ANOVELPROBABILISTICSIMULTANEOUSSEGMENTATIONANDREGISTRATIONUSINGLEVELSET
Melih S. Aslan, Eslam Mostafa, Hossam Abdelmunim
∗
, Ahmed Shalaby, Aly A. Farag, and Ben Arnold
∗∗
Computer Vision and Image Processing Lab., University of Louisville, Louisville, KY, 40299, USA
∗
Computer and Systems Engineering Department, Ain Shams University, Cairo, Egypt
∗∗
Image Analysis, Inc., 1380 Burkesville St., Columbia, KY, 42728, USA
ABSTRACT
We propose a new shapebased segmentation approach usingthe statistical shape prior and level sets method. The segmentation depends on the image information and shape prior.Training shapes are grouped to form a probabilistic model.The shape model is embedded into the image domain takingin consideration the evolution of a contour represented by alevel set function. The evolution of the front gathers information from the image intensities and shape prior. The segmentation approach is applied in segmenting the vertebral bodiesin CT images. Our results shows that the technique is accurate and robust compared with the other alternative in theliterature.
Index Terms
—
Simultaneous segmentation and registration, vertebral body (VB).
1. INTRODUCTION
LevelsetmethodswereﬁrstintroducedbyOsherandSethian[1].The level sets method presents several advantages over theparametric active contours. The contours represented by theLevel sets function may break or merge naturally duringthe evolution, and changes are automatically handled. Another advantage is that the level set contour always remainsa function on a ﬁxed grid, which allows efﬁcient numericalschemes.Level sets method is one of the techniques used in theshape based segmentation which is an important complexproblem in computer vision, computer graphics and medicalimaging. In the shape based segmentation, embedding themodel into the image domain is the key issue and depends onthe registration of the given shape template to the image. Theshape registration problem is formulated such that a transformation that moves a point from a given shape to a targetone according to some dissimilarity measure [2] needs to beestimated. An active contour algorithm that can incorporateshape priors was introduced in [3]. Shape priors, in additionto the image gradient, are embedded into the energy functionof the contour. In [4] the distance function is used to implicitly represent open/closed shapes (structures). The signeddistance function is used for closed shapes.In this paper, our objective is to help to diagnose andtreat osteoporosis by accurate segmentation. Osteoporosis isa bone disease characterized by a reduction in bone mass, resulting in an increased risk of fractures. To diagnose the osteoporosis accurately, the bone mineral density (BMD) measurements of the vertebral bodies (VBs) are required. Therefore, the accurate VB segmentation is an important step toidentify vertebral fractures and to measure BMDs. The vertebrae consists of the VB and spinal processes. In this paper
1
,we are interested in CT images of the vertebral body. Limitedapproaches have been introduced to tackle the segmentationof spine bones such as in [7, 8].There are difﬁcult segmentation challenges in spine computed tomography (CT) images for a good segmentation asshown in Fig 1. Also, exposure levels (Xray tube amperageand peak kilovoltage), slice thickness, and volume of interest (VOI) effect the resolution of CT images. To overcomethose limitations, we propose a new shape based segmentation method using level sets. We use a dissimilarity measurein the shape information which is analytically invariant under afﬁne transformation. In some of previous publications(i.e. [2, 4]), registration parameters and weighting parametersof the shape model are usually estimated numerically usinggradient descent. This iterative optimization requires an appropriate tuning of the time step in order to guarantee a stableevolution. Also, experiments show that the order of updating the different pose parameters and weighting vectors affectthe resulting segmentation process. In this paper, we use theintrinsic registration as in [5], which solves the common disadvantages in the statistical dynamic shape information. Insection 2, we present the proposed method. Section 3 explains the experiments, and compares our results with anotheralternative.
1
This work has been supported by Image Analysis, Inc., Columbia, Kentucky, USA.
2011 18th IEEE International Conference on Image Processing
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(a) (b) (c) (d)
Fig. 1
.
Typical challenges for vertebrae segmentation. (a) Innerboundaries. (b) Osteophytes. (c) Bone degenerative disease. (d)Double boundary.
