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Файл:Ординатура / Хирургия / Библиотека им академика М.И. Перельмана / Книга_5835_Библиотеки_им_академика_М_И_Перельмана.pdf
()
SAB
p
p
q
p
p
q
i
A
q
i
t
i
A
q
it
+=
−
⎛
⎝
⎜
⎞
⎠
⎟
−
+
−
⎛
⎝
⎜
⎞
⎠
⎟
==+
∑∑
1
1
1
1
1
2
1
∂
()
q
q
p
p
q
q
p
i
A
q
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A
−
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⎜
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⎜
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1
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⎟
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q
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v
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q
q
q
q
q
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1
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k
i
B
q
i
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v
i
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q
q
q
q
q
p
p
∑∑
−
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−
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⎝
⎜
⎞
⎠
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−
+
−
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⎜
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−
⎛
⎝
⎜
⎜
⎜
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+−
==+
∑∑
q
i
t
i
A
q
it
q
p
p
q
q
1
2
1
1
1
1
1
∂
⋅
−
⎛
⎝
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−
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==+
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2
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p
p
q
p
p
q
i
A
q
i
t
i
A
q
it
∂
⎟
⎟
⎟
⎟
⎟
⋅
−
⎛
⎝
⎜
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⎠
⎟
−
⎛
⎝
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+
−
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=+
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1
1
1
1
1
2
q
q
q
q
q
i
B
q
i
v
i
B
∂
()
⎟
−
+− ⋅
−
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−
⋅
−
⎛
=+
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∑
∑
q
iv
k
i
B
q
i
v
i
B
q
q
q
q
q
q
q
1
1
1
2
1
1
1
1
1
∂
⎝
⎜
⎞
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−
⎞
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⎟
⎟
⎟
⎟
⎟
=+
∑
q
iv
k
q
1
1
(8.19)
In this case, to find the optimal luminance level, such as the equivalent in (8.10),
we can compute the value for (8.19) by taking the argument that maximizes the next
expression:
204 Automatic Classification of Breast Lesions in 3-D Ultrasound Images
()
()
[
()
()
tSSqSS
SS qSS
opt
AA AA
BB BB
=++−⋅⋅
×++−⋅⋅
arg max
12 12
12 12
1
1
()
()
()
()
()
]
⋅−
× + +− ⋅ ⋅ ⋅ + +− ⋅ ⋅
1
11
12 12 12 12
q
SS qSS SS qSS
AA AA BB BB
(8.20)
In this case, instead of (8.10), t
opt
= {t
opt1
, t
opt2
, t
opt3
} is a set of threshold values for
two recursions, where t
opt1
is the first threshold, and t
opt2
and t
opt3
are new thresholds
for P
A
and P
B
distributions, respectively.
Equation (8.19) is simple, even though it has several terms. Developing a third
recursion would yield a number of terms up to 16, making interpretation extremely
difficult. However, two observations can be done. First, experimental results show
that more than two or three recursions are not necessary in order to obtain results
that are equal to or better than the traditional methods. Second, the growing num-
ber of recursions does not increase the algorithm complexity or computation, since
this growing is accompanied by a reduction in the number of states to be computed
at each recursion, as in the algorithms for binary partitions, which have an O(n lg
2
n) complexity. At each iteration NESRA generates 2
r + 1
regions, where r is the num-
ber of recursions. Note that for r = 0, NESRA is the simple image binarization
algorithm.
The recursive algorithm for the previous formulation as follows:
Algorithmus 1 Nonextensive Segmentation Recursive Algorithm,
NESRA Procedure
Input:
H
= image histogram,
i
= first histogram bin,
k
= last
histogram bin
Call NESRA(
H
,
i
,
k
) procedure
if Histogram
H
is homogeneous then
Goto FIM
end if
for all
t
=
i
until
k
do
compute normalization for background
compute normalization for foreground
compute
q
-entropy for background according to (8.7)
compute
q
-entropy for foreground according to (8.8)
compute composed
q
-entropy according to (8.9) and (8.10)
end for
topt
= argmax of the composed
q
-entropy
Call NESRA(
H
,
i
,
topt
) procedure
Call NESRA(
H
,
topt
+ 1,
k
) procedure
FIM: there is nothing to do, return to calling procedure
After this segmentation, tumor’s boundary must be extracted from the back-
ground. To accomplish this task, we apply a mathematical morphology approach.
