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Файл:Ординатура / Хирургия / Библиотека им академика М.И. Перельмана / Книга_5835_Библиотеки_им_академика_М_И_Перельмана.pdf
seemingly valid transforms, all with alternative derivations and valid performances,
created a confusing situation. This confusion was overcome when Cohen presented
his unified theory for T-F distributions. Cohen proposed that there were in fact an
infinite number of possible transforms that could be derived from the general
formula:
() ()()()
Pt e s u su du d d
jt j ju
,,*ω
π
φθτ θ τ τ θ
θτωθ
=−+
−− +
∫∫∫
1
4
12 12
2
This general formula has the same bilinear structure (s*(u − 1/2θ) s(u + 1/2τ)) as
the Wigner distribution and in the general case suffers from the same cross-term
304 Diagnosis of Heart by Phonocardiography
0
200
400
600
800
1000
Frequency
1200
0 500
Time
Wigner-Ville distribution of a PCG
1000 1500 2000 2500
Figure 11.11 Wigner distribution of a PCG.
55
60
65
70
75
80
85
90
Instantaneous frequency of a PCG
0
1
Time (sec)
Frequency (Hz)
2
3
4
5
Figure 11.12 Instantaneous frequency of PCG for a single cycle.
problems. The important feature of the second equation is the term phi, which is
commonly referred to as the kernel function. The advantage of this kernel notation
is that properties of the kernel determine and in fact can be used to control the prop-
erties of the distribution. The satisfaction of desirable distribution properties has
driven the development of various distributions. One in particular has been devel-
oped to address the cross-term problem. The distribution in question was first stud-
ied by Choi and Williams [26]. They named the new distribution the exponential
distribution after its kernel function, but future work has referred to it as the
Choi-Williams distribution. The CWD kernel is:
()
φθτ
θτ σ
, =
−
e
22
where σ is a constant. Substituting this into the generalized distribution we obtain
the CWD:
()
(
)
()
[]
()()
CW t e s u s u du d
ut j
,*ω
π
τσ
θττ
τσ τω
=−+
−− −
∫∫
1
4
1
12 12
32
2
4
2
2
The term σ controls the reduction of undesirable cross-terms; a low value for σ
results in reduced cross-terms, and a high value for σ forces the distribution to
behave as the Wigner distribution with large cross-terms. A side effect of reducing
the cross-terms is that the distribution loses resolution. Hence a value for σ is used
such that a balance is found where cross-term reduction occurs without a significant
loss of resolution [27]. The CWD distribution of a typical PCG is shown in Figure
11.13.
11.3.11 Autoregressive Modeling
The quadrature Doppler signal can be modeled by a complex AR process. To com-
pute the T-F domain by using AR modeling, an approach similar to spectrogram
analysis is used.More precisely, the time-varying AR model is expressed as follows:
() ( )( ) ()
xn amnxn m en
m
p
=−+
=
∑
,
1
where x(n) is the complex Doppler signal, p is the order of the model, a(m,n) is the
complex time-varying coefficients, and e(n) is the modeling error. The complex
coefficients a(m,n) can be computed by using the Yule-Walker equations together
with the Levinson-Durbin algorithm [28] for an efficient recursive solution. Once
the AR coefficients are found, the power spectrum of each windowed signal seg-
ment is computed by using
()
()
()( )
P
n
amn jm
n
p
m
p
θ
δ
θ
=
+−
=
∑
2
1
2
1,exp
11.3 Phonocardiograph 305
where δ(n) is the variance of the modeling error signal corresponding to model order
p. Thus, the T-F domain of the Doppler signal x(n) is obtained by sliding the win-
dow with a prespecified time increment to compute the time-indexed spectra. For
0 = 2∏k/N, the T-F domain based on AR modeling is
() ()
()
()
AR n k P
n
amn j
nk
N
m
xn
p
m
P
,
,exp
==
+−
⎛
⎝
⎜
⎞
⎠
⎟
=
∑
θ
δ
2
1
2
1
2
11.3.12 Feature Extraction
Morphological feature extraction has been applied extensively to the analysis of
heart sounds using spectrum analysis techniques [29–31]. Joo et al. [29] used the
two most dominant second sound frequencies as diagnostic features. Stein et al. [30]
used the dominant frequency of the second sound as a diagnostic feature, showing
that there is clear correlation between this feature and the valve condition. Durand
et al. [31] initially used nine diagnostic features, but subsequently [32] added more.
