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Файл:Ординатура / Хирургия / Библиотека им академика М.И. Перельмана / Книга_5835_Библиотеки_им_академика_М_И_Перельмана.pdf
model-based initialization and mesh refinement with deformable models. Six points
were required to initialize the outline of the prostate using shape information. The
initial outline was then automatically deformed to better fit the prostate boundary.
Chiu et al. [9] introduced an algorithm based on the dyadic wavelet transform and
the discrete dynamic contours. A spline interpolation was used to determine the ini-
tial contour based on four user-defined initial points. Then the discrete dynamic
contour refined the initial contour based on the approximate coefficients and the
wavelet coefficients generated using the dyadic wavelet transform. Ghanei et al. [10]
proposed a three-dimensional deformable surface model for prostate segmentation
based on a discrete structure, which was made from a set of vertices in the 3-D space
as triangle facets. But this model also needed manual initialization from a few
polygons drawn on different slices.
Pattern recognition—based methods have also been widely used. Richard and
Keen [11] presented a texture-based algorithm, which segments a set of parallel 2-D
images of the prostate into prostate and nonprostate regions. The algorithm utilizes
a pixel classifier based on four texture energy measures associated with each pixel in
the image. Clustering techniques are used to label each pixel in the image with the
label of the most probable class. Prater and Richard [12] use feed-forward neural
networks for segmentation of the prostate in transrectal ultrasound images. These
networks are trained using a small portion of a training image segmented by an
expert and then applied to the entire training image.
Due to the weak boundary information, some researchers exploit statistical
shape knowledge to help more accurate segmentation. Shen et al. [13] introduced a
statistical shape model to segment the prostate in transrectal ultrasound images. The
authors used a Gabor filter bank in both multiple scales and multiple orientations to
characterize the prostate boundaries. Then, a hierarchical deformation strategy was
used. The model focused on the similarity of different Gabor features at different
deformation stages using a multiresolution technique. Although the approach had
very high segmentation accuracy, the complexity and the computational cost were
big barriers. Gong et al. [14] presented an approach based on deformable
superellipses. Model initialization and constraining model evolution were based on
prior knowledge about the prostate shape.
Generally in ultrasound images, prostate segmentation methods based on edge
information have limitations when the image contains shadows with similar gray
levels and textures as the prostate region, or missing boundary segments. The simi-
lar pattern inside and outside the prostate presents more difficulties for the pattern
recognition-based method. Algorithms based on active contours depend on user
interaction to determine the seed points. Although statistical shape information has
been successfully applied, the convergence criterion based on the edge information
makes the approach a nonrobust solution.
In recent years, region based active contour techniques [15–18] have been
widely used in medical image analysis. The method can detect contours both with or
without a gradient, for instance, objects with very smooth boundaries or even with
discontinuous boundaries. The model has a level set formulation with a simple solu-
tion. The initialization of the contour also becomes simpler because the energy func-
tion defined by the prior shape and intensity prior, extracted from the training data
set, will drive the contour to the location close to the object. Compared with
314 Prostate Analysis Using Ultrasound Imaging
edge-based methods, this approach is robust and does not need extra human inter-
vention. In this chapter, we introduce a system for fully automated 3-D prostate seg-
mentation from transrectal ultrasound images, that uses shape and intensity
knowledge in a level set framework. Readers will gain an understanding about the
regional-based active contour in a level set framework; how it solves the segmenta-
tion problem without using image gradient information, which is a major control
factor for other approaches; and how it can be implemented via the level set
method. In addition, readers will be shown how statistic shape information can be
extracted from a training procedure and combined with regional information and
intensity prior to segmentation.
In Section 12.1, a short introduction of the level set method and the theoretical
background is given. Section 12.2 introduces the relationship between active meth-
ods and level set methods. Also, the active contours using level set frameworks are
discussed in more detail, including the mathematical background. Section 12.3
describes a practical application: a system for fully automated 3-D prostate segmen-
tation from transrectal ultrasound (TRUS) images, which uses shape and intensity
knowledge in a level set framework. Finally, Section 12.6 gives a brief conclusion
and a discussion of the pros and cons of the method.
12.1 Level Set Method
12.1.1 Front Evolution
Front evolution is a useful technique in image analysis for object extraction, object
tracking, and so on. The basic idea behind the method is to evolve a curve toward
the lowest potential of a cost function, where its definition reflects the task to be
addressed and imposes certain smoothness constraints. Propagating interfaces
occur in a wide variety of settings, and include ocean waves, burning flames, and
material boundaries. Less obvious boundaries are equally important and include
shapes against backgrounds, handwritten characters, and isointensity contours in
images. Furthermore, there are applications not commonly thought of as moving
interface problems, such as optimal path planning and construction of shortest geo-
desic paths on surfaces, which can be recast as front propagation problems with sig-
nificant advantages [19].
