Добавил:
Sekretar
kiopkiopkiop18@yandex.ru
t.me/Prokururor I Вовсе не секретарь, но почту проверяю
Опубликованный материал нарушает ваши авторские права? Сообщите нам.
Вуз:
Предмет:
Файл:Ординатура / Хирургия / Библиотека им академика М.И. Перельмана / Книга_5835_Библиотеки_им_академика_М_И_Перельмана.pdf
information will increase the accuracy of the energy function definition. Because
we are using a statistic shape model, from the training dataset, we can classify the
image as foreground and background. The corresponding histograms provide the
conditional probability for different gray levels inside and outside the segmenta-
tion. This information can be efficiently combined with the shape prior for final
segmentation (Figure 12.12).
12.3.3 Segmentation in Bayesian Interface
Now that we have collected the statistical shape priors, we need to combine with the
intensity knowledge too. However, how to accurately use this information in seg-
mentation is still an unsolved problem. From (12.24), readers have already seen how
334 Prostate Analysis Using Ultrasound Imaging
Background
Prostate
Prostate
Background
0
0.005
0.01
0.015
0.02
0.025
1
51 101 151 201 251
(b)
(a)
Figure 12.12 In this prostate segmentation project, we have the two intensity distributions corre
-
sponding to the empirical probability of intensities inside and outside the prostate. (a) One exam
-
ple of the total training ultrasound image data sets. Sagittal and Coronal views are shown, with
background and object (prostate) labeled. (b) The probability of intensities inside and outside the
prostate from all the training data sets.
to solve the region-based energy function for segmentation in a level set framework,
without using any statistical shape and intensity information.
Recently, many researchers have tried to utilize the level set method with statis-
tical shape priors. Given a set of training shapes, one can impose information about
which segmentations are a priori more or less likely. Such prior shape information
can improve segmentation results in the presence of noise or occlusion from differ-
ent imaging modalities [15, 54, 65]. However, most of these approaches are based
on the assumption that the training shapes, encoded by their signed distance func-
tion, form a uniform distribution [15] or Gaussian distribution [54]. As introduced
in [16], these assumptions have been proven with the following problems:
1. The space of signed distance functions is not a linear space, therefore, the
mean shape and linear combinations of eigenmodes are typically no longer
signed distance functions.
2. Even if the space were a linear space, it is not clear why the given set of
sample shapes should be distributed according to a Gaussian density. In fact,
they are generally not Gaussian distributed.
Cremers et al. [65] proposed the use of nonparametric density estimation in the
space of level set functions in order to model nonlinear distributions of training
shapes. Although this resolves the above-mentioned problems, one sacrifices the
efficiency of working in a low-dimensional subspace (formed by the first few
eigenmodes) to a problem of infinite-dimensional optimization. This method is an
efficient solution of the two problems just listed, and it defines kernel density esti-
mation in a linear subspace that is sufficiently large to embed all training data. We
therefore choose this approach to use for the shape and intensity priors.
Segmentation in the level set framework can be formulated as the estimation of
the optimal embedding function Φ: Ω→ᑬ given an image I: Ω→ᑬ [16]. In the
Bayesian framework, this can be computed by maximizing the posterior
distribution
() ()
()
PI PIPΦΦΦ∝
(12.29)
Based on the result from (12.28), the new shape used for segmentation is deter-
mined by parameters α. The orientation, location and scale of segmentation are
determined by pose parameter p. Combined with both factors, the Bayesian inter-
face problem stated in (12.29) therefore can be solved in a problem as the following
conditional probability:
()()
()
PpI PIpPpααα,,,∝
(12.30)
which is optimized with respect to the weight a and the pose parameter p. Maximi-
zation of the posterior probability or equivalently minimization of its negative loga-
rithm is able to create the most probable segmentation of a given image. Combined
with the shape and intensity priors we collected from the previous two sections, a
new energy can be defined as
()
()
()
Ep PIp Ppααα, log , log ,=− −
(12.31)
12.3 Fully Automated 3-D Prostate Segmentation from Transrectal Ultrasound (TRUS) Images 335
In [16], the energy is defined as the following equations in more details:
() () () ()
()
()
Ep H pI H pIdx
N
i
αΦ
αα
σ
Ω
σ
,log log
log
=− + −
−
−
∫
Φ
in
out
1
1
Kernel
⎛
⎝
⎜
⎞
⎠
⎟
⎛
⎝
⎜
⎞
⎠
⎟
=
∑
i
N
1
(12.32)
where H() function is the Heaviside function we defined in (12.18), p
in
and p
out
are
the kernel density estimation of the intensity distributions of object and background
which are given by the corresponding smoothed intensity histograms as shown in
Figure 12.12. The Kernel() function is the special invention proposed in [16]. Before
we describe the kernel function and kernel density estimation in more detail, we
define the following notation. Given a set of aligned training shapes represented by
signed distance {Ψ
1
, Ψ
2
, ..., Ψ
N
}, we can represent each of them by their correspond-
ing shape vector {α
1
, α
2
, ..., α
N
}. These shape vectors can be easily extracted by pro-
jection to the basis formed by the eigenmodes.
