Automatic Liver Segmentation Method based on
Maximum A Posterior Probability Estimation
and Level Set Method
Daisuke Furukawa, Akinobu Shimizu, and Hidefumi Kobatake
Tokyo University of Agriculture and Technology, Japan
{dfuru,simiz,kobatake}@cc.tuat.ac.jp
Abstract. This paper describes an automatic liver segmentation method
from three dimensional abdominal X-ray CT images. Our method extracts liver regions based on the following two steps: (1) rough extraction
based on maximum a posterior probability (MAP) estimation, and (2)
precise segmentation using level set method. In the former process, the
segmentation can be performed more accurately by using a combination
of the probability density function approximated by the extended gaussian mixture distribution and a prior probability derived from a probabilistic atlas of liver. In the latter process, we introduce a novel term
to prevent the level set method from extracting muscles as a part of the
liver incorrectly. From the experimental results using ten test datasets
distributed for the competition, it was confirmed that our method segmented the liver regions with volumetric overlap of about 88%.
1
Introduction
Segmentation of the whole liver region from 3D CT image is often the first
step in the computer-assisted diagnosis and surgery systems for liver disease. So
far many automated algorithms have been proposed for the images of different
contrast conditions. Some researchers used non-contrast CT image [1, 2] and
others focused on contrast enhanced image or multi-phase CT images [3–6].
Recently several abdominal organs including liver are extracted simultaneously,
which can boost the segmentation performance by evaluating the relationship
between neighboring organs [7–9] but is time consuming.
In this paper we present an algorithm for single phase CT images that is a
modified version of the algorithm originally developed for multi-phase CT images
by authors in [6, 9] which won the competition in Japan [6]. The proposed method
extracts liver regions based on the following two steps: rough extraction based on
maximum a posterior probability (MAP) estimation, and precise segmentation
using level set method.
T. Heimann, M. Styner, B. van Ginneken (Eds.): 3D Segmentation in The Clinic:
A Grand Challenge, pp. 117-124, 2007.
2
2.1
Methods
Preprocessing and Probabilistic Atlas
A scaling transformation along z axis is first applied to the input abdominal CT
image so that the spatial resolution along z axis is equal to those along x and y
axis. Then, the isotropic image is resized to one half of its original size to reduce
the computation time.
Next, in order to compensate for the tilted body on the bed of CT scanner,
the input image is rotated based on the centers of mass of lungs. We obtain
the lung regions by a simple thresholding operation. The numbers of voxels
are counted for the extracted regions. We denote the largest two regions as V1
and V2 , and their volumes as #(V1 ) and #(V2 ) (#(V1 ) > #(V2 )). If V1 and
V2 meet the condition #(V1 )/#(V2 ) < TV , they are regarded as left and right
lung regions. However, in the case that two lungs are connected by a bronchus
in the image, the condition will not be satisfied because V1 includes both two
lung regions. In this case, left and right lung regions are extracted from V1 by
dividing V1 into two sub regions using a virtual cutting plane parallel to an axial
plane. For lower part of the two regions obtained by the division, the volume
ratio #(Vsub1 )/#(Vsub2 ) is calculated in the same way described above. The
virtual cutting plane is moved from the top end of V1 to the lower end of V1
until the condition #(Vsub1 )/#(Vsub2 ) < TV is met. Finally the obtained two
regions Vsub1 and Vsub2 correspond to left and right lung regions, respectively.
Each of the extracted two lung regions is projected onto the image plane
parallel to the axial plane. For each of the regions generated by the projection,
the center of mass is calculated. Then we also calculate the intersection of each
lung region and the line that passes through the center of mass and is parallel
to z axis. When we denote the two centers of mass obtained by this operation as
(GLx , GLy , GLz ) and (GRx , GRx , GRz ), respectively, the input image is rotated
around the axis parallel to z axis and passing through the center of axial plane
GL −GR
by the angle of θ = arctan( GLy −GRy ).
x
x
After the compensation for tilt, the normalization of the image is performed
to reduce the variation of organs in position. First, the image is translated so that
(GRx , GRx , GRz ) is equal to the corresponding center of mass (ĜRx , ĜRx , ĜRz )
in a reference image selected from a training dataset. The translated image is
Ĝ −Ĝ
Ŵ
scaled along x and y axis so that both GLLx −GRRx and Wyy are equal to ones. Here,
x
x
Wy and Ŵy represent the width of the left lungs along y axis in the input and
the reference images, respectively.
