Image Segmentation using Active Contour

with Level Set

Mandeep Kaur
Electronics & Electrical Engineering Department
Lovely Professional University, Punjab, India. 144402
kaurmandeep1190@gmail.com
model-based methods, edge-based technique,
region-based methods, watershed technique and
Abstract – In this paper a novel algorithm for image
active contour methods.
segmentation has been proposed based on active
contour model and level set. In this we use the signed II.ACTIVE CONTOUR MODEL
pressure force function using local information of the
image to be segmented. First the level set function is We have used the active contour model in oyr
selectively penalised to be binary n then a Gaussian paper for the segmentation process. In this
kernel is applied for smoothing. Thus, this model can technique the user suggest an initial contour. This
work with heterogeneous images. In addition, by framework attempts to minimize an energy
taking the advantages of Geodesic active contour associated to the current contour as a sum of an
(GAC) and Chan-Vese (C-V) model, the method
internal and external energy:
could deal with objects even with discrete blur
boundaries and gives exact results in detecting object
boundaries. One of its other advantages over C-V E Snake=E∫ ¿+E (2)¿
ext
method is that the cost of re initialisation is also
reduced in this method as we do not need re- The external energy is supposed to be minimal
initialisation of the level set function. Experimental when the snake is at the object boundary position.
results demonstrate that the proposed model is
The internal energy is supposed to be minimal
effective in segmenting biomedical images and the
images with blur and weak edges. when the snake has a shape which is supposed to be
relevant considering the shape of the sought object
Keywords: Image Segmentation, Active Contours, [1][2].
Snakes, Level Sets, GAC, C-V method.
Different ACM approaches
I.INTRODUCTION
The basic purpose of segmentation is to identify  Snakes
and isolate the region of interest from the given  Geometric active contours
image. Image segmentation is used to locate
boundaries and objects in image [1] [3][13]. In  Geodesic active contours
other words we can say that it is the process of A Snakes
assigning a label to every pixel in an image such
that pixels with the same label share certain visual Snakes model was introduced by Michael Kass,
characteristics. The problem of image segmentation Andrew Witkin and Demetri Terzopoulos in 1988
can be formulated as follows. [1], for interactive interpretation of images, in
Given image I = {pi}, a complete segmentation which user-imposed constraint forces guide the
problem is to determine connected subset Ri = (Ri snake near features of the interesting image. They
∈ I), such that stated that a snake is an energy-minimizing spline
guided by external constraint forces and influenced
¿ i Ri =I , Ri ∩ R j=φ ¿ by image forces that pull it toward features such as
lines and edges. Snakes are greatly used in
Segmentation is based on homogeneity of the applications like object tracking, shape recognition,
image characteristics such as intensity, colour, segmentation, edge detection, stereo matching. A
texture, or the combination of all these information simple elastic snake is thus defined by
[14].
There are number of different approaches in image  a set of n points
segmentation such as: histogram-based method,  an internal elastic energy term
 an external edge based energy term
The snakes model try to segment the image based The functional state that curves segmenting the
on the following energy: object should try and surround it with a minimal
weighted arc length. This can be given a physical
E Snake=E∫ ¿+E ¿ (3) interpretation: We are looking for the trajectory of
ext
a particle on a map, where the potential energy at
2 2 each point is g(I ), and we assume the particle's
where, ∫ α |c '| + β|c' '| ds trajectory should form a closed simple curve. The
(4) potential energy of the particle is given by