Inferring a statistical distribution from the training shape, in other words, the
shape vectors, is done differently in the work by Tsai et al. [15], Leventon et al. [54],
and Rousson and Cremers [16]. In Tsai’s work, all of the shapes were regarded
equally and, correspondingly, a uniform distribution was utilized. In Leventon’s
work, the authors useda Gaussiandistribution from thetraining shape. However, in
Rousson’s work, the authors made use of a nonparametric density estimation to
approximate the shape distribution within the linear subspace spanned by the train-
ing shapes. The kernel function and the corresponding kernel density estimate are
defined as:
()
()
P
N
xe
i
i
N
x
α
αα
σ
π
σ
−
−
⎛
⎝
⎜
⎞
⎠
⎟
=
=
−
⎛
⎝
⎜
⎜
⎞
⎠
∑
1
1
2
1
2
2
Kernel
Kernel
(
)
⎟
⎟
≠
=
=−
∑
σαα
2
2
1
1
N
ji i j
i
N
min
(12.33)
Eventually, we are able to compute all of the elements defined in (12.32). The
segmentation task is transformed into a parameter adjustment problem in the
Bayesian interface. Many methods can be utilized to adjustparameters forthe model
based segmentation. Artificial neural networks such as multilayer perceptrons
(MLPs) have become standard tools for regression. In general, an MLP with fixed
structure defines a differentiable mapping from a parameter space to the space of
functions. In the case where an MLP has a fixed structure, the adaption of the net-
work to the sample data becomes a regression problem on the parameters of the net-
work itself. This procedure is often referred to as learning. Because MLPs are
differentiable, gradient-based adaptation techniques are typically applied to deter-
mine the weights. The earliest and most straightforward adaptation rule, ordinary
gradient descent, adapts weights proportional to the partial derivatives of the error
functional. Several improvements of this basic adaptation rule have been proposed,
some of them based on elaborated heuristics, others on theoretical reconsideration
336 Prostate Analysis Using Ultrasound Imaging
of gradient-based learning. Resilient backpropagation (Rprop) is a well-established
modification of the ordinary gradient descent method. The basic idea is to adjust an
individual step size for each parameter to be optimized. These step sizes are not pro-
portional to the partial derivatives but are themselves adapted based on some
heuristics. Ordinary gradient descent computes the direction of steepest descent by
implicitly assuming a Euclidean metric on the weight space.
The Rprop algorithms are among the best performing first-order batch learning
methods. They are many advantages for Rprop methods:
1. Fast and accurate;
2. Robust;
3. First-order methods, therefore time and space complexity scales linearly
with the number of parameters to be optimized;
4. Only dependent on the sign of the partial derivatives of the objective
function and not on their amount, therefore they are suitable for
applications where the gradient is numerically estimated or the objective
function is noisy;
5. Easy to implement and not very sensitive to numerical problems.
The Rprop algorithms are iterative optimization methods. Let t denote the cur-
rent iteration (epoch). In epoch t, each weight is changed according to:
() ()
()
()
wt wt
Et
w
t
ii
i
i
+= −
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
⋅1sign
∂
∂
Δ
(12.34)
The direction of the change depends on the sign of the partial derivative, but is
independent of its amount. The individual step sizes Δ
i
(t) are adapted based on
changes of sign of the partial derivatives of E(w) with regard to the corresponding
weight:
•
If
()()
∂
∂
∂
∂
Et
w
Et
w
ii
−
⋅>
1
0
then Δ
i
(t) is increased by a factor
η
+
> 1
.
