Image and Vision Computing 23 (2005) 1029–1040
www.elsevier.com/locate/imavis
Dynamic B-snake model for complex objects segmentation
Yue Wang, Eam Khwang Teoh*
School of Electrical and Electronic Engineering, Nanyang Technological University, Block S2, Nanyang Avenue, Singapore 639798, Singapore
Received 13 May 2004; received in revised form 30 May 2005; accepted 1 July 2005
Abstract
A close-form B-Snake model using statistics information for 2D objects segmentation is presented in this paper. We called it Dynamic
B-Snake Model (DBM). It is able to model the features of the object in training set and guide the B-Snake in the deforming procedure.
Compared to other deformable models, the DBM maintains the smoothness of curve while still remain compact representation. Moreover, a
method of Minimum Mean Square Error (MMSE) is developed to iteratively estimate the position of those control points in the B-Snake
model. As it deforms the segments of the B-Spline at a time, instead of individual points, it is very robust against local minima. Furthermore,
in order to use available statistical information about the desired object shape, the Principal Component Analysis (PCA) is applied to model
the distribution of knot points of training samples. This allows the deformation of B-Snake to synthesize the shape similar to those in the
training set. By applying the proposed B-Snake model to medical images, it is shown that our method is more robust and accurate in
comparing with the traditional Snake and Active Shape Model(ASM).
q 2005 Elsevier Ltd. All rights reserved.
Keywords: Deformable model; Active contour; Snake; B-Spline; B-snake; Object segmentation; Principal component analysis
1. Introduction
Object segmentation is a very important procedure in
image analysis, computer vision, and medical imaging.
Many medial image analysis applications, like the measurement of anatomical structures, require prior segmentation of
the organ from the surrounding tissue. However, it is a
difficult task due to the variations of objects and the
diversities of image types. For example, one important task
in medical imaging is the boundary extraction of the brain
ventricle from magnetic resonance (MR) images. But the
variations of shapes and sizes of the ventricles with the ages
usually make the extraction procedure difficult. Our special
interest is the human brain ventricle segmentation in MR
images for further study.
The Snake was originally developed by M. Kass [1].
It was curve defined within an image domain which can
move under the influence of internal forces from the
curve itself and external forces from the image data.
* Corresponding author. Tel.: C65 67905393; fax: C65 67912687.
E-mail addresses: [email protected] (Y. Wang), eekteoh@ntu.
edu.sg (E.K. Teoh).
0262-8856/$ - see front matter q 2005 Elsevier Ltd. All rights reserved.
doi:10.1016/j.imavis.2005.07.006
Once internal and external forces have been defined, the
Snake can detect the desired object boundaries (or other
object features) within an image. From the original
philosophy of Snake, there are many techniques have
been proposed as an alternative method for presenting a
curve: Fourier descriptors [2], B-Splines [3,7,9,10,21,26],
auto-regressive model [4], moments [5], HMM models
[6] and wavelets [8], etc. Among these methods,
B-Spline representation of the curve stands for an
advantage as such a formulation of a deformable model
allows for the local control and a compact representation.
The flexibility of the curve increases as more control
points are added while still remain less parameters
compared to other model. Moreover, the internal force is
not needed as the smoothness requirement has been
implicitly built into the model.
In this paper, we present a closed-form B-Snake model
called Dynamic B-Snake Model (DBM) with a Minimum
Mean Square Error (MMSE) approach [24]. It possesses
three main novelties:
(1) Internal forces are not required in DBM since the
B-Snake representation maintains smoothness via hard
constrains implicit in the representation while remains
compact representation.
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Y. Wang, E.K. Teoh / Image and Vision Computing 23 (2005) 1029–1040
(2) A method of MMSE is developed to iteratively estimate
the position of those control points in the closed
B-Snake model. As MMSE deforms the segments of the
B-Spline at a time, instead of individual points, it is very
robust to against local minima.
