Journal of Signal and Information Processing, 2013, 4, 36-42
doi:10.4236/jsip.2013.43B007 Published Online August 2013 (http://www.scirp.org/journal/jsip)
Liver Segmentation from CT Image Using Fuzzy
Clustering and Level Set
Xuechen Li1, Suhuai Luo1, Jiaming Li2
1The University of Newcastle, Australia; 2The CSIRO ICT Centre, Australia.
Received April, 2013.
ABSTRACT
This paper presents a fully automatic segmentation method of liver CT scans using fuzzy c-mean clustering and level
set. First, the contrast of original image is enhanced to make boundaries clearer; second, a spatial fuzzy c-mean cluster-
ing combining with anatomical prior knowledge is employed to extract liver region automatically; thirdly, a distance
regularized level set is used for refinement; finally, morphological operations are used as post-processing. The experi-
ment result shows that the method can achieve high accuracy (0.9986) and specificity (0.9989). Comparing with stan-
dard level set method, our method is more effective in dealing with over-segmentation problem.
Keywords: Liver Segmentation; Fuzzy c-Mean Clustering; Level Set
1. Introduction
Liver segmentation from CT images is the key prepara-
tion work for the establishment of three dimension model
of liver, which has great significance on the diagnosis of
liver disease. A variety of liver CT image segmentation
methods have been proposed. These methods include
region growing [1-3], level set [4-6], clustering [7, 8],
statistic shape model [9], neural network [7] and support
vector machine (SVM) [10-12], etc. The following briefs
these progresses.
Ruskó et al. [1] presented an adaptive region growing
method. First, the seed region was determined based on
the intensity of gray level; second, the liver and heart
were separated based on the anatomical feature; thirdly,
an improved region growing was used to segment the
image; finally, a post-processing was employed to deal
with the under-segmentation. Their method can deal with
most cases well, but in some difficult cases (e.g. when
the gray level intensity of live is inhomogeneous), it will
cause under-segmentation. Li et al. [4] presented a dis-
tance regularized level set method. The main advantage
is that it allows the use of more general and efficient ini-
tialization of the level set function. Therefore, relatively
large time steps can be used in the finite difference
scheme to reduce the number of iterations. However, the
method needs user to select seed points, which makes it a
semi-automatic method. In [7] the initial image was seg-
mented by fuzzy c-means clustering (FCM) and smoothed
by morphological processing; then the candidate regions
were classified by neural network; finally, the regions
which belong to liver or node were extracted. However,
only gray level information was used in the original FCM
method, which may cause over-segment when other tis-
sues have similar gray level with liver. In [9], the author
presented an approach for automatic liver segmentation
which was based on statistic shape model (SSM) inte-
grated with an optimal-surface-detection strategy. First
the generalized Hough transform was used to build the
average shape model of liver; then the subspace of SSM
wasinitialize; finally, deform the shape model to adapt to
liver contour through an optimal-surface detection ap-
proach based on graph theory. The method achieves
higher accuracy compare with previous model based
methods. However, all model based methods suffer the
same problem. They need large number of training data
which must cover all shape cases. In [10, 11] wavelet
transform was used to achieve texture feature extraction;
then SVM was employed to make the classification; fi-
nally, region growing [10] ormorphological operations
[11] was utilized as the post-processing. However, region
growing may lead to over-segment when the gray level
intensity of liver and around tissues is similar; when us-
ing morphological operation as post-processing alone,
the parameters should be adjusted carefully, and the ro-
bustness may be a problem. Li et al. [5] presented a new
fuzzy level set algorithm. It begins with spatial fuzzy
clustering which presented by Chuang et al. [8] in 2006.
There salt was utilized to initialize level set segmentation
and estimated the parameters of level set evolution.