•
If
()()
∂
∂
∂
∂
Et
w
Et
w
ii
−
⋅≤
1
0
then Δ
i
(t) is increased by a factor
η
−
∈
[0, 1].
Additionally, some Rprop methods implement a weight-backtracking heuristic.
That is, they partially retract “unfavorable” previous steps. Whether a weight
change was “unfavorable” is decided by a heuristic. In the following an improved
version [66] of the original algorithm is described in pseudocode. The difference
compared to the original Rprop method is that the weight-backtracking heuristic
considers both the evolution of the partial derivatives and the overall error.
Step 1: Enter the iteration n.
Step 2: For each weighting factor w
i
, if
∂
∂
∂
∂
Et
w
Et
w
ii
() ()
,
−
⋅>
1
0
then:
12.3 Fully Automated 3-D Prostate Segmentation from Transrectal Ultrasound (TRUS) Images 337
()
()
() ()
()
()
min ,
max
ΔΔ
Δ
i
ii
i
i
t
wt wt
Et
w
t
−⋅
+= −
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
⋅
+
1
1
η
∂
∂
sign
(12.35)
Step 3: Else if
()()
∂
∂
∂
∂
Et
w
Et
w
ii
−
⋅<
1
0,
then: min(Δ
i
(t − 1)
.
η
−
, Δ
min
)ifE(t) > E(t − 1),
then w
i
(t +1) = w
i
(t − 1),
()
∂
∂
Et
w
i
= 0
Step 4: Else if
∂
∂
∂
∂
Et
w
Et
w
ii
() ()−
⋅
1
= 0, then:
() ()
()
()
wt wt
Et
w
t
ii
i
i
+= −
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
⋅1sign
∂
∂
Δ
(12.36)
Step 5: If w
i
still remain to be updated, go to step 2, otherwise go to Step 6.
Step 6: If no more iteration is needed, exit. Otherwise, n = n +1; go to step 2.
12.4 Evaluation of the System Performance
12.4.1 Transrectal Ultrasound Images
The 3-D TRUS images were obtained from the Imaging Research Laboratories of
the Robarts Research Institute,using an Aloka2000 US machine (Aloka, CN) with a
biplane side-firing TRUS transducer mounted on a motorized scanning mechanism.
The transducer rotated about its axis through a scanning angle, while a series of 120
2-D B mode images were acquired with an 8-bit video frame grabber at 15 Hz. For
each 3-D scan, the acquired 2-D images were reconstructed into a 3-D image, using
reconstruction software developed in the laboratory, with the z-axis parallel to the
transducer axis. The sizes of the images are 860 × 427 × 348 and the voxel size is
mm
3
0.154 by 0.154 by 0.154 mm.
12.4.2 Validation Method
Eleven manually extracted prostates were collected from 11 different patients. We
employed a jackknifing
2
strategy by removing the image of interest from the training
phase.
We used the outlines from manual tracing as an independent standard, which
was done by an experienced radiologist. In Section 12.3, we have collected the 3-D
close surface using the 2-D manual tracing and used these smoothed 3-D surfaces as
the final ground truth.
338 Prostate Analysis Using Ultrasound Imaging
2. Cross-validation is a method for estimating generalization error based on “resampling” [67–70]. In k-fold
cross-validation, the data are divided into k subsets of equal size. The system is trained k times, each time
leaving out one of the subsets from training, but using only the omitted subset to compute error. If k equals
the sample size, this is called “leave-one-out” cross-validation. Sometimes, the “leave-one-out” method is
also called jackknifing.