Consider a boundary, either a curve in 2-D or a surface in 3-D, separating one
region from another. Assume that this curve/surface moves in a direction normal to
itself with a known speedfunction F. The goal is to track the motion of this interface
as it evolves. Here, the motions of the interface in its tangential directions are
ignored. At a specific moment, the speed function F(L, G, I) describes the motion of
the interface in the normal direction.
1. Speed factor L depends on local geometric information (e.g., curvature and
normal direction).
2. Speed factor G depends on the shape and position of the front (e.g., integrals
along the front, heat diffusion).
3. Speed factor I does not depend on the shape of the front (e.g., an
underlying fluid velocity that passively transports the front).
12.1 Level Set Method 315
First, Lagragian techniques are based on parameterizing the contour according
to some sampling strategy and then evolving each element according to the speed
function. Although such a technique can be very efficient, it suffers from various
limitations such as deciding on the sampling strategy, estimating the internal geo-
metric properties of the curve, changing its topology, addressing problems in higher
dimensions, and so on [20].
The level set method was initially proposed to track a moving front by Osher
and Sethian in 1988 [21] and has been widely applied across various imaging
domains in the late 1990s. They can be used to efficiently address the problem of
curve or surface propagation in an implicit manner. The central idea is to represent
the evolving contour using a signed function, which is in a higher dimensional space,
where its zero level corresponds to the actual contour. Then, according to the
motion equation of the contour, one can easily derive a similar flow for the implicit
surface that when applied to the zero level will reflect the propagation of the con-
tour. Basically, there are two kinds of formulations regarding the level set method:
boundary value formulation and initial value formulation.
12.1.2 Boundary Value Formulation
Assume for the moment that F > 0 and the front is moving all the way outward. We
can define the arrival time T(x, y) of the front as it crosses each (x, y). Based on the
fact, distance = rate × time, we have the following expression in 1-D situation:
1 = F
dT
dx
(12.1)
∇T is orthogonal to the level sets of T, and its magnitude is inversely propor-
tional to the speed. So
∇= =TF T10,onΓ
(12.2)
where is the initial location of the interface. Then, the front motion is character-
ized as the solution of a boundary value problem. If the speed F depends only on
position, then the equation reduces to what is known as the Eikonal equation.
The efficient way to solve the Eikonal equation is to use the fast marching
method. For readers who are interested in the details of the method, please refer to
the literature [22].
12.1.3 Initial Value Formulation
Assume that the front moves with a speed that is neither strictly positive nor nega-
tive, so the front can move inward and outward and a point (x, y) can be crossed sev-
eral times. T(x, y) is not a single-value function. The solution is to treat the initial
position of the front as the zero level set of a higher dimensional function Φ (Figure
12.1) The evolution of this function will give the propagation of the front through a
time-dependent initial value problem, and at any time, the front is given by the zero
level set of time-dependent level set function Φ. Taking the 1-D situation as exam
-
ple, with x(t) as the front, we have the following definition of the zero-level set:
316 Prostate Analysis Using Ultrasound Imaging
()
()
Φ xt t, = 0
(12.3)
Taking the partial differentials on the preceding equation and we will have the
following equation:
()
()
()
ΦΦ
t
xt t x t+∇ ⋅ ′ =,0
(12.4)
()
Fxtn n=′ ⋅ =
∇
∇
,where
Φ
Φ
(12.5)
()
ΦΦ Φ
t
Fxt+∇ = =00,,given
(12.6)
So, from (12.6) the time evolution of the level set function Φ can be described
such that the zero-level set of this evolving function is always identified with the
propagating interface.
There is also an efficient way to solve the initial value problem we just intro-
duced, the narrowband method. For readers who are interested in the details of the
method, please refer to the literature [23].
12.1.4 Applications of Level Set Method
Nowadays, level set methods have been widely used in medical image processing.
The diversity of applications of level sets has reached into [24] geometry [25–27],
grid generation [22], fluid mechanics [28, 29], combustion [30], solidification [31],
device fabrication, [32], morphing [33], object tracking/image sequence analysis in
images [34–36], stereo vision [37], shape from shading [38, 39], mathematical mor-
phology [40, 41], color image segmentation [42], 3-D reconstruction and modeling
[43], surfaces and level sets [26, 44], topological evaluations [45], inpainting, and
2-D and 3-D (even 4-D) medical image segmentation. All of these give just a glimpse
of the powerful level set method.
