Research Journal of Applied Sciences, Engineering and Technology 4(24): 5416-5422, 2012
ISSN: 2040-7467
© Maxwell Scientific Organization, 2012
Submitted: March 18, 2012
Accepted: April 13, 2012
Published: December 15, 2012
Evaluation of Level Set Segmentation for Medical Images with
Intensity Inhomogeinity
1
M. Renugadevi, 1V. Vaithiyanathan, 1K.R. Sekar and 2N. Raju
1
School of Computing,
2
School of EEE, SASTRA University, Thanjavur, India
Abstract: Image segmentation is an important area in the medical image guide surgery. Major advances in the
field of medical imaging provide richer information in clinical applications and support the advancement in the
biomedical knowledge. With the growing research on image segmentation, it has become crucial challenge for
the images with inhomogeneity in intensity. Level set is the most powerful and broadly used segmentation
technique in the medical image processing. This study aims at making a review on the current level set methods
such as Region Scalable Fitting (RSF), Statistical and Variational Multiphase Level Set (SVMLS) and Local
Clustering based Variational Level Set (LCVLS) developed for intensity inhomogeneous medical image
segmentation. Experiments that apply these algorithms to segment the medical images are presented to highlight
the distinct characteristics of each method. Results prove that the LCVLS method is most suitable and accurate
for intensity inhomogeneous medical image segmentation.
Keywords: Biomedical imaging, Intensity Inhomogeneity (IIH), level set method, segmentation
INTRODUCTION
Medical image analysis plays a crucial role in many
image processing applications. It aids in developing
computational algorithms and methods that helps to
analyze biomedical data in various clinical applications
such as diagnosis, study of anatomical structure,
computed integrated surgery and treatment planning. The
current research on the biomedical imaging focuses on the
segmentation which facilitates the visualization of
medical data and delineation of anatomical structures.
Various techniques such as threshold based methods,
watershed transformation based methods and partial
differentiation equation based methods have been
employed so far for image segmentation problems.
Among all these techniques, active contour method has
been proved to be an efficient framework in the past two
decades.
Active contour model is a framework for marking out
the object outline from the target image driven by the sum
of internal and external energy (Kass et al., 1988; Blake
and Isard, 1998; Paragios and Deriche, 2000). Edge based
methods (Xu and Prince, 1998; Caselles et al., 1993;
Kichenassamy et al., 1996) and Region based methods
(Ronfard, 1994; Mumford and Shah, 1989; Chan and
Vese, 2001; Vese and Chan, 2002; Paragios and Deriche,
2002; Caselles et al., 1997; Tsai et al., 2001; Li et al.,
2007; Zhang et al., 2010a, b) are the two kinds of the
existing active contour models. Edge based active contour
methods are suitable for segmenting the images with well
defined boundaries by utilizing the image gradient to
evolve the contour. Region based active contour model
builds the region descriptor to guide the motion of the
contour and has better performance than the former model
by segmenting the images having weak boundaries.
Implicit implementation of the active contour model is
represented as level set which is one of the most attractive
and popular region based method (Sethian, 1999). It has
been introduced by Osher and Sethian. (1988) and is
defined as a numerical technique for capturing dynamic
interfaces and shapes. It is widely used because of its
advantageous properties such as topology adaptability and
robustness to initialization.
Nowadays segmentation is a compelling challenge
due to the presence of Intensity Inhomogeneity (IIH) in
the images. It often exists in the ordinary or medical
images due to various factors such as imperfection of
imaging devices, static field inhomogeneity, non-uniform
daylight and artificial illumination. In recent times, many
level set algorithms (Li et al., 2007, 2008, 2011; Zhang
et al., 2010a, b) are used to segment the non-uniform
intensity homogeneous images especially medical images.
Yun-Jen (Chiu et al., 2010) explains the three active
contour models and concludes that RSF model (Li et al.,
2008) serves as good one for heterogeneous medical
image segmentation among them. So in this study, we
review three recently proposed level set methods to
segment the intensity inhomogeneous medical images.