In the next step, the liver region is extracted based on MAP estimation which
uses a probabilistic atlas of liver. The atlas represents the prior probability that
each voxel belongs to a organ in a CT image. The atlas is constructed by applying
the same normalization to the label images obtained by manually segmenting
the images in the training dataset. Then, for each voxel in the reference image,
we compute the fractional percentage of the corresponding voxels which have a
label in the normalized label images.
118
2.2 Rough Extraction of Liver based on MAP Estimation
[Step 1] MAP Based Segmentation
Liver regions are extracted by calculating posterior probabilities p(l|v) of tissues
at each voxel in the normalized image, and assigning the label corresponding to
the class with the maximum a posterior probability.
ˆl = arg max p(l|v).
l
(1)
CT value, x, y, and z coordinates in the image are used as feature vector v.
According to Bayes’ theorem, p(l|v) is derived by the production of probability
density function p(v|l) and a prior probability p(l) as expressed by
p(v|l)p(l)
p(l|v) = ∑
.
l p(v|l)p(l)
(2)
In this study, we assume that there are four kinds of tissues, liver, heart, right
kidney, and the other tissue such as other organs or muscles because we limit the
region to be processed in the surrounding area of liver where the prior probability
of liver is greater than zero. Consequently, p(v|l) is approximated by the four
dimensional extended gaussian mixture distribution [9],
p(v|l) =
4
V
1 ∑∑
αl (n)N (v; µl , Σl ),
V
n=1
(3)
l=1
where n is an index which indicates location of voxel, and V the number of
voxels to be processed. N (v; µl , Σl ) represents a normal distribution with an
average of µl and a covariance matrix of Σl . αl (n) is a mixture ratio of the
normal distributions at the voxel n.
The parameters µl , Σl , and αl (n) of p(v|l) are estimated by using the EM
algorithm [9]. As the initial values of the EM algorithm, we use the average
and covariance matrices of features which are calculated from training datasets.
As the initial value of αl (n), we use the probabilistic atlas which shows the
probability that the voxel n belongs to the organ l. Note that estimated αl (n)
is substituted into p(l) in (2) in the extraction. Iterative estimation is carried
∑4
p̄t−1 (l|v)
out until the condition 14 l=1 p̄t (l|v)−
< TEM is satisfied where p̄t (l|v)
p̄t−1 (l|v)
represents the average posterior probability of organ l in the t-th iteration.
[Step 2] Map Based Segmentation with Different Normalization
Now that we have the rough regions of the organs, it is possible to use the information about the positions of the organs for the normalization. By normalizing
the input images based on a top position of a right kidney in addition to the
centers of mass and the width of the right lung along y axis used in the previous step, the subsequent MAP based process could extract more accurate liver
region than the region obtained in the previous step.
First, we enhance the right kidney in the image by the following equation,
I 0 = I + ka exp(
119
¯2
(I − I)
),
2σ 2
(4)
where I represents the CT value at a voxel in the input image, I 0 the corresponding value in the enhanced image, ka a constant value. From our experimental
results, an average CT value in a right kidney is about 60 H.U. higher than that
in a liver. Therefore, when we denote the average CT value in the liver estimated
by the EM algorithm as I¯liver , I¯ in (4) is calculated by I¯ = I¯liver + 60 [H.U.].
Then, σ in (4) is set to be a standard deviation of σ0 which is calculated from
training datasets where the condition I > I¯ is satisfied, otherwise σ is set to be
5σ0 . The top position of the right kidney is estimated by applying the template
matching to the enhanced image by (4), whose searching range is limited by the
segmentation results of step 1.
After the position with maximum correlation coefficient is determined, the
input image is normalized based on the detected top position of right kidney in
addition to the centers of mass and the width of the right lung along y axis.
From the normalized image, liver regions are extracted once again by the same
way described in the step 1. The obtained regions by this extraction are passed
to the next process.