2. PROPOSED METHOD
Intensity models may not be enough to obtain the optimumsegmentation. Hence, we propose a new shape based iterativesegmentation and registration method. In the ﬁrst step, theMatched ﬁlter [9] is employed to detect the VB region automatically. This procedure eliminates the user interaction andimproves the segmentation accuracy. We tested the Matchedﬁlter using
3000
clinical CT images. The VB detection accuracy is
97
.
6%
. For more information, see our work in [10].In the second phase, we initialize the evolving contour onthe VB. Then, an iterative process which simultaneously doesthe segmentation and registration begins. In the segmentation step, we use an improved level sets approach in whicha probabilistic shape model is integrated. To make the shapeprior to be invariant to the transformation, we register it to theevolving contour at each iteration. Our overall segmentationframework is given in Algorithm 1. The next section presentsthe proposed segmentation and registration framework.
Algorithm 1
Proposed Framework
Given
:Aninputimage(
I
), and80manuallysegmentedtraining images.
Objective
: Minimizing the energy function (
E
) to obtain theoptimum segmentation.
Training Stage
: Aligning training shapes and constructingthe probabilistic shape model.
Testing Stage
: Simultaneous segmentation and registration.
1
.
Detect the VB region in
I
using the Matched ﬁlter.
2
.
Iterative segmentation and registration algorithm to minimize the energy function in Eq. 4.
3
.
End the iteration until the energy minimization is saturatedor the iteration number is reached.
2.1. Shape Prior Reconstruction
In our experiments, we use 80 training CT images (slices)which are obtained from 10 different patients and differentregions such as lumbar and thoracic bones. We obtain a probabilistic shape model we presented in [11]. First, VBs aremanually segmented (under the supervision an expert). Thenthe segmented VBs are aligned using
2
D registration. Thealigned training images are shown in Fig. 2. Finally, a shape
Fig. 2
. Obtaining the shape prior image. Training CT slicesof different data sets. The last column shows the shape priorimage with variability region.
51015202500.10.20.30.40.50.60.70.80.9
Distace (Wave)
P r o b a b i l i t y
ObjectBackground
(a) (b)
Fig. 3
. (a) A sample projection of the shape prior image.The white color represents the object region,
O
, the red contourrepresentstheobject/variabilitysurface,
C
OV
, theyellowwaves represent the isosurfaces,
C
d
x
. (b) The probability of the object and background in the variability region,
V
, respectto each isosurface,
C
d
x
.image represented as
P
s
=
O
B
V
is generated. Thewhite color represents the object or the VB (
O
), the black color represents the background (
B
), and the gray color represents the variability region (
V
).To model the shape variations in
V
, the distance probabilistic model is used. The distance probabilistic model describes the object (and background) in the variability regionas a function of the normal distance
d
x
=
min
x
−
c
(where
c
∈
C
OV
) from a pixel
x
∈ V
to the VB/variability surface
C
OV
. Each set of pixels located at an equal distance
d
p
from
C
OV
constitutes an isosurface
C
d
x
for
C
OV
. To estimate the marginal density of the VB, it is assumed that eachisosurface
C
d
x
is a normally propagated wave from
C
OV
asshown in Fig. 3(a). The probability of an isosurface to be anobject decays exponentially as the discrete
d
x
increases. TheVBdistance histogramisestimatedasfollows. Thehistogramentity of the object region at distance
d
x
is deﬁned as
h
(
d
x
 O
) =
M
i
=1
K
ℓ
=1
x
∈
C
dx
δ
(
x
∈ O
iℓ
)
(1)where the indicator function
δ
(
A
)
equals
1
when the condition
A
istrue, andzerootherwise,
M
isthenumberoftrainingdata sets,
K
is the number of CT slices of each data set, and
O
iℓ
is the VB region. The distance,
d
x
, is changed until thewhole distance domain in the variability region is covered. Inpixelwise, this process can be done by obtaining the outer
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edge of the previous isosurface. Then, the histogram is multiplied with shape prior value which is deﬁned as follows:
π
O
= 1
M
V
x
∈V
δ
(
x
∈ O
)
.
(2)The distance marginal density of the object region is calculated as
P
O
(
d
x
) =
h
(
d
x
 O
)
π
O
M

C
d
x

.