Among the applications just mentioned, segmentation has always been a critical
component in 2-D and 3-D medical imagery since it assists largely in medical ther
-
12.1 Level Set Method 317
Y
Y
Y
X
X
X
t=2
t=1
Φ=(,, 2)xyt
Φ=(,, 1)xyt
2D
moving
front
Figure 12.1 The zero level set of a higher-dimensional function is taken as the moving front.
apy [24]. The applications of shape recovery have been increasing since scanning
methods became faster, more accurate and less artifacted [47]. The recovery of
shapes of the human body is more difficult compared to other imaging fields. This is
primarily due to the large variability in shapes, complexity of medical structures,
several kinds of artifacts and restrictive body scanning methods. In the following
sections, we focus on a family of the most advanced medical image segmentation
methods in level set frameworks and also introduce one important application based
on the method of 3-D prostate segmentation using TRUS images.
12.2 Active Contours in Level Set Framework
The level set method is closely connected with the active contour or deformable
model method, which is a powerful tool for image analysis. Active contours are
object-delineating curves or surfaces that move within 2-D or 3-D digital images
under the influence of internal and external forces and user-defined constraints.
Since their introduction by Kass et al. [48], these algorithms have been at the core of
one of the most active and successful research areas in edge detection, image seg-
mentation, shape modeling, and visual tracking. The first kind of active contour
models is represented explicitly as parameterized contours (i.e., curves or surfaces)
in a Lagrangian framework, which are called parametric active contours. Due to the
special properties of the level set method, active contours can be represented implic-
itly as level set functions that evolve according to a Eulerian formulation; they are
called geometric active contours. Although it is only a simple difference of represen-
tation for an active contour, the level set method–based geometric active contours
have many advantages over parametric active contours:
1. They are completely intrinsic and therefore are independent of the
parameterization of the evolving contour. In fact, the model is generally not
parameterized until evolution of the level set function is complete. Thus,
there is no need to add or remove nodes from an initial parameterization or
adjust the spacing of the nodes as in parametric models.
2. The intrinsic geometric properties of the contour such as the unit normal
vector and the curvature can be easily computed from the level set function,
contrast to the parametric case, where inaccuracies in the calculations of
normals and curvatures result from the discrete nature of the contour
parameterization.
3. The propagating contour can automatically change topology in geometric
models (e.g., merge or split) without requiring an elaborate mechanism to
handle such changes as in parametric models.
4. The resulting contours do not contain self-intersections, which are
computationally costly to prevent in parametric deformable models.
Since its introduction, the concept of using active contours for image segmenta-
tion defined in a level set framework has motivated the development of several fami-
lies of methods including the front evolving geometric model, geodesic active
contours, and region-based level set active contours.
318 Prostate Analysis Using Ultrasound Imaging
12.2.1 Front Evolving Geometric Models of Active Contours
Front evolving geometric models of active contours [49, 50] are based on the theory
of curve evolution, implemented via level set algorithms. They can automatically
handle changes in topology. Hence, without resorting to dedicated contour track-
ing, unknown numbers of multiple objects can be detected simultaneously. Evolving
curve C in the normal direction with speed F amounts to solving the differential
equation:
()
∂
∂
γ
Φ
Φ
Φ
Φt
gu=∇ ∇
∇
∇
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
+
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
0
div
(12.7)
where Φ is the level set function corresponding to the current curve C, g() is the gra-
dient function on the image, and γ is constant to regulate the evolving speed.
12.2.2 Geodesic Active Contour Models
The geodesic model was proposed in 1997 by Caselles et al. [51]. This is a problem
of geodesic computation in a Riemannian space, according to a metric induced by
the image. Solving the minimization problem consists of finaling the path of mini-
mal new length in that metric:
() () ()
()
()
JC C s g u Cs ds=′⋅∇
∫
2
0
0
1
(12.8)
where the minimizer C will be obtained when
guCs((())∇
0
vanishes, that is, when
the curve is on the boundary of the object. The geodesic active contour model also
has a level set formulation as follows:
() ()
∂
∂
ν
Φ
Φ
Φ
Φt
gu g u=∇ ∇
∇
∇
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
+∇div
00
(12.9)
The geodesic active contour model is based on the relation between active con-
tours and the computation of geodesics or minimal distance curves. The minimal
distance curve lies in a Riemannian space whose metric is defined by the image con-
tent. This geodesic approach for object segmentation allows classical “snakes” to be
connected based on energy minimization and geometric active contours based on
the theory of curve evolution. Previous models of geometric active contours are
improved, allowing stable boundary detection when their gradients suffer from
large variations.
