Active Contours Without Edges

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Nov 5, 2013 (3 years and 9 months ago)

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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,MumfordShah 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 MumfordShah Functional
The MumfordShah 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 EulerLagrange 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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Tony F.Chan (M98) 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,(19961997),
the University of Paris IX,Paris,France,(1998),
and the Department of Mathematics,University of California (UCLA),Los
Angeles (19972000).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.