Liver Segmentation from CT Image Using Fuzzy Clustering and Level Set 37
Moreover, the fuzzy level set algorithm was enhanced
with locally regularize devolution which can facilitate
level set manipulation and lead tomorero bust segmenta-
tion. It is efficient when the background is simple and the
boundary between background and object is clear.
In this paper, we present a fully automatic segment-
tion method using fuzzy c-mean clustering combining
with level set. Sections 2 and 3 give introductions of
fuzzy c-mean clustering and level set method; Section 4
is the proposed segmentation method for liver CT scans;
Section 5 gives the result of segmentation and discussion;
Section 6 is conclusion and future work.
2. Fuzzy c-Mean Clustering
The FCM algorithm assigns pixels to each category by
using fuzzy memberships. Let (1,2,,
j)
x
jn
denotes
an image with n pixels to be partitioned into c clusters,
where
j
x
represents features data. The algorithm isan
iterative optimization that minimizes the cost function
isdefined as:
2
11
cn
m
ij ij
ij
J
ud

 (1)
iji j
dcx (2)
where presents the membership of
ij
u
j
x
in the ith
cluster, .i is the ith cluster centre, and m is a
constant. The parameter m controls the fuzziness of the
resulting partition.
[0,1]
ij
uc
The cost function is minimized when pixels close to
the centroid of their clusters are assigned high member-
ship values, and low membership values are assigned to
pixels with data far from the centroid. The membership
function represents the probability that a pixel belongs to
a specific cluster. In the FCM algorithm, the probability
is dependent solely on the distance between the pixel and
each individual cluster centre in the feature domain. The
membership functions and cluster centres are updated by
the following:
2( 1)
1
1
ij m
cij
kkj
ud
d







(3)
and
1
1
nm
ij j
j
inm
ij
j
ux
cu



 (4)
The standard FCM algorithm is optimized when the
feature data of pixels close to their cluster centre are as-
signed high membership values, while those that are far
away areas signed low values.
One of the disadvantages of standard FCM used in
image segmentation is that it only uses the gray level
intensity information for clustering rather than spatial
information of pixels. In fact, the probability that those
neighboring pixels share similar gray level intensity be-
long to the same cluster is great. Chuang et al. [8] pre-
sented a spatial FCM algorithm in which spatial informa-
tion can be incorporated into fuzzy membership func-
tions. The spatial function is defined as:
()
j
ij ik
kNBx
hu


(5)
where ()
j
NB x represents a square window centred on
pixel
j
x
in the spatial domain. Just like the membership
function, the spatial function ij represents the probabil-
ity that pixel
h
j
x
belongs to ith cluster. The spatial func-
tion of a pixel for a cluster is large if the majority of its
neighborhood belongs to the same clusters. The spatial
function is incorporated into membership function as
follows:
1
pq
ij ij
ij c
p
q
kj kj
k
uh
uuh

 (6)
where p and q are two parameters to control the rela-
tiveimportance of and .
ij
uij h
3. Level Set
Level set is a continuous deformable model method with
implicit representation. Its main idea is to embed the de-
formable model in a d+1 dimensional space, to segment
iteratively an object in a d dimensional space, using par-
tial differential equations. The main advantage of level
sets is that it allows changes of surface topology implic-
itly.
The standard level set function is defined as:

0
0
0, ,,
F
t

x
yx







y
(7)
where


denotes the normal direction,0(, )
x
y

is the
initial contour and F represents the comprehensive-
forces. 0(, )
x
y

isusually defined as
 
00
0
0
,
,Cifxyisinside
xy Cotherwise


 



 (8)
here is a constant.
0
The original level set method needs reinitialization
because the level set function (LSF) typically develops
irregularities during its evolution which cause numerical
errors and eventually destroy the stability of the level set
evolution. Although reinitialization as a numerical rem-
edy is able to maintain the regularity of the LSF, it may
incorrectly move the zero level set away from the ex-
pected position Liver segmentation process. Moreover it
is time consuming for its large computation.
0C