They are Region Scalable Fitting (RSF) energy method
proposed by Li et al. (2008), Statistical and Variational
Corresponding Author: M. Renugadevi, School of Computing, SASTRA University, Thanjavur, India
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Res. J. Appl. Sci. Eng. Technol., 4(24): 5416-5422, 2012
2
i =1
⎛
⎛ ∇φ ⎞
⎛ ∇φ ⎞ ⎞
⎟⎟ + µ ⎜ ∇ 2φ − div⎜⎜
⎟⎟ ⎟
div ⎜⎜
⎜
φ
∇
⎝
⎠
⎝ ∇ φ ⎠ ⎟⎠
⎝
Region-scalable fitting model: Li et al. (2008) proposed
the Region-Scalable Fitting (RSF) energy method to
segment the medical images with intensity
inhomogeneities. It performs well for the images with
weak object boundaries. Let K: Un6[0, +4) be a
nonnegative kernel function with a localization property
which is defined as 8(u) $ 8(v), if |u|<|v| and lim |u|÷48(u)
= 0 plays a key role in this model. Let C be a closed
contour in the image domain S dUn, that separates S into
two regions S1 (outside (C)) and S2 (inside (C)). For a
given point x 0 S, the local intensity fitting energy is
defined as:
σ
2
Ωi
F (φ , f 1 , f 2 ) = ε ε (φ , f 1 , f 2 ) + µP(φ )
where, : is a positive constant and
2σ
(2)
2
ε (C, f 1 ( x ), f 2 ( x )) = ∫ ε xFIT (C , f 1 ( x ), f 2 ( x))dx + v C
(3)
Which is converted to a level set formulation with the
Heaviside function H as in Eq. (4) where M1(K) = H(K),
M2(K) = 1! H(K) and the last term in Eq. (4) computes
the length of the zero level contour. Here the Heaviside
function H is approximated:
ε (φ , f 1 , f 2 ) = ∑ λi ∫ ∫ κ σ ( x − y ) I ( y ) − f i ( x ) M i (φ ( y ))dy
i =1
dx + v ∫ ∇ H (φ ( x ) ) dx
(
)
2
1
∇ φ ( x ) − 1 dx .
2
ei ( x ) = ∫ κ σ ( x − y ) I ( x ) − f i ( y ) dy , i = 1, 2 and
2
f i ( x) =
κ σ ( x ) * ⎣ M ε i (φ ( x )) I ( x ) ⎦
κ σ ( x ) * M ε i (φ ( x ))
, i = 1, 2
The first term is the data fitting term which is
responsible for driving the active contour, the second term
has smoothing effect on the zero level contour to maintain
the regularity of the contour and the third term is called
level set regularization term.
n
(
P(φ ) = ∫
(8)
2
The energy functional is defined for a contour C as:
2
(7)
The energy functional in Eq. (4) is now approximated by
replacing H with HJ and it can be expressed as in Eq. (5)
where Mg1(K) = Hg (K) and Mg2(K) = 1 ! Hg (K). The
energy functional along with a level set regularization
term P(N ) is defined as:
where, 81 and 82 are positive constants and f1(x) and f2(x)
are two values that approximate image intensities in S1
and S2 respectively. Equation (1) is called a RegionScalable Fitting (RSF) of the contour C at a point x since
the intensities I(y) in above model are in the local region
centered at the point x and its size is controlled by the
kernel function:
e− u
1⎡
2
1
ε
⎛ x⎞ ⎤
1 + atc tan⎜ ⎟ ⎥ δε = Hε' ( x ) =
⎝ ε⎠ ⎦
2 ⎢⎣ π
π ε 2 + x2
The standard gradient descent method is used to minimize
this energy functional with respect to N by solving the
following gradient flow in Eq. (8) where:
ε xFIT (C, f 1 ( x ), f 2 ( x )) = ∑ λi ∫ κ ( x − y) I ( y ) − f i ( x ) dy (1)
2
(6)
by a smooth function defined as HJ and its derivative is
defined as *J:
Hδ =
1
(2π ) n 2
) (5)
∂φ
= − δε (φ )( λ1e1 − λ2e2 ) + vδε (φ )
δt
This section presents and illustrates the effectiveness
of each method for segmenting the intensity
inhomogeneous medical images. These methods are
mainly based on the level set technique but used in its
distinct way. The level set algorithm of each method is
explained as follows:
κ=
2
dx + v ∫ ∇ H (φ ( x ) ) dx
METHODOLOGY
i =1
(
ετ (φ , f 1 , f 2 ) = ∑ λi ∫ ∫ κ σ ( x − y ) I ( y ) − f i ( x ) M iτ (φ ( y ))dy
Multiphase Level Set (SVMLS) method proposed by
(Zhang et al., 2010b) and Local Clustering based
Variational Level Set (LCVLS) proposed by (Li et al.,
2011).