12.2.3 Tuning Geometric Active Contour with Regularizers
The main problem of boundary-based level set segmentation methods is related to
contour leakage at locations of weak or missing boundary data information. One
approach can be followed to solve these limitations, to fuse regularizer terms in the
speed function.
12.2 Active Contours in Level Set Framework 319
Suri et al. [24] reviewed recent works on the fusion of classical geometric and
geodesic deformable models speed terms with regularizers, that is, regional statistics
information from the image. Regularization of the level set speed term is desirable to
add prior information on the object to the segment and prevent segmentation errors
when using only gradient-based information in the definition of the speed. Four
main kinds of regularizers were identified by the authors of the review:
1. Clustering-based regularizers;
2. Bayesian-based regularizers;
3. Shape-based regularizers;
4. Coupling-surfaces regularizers.
12.2.3.1 Clustering-Based Regularizers
Suri proposed in [52] the following energy functional for level set segmentation:
(
)
∂
∂
ε
Φ
ΦΦ
t
kF F
p
=+∇−∇
ext
(12.10)
where F
p
is a regional force term expressed as a combination of the inside and out-
side regional area of the propagating curve. The term is proportional to a region
indicator taking a value between 0 and 1, derived from a fuzzy membership mea-
sure. The second part of the classical energy model constituted the external force
given by F
ext
. This external energy term depended on image forces that were a func-
tion of image gradient.
12.2.3.2 Bayesian-Based Regularizers
Baillard et al. [53] proposed an approach that is similar to the previous one where
the level set energy functional is expressed as:
()
()
∂
∂
Φ
Φ
t
gIkF=∇ + ∇
0
(12.11)
It uses a modified propagation term F
0
as a local force term. This term was
derived from the probability density functions inside and outside the structure to the
segment.
12.2.3.3 Shape-Based Regularizers
Another application of the fusion of Bayesian statistics into a geometric boundary/
surface to model the shape in the level set framework was done by Leventon et al.
[54]. The authors introduced shape-based regularizers where curvature profiles act
as boundary regularization terms more specific to the shape to extract than standard
curvature terms. A shape model is built from a set of segmented exemplars using
principal component analysis applied to the signed-distance level set functions
derived from the training shapes (analogous to Cootes et al.’s [55] technique). The
principal modes of variation around a mean shape are computed. Projection coeffi
-
320 Prostate Analysis Using Ultrasound Imaging
cients of a shape on the identified principal vectors are referred to as shape parame-
ters. Rigid transformation parameters aligning the evolving curve and the shape
model are referred to as pose parameters. To be able to include a global shape con-
straint in the level set speed term, shape and pose parameters of the final curve Φ*(t)
are estimated using maximum a posteriori estimation. The new functional with a
solution for the evolving surface is expressed as:
()()
(()
()
()
)()()
()
ΦΦ Φ ΦΦΦt t gIkc gI t t+= + ∇ +∇+∇∇⋅∇+ −1
12
λλ*
(12.12)
where λ
1
and λ
2
are two parameters that balance the influence of the gradient curva-
ture term and the shape model term.
In [56], Leventon at al. proposed further refinements of their method by intro-
ducing prior intensity and curvature models using statistical image surface relation-
ships in the regularizer terms. Some clinical validations have been reported to show
efficient and robust performance of the method.
12.2.3.4 Coupling-Surfaces Regularizers
This kind of regularizer was motivated by segmentation of embedded organs such
as the cortical gray matter in the brain. The application tries to use a level set seg-
mentation framework to perform simultaneous segmentation of the inner and outer
organ surfaces with coupled level set functions. This method was proposed by Zeng
et al. [57]. In this framework, segmentation is performed with the following system
of equations:
ΦΦ
ΦΦ
in in in
out out out
+∇ =
+∇ =
⎧
⎨
⎩
F
F
0
0
(12.13)
where the terms F
in
and F
out
are functions of the surface normal direction (e.g., cur-
vature), image-derived information and the distance between the two surfaces.
When this distance is within the desired range, the two surfaces propagate accord-
ing to the first two terms of the speed term. When the distance is out of the desired
range, the speed term based on the distance controls the deformation to correct for
the surface positions.
12.2.4 Region-Based Active Contour Models
These classical snakes and active contour models rely on the edge function, depend-
ing on the image gradient, to stop the curve evolution, these models can detect only
objects with edges defined by gradient. Some of the typical edge functions are illus-
trated in Figure 12.2. In practice, the discrete gradients are bounded and then the
stopping function is never zero on the edges, and the curve may pass through the
boundary. If the image is very noisy, the isotropic smoothing Gaussian has to be
strong, which will smooththe edgestoo. Thisregion-based active contour methodis
a different active contour model, without a stopping edge-function, that is, a model
which is not based on the gradient of the image for the stopping process.
