2008 10th Intl. Conf. on Control, Automation, Robotics and Vision
Hanoi, Vietnam, 17–20 December 2008
MRI Image Segmentation Based on Fast Global
Minimization of Snake Model
Thi-Thao Tran**, Po-Lei Lee*, Van-Truong Pham* and Kuo-Kai Shyu*
* Department of Electrical Engineering
National Central University
Chung-Li, 320, Taiwan, R.O.C
**Faculty of Electrical Engineering
Hanoi University of Technology
Hanoi, Vietnam
Abstract— Snakes, or active contours, have been
widely used to locate boundaries of image segmentation and
computer vision. Problem associated with the existence of the
local minima in the active contour energy function makes snakes
have poor convergence in segmentation process; therefore, the
poor convergence has limited applications. In this work, a fast
minimization of snake model is used for an MRI knee image
segmentation. This method provides a satisfied result. As a result,
it is a good candidate for MRI image segmentation approach.
To overcome above drawbacks, in this paper the fast
global minimization of snake model is used to segment the
MRI Image. The successful implementation of this paper
shows the advantages of the proposed method. It can be
considered as a good candidate for MRI image segmentation.
II. ACTIVE CONTOUR MODELS: SNAKES
1. Classical snakes
Keywords— MRI image segmentation, active contours,
snakes, global minimization
I.INTRODUCTION
Image segmentation is one of the most important steps in
image analysis and has been a key problem in computer
vision. The main aim of this process is to separate an image to
several regions or parts which correlate strongly the interested
objects. The level of segmentation depends on the problem
being solved; this work should stop when the interest objects
in our application have been isolated
In automated medical image processing, image
segmentation plays an important role. It is very useful for
doctor’s diagnoses, because the segmented area gives useful
information so that doctor can make decision easier and
quicker. Accordingly, medical image segmentation has been
obtained much attention.
Active contours, or “snakes”, which were originally
introduced by Kass et al in 1988 [1], are used extensively in
computer vision and image processing application, particular
to locating object boundaries [7]. Snakes are energyminimizing curves that deform under the influence of internal
forces within curve itself and external force derived from the
image to minimize the energy function. They are general
techniques of marching a deform model to an image using
energy minimization. Traditional snakes often converge to the
local minimum of the energy function of active contour. It is
sensitive to the initialization and has
difficult in the progressing into boundary concavities. The
classical snakes have some drawbacks such as the
segmentation result does not converge to the actual objects,
especially with the objects in MRI image.
c 2008 IEEE
978-1-4244-2287-6/08/$25.00 The concept of active contours was introduced in 1988 [1]
and has been developed by the many researchers. A classical
snake is a curve C ( s ) = [ x( s ), y ( s )], s ∈ [0,1] that moves thought
the spatial domain of an image I ( x, y ) to minimize energy
functional:
1
E snake = E snake (C ( s ))ds
³
0
1
= Eint (C ( s )) + Eimage (C ( s )) + E cont (C ( s )) ds
³
0
where:
Eint (C ( s )) : represents the internal energy of the
spline due to bending
Eimage (C ( s )) : represents the image forces
Econ (C ( s )) : represents the external constraint force
Internal Energy Eint
The internal energy can be written as:
2
2
Eint = (α ( s ) C s ( s ) + β ( s ) C ss ( s ) ) / 2
( 2)
or
2
Eint =
1
dC
d 2C
{α ( s)
+ β (s) 2
2
ds
ds
1
2
} = (α ( s) C ′( s ) + β ( s) C ′′( s) )
In this equation, the first-order term makes the snake act
like a membrane and the second-order term makes it act like a
thin plate
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ICARCV 2008
The internal energy is composed of first-order and secondorder terms. The first-order term, which controlled by α (s ) ,
adjusts the elasticity (tension) of the snake. The second-order
term, which controlled by β (s ) , adjusts the stiffness of the
snake
The external energy can be written as:
(3)
Snake drives this energy from the image so that external
forces deform snake toward the object’s borders. For example:
when given a gray-level image I ( x, y ) , typical external
energies are designed to lead snake as [10]:
1
E ext
( x, y ) = − ∇I ( x , y )
2
where Gσ ( x, y ) is a two dimensional Gaussian function with
standard deviation σ and ∇ is the gradient operator
Equation (1) can be rewritten as follow:
1
E snake =
1
³{ 2 (α (s) C ′(s)
2
2
+ β ( s ) C ′′( s ) ) + Eext (C ( s ))} ds (4)
0
A snake that minimizes E snake must satisfy the Euler
equation:
(5)
αC ′′( s ) − βC ′′′′( s) − ∇E ext = 0
We can express (5) as a force balance equation
1
Fint + Fext
=0
1
where Fint = αC ′′( s ) + βC ′′′′( s ) and Fext
= − ∇E ext
The internal force Fint regularizes the contour while the
1
external force Fext
drives the contour towards the desired
object boundary.