2
) (4)
SVMLS model: Zhang et al. (2010b) proposed a novel
Statistical and Variational Multiphase Level Set (SVMLS)
method to simultaneous intensity inhomogeneous image
segmentation and bias correction. With a function I : S ÷
Udefined on a continuous domain S, the given intensity
inhomogeneous image is modeled as:
I(x) = b(x) J(x) + n (x), x , S
(9)
where, I(x) is the measured image, b(x) is spatially variant
bias field, J(x) is the true signal that is assumed as
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piecewise constant and n(x) is assumed to be Gaussian
distributed with zero mean and variance F2 . To accurately
model the statistical characteristics of the image, the
intensity corresponding to the object domain Si is
described as:
(
⎛ ( I ( y ) − b( x ) c ) 2 ⎞
i
⎟,y ∈Ω
exp⎜ −
2
⎜
⎟
σ
2
2πσ i
i
⎝
⎠
)
p I ( y) αi =
i =1 Ω Ωi
1
∑ I ( y σi )
=
mi ( x ) y∈Ω i ∩ Ox
i =1 Ω
⎛
where, d i ( y) = Ω∫ κ p ( x, y )⎜⎝ log( 2πσ i ) +
(
E (α )∆ − ∫ log p( D α ) dx
(13)
which is integrated with the likelihood function as:
i =1
{( )
D = I x α i , i = 1,..., N
N
∏
i = 1 y ∈Ω i ∩ Ox
},
( ( ))
p I y αi
(14)
(
)(
)
(18)
[
]
[
]
I = bJ +n
(20)
Equation (13) is called
Statistical and Variational Multiphase Level Set
(SVMLS). Let us consider the characteristic function of
region Ox as:
⎧⎪ 1, y − x ≤ ρ
else
⎩⎪ 0,
)
LCVLS model: Li et al. (2011) proposed a local
clustering based variational level set framework for
simultaneous heterogeneous image segmentation and bias
correction. Let an image I is considered as a function I :
S ÷U defined on a continuous domain S. And to deal
with intensity inhomogeneities, it is modeled as:
i = 1 Ω Ω i ∩ Ox
( ( )) ∝ ∏
)(
where, *(N) is the Dirac function. The smoothness of the
bias field is ensured by the normalized convolution.
Ω
N
)
⎧ ∂ φ1
⎪ ∂ t = − (d 1 − d 2 − d 3 + d 4 ) H (φ2 ) + d 2 − d 4 δ (φ1 )
⎪
⎨
⎪ ∂ φ2 = − (d − d − d + d ) H (φ ) + d − d δ (φ ) (19)
1
2
3
4
1
3
4
2
⎪⎩ ∂ t
The energy functional is defined as:
p( D α ) = ∏ p I x α i
(
The energy functional E{b, ci, Fi} is minimized with
respect to each variable by fixing other variables. The
derived gradient decent by minimizing the energy
functional ESVLMS K N ," is as follows:
y ∈Ωi ∩ Ox
N
( I ( y ) − b( x )cu ) 2 ⎞
⎟ dx
2σ i2
⎠
⎧ M = H (φ ) H (φ )
M 2 = H (φ1 ) 1 − H (φ2 )
1
2
⎪ 1
⎨
⎩⎪ M 3 = 1 − (φ1 ) H (φ2 ) M 4 = 1 − H (φ1 ) 1 − H (φ2 )
σ2 ⎞
∑ ∫ ∫ log( p( I ( y αi )))dy dx
(17)
In the energy minimization for four-phase case, Mi is
defined with the Heaviside function H(N) as:
(11)
⎛
κ ρ ( x, y) = ⎨
(16)
N
∏ p( I ( y αi )) ≈ p( I ( x αi ))mi ( x ) ∝ N ⎜⎜⎝ bci , mi (i x) ⎟⎟⎠ (12)
where,
( I ( y ) − b( x ) cu ) 2 ⎞
⎟⎟ dydx
2σ i2
⎠
E SVLMS Φ N , α = ∑ ∫ d i ( y ) M i (Φ N ( y ))dy
where, mi(x) = ||Si 1 Ox|| and I ( x σ i ) is normally
distributed. The product of the Gaussian probability
density function is still Gaussian, so that:
= cons tan t −
)
2πσ i +
Let (Mi (KN (.)) be the characteristic function of
region Si , where KN (.)is a function of set{Ki , i = 1 ,...
n}. Then Eq. (13) can be rewritten as:
(10)
mapping T: I ( x α i ) → I ( x σ i ) from the original image
intensity domain D(T) to another domain U(T) is defined
as:
( )
⎛
∫ ∫ κ p ( x − y)⎜⎜⎝ log(
1
where, Fi is a standard deviation of intensity, b(x)ci is
spatially varying local mean and ai = {b ,ci , Fi }. To
formulate the energy functional, a circular neighborhood
center is considered for each position x , S,
D. A
i . e . , Ox = { y y − x ≤ ρ} w i t h
radius
I x σi
N
E (α )∑
(15)
By eliminating the trivial constant term and with Eq.
