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Файл:Ординатура / Хирургия / Библиотека им академика М.И. Перельмана / Книга_5835_Библиотеки_им_академика_М_И_Перельмана.pdf
() ()
{}
() () ()
{}
()
C w xy xy
C w xy xy
Cw
== ∈ =
== ∈ >
=
∂ ,:,
,:,
ΩΦ
ΩΦ
Ω
0
0inside
outside
() ()
{}
=∈ <
⎧
⎨
⎪
⎩
⎪
⎫
⎬
⎪
⎭
⎪
xy xy,:,ΩΦ 0
(12.18)
So, the unknown C could be replaced by Φ. We use Heaviside function H and Dirac
measure δ
0
, which are defined by:
()
()
()
()
Hz
z
z
dH z
dz
=
≥
<
⎧
⎨
⎩
=
10
00
0
,
,
, δ
(12.19)
Now, the energy function F(c
1
, c
2
, C) could be rewritten as:
() ()
()
() ()
()
(
)
Fc c xy xy dxdy v H xy dxdy
u
C
12
1
,,, ,,= ∇ +
+
∫∫
ΦΦΦ Φ
ΩΩ
μδ
λ
inside
() ()
()
(
)
() ()
()()
01
2
202
2
1
xy c H xy dxdy
uxy c H xy dxdy
C
,,
,,
−
+−−
∫
∫
Φ
Φλ
outside
(12.20)
where c
1
(Φ) and c
2
(Φ) are defined as:
()
() ()
()
()
()
c
u x y H x y dx dy
Hxydxdy
1
0
Φ
Φ
Φ
Ω
Ω
=
∫
∫
,,
,
(12.21)
()
() ()
()()
()
()()
c
u x y H x y dx dy
Hxydxdy
2
0
1
1
Φ
Φ
Φ
Ω
Ω
=
−
−
∫
∫
,,
,
(12.22)
Finally, the correspondinglevel set equation whichis minimizing the energy, can
be solved by the following equation:
() ()()
∂
∂
δμ λ λ
Φ
Φ
Φ
Φt
vuc uc=
∇
∇
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
−− − + −
⎡
⎣
⎢
⎤
⎦
⎥
div
10 1
2
20 2
2
(12.23)
The level set equation could be solved iteratively using time step Δt. However,
there are inherent time-step requirements to ensure stability of the numerical scheme
via the CFL
1
condition. In Chan and Vese’s approach, the time step could be set
based on the following equation:
324 Prostate Analysis Using Ultrasound Imaging
1. The domain of dependence of a hyperbolic partial differential equation (PDE) for a given point in the prob
-
lem domain is that portion of the problemdomain that influencesthe value ofthe solution atthe given point.
Similarly, the domainof dependence of an explicitfinite difference scheme for a givenmesh point isthe set of
mesh points that affect the value of the approximate solution at the given mesh point. The CFL condition,
named for its originators Courant,Friedrichs, and Lewy,requires that the domain of dependence of thePDE
must lie within the domain of dependence of the finite difference scheme for each mesh point of an explicit
finite difference scheme for a hyperbolic PDE. Any explicit finite difference scheme that violates the CFL
condition is necessarilyunstable, but satisfyingthe CFL conditiondoes not necessarily guarantee stability.
()
()
Δ
ΔΔΔ
t
xyz
v
≤
++ +
min , ,
μλλ
01
(12.24)
12.3 Fully Automated 3-D Prostate Segmentation from Transrectal
Ultrasound (TRUS) Images
The problem of missing or diffuse boundaries is a very challenging one for medical
image processing. Missing or diffuse boundaries can be caused by patient move-
ments, by a low SNR of the acquisition apparatus, or by being blended with similar
surrounding tissues. Under such conditions, without a prior model to constrain the
segmentation, most algorithms (including intensity- and curve-based techniques)
fail—mostly due to the underdetermined nature of the segmentation process. Simi-
lar problems arise in other imaging applications as well and they also hinder the seg-
mentation of the image. These image segmentation problems demand the
incorporation of as much prior information as possible to help the segmentation
algorithms extract the tissue of interest.