2. New version of snake model – Geometric Active Contour
(GAC)
Based on the classical snakes, Caselles et al in [2] and
Kichenassy et al in [3] proposed a new model of snakeGeometric Active Contour (GAC). The model is defined by
the following minimization problem:
L (C )
min{EGAC (C ) =
C
³ g ( ∇I (C (s)) )ds}
0
³ dl
0
g=
1
2
1 + ∇ Iˆ
ˆ
where I is a smoothed version of image I
In [2] authors used the calculus of variation and the gradient
descent method to find the minimum of EGAC in (6). The
result is:
G
G
∂C
= g ( I )κ N − (∇g.N )
(7)
∂t
G
where κ and N are the curvature and the normal to the curve
C respectively.
They also used level set approach with equation (7); so (7) can
be written in the level set form as follows:
2
2
E ext
( x, y ) = − ∇ (Gσ ( x, y ) * I ( x, y ))
L (C )
L (C ) =
So, the energy function in (6) is actually a new length
obtained by dl -Euclidean element of length and function g
which contains information concerning the boundaries of
object [2]. The function g is an edge indicator function. It has
values close to zero in the regions where the gradient of image
is high. For example in [16], authors choose:
External energy Eext
E ext (v( s )) = E image (C ( s )) + E con (C ( s ))
where ds is the length of the Euclidean element and L(C ) is
the length of the curve C defined:
∂φ
= g ( I ) κ ∇ φ + ∇ g ( I ) .∇ φ
(8)
∂t
where φ is the level set function embedding contour C . Thus
they had:
C (t ) ={ ( x, y ) ∈ R n φ ( x, y, t ) = 0 }
Although it has many good properties when using GAC
model in segmentation but the snake/GAC model is highly
sensitive to the initial of contour. In other words an improper
initial contour can give inaccurate result. The problem with the
initial contour relates to the non-convexity of the energy
function, EGAC , to be minimized and then the existence of the
local minimum. It prevents the segmentation of the objects
(parts) lying in images. So, choosing an image segmentation
model, which gives correct results but is independent of initial
contour, is necessary. Such method is to find a global
minimum of the energy function of GAC model.
III GLOBAL MINIZATION OF THE GAC MODEL
In this section, a new segmentation method based on the
Chan–Vese (C-V) model to determine the global minization of
standard GAC will be considered. The Active contours
without edges model proposed by Chan-Vese [4] is an
important segmentation model based on Mumford-Shah
method and level set approach. This model is not based on the
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image gradient such as classical snake model detecting the
boundaries of the object, but on the homogeneous regions.
The model of Chan-Vese (C-V model) [4] is as follow:
E 1 (φ , c1 , c 2 , λ ) =
E (c1 , c 2 , C ) = μ.length(C ) +ν .area(inside(C ))
+ λ1
+ λ2
2
³
inside (C )
I ( x, y ) − c1 dxdy
This equation is the gradient decent flow of the energy
function:
(9)
dxdy
(12 )
Ω
To carry out the global minimization of the segmentation task,
in [5], authors propose the energy function:
2
outside (C )
1, c 2 ) φ
Ω
I ( x, y ) − c 2 dxdy
³
³ ∇ φ dxdy + λ ³ r ( x, y, c
E 2 (u , c1 , c 2 , λ ) = TV g (u ) + λ r ( x, y , c1 , c 2 )dxdy
(13)
³
Ω
where μ ≥ 0, ν ≥ 0, λ1, λ2 >0 are fixed parameter. In [4],
Chan–Vese had most numerical calculations with λ1 = λ2 = λ
and ν = 0 . I ( x, y ) is the given image, constants c1 and c 2 are
averages of image I ( x, y ) inside and outside C respectively.
Define the curve C as the boundary of an open subset Ω C
of Ω. Denote the region Ω C by inside (C ) and the region
Ω \ Ω C by outside (C ) , let
The difference between (12) and (13) is that: Equation (13)
is based on the weighted total variation energy, TVg(u), which
is the function of u with a weight function g. Equation (13)
gives a link between C-V model and Snake model.
The energy equation (13) is a case of characteristic functions
and is expressed as bellow [5], [12]:
E 2 (u , c1 , c 2 , λ )
= TV g (u ) + λ r1 ( x , y , c1 , c 2 ) udxdy
³
Ω
Per (Ω C ) = μ.length(Ω C )
=
Equation (9) can be rewritten:
E (c1 , c 2 , Ω C ) =
+ λ1
+ λ2
So,
2
ΩC
I ( x, y ) − c1 dxdy
Ω \ ΩC
Note that:
³ ∇H ε (φ ) dxdy +
³ (H ε (φ ) ( I ( x, y) − c )
1
2
GAC (C )
E2
while
is
equivalent
approximating
to
minimizing
I ( x, y )
by
two
regions: Ω C and Ω \ Ω C with c1 and c2.