These features have physical meaning, and they can be used to provide information
about the patient’s condition rather than provide a simple yes/no condition classifi-
cation. Using T-F representations it is possible to extract time, frequency, and
combined T-F features.
AR coefficients [33]modeled the heart sounds as a 12-order all-pole system, and
used the 12 coefficients as features. Classification results using these features were
better than with spectral features. DWT is a representation of a signal using an
orthonormal basis of functions (“wavelets”). DWT-based features [34] compared
the classification performance with nine morphological features from the time–
frequency domain to the performance with a large set of features, extracted using
the discrete wavelet transform (the coefficients of the transform). An optimal subset
306 Diagnosis of Heart by Phonocardiography
0 500
0
200
400
600
800
1000
Frequency
1200
1000 1500 2000 2500
Time
Choi-Williams distribution of a PCG
Figure 11.13 CWD distribution of a PCG.
of features was selected using a search scheme. Results indicated that the
DWT-based feature extraction technique yielded superior performance.
Muruganantham et al. [35] derived average power, total power, mean power
frequency, median frequency, frequency variance, frequency skewness, frequency
kurtosis, and jitter from the frequency domain. The frequency skewness and
kurtosis are the third and fourth moment of the frequency spectrum. They used
these features together with a standard deviation for the detailed coefficient of the
wavelet transform. Also, from the FFT spectrum, they derived a Reimann sum,
which involves averaging the FFT spectrum over a specified window length.
11.3.13 Classification
In general, the methods for feature extraction are crucial to a successful classifica-
tion. A successful classification system depends not only on reliable, stable, and
adaptable classifiers, but also on the strategies and methods used to extract and
select features from the raw data. Classifiers perform better on feature vectors that
can represent observedobjects more efficiently such that the relevant information to
patterns is emphasized the most.
The Bayes classifier is the most popular parametric technique of supervised pat-
tern recognition. It is the optimal technique when the probability density functions
of the patterns in the feature space are known. It is common to assume that a
Gaussian PDF represents the distribution of the features for each class and to esti-
mate the required mean and variance parameters from the training set.
When the probability density functions are difficult to estimate, a nonparametric
approach such as the use of the neural network (NN) might be suitable. This method
is based on distance measurements and, therefore, requires a distance metric. The
nearest neighbors of a pattern of an unknown class are found by computing the dis-
tances between the pattern and a given number of patterns from the training set. The
class of the input pattern is set to be the class of the K nearest neighbors.
The choice of distance metrics is important. In most cases, the Euclidian dis-
tance is used. But other measures, such as the Mahalanobis distance (the distance
between two feature vectors, scaled by the statistical variation in each feature, that
is, the covariance matrix) may produce better results.
ANN has been applied to classification of bioprosthetic valves and sounds clas-
sification of murmurs hidden Markov model (HMM), and fuzzy rule-based meth-
ods have been applied to ECG classification for detecting ectopic beats.
Muruganantham et al. developed NN, fuzzy, and wavelet-based classifiers for
the classification of phonocardiograms and they were found to have better results
with wavelet method, as discussed next.
11.3.13.1 Neural Networks
They designed a back propagation neural network with nine inputs and one output,
which is either 1 for a normal case or 0 for an abnormal case. A single hidden layer
with five neurons was implemented with a sigmoid transfer function at the input
and output side. With the derived features the neural network is trained for about
10,000 epochs with performance criteria of 1.E–10. The mean square error was
11.3 Phonocardiograph 307
traced during each epoch. They found that during training MSE was considerably
reduced, and at the end it reached a value of about 0.15. For testing the network,
separate data were given to the network and the outputs were noted.
11.3.13.2 Fuzzy Systems
Muruganantham et al. signal was divided in to four beats and a FFT was performed
on each of the four beats. The results were averaged and an averaged FFT resulted.
Then a Reimann sum was performed on the averaged FFT data, on 50-point (Hz)
intervals In essence, the data of 50-Hz intervals is binned and averaged into eight
values (Figure 11.14). These eight values of all the signals of the same classes are
grouped and averaged to get an 8 × 2 matrix data matrix, which is used to find the
center values of the membership functions. The fuzzy system with eight inputs, each
having two Gaussian membership functions with the known center value and two
outputs corresponding to the two cases, was framed. Also the degree of fulfillment
of each rule is needed, so two rules are written to determine at what extent the input
at each bin of the sample corresponds to each heart condition.