(3)The same scenario is repeated to obtain the marginal density of the background. An example of the distance marginaldensities of the object and background region is shown inFig. 3(b). In the next section, we explain our proposed segmentation method.
2.2. Segmentation
The level set segmentation framework contains the movingfront, denoted by
C
, which is implicitly represented by thezero level of a higher dimensional function,
φ
, that is:
C
(
t
) =
{
x/φ
(
x,t
) = 0
}
. The energy function of the segmentationcan be written as
E
=
E
cv
(
φ
) +
αE
shape
(
φ
)
(4)where
α
isaconstantwhichcontrolshowmuchwedependonthe probabilistic shape prior. The ﬁrst energy (data penalty)term is based on the intensity of the testing image. The second term is based on the shape prior after registering it to theevolving contour to be invariant to the transformation parameters.
2.2.1. Intensity information
The ﬁrst term is modelled using similar way in [12] as follows:
E
cv
(
φ
) =
Ω
(
I
−
u
+
)
2
Hφ
(
x
)
dx
+
Ω
(
I
−
u
−
)
2
(1
−
Hφ
(
x
))
dx
+
v
Ω
∇
Hφ
(
x
)

dx,
(5)where
φ
represents the signed distance function of the evolving contour,
H
is the Heaviside step function, and
u
−
and
u
+
represent the mean intensity in the two regions.
2.2.2. Embedding shape prior information
In this paper, our contribution is to propose a new probabilistic energy function in the level set method using previouslypresented shape model [11]. To register the shape model tothe evolving contour, we use the similar approach presentedin [5]. Each pixel in the shape prior has two probabilities forbeing
i
) an object and
ii
) a nonobject. Our shape prior isembedded in the level sets function in order to obtain moreaccurate segmentation results and extract the spinal processesautomatically. The shape model is registered into the imagedomain by maximizing the probability of pixels inside thecontour belonging to the object space and the pixels outsidethe contour belonging to the nonobject space. This approachleads to the following energy function:
E
shape
(
φ
) =
Ω
(1
−
P
O
(
σ
φ
d
x
+
µ
φ
))
Hφ
(
x
)
dx
+
Ω
(1
−
P
B
(
σ
φ
d
x
+
µ
φ
))(1
−
Hφ
(
x
))
dx.
(6)where the translation and scaling parameters can be estimatedusing a similar method shown in [5] as
µ
φ
=
Ω
xHφ
(
x
)
dx
Ω
Hφ
(
x
)
dx ,
(7)
σ
2
φ
=
Ω
(
x
−
µ
)
2
Hφ
(
x
)
dx
)
Ω
Hφ
(
x
)
dx .
(8)
2.2.3. Gradient Descent ﬂow of
φ
The change of the level set function with time using the twoenergy function is calculated by the EulerLagrange with thegradient descent as :
∂φ∂t
=
−
∂E ∂φ
=
−
∂E
cv
∂φ
−
∂E
shape
∂φ .
(9)The gradient of two energy terms is obtained as follows:
∂E
cv
∂φ
=
δ
(
φ
)[(
I
−
u
+
)
2
−
(
I
−
u
−
)
2
−
vdiv
(
∇
φ
∇
φ

)]
,
(10)
∂E
shape
∂φ
=
δ
(
φ
)[
P
O
(
σ
φ
d
x
+
µ
φ
)
−
P
B
(
σ
φ
d
x
+
µ
φ
)]
.
(11)
3. EXPERIMENTS AND DISCUSSION
To assess the accuracy and robustness of our proposed framework, we tested it using clinical data sets. The clinical datasets were scanned at
120
kV and
2
.
5
mm slice thickness. Allalgorithms are run on a PC 3Ghz AMD Athlon 64 X2 Dual,and 3GB RAM. All implementations are in C++.In this experiment, we use 40 testing CT images. Tocompare the proposed method with other alternative, VBs aresubsequently segmented using the active appearance model(AAM) [13]. Finally, segmentation accuracy is measured foreach method using the ground truths (expert segmentation).To evaluate the results we calculate the percentage segmentation accuracy (
A
) as follows:
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Table 1
. Accuracy of our VB segmentation on 40 CT images.