I ( x, y ) − c 2 dxdy
Ω
+
minimizing
C
By using Heaviside function with the regions:
Ω C and Ω \ Ω C the energy function E can be written,
according to level set function φ , as follow:
E ε (c1 , c 2 , Ω C ) =
− I ( x , y )) 2 − ( c 2 − I ( x , y )) 2 ) u dxdy (14 )
Ω
³ g.ds = G
(10)
2
³
1
C
Per (Ω C )
³
³ gds + λ ³ ( (c
+ (1 − H ε (φ ) ( I ( x, y ) − c 2 )) 2 )dxdy
Ω
where Ω is image domain and H ε is a slightly regularization
of Heaviside function H . Minimizing energy of (11) has the
form:
ª
º
∂φ
∇φ
= H ε' (φ ) « div (
) − λ (( I ( x, y ) − c1 ) 2 − ( I ( x, y ) − c 2 ) 2 )»
«
»
∂t
∇φ
r ( x , y .c1 , c 2 )
¬«
¼»
Using the result in [12], if we choose non-compactly
supported smooth approximation of the Heaviside function,the
steady state solution of the gradient flow (11) is the same as:
³ g.ds = G
GAC (C )
is the snake energy equation (6)
C
Energy function E 2 also provides a global minimum for
the active contour model [5]. It has a stationary solution if u
satisfies the condition: 0 ≤ u ≤ 1 . Minimization problem with
E2 for 0 ≤ u ≤ 1 has the same set of minimizers as
minimization of problem with E3 , where
E 3 (u , c1 , c 2 , λ , α ) = TV g (u ) + λ r1 ( x, y , c1 , c 2 ) + αν (u ) dxdy
³
Ω
(15)
E3 is not strictly convex that means any minimizer of E3 is
the global minimizer. We can use standard Euler-Lagrange
equation technique and explicit gradient descent based
algorithm to compute a global minimizer of E3 [13]. But this
way takes much computation since the regularization of the
TV-norm.
Based on [6], [7], [8], [9] in [5], the fast minimization,
based on Dual Formulation of the TV-norm, was proposed.
Consider the variational model:
∂φ
∇φ
= div(
) − λr ( x, y, c1 , c 2 )
∂t
∇φ
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V. CONCLUSION
min{ E3 (u, c1 , c 2 , λ , α ) = TV g (u ) + λ r1 ( x, y, c1 , c 2 ) +
³
u
Ω
+ αv(u ) dxdy }
The convex regularization of this model is as follow:
min{E 3r (u , v, c1 , c 2 , λ , α ) = TV g (u ) +
u ,v
+ λ r1 ( x , y , c1 , c 2 ) + αν (u ) dxdy }
1
u−v
2θ
2
L2
(16 )
³
Ω
θ is chosen to be small. Because E 3r is convex, its minimizer
can be computed by u, v separately. In [10], u and v are
determined as:
u = v − θdiv ( p )
p n + δ t∇(div ( p n ) − v / θ )
δt
1+
∇(div( p n ) − v / θ )
g ( x, y )
(17)
REFERENCES
and
v = min {max{u ( x, y ) − θ .λ.r1 ( x, y, c1 , c 2 )},1}
(18)
(16), (17), (18) are equations of fast global minimization of
snake model. This is a new numerical model that defines a fast
segmentation algorithm. And the following will give an
application in MRI image segmentation of this model.
IV. MEDICAL IMAGE SEGMENTATION BASED ON
FAST GLOBAL MINIMIZATION OF SNAKE MODEL
This section will show the possible application of the
segmentation, based on fast global minimization of snake
model, to an MRI knee image. The method was implemented
in Matlab program package. The successful segmentation
result given by Fig.1 shows the possibility in MRI image
segmentation.
(a)
(b)
ACKNOWGLEMENT
This study was supported by National Science Council of
Taiwan, R.O.C., under the grant (NSC) 96-2622-E-008-015CC3.
where
p n +1 =
In this paper, we have demonstrated the applicability of
global minimization of the snake model, for medical image
segmentation. This approach has some advantages over the
standard snake approach:
- Overcoming the existence local minimum in energy
function
- Increase the converge of the segmentation process
- Reducing time need to converge to solution
Experimental result shows that this approach is promising and
shows its ability in medical image application.
(c)
Fig. 1. Segmentation of a knee MR image
(a) the original image; (b) the results after 10 iterations;
(c) the result after 100 iterations.
Image Source: http://www.billingsmricenter.com
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