Because this approach may generate a coarse contour of the tumor, we apply a level
set framework to smooth the contour. The level set theory is briefly presented in the
next section.
8.3 Theoretical Background 205
8.3.2.3 The Level Set Formulation
A level set method, also called an implicit snake, is a numerical technique developed
first by Osher and Sethian [46] to track the evolution of interfaces. These interfaces
can develop sharp corners, break apart, and merge together. The technique has a
wide range of applications, including problems in fluid mechanics, combustion,
manufacturing of computer chips, computer animation, structure of snowflakes, the
shape of soap bubbles, and image processing, including medical images.
Informally speaking, consider a boundary image interface separating one region
of interest (the inside) from another (the outside), and a speed function F that gives
the speed of each point of the boundary. This speed can depend on several physical
effects such as temperature if the environment is ice (ROI) inside water. In this case,
the boundary can shrink as the ice melts, or grow as the ice freezes. Then, the speed
function F depends on the adopted model. However, the original level set formula-
tion proposed by Osher and Sethian [46] takes a curve and builds it into a surface.
The curve is evaluated in the surface, which intersects the xy plane exactly where the
curve sits. The surface is called the level set function because it accepts as input any
point in the plane and hands back its height as output. The curve that intersects the
surface is called the zero level set, because it is the collection of all points that are at
height zero. A complete level set study can be found in [47].
Generically speaking, a level set formulation is useful for image segmentation
because, among other advantages, it handles naturally complex topologies, permits
flexible models representingthe boundaryof ROI, can be robustto noise, and can be
computationally efficient.
In our work, we use a level set formulation to smooth the ROI before the match-
ing phase. To enhance the computational overhead, we compute the level set only
inside a narrowband andthe initial curve is given by the previous segmentation with
Tsallis entropy.
More formally, the main idea of the 2-D level set method is to represent the
deformable surface (or curve) as a level set {x ∈ℜ
2
|G(x) = 0} of an embedding
function:
G:ℜ×ℜ→ℜ
+2
(8.21)
such that the deformable surface (also called front in this formulation), at t = 0, is
given by a surface S:
() ( )
{}
St x Gxt== ∈ℜ ==000
2
,
(8.22)
The next step is to find a Eulerian formulation for the front evolution.
Following Malladi et al. [48], let us suppose that the front evolves in the normal
direction with velocity
r
F
, where
r
F
may be a function of the curvature, normal direc-
tion, and so forth.
We need an equation for the evolution of G(x, t), considering that the surface S
is the level set given by:
() ( )
{}
St x Gxt=∈ℜ =
2
0,
(8.23)
206 Automatic Classification of Breast Lesions in 3-D Ultrasound Images
Let us take a point x(t), t ∈ℜ
+
of the propagating front S. From its implicit defi
-
nition, given above, we have:
()
()
Gxt t, = 0
(8.24)
Now, we can use the chain rule to compute the time derivative of this
expression:
GFG
t
+∇ =0
(8.25)
where
FF=
r
is called thespeed function. An initial conditionG(x, t = 0) is required.
A straightforward (and expensive) technique to define this function is to compute a
signed-distance function as follows:
()
Gxt d, ==±0
(8.26)
where d is the distance from x to the surface S(x, t = 0) and the signal indicates
whether the point is interior (−) or exterior (+) to the initial front.
Finite difference schemes, based on a uniform grid, can be used to solve (8.25).
The same entropy condition of T-Surfaces (once a grid node is burnt, it stays burnt)
is incorporated to drive the model to the desired solution (in fact, T-Surfaces was
inspired by the level sets model [49]).
In this higher-dimensional formulation, topological changes can be efficiently
implemented. Numerical schemes are stable and the model is general in the sense
that the same formulation holds for two and three dimensions, as well as for merges
and splits. In addition, the surface geometry is easily computed. For example, the
front normal and curvature are given by:
()
()
()
r
nGxtK
Gxt
Gxt
=∇ =∇⋅
∇
∇
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
,,
,
,
(8.27)
respectively, where the gradient and the divergent (
∇⋅
) are computed with respect to
x.