The validation indices are defined as the following two parameters: correct seg-
mentation rate (CSR) and incorrect segmentation rate (ISR). CSR is defined as the
ratio of correct segmentation voxel number and the total voxel number of the
ground truth. ISR is defined as the ratio of the incorrect (the nonprostate voxel is
classified as prostate voxel) segmentation voxel number and the total voxel number
of the ground truth.
12.5 Results
Our segmentation method successfully determined the prostate in all 20 ultrasound
images. As introduced in the previous section, we used the jackknifing method for
validation. For every ultrasound image, we use the remaining 19 to extract the
shape and intensity priors. Using the computed shape model and intensity knowl-
edge, we initialized the mean shape at the center of the image and started the gradi-
ent descent method to final the best pose parameters and weight factors. On
12.5 Results 339
Figure 12.13 Sagittal, coronal, and transversal views of the prostate segmentation, a patient, and
the overlaid manual segmentation.
average, the CSR was 0.84 and the ISR was 0.15. Also, the standard deviation of the
CSR and ISR were 0.06 and 0.08, respectively. Figure 12.13 shows a detailed result
after using this method. The manual and automated segmentation results are both
overlapped onto theoriginal prostate ultrasound image data. The CSR was 0.89 and
the ISR was 0.15. Figures 12.14 and 12.15 show the surface rendering of the 2-D
and 3-D imagery, respectively, of the same image sequence as in Figure 12.13.
12.6 Conclusions and Future Work
The segmentation of prostate from ultrasound images is an important and challeng-
ing problem. In the context of ultrasound imagery, the main challenges include
these: The edge information of the objects is too weak; speckle noise degrades the
images. Existing algorithms that use edge information depend on extra image
enhancement methods and even need user interaction for initialization or editing.
340 Prostate Analysis Using Ultrasound Imaging
Figure 12.14 The surface rendering of the manual segmentation from 2-D tracing, correspond
-
ing to the close surface in Figure 12.13.
In this chapter, we reported on a system, that incorporates learned shape mod-
els and intensity prior for the objects of interest using level set methods. The algo-
rithm allows the shape to evolve until the optimal segmentation is found by
minimizing a region-based energy factor without special initialization. The quanti-
tative analysis proved that the method is a robust solution for fully automated pros-
tate segmentation tasks in ultrasound images.
The 2-D manual tracing for training was not created at every slice from the orig-
inal ultrasound image. Although we employed the Min/Max flow method to make a
smooth interpolation, the resulting 3-D surface was still not accurate enough. Add-
ing manual tracing in more slices is a possible solution.
Although there are many advantages to using the proceeding methods in a level
big barrier. In this chapter, we exploited kernel density estimation [16] to increase
the efficiency. The total convergence time is still more than 70 seconds. (The hard
-
ware environment is a PC with a Pentium 4, 2.80-GHz CPU and 1.00 GB of RAM.)
12.6 Conclusions and Future Work 341
Figure 12.15 The surface rendering of the automated segmentation, corresponding to the close
surface in Figure 12.13.
Optimization work will be done in the future to make the method more practical for
prostate segmentation from ultrasound images.
We provided a 3-D validation result in this chapter too. For readers who are still
interested in this performance of themethod compared with other methods,segmen-
tation accuracy and computational and memory cost evaluations can be made
through your own attempts.
References
[1] Garfinkel, L., and M. Mushinski, “Cancer Incidence, Mortality, and Survival Trends in
Four Leading Sites,” Stat. Bull. Metrop. Insur. Co., Vol. 75, 1994, pp. 19–27.
[2] Silverberg, E. C. C., Boring, and T. S. Squires, “Cancer Statistics,” CA Cancer J. Clin.,
Vol. 40, 1990, pp. 9–26.
[3] Aarnink, R., et al., “A Practical Clinical Method for Contour Determination in
Ultrasonographic Prostate Images,” Ultrasound in Medicine and Biology, Vol. 20, 1994,
pp. 705–717.