(3) In order to use available statistical information about
the desired object shape, the Principal Component
Analysis (PCA) has been applied to our B-Snake model.
This approach allows the deformation of B-Snake to
synthesize the shape similar to those in the training set.
The structure of this paper is arranged as follows. In
Section 2, a review of the existing B-Snake model is
presented. Section 3 introduces the B-Snake Model with the
algorithm for estimating the parameters of B-Snake. In
Section 4, the statistic model is given to model the samples
of the training set and guide the B-Snake deformation. The
simulation results are shown in Section 5. This paper
concludes in Section 6.
2. Related works
A deformable B-Spline algorithm was presented in [9]
for determining vessel boundaries and enhancing their
centerline features. A bank of even and odd S-Gabor filter
pairs of different orientations are convolved with vascular
images. The resulting responses across filters of different
orientations are combined to create an external energy field
for Snake optimization. Vessels are represented by cubic
B-Snake, and are optimized on filter outputs with dynamic
programming. In this algorithm, the starting and the end
points have to be fixed, and the dynamic programming is
very slow for the deformation of B-Snake.
Stammberger [19] proposed a B-Snake algorithm for the
segmentation of the knee joint cartilage from MR images by
using a multi-resolution approach. As the external forces are
generated by image edges and the distance transformation of
a standard model, the B-Snake may not deform to a desired
object, because there is no statistical information has been
included in this algorithm.
Mário presented an approach to unsupervised contour
representations and estimations by using B-Spline [20]. The
problem is formulated in a statistical framework with the
likelihood function being derived from a region-based
image model. However, no any priori knowledge of the
shape is used in this model.
Wang [21–23] presented a B-Snake based lane model for
lane detection. The external forces in this model are
designed based on the perspective relationship of lane
boundaries on the image plane. However, although the
results are good in lane detection, the number of control
points is fixed to three, this limits the capability to describe
the complex shape. In their later paper [26], a B-Snake
model with a strategy of adaptive control point insertion was
proposed for segmenting the complex structures in medical
images. However, in all of above methods, no any priori
knowledge of the shape is used.
For the case that the priori information is available,
current researches make it possible to be implemented into
the Snake model. Cootes presented a Point Distribution
Model (PDM) [11] for building flexible shape models. The
shape is represented by a set of landmark points. The shapes
are aligned and the deviations from the mean are analyzed
using principal component analysis. As an extension
research work to the point distribution model, Baumberg
proposed a cubic B-Spline model [10] for detecting and
tracking the walking pedestrians. The control points of
B-Spline are treated exactly the same way as the labeled
points of point distribution model [11]. This method has
been applied to a real-time processing system.
The Active Shape Model (ASM) was introduced by
Cootes et al. [12] to improve the PDM for object
segmentation. The object’s contour is presented by a set
of fix number of landmark points. PDM is constructed to
modeling the landmark points’ distribution of these shapes.
The local intensity structure of each landmark point is
constructed as well. ASM segmentation method involves
finding the new landmark points with the similar intensity
value and the new shape of landmark points is generated
that is similar to the shapes in the training set. ASM
parameters are then updated by fitting the shape to the target
points, a new contour is generated and new target points
searched. This process is repeated until convergence. ASM
has been used for several segmentation tasks in medical
images [13–18]. The shortcomings of ASM are, firstly these
landmark points have to be manually chosen for describing
the whole shape. That means these landmark points are not
only those key points of shape, but also other points between
them. They may not possess the affine-invariant property,
this will lead to the problem that the affine transformation
may not be recovered properly. Secondly, as the method
works by modeling how different landmark points tend to
move together as the shape varies, if the labeling is
incorrect, with a particular point placed at different sites
on each training shape, the method will fail to capture shape
variability. Thirdly, as the deformation for each model
points on the contour is independently, the gross error may
rise and the contour may be pulled away from the real
boundary position.