The initialization of the model through (8.26) is computationally expensive and
not efficient if we have more than one front to initialize [50].
The narrowband technique ismuch more appropriate forthis case. The key idea
of this technique comes from the observation that the front can be moved by updat-
ing the level set function at a small set of points in the neighborhood of the zero set
instead of updating it at all the points on the domain (see [48, 51] for details).
To implement this scheme, we need to preset a distance Δd to define the narrow
band. The front can move inside the narrow band until it collides with the
narrowband frontiers. Then, the function G should be reinitialized by treating the
current zero set configuration as the initial one.
Also, this method can be made cheaper by observing that the grid points that do
not belong to the narrow band can be treated as sign holders [48]; in other words,
points belonging inside the narrow band are positive, and negative otherwise.
8.3 Theoretical Background 207
To set ideas, consider Figure 8.2(a), which shows the level set bounding the
search space, and Figure 8.2(b), which shows a bidimensional surface in which the
zero level set is the contours just presented.
The surface evolves such that the contour gets closer to the edge. To accomplish
this goal, we must define a suitable speed function and an efficient numerical
approach. For simplicity, we consider the 1-D version of the problem pictured in
Figure 8.2. In this case, the evolution equation can be written as:
G
G
x
F
t
+=
∂
∂
0
(8.28)
The main point is to design the speed function F such that the desired result can
be obtained if G
t
> 0. For instance, if we set the sign of F opposite to the one of Gx
we get G
t
> 0, once:
G
G
x
F
t
=−
∂
∂
(8.29)
Hence, the desired behavior can be obtained by the distribution of the sign of F,
as shown in Figure 8.3.
Once our application focus is shape recovery in an image I, we must choose a
suitable speed function F as well as a convenient stopping term S to be added to the
right-hand side of (8.28). Among the possibilities [52], the following were suitable
for our case:
208 Automatic Classification of Breast Lesions in 3-D Ultrasound Images
+
−−
(a) (b)
Figure 8.2 (a) Level set bounding the search space. (b) Initial function with the zero level set as
the contour presented.
F
G
>0
<0
x
F
G
<0
>0
x
Figure 8.3 Sign of speed function.
F
k
I
=
+
−∇
1
1
2
α
(8.30)
SIG=∇⋅∇β
(8.31)
where β is a scale parameter. Therefore, we are going to deal with the following
level set model:
G
k
I
GPG
t
=
+
+∇
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
∇+∇⋅∇
1
1
2
α
β
(8.32)
where P =−|∇I|
2
. After initialization, the front evolves following (8.32).
Given the smooth contour of the level set output, we extract the region features
mentioned in Section 8.1. These features are then input to an SVM classifier, which
classifies the region as a malignant or benign tumor. The SVM theory is briefly dis-
cussed in the next section.
8.3.2.4 The Support Vector Machine Theory
SVM is a very good classification tool. It has been proven to be as a very effective
method for many applications such as pattern recognition, machine learning, and
data mining. The general idea is as follows. Let {x
i
, y
i
}, i = {1, ..., N} be a training
example set S; each example x
i
∈ R
n
belongs to a class labeled y
i
∈ {−1, 1}.The goal is
to define a hyperplane that divides S, such that all of the points with the same label
are on the same side of the hyperplane, while maximizing the distance between the
two classes and thehyperplane. This means we need tofind a pair (w, b)such that
()
ywx b i N
ii
⋅+> =11,,,K
(8.33)
where w ∈ R
n
and b ∈ R. After finding the pair (w, b), we can achieve a separating
hyperplane (see Figure 8.4) and its equation is defined as follows:
wx b⋅+=0
(8.34)
8.3 Theoretical Background 209
Figure 8.4 Separating hyperplane. The dashed lines identify the margin.
Maybe there are many separating hyperplanes; the optimal separating
hyperplane (OSH) is a hyperplane for which the distance to the closest point is maxi-
mal. Hence, to find the OSH, we must minimize ||w||
2
under constraint (8.33).