A number of model-based image segmentation algorithms in the literature were
used to deal with cases when boundaries in medical images are smeared or missing.
In 1995, Cootes et al. [55] developed a parametric point distribution model (PDM)
for describing the segmenting curve that uses linear combinations of the
eigenvectors to reflect variations from the mean shape. The shape and pose parame-
ters of this PDM are determined to match the points to strong image gradients.
Wang and Staib [60] developed a statistical point model in 1998 for the segmenting
curve by applying principal component analysis (PCA) to the covariance matrices
that capture the statistical variations of the landmark points. They formulated their
edge detection and correspondence determination problem in a maximum a posteri-
ori Bayesian framework. An image gradient is used within that framework to calcu-
late the pose and shape parameters that describes their segmenting curve.
Leventon et al. [56] proposed a less restrictive model-based segmenter. They
incorporated shape information as a prior model to restrict the flow of the geodesic
active contour. Their prior parametric shape model is derived by performing PCA
on a collection of signed distance maps of the training shape. The segmenting curve
then evolves according to the gradient force of the image and the force exerted by
the estimated shape. In 2002, Mitchell et al. [61] developed a model-based method
for fixedmage segmentation. Comprehensive design of a 3-D active appearance
model (AAM) is reported for the first time as an involved extension of the AAM
framework introduced by Cootes et al. [62]. The model’s behavior is learned from
manually traced segmentation examples during an automated training stage. Infor-
mation about the shape and image appearance of the cardiac structures is contained
in a single model. The clinical potential of the 3-D AAM is demonstrated in
short-axis cardiac MR images and four-chamber echocardiographic sequences. The
AAM method showed good agreement with the independent standard using quanti-
tative indexes of border positioningerrors, endocardial and epicardial volumes, and
left ventricular mass. The AAM method shows high promise for successful applica-
tion to MR and echocardiographic image analysis in a clinical setting. The reported
method combined the appearance feature with the shape knowledge and it provided
12.3 Fully Automated 3-D Prostate Segmentation from Transrectal Ultrasound (TRUS) Images 325
robust matching criterion for segmentation. However, this method needs to set up
point correspondences, which makes the procedure complicated.
In [15], the authors adopted implicit representation [54] of the proposed seg-
menting curve and calculated the parameters of the implicit model to minimize the
region-based energy based on the Mumford-Shah functional for image segmenta-
tion [59]. In [16], the authors combined intensity prior with the shape knowledge
and transferred the segmentation problem in a Bayesian interface. Efficient kernel
functions were utilized, which made the method more robust and more efficient.
Motivated by these state-of-the-art medical imaging techniques, we designed and
implemented a system for fully automated 3-D prostate segmentation from TRUS
images that uses shape and intensity knowledge in a level set framework. The
method used in this system includes fixed important steps:
1. 3-D shape prior extraction;
2. Intensity prior extraction;
3. Segmentation in Bayesian interface;
A brief flow chart of the method is shown in Figure 12.4.
326 Prostate Analysis Using Ultrasound Imaging
Generate
3-D close
surfaces
from
2-D manual
tracing
Training
binary
image
set B
Aligned
training
image
set B
~
Level
sets of
aligned
images
Ψ
Mean
shape
Eigen
shapes
Φ+
Φ
3-D original
transrectal
ultrasound
images
Use image
as mask
to extract
intensity
prior
B
Intensity
mode
Shape
model
Shape
vector
( , ,... )
for every
training
shape
αα α
12 N
Initial
pose–
weighting
factor–
p
α
Update
pose–
weighting
factor–
p
α
Generate
new shape
Compute
the posterior
probability in
Bayesian
inference
If the
posterior
probability
maximized
Output
segmentation
result
Segmentation
Prior
knowledge
extraction
Yes
No
Figure 12.4 The flow chart of the 3-D prostate analysis system using ultrasound images. Basically
there are two stages. The first stage is system training to extract shape and intensity prior and the
second is to segment image using the statistical information from first step.