(10) and (15) the energy functional E (") can be rewritten
as:
where, J is the true image which is assumed to be
piecewise constant since it measures an intrinsic physical
property of the objects, b is a bias field component that
denotes the intensity inhomogeneity and assumed to be
varying slowly and n is additive noise that is assumed to
be zero-mean Gaussian noise.
To derive a local intensity clustering property, a
circular neighborhood with a radius D centered at each
point y , S is defined by Oy ∆ { x: x − y ≤ ρ} . The partition
of the neighborhood Oy is induced by the
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Res. J. Appl. Sci. Eng. Technol., 4(24): 5416-5422, 2012
partition{Si}Ni = 1 of the entire domain S.Then specifically
for classifying the intensities I(x) in the neighborhood Oy
, a clustering criterion is defined as:
ε y = ∑ ∫ K ( y − x ) I ( x ) − b( y )ci dx
N
⎛
N
⎞
2
N
(
)
i =1
2
(27)
where, L(N) and Up(N) are the regularization terms
defined below. The term L(N) that computes the arc
length of the zero level contour of N is defined by:
(28)
The term Up(N ) is defined by:
ℜ
p
(φ ) = ∫ p( ∇ φ )dx
(29)
(23)
(24)
that forms a partition of the domain S. The level set
formulation of the energy with N = 2 and N>2, called two
phase and multiphase formulation, respectively. Here only
the two phase formulation is considered.
In the two case level set formulation, the regions S1
and S2 can be represented with their membership function
defined by M1(N) = H(N) and M2(N) = 1!H(N)
respectively, where H is the Heaviside function [19].Thus,
the energy functional can be expressed as:
ε = ∫ ∑ ∫ K ( y − x ) I ( x ) − b( y)ci dy Mi (φ ( x ) )dx
F (φ , c, b) = ε (φ , c, b) + vL(φ ) + µℜ p (φ )
L( φ ) = ∫ ∇ H (φ ) dx
where, a is a normalization constant such that ∫ K (u) = 1 ,
F is the scale parameter of the Gaussian function and D is
the radius of the neighborhood Oy. The above defined
energy functional is in terms of the regions S1, ..., SN. To
derive the solution for the energy minimization problem,
the energy is converted to a level set formulation. Let N:S
6 U be a level set function with two disjoint region:
Ω 1 { x:φ ( x ) > 0} andΩ 2 { x:φ ( x ) < 0}
(26)
The variational level set formulation uses the above
defined energy as the data term which is defined by:
(22)
Here the kernel function K is truncated Gaussian
function defined by:
⎧1
2
⎪ e − u / 2σ 2 , for u ≤ ρ
K ( u) ⎨ a
⎪⎩ 0, otherwise
i =1
2
(21)
where, K(y-x) is kernel function (nonnegative window
function), such that K(y-x) = 0 for x ó Oy . This is called
local clustering criterion function that evaluates the
classification of the intensities in the neighborhood Oy .
The smaller value of gy yields the better classification. So
to minimize the gy with respect to y over the image
domain S , an energy is defined as:
ε ∆ ∫ ⎜ ∑ ∫ K ( y − x ) I ( x ) − b( y )ci dx⎟ dy
⎝ i =1 Ωi
⎠
N
where, ei ( x ) = ∫ K ( y − x ) I ( x ) − b( y )ci dy
2
i =1 Ω
ε (φ , b, c) = ∫ ∑ ei ( x ) M i (φ ( x ) ) dx
(25)
For convenience, the constants c1, ..., cN can be
represented as a vector c = (c1, ..., cN ). Thus the variables
of the energy functional g are the level set function g, the
vector c and the bias field b. So the energy functional g
can be written as:
where, p is a potential or energy density function p:
[0,4)6U such that p(s) = (1/2)(s!1)2. With this potential
p, the energy term Up(N ) is minimized by maintaining
∇ φ = 1 which is called signed distance property. And so
Up(N ) is called distance regularization term (Li et al.,
2010).
