270 CHINESE OPTICS LETTERS / Vol. 5, No. 5 / May 10, 2007
Automatic detection of the belt-like region in an image
with variational PDE model
Shoutao Li (
)
1,2
, Xiaomao Li (
)
1,2
, and Yandong Tang (
üüü
)
1
1
Shenyang Institute of Automation, Chinese Academy of Sciences, Shenyang 110016
2
Graduate School of the Chinese Academy of Sciences, Beijing 100039
Received December 12, 2006
In this pap er, we propose a novel method to automatically detect the belt-like object, such as highway,
river, etc., in a given image based on Mumford-Shah function and the evolution of two phase curves. The
method can automatically detect two curves that are the b oundaries of the belt-like object. In fact, this is
a partition problem and we model it as an energy minimization of a Mumford-Shah function based minimal
partition problem like active contour model. With Eulerian formulation the partial differential equations
(PDEs) of curve evolution are given and the two curves will stop on the desired boundary. The stop term
does not depend on the gradient of the image and the initial curves can be anywhere in the image. We also
give a numerical algorithm using finite differences and present various experimental results. Compared
with other methods, our method can directly detect the boundaries of belt-like object as two continuous
curves, even if the image is very noisy.
OCIS codes: 100.0100, 150.0150.
The automatic detection of a belt-like region in an image
is important. Many important object features in natural
scenes or tissues of human body, such as rivers or blood
vessels, correspond to such regions. Several approaches,
like edge collection
[1]
, active contour methods
[2,3]
,andso
on, are widely applied in the object segmentation and
recognition. Approaches based on edge detection have
some drawbacks. In fact, up to date, the edge detec-
tion is most frequently used to extract thin-belt-like road,
river or other objects
[4,5]
. Edge detection as the essen-
tial part of these algorithms is sensitive to noises in an
image. Denoising and object enhancing of the noisy im-
ages, as image pre-processing in these approaches, are
always necessary for the correct object detection. When
the image edges are extracted based on image gray gra-
dient, they contain not only the boundaries of the object
to be detected but also the boundaries of the other ob-
jects. The belt-like object must be identified among the
detected edges, which usually include pseudo-edges and
much more other edges that do not belong to the object
to be detected. Due to the noises and the complex tex-
ture, the belt-like object usually cannot be detected as
a continuous edge. Therefore, the post-processing, such
as mathematical morphology transforms
[6]
or curve ap-
proximation, is also needed for the connectivity of edges
to form the whole boundary of the belt-like object to be
detected.
In 2001, based on Mumford-Shah segmentation
techniques
[7]
and the level set method, Chan and Vese
[3]
proposed the active contour model without edge, also
called C-V model, that can detect contours both with
[8]
or without gradient. In addition, in that model the initial
curve can be anywhere in an image. In the active contour
model, the contour to be detected is usually supposed to
be a closed curve, and with level set method it is repre-
sented as the zero level set of a function defined in higher
dimensions. It overcomes the drawbacks of the classic ac-
tive contour models (or snake models) that are based on
edge-detector, dependent on initial curve location and
sensitive to noise. Because the belt-like objects appear
in an image as two open curves that are not intersected,
with the C-V model some contours of other objects, es-
pecially some closed contours, are also detected. After
contour detection it is also required to recognize that
curve is the object boundary.
Based on Mumford-Shah model
[7]
and variational PDE
model and inspired by C-V model, with two phase curves
we proposed a novel method to automatically detect the
boundaries of the belt-like object in a given image. The
evolving curves are two open curves and do not need to be
represented as the zero level set of a function defined in
higher dimensions. It is a development of the technique
used in Ref. [9]. The initial curves, which are usually
chosen as two parallel straight lines, can be anywhere in
the image and they evolve and stop on the boundaries of
the belt-like object in an image, even for a noisy image.
In this way the belt-like object can be very well detected
and preserved. Compared with classic snake models
[10]
,
which can also be used in the detection of an open curve
boundary with their initial curves being near the bound-
aries of the object in noisy image, our model is more
robust against image noise and its initial curve can be
anywhere.
The Mumford-Shah piecewise smooth segmentation is
defined by
[7]
inf
u,Γ
E
MS
[u, Γ|u
0
]=
Ω
|u − u
0
|
2
dx
+μ
Ω\Γ
|∇u|
2
dx + ν · L(Γ), (1)
where u
0
: Ω →is a given image, μ and ν are the
positive parameters. The solution image u obtained by
minimizing this function is formed by smooth disjoint re-
gions R
i
and with their boundaries denoted by Γ, where
i
R
i
=Ω\Γ. L(Γ) represents the length of Γ. It allows
1671-7694/2007/050270-04
c
2007 Chinese Optics Letters
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