Snakes are autonomous and self-adapting in their

search for a minimal energy state. They can be
easily manipulated using external image forces. u (c) = - λ g (l)2 (9)
They can be made sensitive to image scale by
incorporating Gaussian smoothing in the image Snakes is the classical model in active contour
energy function. They can be used to track dynamic method [1], [3]. It gives an efficient framework for
objects in temporal as well as the spatial image segmentation, but it cannot change the
dimensions. But some of the drawbacks are that topology in the process of segmentation process.
they can often get stuck in local minima states; this To overcome this drawback, the level set method
may be overcome by using simulated annealing was proposed in order to isolate shapes from their
techniques at the expense of longer computation background. Since then, the active contour with
times. They often overlook minute features in the level set methods has been widely applied to image
process of minimizing the energy over the entire segmentation in the fields of computer vision and
path of their contours. image processing. GAC is an edge based active
contour model that works well when the objects
B. Geometric active contours and background in segmented image are
heterogeneous, but we cannot get satisfied results
Geometric active contours attempt to segment an when dealing with object with discrete/ blur
object based on its edges, in a level-set framework boundaries or noise. Whereas Chen-Vese (C-V)
[4]. The initial contour is chosen to include the model is a region-based active contour models,
object. The contour evolves according to which instead of using the image gradient, use the
statistical information inside and outside the initial
C t=g ( l )k N (5) curve to evolve the contour towards the boundary
of desired object. It give better performance while
compared with the edge-based model such as the
Where g ( … … .. ) is a function which should ability to work well with the object having
drop to zero at edges. The contour evolution tends blur/weak boundary, and less sensitive to the initial
to smooth the contour, if no other information is position of the contour. In order to get the
available. The contour according to this evolution advantages of both GAC and C-V model, some
will shrink to a point. Hence, a balloon force may hybrid model are proposed. Recently, Zhang [12]
be added proposed the sign pressure force in an active
contour model. This model takes the advantages of
GAC and C-V models and could optionally select
C t=( g(l)¿ ¿ k−β )N (6)¿
local or global segmentation.
C. Geodesic active contours III. REFERENCE MODELS
The choice of a balloon force is arbitrary. It is not A.GAC Model
clear if we actually minimize some functional, and
the global minimizer is not clear either [5]. The Geodesic Active Contour model is an enhanced
geodesic active contour tries to remedy this by version of the classical snakes in [1]. This model
minimizing the following weighted length depends on the image gradient to stop the evolution
functional: of the curve. Thus it surpasses the regions
∫ g( l) ds( 7) having blur or weak boundaries. Let Ω be a
bounded open subset of R2 and I: [0,a] x [0,b] →
g(l) constitutes an (inverse) edge indicator. For R+ be a given image. Let C(q):[0,1] → R2 be a
example, parameterized planar curve in Ω. The GAC model
1 is formulated by minimising the energy term given
g ( l )= 2
(8) below:
√|∇ l| +∈
EGAC ( C )=∫ g (|∇ I ( C ( q ) )|)|C ' ( q )|dq ¿10)
where g is the ESF. The corresponding level set ∂φ
formulation for GAC is given as: =∂ ( φ ) [ μ ∇ к−v− λ1 ( I −c1 )2 + λ 2 ( I −c2 )2 ] ,(15)
∂t
∂φ where μ ≥ 0 , v ≥ 0 , λ1 >0 , λ2 >0 are fixed
=g ( к + α )|∇ φ|+ ∇ g . ∇ φ(11)
∂t parameters, μ controls the smoothness of zero level
where α is a real constant called balloon force to set, v increases the propagation velocity and λ 1 and
control the expanding and velocity, к is the λ2 controls the image data drive force inside and
euclidean curvature of curve C, and g is the edge outside the contour respectively.∇ is the gradient
based function. operator. H(φ ) is the heaviside function and ∂( φ)
is the dirac function. The regularised version is
B. C-V Model selected as follows:

Chan and Vese proposed an active contour without

1 2 z
( ( ))
{
edges that can be seen as a special case of H ε ( z )= 1+ arctan
Munford-Shah problem. This model utilizes the 2 π ε
homogeneity information of the object as a term in (16)
1 ε
the energy function unlike the GAC model which
relies on the image gradient. In this method the
image is considered to be including two regions i.e
(
δ ε ( z) = . 2 2 , z ∈ R
π ε +z )
c1 and c2 inside and outside the curve C
IV. HYBRID MODEL
respectively. For a given model Ω, the C-V model
is formulated by minimizing the following energy In our hybrid model first of all we calculate the
functional: SPF (Signed Pressure Force) function. Its value
❑ ❑ always lies in the range [-1,1]. It is used to control
2 2 the motion of the contour by modulating the sign of
ECV =λ 1 ∫ |I ( x )−c 1| d+ λ2 ∫ |I ( x ) −c 2| dx
¿(C) out (C)
the pressure force, so that the contour shrinks when
outside and expands when it is inside the region of
(12) interest.

where c1 and c2 are the average intensities inside c1 + c2

and outside the contour, respectively. With the I ( x )−
2
level set method we assume spf ( I ( x ) )= , x ∈Ω
c 1+ c2
C={ x ∈ Ω :φ ( x )=0 } , |
max ⁡( I ( x )−
2
) |
inside ( C )= { x ∈Ω :φ ( x )> 0 } , (17)

where c1 and c2 are defined in Eq.(13) and (14),

outside (C )= { x ∈Ω :φ ( x ) <0 } , respectively.
By minimizing Eq.(6), we get the values for c 1 and The significance of Eq. (17) can be explained as
c2 as: follows. Refer to Figure 1, we assume that the
❑ intensities inside and outside the object are
homogeneous. It is intuitive that
∫ I ( x ) . H ( φ ) dx Min ( I ( x ) ) ≤ c 1 ,c 2 ≤ Max (I ( x)) and the equal
c 1 ( φ )= Ω ,(13)
❑ signs cannot be obtained simultaneously wherever
∫ H ( φ ) dx the contour is. Hence, there is
Ω