Liver Segmentation from CT Image Using Fuzzy Clustering and Level Set
38
To deal with the problem, Li et al. [4] developed a
distance regularized level set evolution (DRLSE). Due to
the distance regularization term, the DRLSE can be im-
plemented with a simpler and more efficient numerical
scheme in both full domain and narrowband implementa-
tions than standard level set formulations. Moreover,
relatively large time steps can be used to significantly
reduce the number of iterations and computation time,
while ensuring sufficient accuracy.
Let :ΩR

 be a LSF defined on a domain Ω. The
energy function ()


is define as
 
()
pext
R

 
(9)
where ()
p
R

is the level set regularization term defined
as Equation (10), 0

 is a constant, ()
ext

 is the
external energy.
()( )
p
Rp 


dx
(10)
Here p is a potential function. The purpose of ()
p
R

is
to smooth the level set function and maintain the signed
distance property1  
, at least at least in a vicinity of
the zero level set, in order to ensure accurate computa-
tion for curve evolution.
The gradient flow of the energy function is:


pext ext
p
Rdiv d
t

 



 
  

(11)
where

()
p
ds pss

.
The level set evolution an equation is discredited as
the following finite difference equations:


1( kk ext
p
tdivd )

 


 
 (12)
In our work, the potential function paned external en-
ergy


ext
ε

are defined as:
  

2
2
11coscos2, 1
2
1(1) 1
2
s
if s
ps
si








fs
(13)
and



()
ext g g
LA

  (14)
where




g
Lg 


dx
(15)

()
g
A
gH dx 




(16)
1
1| (*)|
gGI

 (17)
here

and H are the Dirac delta function and the
Heaviside function respectively.
g
is edge indicator func-
tion and G

is a Gaussian kernel with a standard devia-
tion σ. In practice, we use


and
H

to approximate
δ and H in Equation (15) and (16) (their definition see
[4]). The final form of the energy function is:






()
p
()
dx gdx
gH


 



 

dx



(18)
This energy function can be minimized by solving the
following gradient flow:
 

()
p
d g
t



div
g
div






 





(19)
Given an initial LSF


0( ),0
x
x 
)

(
. The first term
on the right hand side is associated with the distance
regularization energyp R

, while the second and third
terms are associated with the energy terms ()
g
L

and
()
g
A

, respectively.
4. Methodology
In developing an automatic segmentation of liver in CT
images, we have focused our attention mainly on three
aspects (see Figure 1): First, a contrast enhance cement
based on histogram operation is employed as preproc-
essing to make the boundary clearer; second, spatial
fuzzy clustering [8] combining with anatomical prior
knowledge are employed to extract liver region auto-
matically, and a distance regularized level set [4] is used
to refine the segmentation result; finally, morphological
filter is used as post-processing to fill holes and smooth
the final segmentation result.
Figure 1. The overview of the proposed liver segmentation.
4.1. Pre-Processing
The pre-processing contains three main steps: the range
of liver, contrast enhancement and median filtering
smoothing. First, the gray level range of live is deter-
mined by analyzing the histogram of CT slice. The prior