12.3.1 3-D Shape Prior Extraction
Extraction of 3-D shape priors is more complicated than in 2-D space. Consider the
3-D close surface from 2-D contours method: A set of binary 3-D images {B
1
, B
2
, ...,
B
n
}, has 1 as an object and 0 as the background. The 3-D binary shape is from man-
ual segmentation in each 2-D slice. The first step toward 3-D shape model creation
is to stack the 2-D segmentation. However, the stacked version has a zigzagged sur-
face and so its accuracy is not guaranteed. We apply the Min/Max flow [63] method
to smooth the 3-D surface and increase the accuracy between every two slices.
Min/Max flow uses the curvature-based force to fix the surface and the patch on the
surface with large positive or small negative curvatures will be smoothed, which can
not be done by simply creating meshes based on adjacent contours.
To show the effect of using curvature as the driving force in the level set method,
we give the following 2-D example. In (12.6), we define the F as:
()
()
Fk
Fk
min
max
min , .
max , .
=
=
00
00
(12.25)
Here, we have chosen the negative of the signed distance in the interior, and the
positive sign in the exterior region. As shown in Figure 12.5, the effect of flow under
F
min
is to allow the inward concave fingers to grow outwards, while suppressing the
motion of the outward convex regions. Thus, the motion halts as soon as the convex
hull is obtained. Conversely, the effect of flow under F
max
is to allow the outward
12.3 Fully Automated 3-D Prostate Segmentation from Transrectal Ultrasound (TRUS) Images 327
Figure 12.5 Motion of Curve under Min/Max Flow. The first column are the five intermediate
results using F
min
= min(k, 0.0). The second column are the five intermediate results using F
max
=
max(k, 0.0).
regions to growinward while suppressing the motion of the inward concave regions.
However, once the shape becomes fully convex, the curvature is always positive
and, hence, the flow becomes the same as regular curvature flow; hence, the shape
collapses to a point.
We can summarize the preceding paragraphs by saying that, for the preceding
case, flow under F
min
preserves some of the structure of the curve, whereas flow
under F
max
completely diffuses away all of the information. Similar effects can be
achieved in 3-D space. By using Min/Max flow in a small number of iterations, the
3-D surface will be smoothed. In Figure 12.6, one example is shown of how
Min/Max flow smooths the 3-D surface for 3-D prostate shape extraction.
After using the Min/Max flow method for each 3-D surface, in order to extract
accurate shape information, alignment has to be applied. Alignment is a task to cal-
culate pose parameter p, which includes translation in the x, y, and z directions,
rotation around x, y, and z axis and scale in x, y, and z directions. In Figure 12.7, we
listed the original 3-D shape after Min/Max flow. The example includes only eight
data sets for illustration.
The strategy to compute the pose parameters for n binary images is to use the
gradient descent method to minimize the specially designed energy functional
E
j
align
align for each binary image corresponding to the fixed one, say, the first binary
image B
1
. The energy is defined by the following equation:
328 Prostate Analysis Using Ultrasound Imaging
(a) (b)
(c) (d)
Figure 12.6 Using Min/Max flow to smooth 3-D surface. (a) The image is the 2-D contours from
manual tracing in 3-D space. (b) Three views of after overlay the stacked 2-D contours onto the
original 3-D ultrasound image data. (c) Three views of the surface rendering of the stacked 2-D
contours. (d) Three views of the surface rendering of after Min/Max flow.
