266 IEEE TRANSACTIONS ON IMAGE PROCESSING,VOL.10,NO.2,FEBRUARY 2001

Active Contours Without Edges

Tony F.Chan,Member,IEEE,and Luminita A.Vese

Abstract In this paper,we propose a newmodel for active con-

tours to detect objects in a given image,based on techniques of

curve evolution,MumfordShah functional for segmentation and

level sets.Our model can detect objects whose boundaries are not

necessarily defined by gradient.We minimize an energy which can

be seen as a particular case of the minimal partition problem.In

the level set formulation,the problembecomes a mean-curvature

flow-like evolving the active contour,which will stop on the de-

sired boundary.However,the stopping term does not depend on

the gradient of the image,as in the classical active contour models,

but is instead related to a particular segmentation of the image.We

will give a numerical algorithmusing finite differences.Finally,we

will present various experimental results and in particular some

examples for which the classical snakes methods based on the gra-

dient are not applicable.Also,the initial curve can be anywhere in

the image,and interior contours are automatically detected.

Index Terms Active contours,curvature,energy minimization,

finite differences,level sets,partial differential equations,segmen-

tation.

I.I

NTRODUCTION

T

HE BASIC idea in active contour models or snakes is to

evolve a curve,subject to constraints from a given image

be a bounded open subset of

,with

its boundary.

Let

be a given image,and

be a

parameterized curve.

In the classical snakes and active contour models (see [9],[3],

[13],[4]),an edge-detector is used,depending on the gradient

of the image

,depending on the gradient of the image

For instance

A geometric active contour model based on the mean curva-

ture motion is given by the following evolution equation [3]:

CHAN AND VESE:ACTIVE CONTOURS WITHOUT EDGES 267

Its zero level curve moves in the normal direction with speed

vanishes.The constant

is a correction term

chosen so that the quantity

div

,depending on the image gradient

is never zero on the edges,and the curve may pass through

the boundary,especially for the models in [3],[13][15].If the

image

,as the boundary of an

open subset

of

(i.e.

,and

).In what follows,

denotes the region

,and

denotes the

region

.

Our method is the minimization of an energy based-segmen-

tation.Let us first explain the basic idea of the model in a simple

case.Assume that the image

268 IEEE TRANSACTIONS ON IMAGE PROCESSING,VOL.10,NO.2,FEBRUARY 2001

Fig.1.Consider all possible cases in the position of the curve.The fitting term

is minimized only in the case when the curve is on the boundary of the object.

length of the curve

,and (or) the area of the region inside

.

Therefore,we introduce the energy functional

,de-

fined by

Length

Area

:

Area

Length

where

is a constant depending only on

.

A.Relation with the MumfordShah Functional

The MumfordShah functional for segmentation is [18]

is a given image,

and

.Therefore,as it was also

pointed out by D.Mumford and J.Shah [18],

CHAN AND VESE:ACTIVE CONTOURS WITHOUT EDGES 269

(in the sense of distributions),we express the terms in the energy

in the following way (see also [7]):

Length

Keeping

fixed and minimizing the energy

with respect to the constants

and

,it is easy to express

these constants function of

by

),and

).For the corresponding degenerate

cases,there are no constrains on the values of

and

.Then,

and

are in fact given by

average

.Here,

means almost everywhere with respect to the

Lebesgue measure.

We expect,of course,to have existence of minimizers of the

energy

,due to several general results:our model

is a particular case of the minimal partition problem,for which

the existence has been proved in [18] (assuming that

),and also in [16] and [17],for more general data

,denoted here by

and

,

as

.Let

beany

regularization of

,and

.We will give further examples of such approximations.Let

us denote by

the associated regularized functional,defined

by

270 IEEE TRANSACTIONS ON IMAGE PROCESSING,VOL.10,NO.2,FEBRUARY 2001

Fig.3.Two different regularizations of the (top) heaviside function and

(bottom) delta function

.