[4] Abolmaesumi, P., and M. Sirouspour, “Segmentation of Prostate Contours from Ultra-
sound Images,” Proc. of IEEE Inter. Conf. on Acoustics, Speech and Signal Processing,
Vol. 3, 2004, pp. 517–520.
[5] Pathak, S. V., et al., “Edge-Guided Boundary Delineation in Prostate Ultrasound Images,”
IEEE Trans. on Medical Imaging, Vol. 19, 2000, pp. 1211–1219.
[6] Ladak, H., et al., “Prostate Boundary Segmentation from 2D Ultrasound Images,” Medical
Physics, Vol. 27, 2000, pp. 1777–1788.
[7] Wang, Y., et al., “Semiautomatic Fixed-Dimensional Segmentation of the Prostate Using
Two-Dimensional Ultrasound Images,” Medical Physics, Vol. 30, September 2003,
pp. 887–897.
[8] Hu, N., et al., “Prostate Boundary Segmentation from 3D Ultrasound Images,” Medical
Physics, Vol. 30, 2003, pp. 1648–1659.
[9] Chiu, B. et al., “Prostate Segmentation Algorithm Using Dyadic Wavelet Transform and
Discrete Dynamic Contour,” Phys. Med. Biol., Vol. 49, 2004, pp. 4943–4960.
[10] Ghanei, A., et al., “A Fixed-Dimensional Deformable Model for Segmentation of Human
Prostate from Ultrasound Image,” Medical Physics, Vol. 28, 2001, pp. 2147–2153.
[11] Richard, W., and C. Keen, “Automated Texture-Based Segmentation of Ultrasound Images
of the Prostate,” Computerized Medical Imaging and Graphics, Vol. 20, 1996, pp. 131–140.
[12] Prater, J. S., and W. Richard, “Segmenting Ultrasound Images of the Prostrate Using Neural
Networks,” Ultrasound Imaging, Vol. 14, 1992, pp. 159–185.
[13] Shen, D., Y. Zhan, and C. Davatzikos, “Segmentation of Prostate Boundaries from Ultra-
sound Images Using Statistical Shape Model,” IEEE Trans. on Medical Imaging, Vol. 22,
No. 4, 2003, pp. 539–551.
[14] Gong, L., et al., “Parametric Shape Modeling Using Deformable Superellipses for Prostate
Segmentation,” Medical Physics, Vol. 23, 2004, pp. 340–349.
[15] Tsai, A., et al., “A Shape-Based Approach to the Segmentation of Medical Imagery Using
Level Sets,” IEEE Trans. on Medical Imaging, Vol. 22, 2003, pp. 137–154.
[16] Rousson, M., and D. Cremers, “Efficient Kernel Density Estimation of Shape and Intensity
Priors for Level Set Segmentation,” Proc. of Inter. Conf. on Medical Image Computing and
Computer Assisted Intervention, Vol. 1, 2005, pp. 757–764.
[17] Yang, F., J. S. Suri, and M. Sonka, “Volume Segmentation Using Active Shape Models in
Level Set Framework,” in Parametric and Geometric Deformable Models: An Application
in Biomaterials and Medical Imagery, J. S.Suri and A. Farag, (eds.),Berlin: Springer, 2006.
342 Prostate Analysis Using Ultrasound Imaging
[18] Suri, J. S., D. L. Wilson, and S. Laxminarayan, Handbook of Medical Image Analysis,
Berlin: Springer, 2006.
[19] Sethian, J. A., Level Set Methods and Fast Marching Methods Evolving Interfaces in Com
-
putational Geometry, Fluid Mechanics, Computer Vision, and Materials Science, Cam
-
bridge: Cambridge University Press, 1999.
[20] Osher, S., and N. Paragios, Geometric Level Set Methods in Imaging Vision and Graphics,
Berlin: Springer, 2003.
[21] Osher, S., and J. A. Sethian, “Fronts Propagating with Curvature-Dependent Speed: Algo-
rithms Based on Hamilton-Jacobi Formulation,”Comput. Phys., Vol. 79, 1988,pp. 12–49.