Motivated by the above problems, we here present a
knowledge based B-Snake model, Dynamic B-Snake
Model, for object segmentation. It is the extension of our
previous research work [26]. This model is combined with
the robustness of B-Spline and ASM. It consists with a reconstruction method of B-Spline curve and an affine
invariant alignment strategy for PCA to synthesize the
shape of B-Snake similar to those in the training set, and a
MMSE to perform a fine define deformation for matching
the object boundaries more precisely. The details are given
in the followings.
Y. Wang, E.K. Teoh / Image and Vision Computing 23 (2005) 1029–1040
1031
Where
3. On B-Snake model
The traditional Snake is a point-based deformable model.
The curve is represented by a set of discrete points. Its
deformation is driven by the external and internal forces that
are calculated from an energy function. From the original
philosophy of Snake, an alteration is using a parametric
B-Spline representation as the curve descriptor. Compared
to traditional point-based Snake, the B-Snake greatly
reduces the number of state variables required for
a Snake. We have implemented it successfully in
lane detection [21–23] and modeling complex object
contour [26].
B-Snake can be either open or close with any order. In
this paper, we are concerning to implement the close cubic
B-Spline to B-Snake.
3.1. Close cubic B-Splines
2
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s2
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3.2. The B-Snake model
The cubic B-Snake is defined as follows:
X
rðsÞ Z
gi ðsÞ; where 0% s% 1:
(2)
i
A close cubic B-Spline (Fig. 1) has control nC1 points
{QiZ[xiyi]T, iZ{0,1,.,n}, and nC1 connected curve
segments {gi(s)Z(xi(s),yi(s)), iZ1,2,.,nC1}. The connection points between curve segments are called the knot
points Pi(iZ1,2,.,nC1), where the B-spline bases are tied
together, PiZgi(0), (iZ1,2,.,nK1). Each curve segment is
a linear combination of four cubic polynomials by the
parameter s, where s is normalized between 0 and 1
(0%s%1). That is,
For a cubic B-Snake, gi(s) is defined by Eq. (1). The
control points are positioned so that the curve locates the
desired object contour.
The external energy term on r(s) is defined as E(r(s)).
There are no internal forces since the B-Spline representation maintains smoothness via hard constraints implicit in
the representation. Therefore, the total energy function of
the B-Snake EB-snake can be defined by integrating E(r(s))
along the B-Snake. That is,
ð1
EB-snake Z EðrðsÞÞ ds:
3
2
QðiK1ÞmodðnC1Þ
7
6
6 QimodðnC1Þ 7
7; i Z 1; 2; .; n C 1:
6
gi ðsÞ Z MR ðsÞ6
7
4 QðiC1ÞmodðnC1Þ 5
QðiC2ÞmodðnC1Þ
(3)
0
(1)
To identify the edge features in the image, the above
expression must be minimized along the B-Spline curve. So
that, at equilibrium (when image forces vanish), r(s)
stabilizes close to a contour of object.
Fig. 1. A closed cubic B-spline curve.