Keeping

and

fixed,and minimizing

with respect to

,we deduce the associated EulerLagrange equation for

.

Parameterizing the descent direction by an artificial time

,

the equation in

div

on

(9)

where

denotes the exterior normal to the boundary

,and

denotes the normal derivative of

at the boundary.

III.N

UMERICAL

A

PPROXIMATION OF THE

M

ODEL

First possible regularization of

by

functions,as pro-

posed in [27],is

if

if

if

Fig.4.Detection of different objects from a noisy image,with various

shapes and with an interior contour.Left:

and the contour.Right:

the piecewise-constant approximation of

.Size

,

CHAN AND VESE:ACTIVE CONTOURS WITHOUT EDGES 271

Fig.5.Detection of three blurred objects of distinct intensities.Size

,

64,

272 IEEE TRANSACTIONS ON IMAGE PROCESSING,VOL.10,NO.2,FEBRUARY 2001

Fig.9.Object with smooth contour.Top:results using our model without edge-function.Bottom:results using the classical model (2) with edge-func tion.

compute

by the following discretization and linearization

of (9) in

sign

of the Heaviside and Dirac delta

functions (

),in order to automatically detect interior

contours,and to insure the computation of a global minimizer.

Only the length parameter

,which has a scaling role,is not

the same in all experiments.If we have to detect all or as many

objects as possible and of any size,then

should be small.

If we have to detect only larger objects (for example objects

formed by grouping),and to not detect smaller objects (like

CHAN AND VESE:ACTIVE CONTOURS WITHOUT EDGES 273

Fig.10.Detection of a simulated minefield,with contour without gradient.

Size

,

274 IEEE TRANSACTIONS ON IMAGE PROCESSING,VOL.10,NO.2,FEBRUARY 2001

Fig.12.Spiral from an art picture.Size

,

,five iterations of reinitialization,cpu

s.

the geometric model (2)],by which the curve cannot detect the

smooth boundary.

In Fig.10,we validate our model on a very different problem:

to detect features in spatial point processes in the presence of

CHAN AND VESE:ACTIVE CONTOURS WITHOUT EDGES 275

Fig.13.Detection of the contours of a plane froma noisy image.Size

,

,

,

276 IEEE TRANSACTIONS ON IMAGE PROCESSING,VOL.10,NO.2,FEBRUARY 2001

Fig.15.Grouping based on shape identity.In our model,we replaced

from

Fig.14 top left,by the curvature of the level curves of

(Fig.14 top right).Size

,

,

CHAN AND VESE:ACTIVE CONTOURS WITHOUT EDGES 277

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[21]

,Geodesic active regions for motion estimation and tracking,

INRIA RR-3631,1999.

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Tony F.Chan (M98) received the B.S.degree in

engineering and the M.S.degree in aerospace engi-

neering,both in 1973,from the California Institute

of Technology,Pasadena,and the Ph.D.degree

in computer science from Stanford University,

Stanford,CA,in 1978.

He is currently the Department Chair of the De-

partment of Mathematics,University of California,

Los Angeles,where he has been a Professor since

1986.His research interests include PDEmethods for

image processing,multigrid,and domain decomposi-

tion algorithms,iterative methods,Krylov subspace methods,and parallel algo-

rithms.

Luminita A.Vese received the M.S.degree in mathe-

matics fromthe University of Timisoara,Romania,in

1993 and the M.S.and Ph.D.degrees in applied math-

ematics,both fromthe University of Nice-Sophia An-

tipolis,France,in 1992 and 1996,respectively.Her

Ph.D.subject was variational problems and partial

differential equations for image analysis and curve

evolution.

She held teaching and research positions at the

University of Nice,Nice,France,(19961997),

the University of Paris IX,Paris,France,(1998),

and the Department of Mathematics,University of California (UCLA),Los

Angeles (19972000).Currently she is computational and applied mathematics

Assistant Professor at UCLA.Her research interests include problems of curve

evolution and segmentation using PDEs,variational methods,and level set

methods.

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