[Step 3] Removal of Heart Region using Depth Map of Lungs
First, we generate a two dimensional depth map of the lungs where the gray
value of each pixel shows the z coordinate of the voxel belonging to the bottom
surface of the lung regions. Once the depth map is generated, the depth values
are interpolated to estimate the bottom surface of heart. In each row of the
depth map, two reference pixels are determined based on the criterion Ez =
(z(i) − z̄(i))/σz (i). Here, i represents the index which indicates a pixel in each
row, z(i) the depth value at the pixel i. z̄(i) and σz (i) are an average and a
standard deviation of the depth values calculated with range from i − N to i − 1
for the right lung, and with range from i + 1 to i + N for the left lung. Then,
the linear interpolation is performed at the pixels between two reference pixels.
Finally, the voxels whose z coordinate is equal to or less than the corresponding
depth value are removed from the extracted regions in the previous step, because
such voxels would belong to the regions, typically the heart, other than the liver.
[Step 4] Extraction of Metastatic Carcinoma and Cyst using Likelihood Images and Fusion of the Extracted Regions
This step is performed only in the case that the estimated average CT value
of the liver region is higher than Tliver [H.U.]. First, we generate the likelihood
image of abnormal regions. Our preliminary experimental results showed that we
could not enhance the abnormal regions well by using only one set of an average
and a standard deviation of CT value, because they have a variety of CT values.
Therefore, we use two sets of of averages and standard deviations, (µ1 , σ1 ) and
(µ2 , σ2 ) to enhance them. The likelihood of lesions are multiplied by a prior
probability to eliminate false positives extracted in the outside of liver. The
candidate regions are determined by simple thresholding with the thresholds of
Tmeta1 and Tmeta2 followed by the morphological opening and closing operations
with a sphere of radius 6. The regions with more than Vmeta voxels are regarded
as abnormal regions, and are added to the regions decided in the previous steps.
Tmeta1 , Tmeta2 , and Vmeta can be determined from training datasets.
120
2.3
Precise Segmentation using Level Set Method
In our method, the evolution function consists of two terms, the conventional
geodesic term proposed by Caselles [10] and our original term D which is defined
based on the distance from the contour of a human body,
]
∂φ [
= g(I)(kb + kc κ)|∇φ| + kd ∇φ · ∇g(I) D,
(5)
∂t
where φ represents a level set function. g(I) is calculated using a gradient of CT
value convolved by the gaussian filter with a standard deviation of σ.
g(I) =
1
.
1 + |∇Gσ ∗ I|
(6)
κ is the mean curvature, and kb , kc , and kd are constant values.
When the level set is calculated based on only the term proposed by Caselles,
the muscles adjacent to a liver are incorrectly extracted as a liver, because there
are few differences between the CT values in the liver and its neighboring muscles. Therefore, we introduce the term D which is possible to suppress such over
extraction as expressed by
(
)2
√

 √ds
(if ds ≤ 2Ds ),
2Ds
D=
(7)


1
(otherwise),
where ds is the Euclidean distance between a voxel and a surface of the human
body in the image. Ds is calculated by Ds = µD + kD σD . µD and σD are the
average and the standard deviation of the distances between ribs and the contour
of the human body measured from training datasets.
Based on the evolution equation described above, the level set function is
calculated iteratively by the following update equation,
∂φ
∆t.
(8)
∂t
As the initial level set, we use the signed distance which has negative distance
for the voxels belonging to the liver regions, and positive distance for any other
voxels. In our implementation, an extension velocity field is used to update
the level set function. The iterative calculation
is performed until either of the
P
i∈S ∇φ·∇g(I)
following conditions are satisfied: (1)
≤ TLSM and Niter ≥ NLSM1 ,
Ns
and (2) Niter ≥ NLSM2 . In these conditions, S represents the zero level set, and
NS the number of the voxels on the zero level set. Niter is the number of the
iteration, TLSM , NLSM1 , and NLSM2 are constant values, respectively.
After completing the evolution, we binarize the level set function at zero
threshold. Morphological opening and closing with a sphere of radius 12 are
applied to compensate for the irregular shape. Then the dilation with a sphere
of radius 2 is performed because the level set method tends to under-segment
the liver region, in particular, around its boundary due to the geodesic term in
the evolution function. The resulting regions are output as liver region.
φ(x, y, z, t + ∆t) = φ(x, y, z, t) +
121
Table 1. Results of the comparison metrics and corresponding scores for all ten test
cases.