1032
Y. Wang, E.K. Teoh / Image and Vision Computing 23 (2005) 1029–1040
2
3.3. Minimum mean square error approach
Gradient Vector Flow (GVF) [27] is selected here as the
definition of external forces for B-Snake, since GVF has a
larger capture range. It is computed as a diffusion of the
gradient vector of a gray-level or binary edge map derived
from the image. When the B-Snake is on the boundaries of
object, its total external force Fext that is summation of the
localized external force Fext(r(s)) should be zero. That is,
ð1
Fext Z fext ðrðsÞÞ ds Z 0:
(4)
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On the other hand, if the total external force of the
B-Snake is zero, then there will be no change for the
position of B-Snake. We assume that at time t the localized
external force fext(rt(s)) is connected with the position
change of B-Snake rt(s)KrtK1(s) in the following way,
fext ðr t ðsÞÞ Z gðr t ðsÞKr tK1 ðsÞÞ;
2
6 3
6 snK1;2 s2nK1;2 snK1;2
MnK1 Z 6
6 $
$
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s3nK1;m s2nK1;m snK1;m
(5)
where g is a step-size. The position change of the B-Snake
can be mapped to the adjustment of the control points at Q
time t. For example, DQ(t) is defined as the adjustment at
time t, then we have
QðtÞ Z QðtK1Þ C DQðtÞ:
(6)
2
K1 T
M M F
(7)
where g is the step size, and M is a matrix [26] calculated
from Eq. (1):
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Mn Z 6
6 $
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The localized external force fext(rt(s)) can be sampled
along the B-Snake. In this way, we can solve the Eq. (5)
digitally. Here, the Minimum Mean Square Error solution
for the digital version of the Eq. (5) is given as a matrix form
[24],
DQðtÞ Z gK1 ½ M T
s3nK1;1 s2nK1;1 snK1;1
2
s3n;1 s2n;1 sn;1
6 3
6 sn;2 s2n;2 sn;2
Mn0 Z 6
6 $
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s3n;m s2n;m sn;m
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Y. Wang, E.K. Teoh / Image and Vision Computing 23 (2005) 1029–1040
1033
Fig. 2. B-Snake in human brain ventricle contour extraction.
2
2
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6 3
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MnC1 Z 6
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knowledge-based strategy for B-Snake deformation using
B-Spline re-construction and statistical information of a
training set.
In order to use statistical information to guide the
B-Snake deformation, the correspondence problem between
two shapes should be solved. First we have to fix the number
of control points of B-Snake, and then find the corresponding knot point between the training sets. Every B-Spline
shape in the training set has to be aligned before extracting
the statistical geometric information by using PCA. PCA
will then guide the B-Snake deformation in the new image
with a similar shape in the training set.
4.1. Re-construction of B-Spline curve
Re-construction of the B-Spline with a fix number of
control point is based on re-assigning the knot points for
each segment with equal affine-arclength [29], since the
equal arclength is not preserved under affine transformation, but affine-arclength will do. Affine-arclength is
defined as:
and si,m is the parameter s to form the sampling point m in ith
segment, FZ[.fext(rt(si,m)).].
In real practice, B-Snake is initialized by four control
points shown in Fig. 2(a). Associated with an adaptive
control point insertion strategy presented in [26], the final
result of B-Snake in human brain ventricle contour
extraction is shown in Fig. 2(b), the number of control
points is increased to 17.
tl ðsÞ Z
4. B-Snake model using statistical geometric information
Here we chose m as the number for the new knot
points of re-constructed B-Spline curve, the length
between two conscious knots is equal affine arc-length.
Suppose the new knots are PZ(P0,P1,P2,.,PmK1) the
corresponding control points are defined as:
PCA is a classical statistical method [28]. This linear
transform has been widely used in data analysis and
processing. We have implemented PCA to B-Snake as an
effect approach to the data. In this section, we suggest a
Ðs
ðx_l y€ l Kx€ l y_l Þ1=3
0
P Ðs
i
(9)
ðx_i y€ i Kx€ i y_i Þ1=3
0
Q Z AK1 P
(10)
1034
Y. Wang, E.K. Teoh / Image and Vision Computing 23 (2005) 1029–1040
Fig. 3. A B-Snake with 40 control points.
where
2
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affine transformation Aalign between each two alignment
could be estimated. Please see the triangles’ area in Fig. 4 as
a demo example on the attribute vector. Some results of
shape alignment based on this method are shown in Fig. 5.
4.3. Modeling the shape distribution from the training set
(11)
Fig. 3 shows an example of 40 control points of
B-Snake, it is re-constructed from Fig. 2(b) which has 17
control points.
4.2. The shape alignment strategy
To determine the correspondence between two
B-Snakes, a shape algorithm [25] which is using an affineinvariant feature has been implemented in B-Snake model.
