Computing. He also servedas a program committeemember for the IEEE Conference on CVPR in1993,and was the chairman ofthe SPIE-sponsored conferenceson Geometric Methods in Computer Visionin 1991 and 1993,respectively.

Dr. Vemuri is a member of the IEEE Computer and Signal ProcessingSo-cieties,, ACM SIG GRAH and the SIAM.

175