❑
c1 +c 2
∫ I ( x ) . ( 1−H ( φ ) ) dx Min ( I ( x ) )< < Max ( I ( x ) ) , x ∈Ω
c 2 ( φ )= Ω ,(14) 2
❑

∫ ( 1−H ( φ ) ) dx (18)
Ω

The corresponding variational level set formulation

for this is:
 −ρ x ∈Ω 0−∂ Ω0
φ ( x ,t=0 )=
{ 0 x ∈∂ Ω0
ρ x ∈ Ω−Ω 0
(21)

where ρ > 0 is a constant, Ω0 is a subset in the

image domain Ω and ∂ Ω0 is the boundary of Ω0 .
2. Compute c 1 ( φ )∧c2 (φ) using Eqs. (13) and
(14), respectively.
Fig 1 3. Evolve the level set function according to Eq.
(20).
Substituting the SPF function in Eq. (17) for the 4. Let φ = 1 if φ > 0; otherwise, φ = -1.
ESF in Eq. (11), the level set formulation of the This step has the local segmentation property. If we
proposed model is as follows: want to selectively segment the desired objects, this
step is necessary; otherwise, it is unnecessary.
∂φ ∇φ 5. Regularize the level set function with a Gaussian
∂t (( ) )
=spf ( I ( x ) ) ¿
|∇ φ|
+ α |∇ φ| filter, i.e φ=φ∗G σ .
6. Check whether the evolution of the level set
function has converged. If not, return to step 2.
+ ∇ spf ( I ( x ) ) . ∇ φ , x ∈ Ω(19)
V. EXPERIMENTS AND RESULTS
In the traditional level set methods, the level set We have done experiments with both , the natural
function is initialized to be an SDF to its interface as well as biomedical images. Fig 1 shows the
in order to prevent it from being too steep or flat results on natural image with both the reference
near its interface, and re-initialization is required in model and the hybrid model.
the evolution. Unfortunately, many existing re-
initialization methods have an undesirable side
effect of moving the zero level set away from its
interface. Furthermore, it is difficult to decide when
and how to apply the re-initialization. In addition,
re-initialization is a very expensive operation. To
solve these problems, we propose a novel level set
method, which utilizes a Gaussian filter to
regularize the selective binary level set function
after each iteration. The procedure of penalizing
level set function to be binary is optional according
to the desired property of evolution. If we want
(a)
local segmentation property, the procedure is
necessary; otherwise, it is unnecessary.
Since we utilize a Gaussian filter to smooth the
level set function to keep the interface regular, the
regular term ¿( ∇ φ /|∇ φ|)∨∇ φ∨¿ is
unnecessary. In addition, the term ∇ spf . ∇ φ in
Eq. (19) can also be removed, because our model
utilizes the statistical information of regions, which
(b) (c)
has a larger capture range and capacity of anti-edge
leakage. Finally, the level set formulation of the
proposed model can be written as follows:

∂φ
=spf ( I ( x ) ) . α |∇ φ| , x ∈Ω(20)
∂t
The main procedures of the proposed algorithm are
summarized as follows: (d) (e)

1.Initialize the level set function / as Fig 2.Segmentation results on a natural image, (a)
original image (b) & (c) results by the reference
model after 200 and 500 iterations respectively, (d)
& (e) results from the hybrid model after 100 and Contour Snake and Level Set in Biomedical Applications”,
2011 IEEE International Conference on Bioinformatics
200 iterations. and Biomedicine.
[11] Arie Nakhmani and Allen Tannenbaum “Self-Crossing
Detection and Location for Parametric Active Contours”,
IEEE Transactions on Image Processing,VOL. 21, NO. 7,
July 2012.
[12]Satoshi Urata, Hiroshi Yasukawa “Improvement of
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[15] Shuqian He, Jiangqun Ni, Lihua Wu, Hongjian Wei ,
VI. CONCLUSION Sixuan Zhao.," Image threshold segmentation method with
In this paper a hybrid region based model is 2-D histogram based on multi-resolution analysis",
proposed with selective local and global image Computer Science & Education, ICCSE, 25-28 July 2009,
segmentation. By taking the advantage of the GAC PP.753 – 757, Nanning, China
and C-V model, this method can effectively detect [16]Gang Liu, Robert M. Haralick, "Assignment Problem in
and segment the regions with weak or blur edges. Edge Detection Performance Evaluation," cvpr, vol. 1,
pp.1026, 2000 IEEE Computer Society Conference on
The model is able to change the topology as the Computer Vision and PatternRecognition(CVPR'00) –
evolution flow of model is derived using level set Volume1, 2000.
method. Experiments are performed on both the
natural as well as biomedical images.

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