In this algorithm, for every knot point Pi obtained in last
Section, an attribute vector Fi which is calculated by the
areas formed by adjacent knot points has been assigned to it.
As the attribute vector holds parameters that describe the
shape characteristics of the adjacent knot points around each
knot point, the far adjacent knot points represents more
global properties of the contour of knot points. Therefore,
the attribute vector corresponding to Pi can be expected to
be different from attribute vectors of other points. As the
area is related affine invariant, the attribute vectors are
affine-invariant as well. Hence, shape alignment could be
achieved by an error minimization process and the assumed
An effective approach is to apply PCA to the training data.
The data form a cloud of points in the n-d space. PCA computes
the main axes of this cloud, allowing one to approximate any of
the original points using a model with fewer than n-d
parameters. Given a set of the training data of aligned knot
points of B-Snake, {Pi}, the approach is as follows:
(1) Compute the mean of data, P T and the covariance of the
data, C.
(2) Compute the eigenvectors, fi and corresponding eigenvalues li of C, and sort fi by the size of their corresponding
eigenvalues.
Fig. 4. Illustration of the concept of the ‘attribute vector’ on the ith knot
point. The area of a triangle P[iKvs]P[i]P[iCvs] is used as the vsth element of
the ith attribute vector Fi. Here, 1%vs%n/2.
Y. Wang, E.K. Teoh / Image and Vision Computing 23 (2005) 1029–1040
1035
Fig. 5. Some aligned results of B-Snake model.
(3) Let F contains the m eigenvectors corresponding to
largest eigenvalues,0 then use below formulas for fitting the
knot point vector P of the model to the knot point vector P
of the aligned B-Snake shape:
P 0 Z P T C FH
(12)
where H is a m dimensional vector given by
H Z FT ðPKP T Þ
(13)
The vector H defines a set of parameters of a B-Snake
model.
By varying the elements of H, the shape of knot points,
0
P , can be varied as well by using Eq. (12). The number of
eigenvectors to retain, m, can be chosen so that the model
represents some proportion (98%) of the total variance of the
data, or so that the residual terms can be considered
noise.0
0
Once obtaining the best knot point vector P , we transform P
back to update the positions of knot points in the current
B-Snake by using the affine-transformation matrix Aalign.
4.4. The complete DBM algorithm
The complete algorithm using both affine invariant
alignment and statistical information is as follows:
; iZ 0; 1; .; Ng from the
(1) Get a reference model fPmodel
i
training set.
(2) Initialize the B-Snake as fPmodel
; iZ 0; 1; .; Ng.
i
(3) Deform the B-Snake by using MMSE to minimize
external force (Section 3.3). If the iteration number
exceeds a predefined number or external force of
B-Snake is below a define value, go to step 6.
(4) Align the current B-Snake configuration with the
standard model contour by using the affine-transformation matrix Aalign calculated from the B-Snake to the
model [25]. Then, stack the aligned B-Snake as a knot
point vector
align T
P Z ½xalign
; yalign
; .; xalign
N ; yN 0
0
Fig. 6. Deformation of DBM in ventricle segmentation.
1036
Y. Wang, E.K. Teoh / Image and Vision Computing 23 (2005) 1029–1040
Fig. 7. Some contour extraction results of different MR brain images using DBM. (a)–(l): the green contour denotes the initial shape of B-Snake while the bright
one denotes the final result.
(5) Map the knot point vector into the new vector P into the
new vector P 0 using the statistical model, P 0 Z P T C FH,
which is described in Section 4.3. Then, transform P 0
back into the original coordinate space of the B-Snake
via the inverse matrix of Aalign and update the B-Snake.
Go to step 3.
(6) Stop.
5. Experimental results and discussion
The algorithm presented above has been implemented
in medical images for object segmentation. Two experiment results are presented. First, the performance of
DBM for ventricle segmentation in MR images are
presented. Second, the qualitative and quantitative
Y. Wang, E.K. Teoh / Image and Vision Computing 23 (2005) 1029–1040
1037
Fig. 8. A failed case of DBM.
comparisons of DBM with the traditional Snake and ASM
are presented as well.