Dataset Overlap Error Volume Diff. Avg. Dist. RMS Dist. Max. Dist. Total
[%] Score [%] Score [mm] Score [mm] Score [mm] Score Score
1 10.8
58 -3.7
81
1.7 57
3.3 55 26.8 65
63
2 11.0
57 -6.8
64
1.5 62
2.2 69 15.6 80
66
3 13.5
47 -12.0
36
3.3 17
6.2 14 36.5 52
33
4 11.5
55
3.9
79
2.7 32
6.4 11 52.4 31
42
5 12.9
49
3.0
84
2.4 39
4.0 44 36.5 52
54
6 15.1
41 -9.5
49
3.0 26
5.5 23 48.3 36
35
7 13.3
48 -11.6
38
2.1 46
3.3 54 26.8 65
50
8 12.2
52
5.2
72
2.3 43
4.1 43 40.0 47
52
9 9.5
63
0.5
97
1.3 67
2.5 66 20.8 73
73
10 12.5
51 -5.9
68
2.2 46
4.2 42 42.0 45
50
Average 12.2
52 -3.7
67
2.3 44
4.2 42 34.6 55
52
3
Experimental Results
To evaluate accuracy of the segmentation results, we applied the proposed method
to 10 test datasets distributed for the competition. In this experiment, the probabilistic atlas was constructed from 20 training datasets which are also distributed
for the competition. The parameters such as the averages, the standard deviations, and the covariance matrices were calculated from these training datasets.
The other parameters in the process such as TEM , TLSM , and Tliver were determined experimentally.
Figure 1 shows the examples of the segmentation results. The segmentation
results are evaluated quantitatively with respect to the following five criteria,
(1) volumetric overlap, (2) relative absolute volume difference, (3) average symmetric absolute surface distance, (4) symmetric RMS surface distance, and (5)
maximum symmetric absolute surface distance. For more details for these criteria, refer to [11]. The evaluation results are summarized in Table 1.
The experiments were carried out on a conventional workstation (CPU: Xeon
2.8 GHz × 2, Memory: 12 GB). The total computation time was about 36 minutes
per one test dataset for use of one CPU core, and about 15 minutes for use of
four CPU cores. In both cases, about half of the computation time was spent for
the rough extraction based on the MAP estimation, and the rest was used for
the level set method.
4
Discussion and Conclusions
We can see that the muscles adjacent to the liver regions were correctly segmented as shown at top right and middle right in Fig. 1. The term D expressed
by (7) prevents the level set method from segmenting the adjacent muscles as the
122
Fig. 1. From left to right, a sagittal, coronal and transversal slice from a relatively easy
case (1, top), an average case (4, middle), and a relatively difficult case (3, bottom). The
outline of the reference standard segmentation is in red, the outline of the segmentation
of the method described in this paper is in blue. Slices are displayed with a window of
400 and a level of 70.
liver region incorrectly. It would be difficult to extract the border between these
two tissues due to the low contrast when using only the conventional geodesic
term.
The portal vein and the inferior vena cava was over extracted as a part of
liver at top right and middle right in Fig. 1. Moreover, in some cases, parts of
kidney and heart are also segmented incorrectly. This is because this kind of
organs has similar CT value to the liver. One possible way to reduce this type of
over extraction is to use a method for segmenting several organs around a liver
simultaneously [7–9]. Although a simultaneous segmentation takes much more
computation time than that we proposed, it would be solved in near future due
to the rapid progress of computer hardware.
In the test dataset 3, there are lesions widely spread in the liver as shown
at bottom row of Fig. 1. Our method segmented some part of such abnormal
regions successfully. This is mainly because of the likelihood-based operations in
123
the step 4 described in Sect. 2.2. Although the procedures in the step 1 and 2
failed to extract almost all of the abnormal regions, a part of lesion was correctly
extracted in the step 4, and then the subsequent level set method evolved the
boundary of the obtained regions toward the correct position. This result showed
that the abnormal region extraction process of the step 4 was effective. In future
work, however, we have to improve the detection scheme to achieve more accurate
segmentation since some part of the abnormal region is still under extraction.
Because the lesion such as metastatic carcinoma and cyst has a variety of CT
values, it would be difficult to extract such abnormal region more accurately
by the simple method adopted in this study. Therefore, we plan to introduce a
sophisticated method based on a statistical pattern classification technique with
other feature values in addition to CT value.
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