Our AAM [13]
Mean accuracy, % 95.59 86.15Min. accuracy, % 92.92 83.21Max. accuracy, % 99.40 89.15Stand. dev.,% 2.11 2.68
A
% = 100
∗
(
number of correctly segmented voxels
)
Total number of
VB
voxels .
(12)The statistical analysis of our method is shown in the Table 1.In this table the results of the proposed segmentation methodand the other alternative are shown. Figure 4 shows the segmentation results of the proposed framework. As we show inthe results, the spinal processes which are not required in theBMD measurements are eliminated automatically. We obtainvery high segmentation accuracy using our new probabilisticshape energy function.
4. CONCLUSION
In this paper, we have presented a new simultaneous segmentation and registration framework. We tested our method onVBs in CT images. The probabilistic shape model has twoadvantages: i) the spinal processes is eliminated, ii) the registration and segmentation errors are reﬁned. We comparedthe results with the AAM method. Experiments on the datasets show that the proposed segmentation approach is moreaccurate and robust than other known alternative.
5. REFERENCES
[1] S. Osher and J. A. Sethian, Fronts propogating with curvaturedependent speed: algorithms based on HamiltonJacobi formulations, J. Comp. Phys. vol. 79, pp. 1249, 1988.[2] N. Paragios, M. Rousson, and V. Ramesh, Matching DistanceFunctions: A ShapetoArea Variational Approach for Globalto Local Registration Proc. Seventh European Conf. ComputerVision 2002.[3] Y. Chen, H.D. Tagare, S. Thiruvenkadam, F. Huang, D. Wilson, K.S. Gopinath, R.W. Briggs, and E.A. Geiser, Using PriorShapes in Geometric Active Contours in a Variational Framework Intl J. Computer Vision vol.50, no. 3, pp.315328, 2002.[4] X. Huang, N. Paragios, and D. Metaxas, hape Registration inImplicit Spaces Using Information Theory and Free Form Deformations IEEE Trans. Pattern Analysis and Machine Intelligence vol. 28, no. 8, pp. 1303 1318, 2006.[5] D. Cremers, S. J. Osher, S. Soatto, Kernel density estimationand intrinsic alignment for shape priors in level set segmentation, International Journal of Computer Vision, vol. 69(3), pp.335351, 2006.[6] www.cedarssinai.com
Fig. 4
. Segmentation results of some clinical CT images. Theyellow color shows the contour of the segmented region.
[7] A. Mastmeyer and K. Engelke and C. Fuchs and W. A. Kalender, A hierarchical 3D segmentation method and the deﬁnitionof vertebral body coordinate systems for QCT of the lumbarspine Medical Image Analysis. vol. 10, no. 4, pp. 560577,2006.[8] T. Klinder, J. Ostermann, M. Ehm, A. Franz, R. Kneser, C.Lorenz, Automated modelbased vertebra detection, identiﬁcation, and segmentation in CT images, Medical Image Analysis,vol. 13, pp. 471482, 2009.[9] B. V. K. V. Kumar, M. Savvides, and C. Xie, Correlation patternrecognition for face recognition, Proceedings of the IEEE, vol.94, no. 11, pp. 19631976, 2006.[10] M. S. Aslan, A. Ali, H. Rara, B. Arnold , A. A. Farag, R.Fahmi, and P. Xiang, A Novel 3D Segmentation of Vertebral Bones from Volumetric CT Images Using Graph Cuts,ISVC’09, 2009.[11] M. S. Aslan, A. Ali, D. Chen, B. Arnold , A. A. Farag, andP. Xiang, 3D Vertebrae Segmentation Using Graph Cuts WithShape Prior Constraints, Proc. of 2010 IEEE International Conference on Image Processing, pp. 21932196, 2010.[12] T. F. Chan and L. A. Vese, A level set algorithm for minimizing the MumfordShah functional in image processing, IEEEWorkshop Proceeding on and Level Set Methods in ComputerVision, 2001.. pp. 161168, 2001.[13] T. F. Cootes and G. J. Edwards and C. J. Taylor, Active Appearance Models, IEEE Transactions on Pattern Analysis andMachine Intelligence, vol.23(6), pp.681685, 2001.
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