5.1. Experiment results on object segmentation
We are using 105 shapes of ventricle contour with 40
control points to form the training sets in this application. One
of the shapes in the training set was chosen as the
initialization of B-Snake.
Fig. 6(a) shows an initialization of B-Snake, the initial
shape is taken from the training set. The deformation of DBM
is showed in Fig. 6(b)–(e), there are a total of 17 iterations
that have been involved. The final result superimposed on the
original image is shown in Fig. 6(f). In this experiment, the
B-Snake was firstly attracted by the external force which is
generated by the significant edges of object, and then the
deformed shape of B-Snake was mapped to the PCA to get
the most close shape in the training set. The final shape of
B-Snake fitting to the ventricle boundaries is quite close.
Fig. 7(a)–(l) show more contour extraction results for
different MR brain images using DBM. The green contours
are the initial shapes of B-Snake in each image, while the
white contours denote the final results after B-Snake
deformation. Please notice that the deformation from the
initialization to the final result is large. Since the distribution
of knot points of the shapes included in the training set has
been modeled in our model, therefore, the proposed method
intends to maintain the geometric shape of the B-Snake
model during the shape deformation procedure. More
importantly, our model doesn’t deform Snake points
individually, but it deforms segments of the B-Snake at a
time. It is shown that this procedure helps B-Snake to be very
robust to local minima. Further more, as the external force
field is generated by the edge image, the MMSE makes sure
the B-Snake curve matches the object boundaries more
accurately.
However, as the presented B-Snake is attracted strongly
by the local edges, therefore, we can’t guarantee that all
global minima are found, and this will lead to some failed
cases. Fig. 8 gives one example of failure of our algorithm
after convergence to a local minima. We can observe that, on
the bottom left of B-Snake, the B-Snake was deformed to the
wrong edge. This is because the proposed B-Snake model
does not use the additional information to form the capability
to distinguish whether it is the desired object’s edge or not.
To overcome this problem, more available information about
the studied object should be used, such as intensity and
texture.
5.2. Experiments on comparing DBM with the traditional
snake and ASM
To evaluate the robustness of DBM, a comparative study
of DBM with the traditional Snake and ASM have been
carried out. The results obtained are presented in this Section.
Fig. 9 qualitatively compares the performance of our
DBM, the traditional Snake and ASM in detecting ventricles
from the MR brain images. Initialization for three different
MR images are shown in Fig. 9(a). The three models are
initialized with the same location and contour. Fig. 9(b)
shows the final results of traditional Snake, we can observe
that the traditional Snake lacks the capability to deform
largely to overcome the local erroneous edges. In addition, it
did not preserve any anatomical homology during the
deformation. ASM is much better than the traditional Snake
in preserving the model shape in the deformation procedure.
This is as shown in Fig. 9(c). However, we observed that
some parts of the model curve are either trapped in local
minimum or not matching the desired object boundaries
precisely. This is because ASM is not using the object edges
information as an assistant to help it approach the object
boundaries accurately. Hence, its final results in Fig. 9(c) are
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Y. Wang, E.K. Teoh / Image and Vision Computing 23 (2005) 1029–1040
Fig. 9. Qualitative Comparison of DBM with the traditional Snake and ASM. (a1)–(a3) indicate the initialized contours; (b1)–(b3) indicate the results of
traditional Snake; (c1)–(c3) indicate the results of ASM; (d1)–(d3) indicate the results of DBM.
still unsatisfactory. On the contrary, DBM achieved good
results as shown in Fig. 9(d). We can observe that, in addition
to preserve the model shape, DBM can match the ventricles’
boundaries more precisely. More importantly, compared to
the traditional point based Snake and ASM, DBM deforms
the segments of the B-Snake at a time rather than the
individual points. This procedure is shown to help the
B-Snake to be very robust to overcome the local minima.