12.2 Active Contours in Level Set Framework 321
12.2.4.1 Mumford-Shah (MS) Function
One kind of stopping term is based on Mumford-Shah [58] segmentation tech-
niques. In this way, the model can detect contours both with or without gradients,
for instance, objects with very smooth boundaries or even with discontinuous
boundaries. In addition, the model has a level set formulation, interior contours are
automatically detected, and the initial curve can be anywhere in the image.
The original MS function is defined as:
() ()
()()
()
FuC LengthC
u x y u x y dx dy u x y dx dy
MS
C
,
,, ,
=+
−+∇
∫∫
μ
λλ
ΩΩ0
22
(12.14)
where u
0
approximates original image u, u
0
does not vary too much on each seg-
mented region and theboundary C is becomingas shortas possible. In fact, u
0
is sim-
ply a cartoon version of the original image u. Basically u
0
is a new image with edges
drawn sharply. The objects are drawn smoothly without texture. Such cartoons are
perceived correctly as representing a simpler version of the original scene containing
most of its essential features. In Chan and Vese’s approach [59], a level set approach
is proposed to solve the modified Mumford-Shah functional.
12.2.4.2 The Level Set Approach to Solving the Mumford-Shah Function
To explain the model clearly, let us first define the evolving curve C in Ω as the
boundary of an open subset w of Ω (i.e., w ⊂Ω, and C =∂w). In what follows, inside
(C) denotes region w, and outside (C) denotes region
Ω / w
. The method is the
minimization of an energy-based segmentation. Assume that image u
0
is formed by
two regions of approximatively piecewise-constant intensities, of distinct values
322 Prostate Analysis Using Ultrasound Imaging
Outside Inside
|I|∇
e
|I|∇
−α
Figure 12.2 Typical definition of the edge-functions to control the propagation of the moving
front.
u
i
0
u
i
0
and
u
o
0
. Assume further that the object to be detected is represented by the region
with the value
u
i
0
. Let denote its boundary by C
0
. Then we have
uu
i
00
≈
inside the
object [or inside (C
0
)], and
uu
o
00
≈
outside the object [or outside (C
0
)]. The fitting
term is defined as:
() ()
(
)
()
(
)
()
FC F C
u x y c dx dy u x y c dx
CC
12
01
2
02
2
+=
−+ −
∫∫
inside outside
,,dy
(12.15)
where C is any other variable curve, and the constants c
1
, c
2
, depending onC, are the
averages of u
0
inside C and outside C, respectively. In this simple case, it is obvious
that C
0
, the boundary of the object, is the minimizer of the fitting term:
() ()
{}
() ()
inf
C
FC FC FC FC
12 1020
0+≈≈+
(12.16)
If curve C is outside the object, then F
1
(C) > 0 and F
2
(C) ≈ 0. If curve C is inside
the object, then F
1
(C) ≈ 0 and F
2
(C) > 0. If curve C is both inside and outside the
object, then F
1
(C) > 0 and F
2
(C)> 0. The fitting term is minimized when C = C
0
, that
is, curve C is on the boundary of the object (Figure 12.3).
To solve more complicated segmentation, we regularize terms, such as the
length of curve C, or the area of the region inside C. A new energy functional F
(c
1
,c
2
,C) is defined as:
( ) () ()
()
(
)
()
Fc c C C v C
uxy c
C
12
101
,
,
=+
+−
∫
μ
λ
Length Area inside
inside
(
)
()
2
202
2
dx dy
u x y c dx dy
C
+−
∫
λ
outside
,
(12.17)
where μ ≥ 0, v ≥ 0, λ
1
, λ
2
≥ 0. (In Chan-Vese’s approach, λ
1
, = λ
2
= 1 and v = 0.) Corre-
spondingly, the level set based on C is defined as:
12.2 Active Contours in Level Set Framework 323
(c) (d)
(a) (b)
Figure 12.3 Consider all possible cases in the position of the curve. The fitting term is minimized
only in the case when the curve is on the boundary of the object (a) The curve C is outside the
object, and F
1
(C) > 0 and F
2
(C) ≈ 0. (b) The curve C is inside the object, then F
1
(C) ≈ 0 and F
2
(C) >
0. (c) The curve C is both inside and outside the object, then F
1
(C) > 0 and F
2
(C) > 0. (d) The fitting
term is minimized when the curve C is on the boundary of the object