()
()
E
BB dA
BB dA
j
j
j
align
=
−
+
∫∫
∫∫
Ω
Ω
~
~
1
2
1
2
(12.26)
where Ω denotes the image domain,
~
B
j
denotes the transformedimage of B
j
based on
the pose parameters p. We term the aligned 3-D shapes as {
~
B
1
,
~
B
2
, ...,
~
B
n
}. Minimiz-
ing this energy is equivalent to minimizing the difference between current binary
image B
1
and the fixed image in the training database. The normalization term in
the denominator is employed to prevent the images from shrinking to improve the
cost function. The hill climbing or Rprop [64] method could be applied for the gra-
dient descent. Figure 12.8, shows the 3-D shapes after 3-D alignment. The example
includes only eight data sets for illustration and they correspond to the original
shapes shown in Figure 12.7.
Based on the 3-D binary images from the training datasets, we collected a set of
3-D training shapes encoded by their signed distance functions. The popular and
natural approach to represent shapes is via point models where a set of marker
12.3 Fully Automated 3-D Prostate Segmentation from Transrectal Ultrasound (TRUS) Images 329
Figure 12.7 The three views of the surface rendering of the 3-D surface after Min/Max flow for
the original 3-D shapes. The last image is the overlapping of the eight images.
points is often used to describe the boundaries of the shape. This approach suffers
from problems such as numerical instability, inability to accurately capture high
curvature locations, difficulty in handling topological changes, and the need for
point correspondences. To overcome these problems, Eulerian approach to shape
representation based on the level set methods [21] of Osher and Sethian can be
utilized.
The signed distance function is chosen as the representation for shape. In partic-
ular, the boundaries of each of the aligned shapes in {
~
B
1
,
~
B
2
, ...,
~
B
n
}, are embedded as
the zero-level set of separate signed distance functions {Ψ
1
, Ψ
2
, ..., Ψ
n
} with negative
distances assigned to the inside and positive distances assigned to the outside of the
object. The mean level set function of the shape database as the average of these
signed distance functions can be computed as
ΦΨ=
=
∑
1
1
n
i
i
n
330 Prostate Analysis Using Ultrasound Imaging
Figure 12.8 The three views of the surface rendering of the 3-D surface after 3-D alignment. The
last image is the overlapping of the eight images after alignment.
To extract the shape variabilities,
Φ
is subtracted from each of the n signed dis-
tance functions to create n mean-offset functions {
∼
Ψ
1
,
∼
Ψ
2
, ...,
∼
Ψ
n
}. These mean-offset
functions are analyzed and then used to capture the variabilities of the training
shapes.
Specifically, n column vectorsare created,
∼
ψ
I
, from each
∼
Ψ
1
. A naturalstrategy is
to utilize the N
1
× N
2
× N
3
grid of the training images to generate N = N
1
× N
2
× N
3
lexicographically ordered samples (where the columns of the image grid are sequen-
tially stacked on top of one other to form one large column). Next, define the
shape-variability matrix S as: [
∼
ψ
1
,
∼
ψ
2
, ...,
∼
ψ
n
]. The procedure for creation of the
matrix S is shown in Figure 12.9.
An eigenvalue decomposition is employed:
1
n
TT
SS U U=Σ
(12.27)
where U is an N = N × n matrix whose columns represent the orthogonal modes of
variation in the shape and Σ is an n × n diagonal matrix whose diagonal elements
represent the corresponding nonzero eigenvalues. The N elements of the ith column
of U, denoted by U
i
, are arranged back into the structure of the N = N
1
× N
2
× N
3
image grid (by undoing the earlier lexicographical concatenation of the grid col-
umns) to yield Φ
i,
the ith principal mode or eigenshape. Based on this approach, a
maximum of n different eigenshapes {Φ
1
, Φ
2
, ..., Φ
n
} are generated. In most cases,
the dimension of the matrix
1
n
T
SS
is large so the calculation of the eigenvectors and
eigenvalues of this matrix
1
n
T
SS
is computationally expensive. In practice, the
eigenvectors and eigenvalues of can be efficiently computed from a much smaller n
× n matrix W given by
1
n
T
SS
. It is straightforward to showthat ifd isan eigenvector
12.3 Fully Automated 3-D Prostate Segmentation from Transrectal Ultrasound (TRUS) Images 331
N
N
3
N
1
Ψ
i
∼
Ψ
i
∼
S
Figure 12.9 Create the shape-variability matrix S from 3-D mean-offset functions
~
Ψ
i
.
of W with corresponding eigenvalue λ, then Sd is an eigenvector of
1
n
T
SS
with
eigenvalue λ.