Liver Segmentation from CT Image Using Fuzzy Clustering and Level Set 39
knowledge shows that the gray level of live is between
130 and 150. Therefore, the peak of histogram in this
range is extracted and other parts will be set to 0. Second,
the liver peak is uniformed by contrast stretching. The
original and enhanced image and their histogram are
shown inFigure 2.After this step, most non liver tissues
are separate from liver and the boundary between liver
and the around tissues become clearer. It is benefit for
FCM and LSM. Thirdly, a median filter (10×10) is em-
ployed to reduce noise and smooth image.
050100 150 200 25030
0
0
2000
4000
6000
8000
10000
12000
14000
16000
(a) (b)
0
0.5
1
1.5
2
2.5
x 10
4
050100 150 200 250
(c) (d)
Figure 2. The original and enhanced image and their histo-
gram (a is original image; b is histogram of a; c is contrast
enhanced image; d is histogram of c. Note: ∆ in b are the
thresholds of liver peak).
4.2. Fuzzy Clustering with Level Set
Segmentation
Our work is based on spatial fuzzy c-mean clustering [8]
and a distance regularized level set [4]. The problem of
both methods above is that they are semi-automatic
methods. The spatial fuzzy clustering needs user to
choose which class is the liver, so the level set refine-
ment can continue on that class. The distance regularized
level set needs user to select seed points. Moreover, the
number of cluster group is fixed. In some cases, liver
tissue may be clustered into different groups. For these
reasons, human participate is needed. To make the
method fully automatic and achieve better segment result,
we proposed the following improvements:
1. The image is cluster edict 4 groups based on Equa-
tion (1-6). One for liver, one for background, one for
other tissues brighter than liver (such as bones), and one
for tissues darker than liver (such as muscles). The
membership matrix u is initialized by random number;
repeat Equation (3-6) until J in Equation (1) less than
0.01. In our work,
j
x
represents the gray level of each
pixels, m = 2, p = 1, q = 1 and ()
j
NB x represents a (5 ×
5) window. The selection of liver group is done auto-
matically by using prior knowledge. Liver is the largest
organ located at upper left of CT slice. So we locate the
abdomen in the CT slice and divide it into 4 equal areas
(see Figure 3(a)). The group with most pixels locate at
the region 1(except background group) will most proba-
bly be the group contains liver. The background group
can be rejected by analysing the gray level intensity of
each group (it has the lowest average gray level inten-
sity).
2. In some cases, there are only a few other tissues in
the CT slice. The liver may be clustered into separate
groups. Therefore, merging similar groups is necessary.
Calculate average gray level intensity of each group on
original image instead of contrast enhanced image be-
cause the difference in enhanced image will always be
large even if the intensity is similar in original image. If
there is another group whose average intensity is similar
enough (difference less than 2.5) to liver group, add it to
the liver group.
3. Calculate the average intensity and standard devia-
tion of the liver group and reject tissues which are not
belong to liver based on statistic theory. Letk
x be the
gray level intensity of pixels of liver group,x be the
average intensity of liver group, and be the standard
deviation. The pixels which belong to liver are defined
as:
x(x3δ,x 3δ)
k

 (20)
4. The largest connected region of selected group
works as initialization of level set. In Li’s work [4], they
presented two kinds of application. One is the initial re-
gion contains all object and the other is the initial region
is all inside the object. In our work we use the first cases.
The reason is that there are less fake boundaries outside
liver. To make initial region contains all liver, we extend
the region out for 5 pixels. Then the level set is use on
the image after contrast enhancement which has clearer
boundary to avoid leakage. In our work, 0
C=2; ∆t=5 is
used in Equation (12); μ=0.04, λ=5, α=1.5 are used in
Equation (19) and the number of iterations is 40.
4.3. Post-Processing
There are three main problems in the output of level set.
One is that there may still be other tissues in the output
(such as heart tissues); another problem is that the con-
tour is not smooth where the boundary is not clear; the
final one is that there are holes in the liver area because
of contrast enhancement and statistic rejection. Therefore,
a post-processing based on morphology is needed.
There are two main morphological operations: dilate
and erode. By using them together, operations open and