11.3.13.3 Wavelet Method
Using “Daubechies 4” as the mother wavelet, the signals are decomposed in to six
stages. The variances of the detailed coefficients at each stage are calculated for all
signals and the average taken for both cases. The slopes of the curves for both cases
are obtained by plotting log2 of these values against the log2 scales. While testing,
this slope was calculated and compared with the standard from which the results
were concluded.
They found that, via the wavelet method, 92% of signals were classified correctly,
whereas a neural network correctly classifies 85% and a fuzzy system classifies 88%
308 Diagnosis of Heart by Phonocardiography
0
0 500
Frequency
1000 1500 2000 2500
3000
20
40
60
0
0 500
Frequency
1000 1500 2000 2500
3000
10
20
30
Power spectrum of abnormal
Power spectrum of normal
0
0
100
50
200
100
300
150
400
200
0 50 100 150 200 250
300 350
400
0 50 100 150 200
Frequency
Reimann sums of abnormal
Reimann sums of normal
Reimann sum
Frequency
250
300 350
400
Reimann sum
Figure 11.14 FFT spectrum and Reimann sum of a normal and abnormal PCG.
of the test signals. This is due to the fact that in a wavelet transform the signal is ana-
lyzed at a higher resolution in the frequency domain at varying window lengths,
whereas in the neural network and fuzzy methods the parameters are extracted from
the fixed-size window frequency representation, the Fourier transform.
Manivannan et al. [36] proposed three types of statistical classifier and com-
pared their results. They proposed methods for classification of phonocardiograms
using the Ho-Kashyap learning algorithm (HKL), Fisher discriminant analysis
(FDA), and support vector machine (SVM). They found that both the FDA and
HKL algorithm classifiers had almost equal efficiency of about 91% and 89.98%,
respectively, whereas the efficiency of the SVM classifier was only 70%, and it can
be increased by using larger amounts of data.
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310 Diagnosis of Heart by Phonocardiography
PART V
3-D Prostate Ultrasound Imaging
CHAPTER 12
Prostate Analysis Using Ultrasound
Imaging
Fuxing Yang, Jasjit Suri, and Aaron Fenster
Prostate cancer is the most commonly diagnosed malignancy in men over the age of
50 [1]. It is the second leading cause of death due to cancer in men worldwide [2].
Ultrasound is a widely used imaging modality for prostate biopsy. The accurate
detection of the prostate boundary in ultrasound images is of great importance for
volume measurement and disease monitoring. However, in ultrasound images, the
contrast is usually low and missing or the image has diffuse boundaries. These are
very challenging problems for accurate segmentation of the prostate from the
background.
Currently, many methods have been introduced to facilitate more accurate seg-
mentation of the prostate boundaries from ultrasound images. Many of the meth-
ods are based on edge information. Aarnink et al. [3] proposed a method for
determining of the contour of the prostate in ultrasound images via an edge detec-
tion technique based on nonlinear Laplace filtering. The authors combined the
information about edge location and strength to construct an edge intensity image.
Then edges representing a boundary are selected and linked to build the final out-
line. Abolmaesumi and Sirouspour [4] used a Sticks filter to reduce the speckle, and
the problem isthen discretized by projecting equispaced radii from an arbitrary seed
point inside the prostate cavity toward its boundary. Candidate edge points
obtained along each radius include the measurement points and some false returns.
Pathak et al. [5], proposed an algorithm for guided edge delineation, which pro-
vided automatic prostate edge detection as a visual guide to the user. The edge
detection contained contrast enhancement and an anisotropic diffusion filter for
smoothing the images. However, the method needed a manual linking procedure on
the detected edges.
Another group of methods require human interactivity for initialization. For
example, Ladak et al. [6] proposed an algorithm, that used model-based initializa-
tion and a discrete dynamic contour. The user selected four points around the pros-
tate. Then the outline of the prostate was estimated using cubic interpolation
functions and shape information, and the estimated contour was deformed auto-
matically to better fit the prostate. Wang et al. [7] used a similar 2-D method. How-
ever, the authors segmented the remaining slices by iteratively propagating the
result to the other slices and implementing the refinement. Hu et al. [8] used a
313