For segmentation, it is not necessary to use all of the shape variabilities after the
preceding procedure. Let k ≤ n, which is selected prior to segmentation, be the num-
ber of modes to consider. In general, k should be chosen large enough to be able to
capture the main shape variations present in the training set. One way to choose the
value of k is by examining the eigenvalues of the corresponding eigenvectors. In
some sense, the size of each eigenvalue indicates the amount of influence or impor-
tance its corresponding eigenvector has in determining the shape. By looking at a
histogram of the eigenvalues, one can estimate the threshold for determining the
value of k. However, an automated algorithm is hard to implement as the threshold
value for k varies for each different application. It is hard to define a universal k that
can be set. The histogram of the eigenvalues for the preceding examples are shown
in the Figure 12.10. In total, there are eight eigenshapes, corresponding to the eight
eigenvalues. However, only the first seven are large enough to be used to represent
all of the shape variations.
The preceding processing method is based on work from Tsai et al. [15], who
proposed an efficient method to reduce the segmentation problem to one of
finite-dimensional optimization by constraining the optimization problem to the
finite-dimensional subspace spanned by the training shapes. Then a new level set
function is defined to represent the 3-D shape as:
[]
ΦΦ Φα =+
=
∑
a
ii
i
k
1
(12.28)
where α = {a
1
, a
2
, ..., a
k
} are the weights for the k eigenshapes. Now we can use this
newly constructed level set function Φ as the implicit representation of shape. specif-
332 Prostate Analysis Using Ultrasound Imaging
The histogram of the eigenvalues
The index of the eigenvalues
The value of the eigenvalues
0.00E 00+
2.00E 06+
4.00E 06+
6.00E 06+
8.00E 06+
1.00E 07+
1.20E 07+
1.40E 07+
1.60E 07+
1.80E 07+
2.00E 07+
12
5498003
4966346
2674709
1886984
1380188
1254552
0.000026
18972352
3
4
56
7
8
Figure 12.10 The histogram of the eight eigenvalues after PCA analysis on the eight 3-D prostate
shapes.
ically, thezero-level set of Φ describes the shape, with the shape’s variability directly
linked to the variability of the level set function. Therefore, by varying α, Φ will be
changed, which indirectly varies the shape. The average shape and several shape
variances are illustrated in Figure 12.11.
12.3.2 Intensity Prior Extraction
From the original energy definition for segmentation in (12.17), we found that the
intensity of each pixel inside the image is taken with the same probability. In other
words, the energy function never uses the intensity prior based on the shape model.
Although for certain imaging modalities, such as ultrasound or CT, the struc-
tures of interest do not differ much from their background, the intensity informa-
tion in the image can be utilized by probabilistic intensity models. This extra
12.3 Fully Automated 3-D Prostate Segmentation from Transrectal Ultrasound (TRUS) Images 333
(f)
(c)
(a)
(g)
(d) (e)
(b)
Figure 12.11 Three views and surface rendering of the average shape and several shape vari
-
ances. (a) The mean shape + 0.5 of the first eigenshape. (b) The mean shape −0.5 of the first
eigenshape. (c) The mean shape + 0.5 of the second eigenshape. (d) The mean shape. (e) The
mean shape −0.5 of the second eigenshape. (f) The mean shape + 0.5 of the third eigenshape.
(g) The mean shape −0.5 of the third eigenshape.