If we denote α = (α
1
, α
2
, ..., α
N
), where N is the nonnegative Lagrange multipliers
associated with the constraints of (8.33), the problem of finding the OSH is equiva-
lent to maximization of the function:
()
Wyyxx
iijijij
ij
N
i
N
αα αα=− ⋅
==
∑∑
1
2
11,
(8.35)
where α
i
≥ 0 and, under constraint,
y
ii
i
N
α =
∑
0
(8.36)
Once we achieve the vector
ααα α=
12
,,,K
N
, the solution of (8.34), the OSH
(,w
b), has the following expansion:
wyx
ii i
i
N
=
=
∑
α
1
(8.37)
while
b
can be determined from
α
and from the Kuhn-Tucker condition [53]:
()()
α
ii i
ywx b i N⋅+−= =10 12,,,,K
(8.38)
The decision function of classifying a new data point can be written as:
()
fx yx x b
ii i
i
N
=⋅+
⎛
⎝
⎜
⎞
⎠
⎟
=
∑
sgn α
1
(8.39)
If the S is not linearly separable, we must introduce N nonnegative slack vari-
ables ξ = (ξ
1
, ξ
2
, ..., ξ
N
), such that
()
ywx b i N
ii i
⋅+≥− =112ξ ,,,,K
(8.40)
Hence, we can regard the generalized OSH as the solution of the following
minimization problem:
1
2
1
ww C
i
i
N
⋅+
=
∑
ξ
(8.41)
where C is a regularization parameter. When the parameter C is small, the OSH
tends to maximize the distance 1/||w||; in contrast, a larger C will lead the OSH to
minimize the number of misclassified points.
The training example set that we want to classify is usually linearly
nonseparable. To achieve better generalization performance, the input data can first
be mapped into a high-dimensional feature space. Then the OSH is constructed in
210 Automatic Classification of Breast Lesions in 3-D Ultrasound Images
the feature space. If φ(x) denotes a mapping function that maps x into a high-dimen-
sional feature space, (8.34) can then be rewritten as follows:
() ()
(
)
Wyyxx
iijijij
ij
N
i
N
αα ααφφ=− ⋅
==
∑∑
1
2
11,
(8.42)
Now, let
Kx x x x
ij i j
(,) ()(),=⋅φφ
we can rewrite (8.40) as
()
(
)
WyyKxx
iijijij
ij
N
i
N
αα αα=−
==
∑∑
1
2
11
,
,
(8.43)
where K is a kernel function and must satisfy Mercers theorem [54]. Finally, the
decision function becomes:
() ()
fx yKx y b
ii i i
i
N
=+
⎛
⎝
⎜
⎞
⎠
⎟
=
∑
sgn ,α
1
(8.44)
In this study, we use nonlinear SVM with a B-spline basis kernel as our classi-
fier. The five tumor features discussed earlier are used as the inputs to an n-dimen-
sional vector to find the OSH for distinguishing benign tumors from malignant
ones.
In the next section we put to use all of the theories presented in this section in
order to summarize our five-step methodology for breast image classification.
8.3.3 Morphological Chains: Mathematical Morphology Definitions
The use of mathematical morphology for initializing deformable models is a com-
mon practice with promising results and general application. Nevertheless, it is an
open area of research and there is room for improvement. For the particular case of
medical images, the general idea is to isolate objects of interest—such as lungs,
arteries, heart, or bones—in the scene and to work with them individually while
avoiding neighboring interference from other objects, noises, spurious artifacts, and
background noise.
Mathematical morphology is a set of mathematical tools used in the digital
image processing area to perform linear transformations on the shapes of image
regions to obtain results such as merging, splitting, noiseless, or smoothing of an
ROI. These results are accomplished throughout several sets of operations between
the original image and specific sets, called structuring elements. However, two basic
operations are always used: erosion and dilation. Suppose the image X and the
structuring element B are represented as sets in 2-D Euclidian space. Let B
x
denote
the translation of B so that its origin is located at x. Then the erosion of X by B is
defined as the set of all points x such that B
x
{}
Erosion: :XB xB X
x
䊞
=⊂
(8.45)
Similarly, the dilation of X by B is defined as the set of all points x such that B
x
hits X, that is, they have a nonempty intersection:
8.3 Theoretical Background 211
{}
Dilation: :XB xB X
x
⊕= ∩ ≠φ
(8.46)
These two operations are the base for all other, more complex operations. As an
example, a well-known operation, directly derived from (8.45) and (8.46), is the
opening, which consists of an erosion on a given image followed by a dilation of the
result. As an effect of an opening, small connections between regions can be accom-
plished. This is particularly useful when we wish to disconnect two different regions
to treat them separately. The dual of the opening is the close operation, which con-
sists of an erosion over the dilation’s result. The effect of closing an image is, cor-
rectly, the opposite of an opening: It connects weak separated regions.