Liver Segmentation from CT Image Using Fuzzy Clustering and Level Set
40
close are made. Open is separating whole object apart by
using erode first and then dilate; close is organising
smaller pieces together as one object by using dilate first
and then erode. To reject other tissues around liver, the
output of level set is processed by operation open (radius
is 2). After that, the liver should be separated from tis-
sues around. The liver can be extracted by selecting the
largest connected region. At the third step, a median filter
(3×3) is employed to smooth the boundary. The final step
is to fill holes inside liver. There are still some black
holes inside liver because some vessel or other tissues are
rejected in the processing fuzzy clustering and statistic
rejection. To refine the result, operation “hole-filling” is
employed. The final output is shown in Figure 3(f),
where the white curve is our segment result, black curve
is the manually segment result.
(a) (b)
(c) (d)
(e) (f)
Figure 3. Liver segmentation processing(a is region defini-
tion for liver group selection; b is the selected liver group; c
is the initial region of level set; d is the output of level set; e
is the smoothed liver region before hole-filling; f is the final
segmentation. Note: white curve in f is result of our method,
black curve in f is ground truth).
5. Result and Discussion
We use three metrics to evaluate the segmentation result:
accuracy, sensitivity and specificity. The first one is used
to evaluate the general performance of our method; the
second one is used to evaluate the acceptance capability
of liver tissues and the last one focus on the rejection
capability of non-liver tissues.
Accuracy is a widely used metrics to evaluate per-
formance of segmentation methods. It is defined as:
TP TN
accuracy TP TNFPFN


where TP is the number of true positive cases; TN is true
negative cases; FP is false positive cases and FN is false
negative cases. The definition of TP, TN, FP and FN is
shown in Table 1.
Table 1. The definition of TP, TN, FP and FN.
Ground truth label
positive negative
positive TP FP
Test
outcome negative FN TN
Sensitivity is defined as:
TP
sensitivity TP FN 
It means how many liver tissues are accepted in the
outcome compare with ground truth.
Specificity is defined as:
TN
specificity TN FP 
It shows how many non-liver tissues are rejected in the
outcome.
We use two groups of data to test the performance of
our method. The data are from two different patients.
Each group has 64 slices and the size of each slice is
512×512 pixels. Table 2 shows the segmentation result.
The result shows that our method has high accuracy
and specificity. Compare with standard level set method,
our method shows more effectiveness on the unclear
boundary cases. Figure 4 is the comparison of standard
level set (left) and our method (right). It shows that when
the boundary between liver and around tissues is not
clear, standard level set will lead to over-segmentation
and our method will not. However, the sensitivity is
lower compare with other metrics. The under segmenta-
tion cases are shown in Figure 5. The main problem is
that the gray level intensity of vessels is much higher
than liver parenchyma. When vessels are completely
inside the liver, the under-segmentation can be corrected

Liver Segmentation from CT Image Using Fuzzy Clustering and Level Set 41
by hole-filling operation. However, when vessels and
other inhomogeneous tissues are at the edge of liver, the
under-segmentation would show as indentations instead
of holes. Our method shows less effectiveness to deal
with these cases.
Table 2. Performance metrics of the proposed liver seg-
mentation algorithm.
Data1 Data2
Average
(standard deviation)
0.9887
(0.0050)
0.9886
(0.0088)
Max 0.9977 0.9991
Accuracy
Min 0.9668 0.9195
Average
(standard deviation)
0.9330
(0.0258)
0.8703
(0.1024)
Max 0.9726 0.9766
Sensitivity
Min 0.8389 0.6397
Average
(standard deviation)
0.9987
(0.0006)
0.9991
(0.0022)
Max 0.9999 1
Specificity
Min 0.9959 0.9937
Figure 4. Comparison between standard level set and pro-
posed method.
Figure 5. Under-segmentation cases of proposed method.
6. Conclusions
In our work, a fully automatic fuzzy clustering segmen-
tation method combining with level set has been pre-
sented. It employed spatial fuzzy c-mean clustering and
anatomical prior knowledge to extract liver area from CT
scan automatically. The distance regularized level set
was used for refinement. The experiment result shows
high accuracy (average 0.9986) and specificity (average
0.9989) in all testing data. Compared with standard level
set method, our method is fully automatic and can
achieve better segmentation result even if the boundary is
not clear. The future work will focus on the under seg-
ment problem when there are vessels or other in homo-
geneous tissues on the edge of liver.
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