[22] Sethian, J. A., and R. Malladi, “An O(N log N) Algorithm for Shape Modeling,” Proc. of
the National Academy of Sciences, Vol. 93, 1996, pp. 9389–9392.
[23] Adalsteinsson, D., and J. A. Sethian, “A Fast Level Set Method for Propagating Interfaces,”
Journal of Computational Physics, Vol. 118, 1995, pp. 269–277.
[24] Suri, J. S., et al., “ShapeRecovery Algorithms Using Level Sets in 2-D/3-DMedical Imagery:
A State-of-the-Art Review,” IEEE Trans. on Information Technology in Biomedicine, Vol.
6, 2002, pp. 8–28.
[25] Angenent, S., D. Chopp, and T. Ilmanen, “On the Singularities of Cones Evolving by Mean
Curvature,” Commun. Partial Differential Equations, Vol. 20, 1995, pp. 1937–1958.
[26] Chopp, D. L., “Computing Minimal Surfaces Via Level Set Curvature Flow,” Journal of
Computational Physics, Vol. 106, 1993, pp. 77–91.
[27] Sethian, J. A., “Numerical Algorithms for Propagating Interfaces: Hamilton Jacobi Equa-
tions and Conservation Laws,” J. Differential Geometry, Vol. 31, 1990, pp. 131–161.
[28] Sethian, J. A., “Algorithms for Tracking Interfaces in CFD and Material Science,” Annu.
Rev. Comput. Fluid Mechanics, 1995.
[29] Sussman, M., P. Smereka, and S. J. Osher, “A Level Set Method for ComputingSolutions to
Incompressible Two-Phase Flow,” J. Comput. Phys., Vol. 114, 1994, pp. 146–159.
[30] Rhee, C., L. Talbot, and J. A. Sethian, “Dynamical Study of a Premixed V-Flame,” J. Fluid
Mechanics, Vol. 300, 1995, pp. 87–115.
[31] Sethian J. A., and J. D. Strain, “Crystal Growth and Dentritic Solidification,” J. Comput.
Phys., Vol. 98, 1992, pp. 231–253.
[32] Adalsteinsson, D., and J. A. Sethian, “A Unified Level Set Approach to Etching, Deposition
and Lithography I: Algorithms and Two-Dimensional Simulations,” J. Comput. Physics,
Vol. 120, 1995, pp. 128–144.
[33] Breen, D. E., and R. T. Whitaker, “A Level-Set Approach for the Metamorphosis of Solid
Models,” IEEE Trans. on Visualization and Computer Graphics, Vol. 7, April 2001,
pp. 173–192.
[34] Mansouri, A. R., and J. Konrad, “Motion Segmentation with Level Sets,” IEEE Int. Conf.
Image Processing, Vol. 2, October 1999, pp. 126–130.
[35] Paragios, N., and R. Deriche, “Geodesic Active Contours and Level Sets for the Detection
and Tracking of Moving Objects,” IEEE Trans. on Pattern Analysis and Machine Intelli-
gence, Vol. 22, No. 3, 2000, pp. 266–280.
[36] Kornprobst, P., R. Deriche, and G. Aubert, “Image Sequence Analysis Via Partial Differen-
tial Equations,” J. Math. Imaging Vision, Vol. 11, 1999, pp. 5–26.
[37] Faugeras, O., and R. Keriven, “Variational Principles, Surface Evolution, PDEs Level Set
Methods and the Stereo Problem,” IEEE Trans. on Image Processing, Vol. 7, May 1998,
pp. 336–344.
[38] Kimmel, R., “Tracking Level Sets by Level Sets: A Methodfor Solving the Shape from Shad
-
ing Problem,” Comput. Vision Image Understanding, Vol. 62, 1995, pp. 47–58.
[39] Kimmel, R., and A. M. Bruckstein, “Global Shape from Shading,” Comput. Vision Image
Understanding, Vol. 62, 1995, pp. 360–369.
[40] Sapiro, G., et al., “Implementing Continuous-Scale Morphology Via Curve Evolution,”
Pattern Recognition, Vol. 26, 1997, pp. 1363–1372.
12.6 Conclusions and Future Work 343