Other operations based on opening and closing are defined in the specific litera-
ture. For example, we can find boundary, convex hull, skeleton, and thinning opera-
tions. For a complete approach to the issue, see [55].
Specific problems, such as those encountered in medical images, can require spe-
cific solutions, with different morphological operations and several structuring ele-
ments. A sequence of morphological operations is called a morphological chain. The
chain to use depends on specific factors, such as the quality of the images (noise, illu-
mination, size, resolution, and so forth), objects of interest (bones, lungs, heart, and
so forth), computational time consumed (real-time applications or off-line), and our
goals.
In our proposed five-step methodology, we use mathematical morphology after
the initial segmentation of the image in order to extract the ROI from the back-
ground. This approach is described in detail next.
8.4 The Proposed Five-Step Methodology
As outlined, we put all of the theoretical background discussed in the preceding sec-
tions together to generate a five-step protocol that can be used to classify 3-D breast
images. After application of NESRA, we extract the lesion area from the back-
ground. Figure 8.5 showsan original benign image example and the NESRA result.
This result was obtained with two NESRA recursions, which means that the
gray-scale distribution was partitioned into two new distributions and each one was
partitioned into two others, generating four distinct regions. Although we wish to
split the image into two regions only—and in this case no recursion will be
needed—the use of one recursion in this case helps to delimit a tumor’s nucleus and
also a boundary as a narrow region around the nucleus. Note in Figure 8.5(b) the
white region around the image center (tumor) and the intermediate gray-level region
around the tumor’s region (the transition region between the tumor’s region and
background). This transition region can be isolated and help to delimit accurately
the lesion boundary. As mentioned in Section 8.1, this transition region is made of a
mix of dead and alive cells. These dead cells are generally swallowed to inside region
and the tumor grows. For the result of Figure 8.5(b) we used q = 0.5. In this case, the
ROI is the tumor’s nucleus and its transition region.
In the second step we use a morphological chain approach to extract the ROI
from the background. This is accomplished through the following rule. Considering
the binary image generated by NESRA [e.g., Figure 8.5(b)], let α and β be the total
212 Automatic Classification of Breast Lesions in 3-D Ultrasound Images
ROI’s area and the total image area, respectively. If α ≥ ξβ an erosion is carried out;
and if α ≤ δβ, a dilation is carried out. After, assuming that the ROI has a geometric
center near the image center, we apply a region growing algorithm that defines the
final ROI’s boundary. In our experiments, we fixed ξ = 0.75 and δ = 0.25 to cor-
rectly extract most of the ROIs. The result of this morphological rule applied in the
image of Figure 8.5(b) is shown in Figure 8.6(a).
The region generated by the morphological chain rule is a coarse representation
of the lesion region. Then, as a third step of our method, we apply a level set frame-
work using as the initialization this region’s boundary [52]. However, because this
initial ROI is near the real boundary, there is no need for several iterations, as nor-
mally occurs in several level set applications. Then, in our experiments, we have
applied 10 iterations only. It was sufficient to get good results, which generate low
computational overhead. Figure 8.6(b) shows the final lesion boundary after the
level set computation.
After the preceding boundary tracing process, the next step is feature extraction
of the ROI. Three radiologists have defined five features that have a high probabil-
ity of working well as discriminators between malignant and benign lesions. In our
work, we have used these five features and tested them in order to achieve a combi-
nation that yields good empirical results.
•
AR: The first feature is the lesion area. Because malignant lesions generally
have large areas in relation to benign ones, this characteristic is a power
discriminant. We have normalized it by the total image area.
8.4 The Proposed Five-Step Methodology 213
(a) (b)
Figure 8.5 (a) Original ultrasound benign image and (b) the NESRA results with two recursions
and q = 0.5.
(a) (b)
Figure 8.6 (a) ROI after morphological chain application. (b) ROI after level set application.