Table 1 gives the quantitative comparison of DBM with
the traditional Snake and ASM using the results presented in
Fig. 9. The average and maximum distance of the final
contour marked by experts is shown Table 2, using units of
Y. Wang, E.K. Teoh / Image and Vision Computing 23 (2005) 1029–1040
1039
Table 1
Quantitative comparison in term of DBM with the traditional Snake and ASM based on the results presented in Fig. 9
Traditional Snake
ASM
DBM
Results for MR images shown
in Fig. 9 (b1,c1,d1)
Results for MR images shown
in Fig. 9 (b2,c2,d2)
Results for MR images shown
in Fig. 9 (b3,c3,d3)
Avg. dist.
Max dist
Avg. dist.
Max dist
Avg. dist.
Max dist
8.2
4.5
2.7
23.5
10.4
6.8
6.2
3.1
2.4
20.1
8.9
6.5
5.1
3.7
2.2
19.3
16.7
5.3
Table 2
Quantitative comparison of computation time taken by DBM with the traditional Snake and ASM based on the results presented in Fig. 9
Traditional Snake
ASM
DBM
Results for MR images shown
in Fig. 9 (b1,c1,d1) (ms)
Results for MR images shown
in Fig. 9 (b2,c2,d2) (ms)
Results for MR images shown
in Fig. 9 (b3,c3,d3) (ms)
220
430
560
250
410
490
210
470
520
pixel. DBM has the least average distances at 2.2–2.7 pixels
and least maximal distance at 5.3–6.8 pixels for all three
examples. In view of above comparison results, DBM
demonstrates its superior robustness to the rest two
deformable models against of local minima.
Table 2 shows the comparison of processing time of DBM
with the traditional Snake and ASM based on the results
presented in Fig. 9. The traditional Snake is the fastest
because it uses the edge image for deformation only. DBM is
slightly slower than ASM. This is because more time is
required in process of MMSE. This, in term, will guarantee a
more precise matching of the object’s edge by DBM.
6. Conclusion
In this paper, a Dynamic B-Snake Model (DBM) is
presented for complex shape segmentation. The proposed
algorithm is applied for the segmentation of 2D ventricle in
the MR brain images. In this model, a statistical framework is
embedded in order to use the prior knowledge of the studied
object. This is done by combining the PCA for modeling the
shape distribution in the training samples while taking into
account the case of affine transformation. Constrained by the
prior geometric knowledge that reflects the shape, DBM can
keep its geometric shape during the procedure of segmentation. This helps to avoid the local minima. The final positions
of the control points are refined by MMSE method during the
B-Snake deformation procedure. It provides fine deformation
for allowing the B-Snake to accurately adapt to the desired
object boundaries in the image. The experimental results
show that the proposed DBM has a robust capability to
describe complex object contours and can achieve more
accurate object segmentation when compared to the
traditional Snake and ASM.
Several extensions of this approach are possible. In this
paper, the main focus is on the construction of 2-D
deformable model for object segmentation. However, from
a 2-D image, it is not possible to fully recover the 3-D shape
due to projection. Although our model can be employed to
segment 3-D objects by slice-to-slice segmentations, the
segmentation results will be definitely worse than those by
the actual 3-D B-Surface model. Using B-Surface to
construct and describe the object surface remains a challenge.
Two problems need to be solved. First, how to define the
network of B-Surface patches that are continuously
connected at their boundaries without any overlap or leakage
to form a closed object surface is a difficult task. The
technique of multiple control point may be used here to form
a particular shape of patch or using different orders of
Bsurface patched to form one surface. A triangle patch
definition has been proposed in [30]. Second, the MMSE
method in this thesis is for 2-D only, but it is possible to be
extended to 3-D B-Surface model for estimating the 3-D
object. Here, the principle for deforming the B-Surface is the
same as the B-snake model, but more calculation is expected
for B-Surface deformation.
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