The energy F(N, c, b) is minimized by an iteration
process with respect to each of its variables (Li et al.,
2011). Thus solution for the energy minimization problem
is achieved by minimizing this energy term that results in
the image segmentation given by the level set function N
and estimation of the bias field b.
EXPERIMENTAL RESULTS
The performance evaluation of the three IIH level set
segmentation methods are demonstrated in this section.
The intensity inhomogeneous medical images are
employed to compare the ability of these methods. The
comparison is based on the segmented result, number of
iterations desired to complete the segmentation and the
CPU processing time for the contour evolution. These
measures are commenced to pick up the suitable model
for the IIH medical image segmentation.
The segmentation results obtained using RSF,
SVMLS and LCVLS methods on the heterogeneous MRI
slice image of the brain are shown in Fig. 1. For the level
set evolution, the initial contour is located at the same
position in the image for all the three methods. The curve
evolution of the RSF model is shown in Fig. 1a. From
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Res. J. Appl. Sci. Eng. Technol., 4(24): 5416-5422, 2012
Processing time
25
(a)
20
15
10
5
0
RSF
SVMLS
LCVLS
Models
(a)
Processing time
(b)
(c)
Fig. 1: Shows the segmented results of three models in MRI
brain image. Contour evolution of RSF, SVMLS and
LCVLS methods are shown in, (a), (b) and (c),
respectively.
50
45
40
35
30
25
20
15
10
5
0
SVMLS
RSF
LCVLS
Models
(b)
Fig. 3: Shows the processing time and number of iterations
charts for MRI image
(a)
Table 1:Shows the performance analysis of three models using MRI
brain image
Parameters
---------------------------------------------------------Models
Processing
Time Iteration
RSF
160.975
400
SVML
S72.493
70
LCVLS
9.563
30
(b)
Table 2: Shows the performance analysis of three models using human
vertebra phantom image
Models
Parameters
---------------------------------------------------------Processing
Time Iteration
RSF
22464
50
SVMLS
23.7434
20
LCVLS
0.7176
10
(c)
Fig. 2: Shows the segmented results of three models in cross
sectional image of a human vertebra phantom. Contour
evolution of RSF, SVMLS and LCVLS methods are
shown in (a), (b) and (c) respectively.
the Fig. 1, we analyze that the RSF model can’t complete
its contour evolution even after 2000 iterations. Also, it
does not segment the inner regions of the image. The
SVMLS segmentation result shown in the Fig. 1b is not
perfect since it fails to capture the interior parts of the
MRI image. The satisfied and accurate segmented result
of the LCVLS model is shown in Fig. 1c.
Figure 2 is another example to validate the
effectiveness of three models using the cross sectional
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Res. J. Appl. Sci. Eng. Technol., 4(24): 5416-5422, 2012
ACKNOWLEDGMENT
Iteration
180
160
The authors thank Dr. B. Shanthi, Professor of CSE
Department, SASTRA University for her imputing
discussions on this manuscript. We thank Dr. N. Sairam,
Professor of CSE Department, SASTRA University for
his valuable suggestions. Also we wish to thank Prof. T.
S. Varadharajan of English Department, SASTRA
University for his linguistic support.
140
120
100
80
60
40
20
0
RSF
SVMLS
REFERENCES
LCVLS
Models
(a)
400
350
Iteration
300
250
200
150
100
50
0
SVMLS
RSF
LCVLS
Models
(b)
Fig. 4: Shows the processing time and number of iterations
charts for human vertebra phantom image
image of a human vertebra phantom. As shown in
Fig. 2a, the RSF model segments the image but it also
detects the uninterested regions of the image. Though the
SVMLS model segments the image, it is not satisfied and
not smooth as shown in Fig. 2b
One can see the satisfied and desired segmented
result of the LCVLS model in Fig. 2c. The total time and
the number of iterations taken for the curve evolution of
MRI image and vertebra image are reported in the Table
1 and 2 and graphically depicted in the Fig. 3 and 4,
respectively.
CONCLUSION
This study has presented the comparison review on
the recent level set segmentation methods that have been
developed for addressing the challenge of the
heterogeneous medical image segmentation. The goal of
this common approach is to identify the suitable level set
method for segmenting the medical images with intensity
inhomogeneous. The advantage and the effectiveness of
each method is evaluated and the experimental results
show that the LCVLS model is the promising one for the
IIH medical image segmentation.
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