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_5837_Библиотеки_им_академика_М_И_Перельмана.pdf
4.1 Backgrounds and Introductions 79
4.1.1 Speckle Noise Models for Ultrasonic Medical Images
We still use the same ultrasound image model as we have introduced in the previous
chapters:
g(x , y) = ω(x , y)n(x , y) + ξ(x, y) (4.3)
Due to the very limited influence of additive noise can be, this model can be
simplified as follows:
g(x , y) = ω(x , y)n(x , y) (4.4)
The speckle noise has widely been assumed to obey the Rayleigh distribution,
and this is a common assumption for the model. In the process of ultrasonic imaging,
the ultrasonic system receives scattered signals, i.e. complex signals consist of the
amplitude and phase information. In this chapter, we will discuss the detail of model
of scattered signal. Every pixel in ultrasonic medical images not only reveals the
reflected magnitude of internal organs and tissues (gray values), but also contains
the phase information in relevance to the tissue movements. Any scattered signal
reflected by a scattering point can be represented as:
A
r
+ jA
j
=
A
2
r
+ A
2
j
e
jθ
= xe
jθ
(4.5)
where A
r
,A
j
denote the real and imaginary part of scattered signal, respectively, and
θ denote the phase shift.
In the ultrasonic imaging system, backward signals in scattered medium are the
superposition of scattered signals reflected by lots of scattering points:
X =
N −1
i =0
x
i
e
jθ
i
(4.6)
or
¯
X =
1
√
N
N −1
i =0
ς
i
e
jθ
i
,ς
i
=
x
i
√
N
(4.7)
If ς
i
and θ
i
are independent and identically distributed, then real part X
R
and
imaginary part X
J
of X are
E(X
R
) =
1
√
N
N −1
i =0
E(ς
i
)E(cos θ
i
) (4.8)
80 4 Despeckle Filtering of Medical Ultrasonic Images …
E(X
J
) =
1
√
N
N −1
i =0
E(ς
i
)E(sin θ
i
) (4.9)
If θ
i
is an uniform distribution on the interval
[
−π, π
]
, then E (cos θ
i
), E(sin θ
i
)
are all equal to zero, and hence E ( X
R
), E( X
J
) are all equal to zero.
The second moment of X
R
and X
J
is
E(X
2
R
) =
1
√
N
N −1
i, j=0
E(ς
i
ς
j
)E(cos θ
i
, cos θ
j
) (4.10)
E(X
2
J
) =
1
√
N
N −1
i, j=0
E(ς
i
ς
j
)E(sin θ
i
, sin θ
j
) (4.11)
respectively.
For ς
i
and θ
i
are independent and identically distributed, we have sin
2
θ
i
+
cos
2
θ
i
= 1, and hence
E(X
2
R
) = E( X
2
J
) =
E(X
2
)
2
= σ
2
(4.12)
provided that E ( X
R
, X
J
) = 0 when X
R
and X
J
are not correlated.
If the number of scattering bodies in each resolution unit is sufficient large
(N →∞), and the phase of backward signals subjects to the uniform distribu-
tion in
[
−π, π
]
, then joint distribution of the real part X
R
and imaginary part X
J
will obey the Gaussian distribution according to the central limit theorem (CLT). Its
joint probability density function is:
P(X
R
, X
J
) =
1
2πσ
2
exp(−
X
2
R
+ X
2
J
2σ
2
) (4.13)
that is, the product of two normal distributions with zero means and σ
2
variances.
Hence
X =
X
2
R
+ X
2
J
(4.14)
has its probability density function as follows:
P(I ) =
1
2σ
2
exp( -
I
2σ
2
), I ≥ 0
0, Others
(4.15)
where I = X
2
, and the formula (4.14) is the Rayleigh distribution f unction.
In order to get adaptations to the gray show circumstance for the screen of an
ultrasonic scanning device, we have to do logarithmic process on (4.4), and transform
4.1 Backgrounds and Introductions 81
Fig. 4.1 Rayleigh
distribution approximately
obeys the Gaussian
distribution after the
logarithmic transform
the multiplicative noise into the additive noise. After the logarithmic process, g, ω
and n are denoted by g
l
, ω
l
and n
l
, respectively. Hence the additive model is as
follows:
g
l
(x , y) = ω
l
(x , y) + n
l
(x , y) (4.16)
The signal g
l
(x , y) undergone the logarithmic transformation is just the normal
ultrasonic medical image. From the analyses of many scholars, n
l
can be approxi-
mately treated as in conformation to the Gaussian distribution after the logarithmic
transform. The tail part of the left high density of a Rayleigh curve has underwent
more influence than the tail part of the right low density, which is shown in Fig. 4.1:
4.2 Speckle Reduction of Ultrasound Images Based
on Wavelet and Guided Filter
It is desirable to develop a filter, which improves the performances of strong de-
noising ability and better edge-preservation, as well as non-iterative and low algo-
rithm complexity. In the previous chapter, we knew the shortcomings of wavelet
de-noising in low-frequency components and solved it by combining bilateral filter-
ing. A new de-noising method based on wavelet de-noising and guided filter will be
introduced in this chapter, where the low algorithm complexity of the wavelet de-
noising method and the strong de-noising ability of the guided filter are combined.
4.2.1 The Guided Filter
The main idea of wavelet de-noising method is to retain the wavelet coefficients of
the low-pass component (LL), while wavelet coefficients of high-pass component
(LH, HL, HH) are shrunk with the wavelet threshold function. However, the speckle
noise in medical ultrasound images appears as a larger “granular”, and it is observed
in the experiment that the low-pass sub-band in wavelet domain still exists speckle
noise. Thus we are going to take steps to remove the noise at low frequencies while
preserving the edge information of the image.
In the previous chapter, we used a combination of wavelet and bilateral filtering
to filter the noise in the low-frequency part. But we also realized that to remove the
82 4 Despeckle Filtering of Medical Ultrasonic Images …
speckle noise by using bilateral filtering will cost a great deal of time, and thus this
method cannot be used in real-time processing tasks. In order to find ways that has
similar effects to bilateral filtering and at the same time takes less time, the guided
filter is adopted to filter the speckle noise in low-pass sub-band while keeping the
edge information.
Guided filter is a filter that requires a guided image, which can be a single image or
an input image. When the guided image is an input image, the guided filter becomes
an edge preserving filter, which can be used for i mage reconstruction. The guided
filter is developed by the local linear model, and the filtering output at a pixel i is
expressed as a weighted average:
q
i
=
j
W
ij
(I ) p
j
(4.17)
where I is guided image, p is input image and q is output image. W
ij
represents a
function of guided image I and independent with p, i and j are pixel indexes. I is
determined by the specific problems, I can be equal to p like bilateral filter.
We assumed that in a window w
k
centered at the pixel k, q is a linear transform
of I :
q
i
= a
k
I
i
+ b
k
, ∀i ∈ w
k
(4.18)
where a
k
and b
k
are linear coefficients in w
k
. In image filtering, a goal is to minimize
the difference between input and output images to reduce the loss of original image
details. So the following cost function is defined to determine a
k
and b
k
in w
k
:
E(a
k
, b
k
) =
i ∈w
k
[(a
k
I
i
+ b
k
− p
i
)
2
+ εa
2
k
] (4.19)
where ε is a regularization parameter to prevent a
k
to be large. Calculating Eq. (4.19),
we have:
a
k
=
1
|
w
|
i ∈w
k
I
i
p
i
− μ
k
¯p
k
σ
2
k
+ ε
b
k
=¯p
k
− a
k
μ
k
¯p
k
=
1
|
w
|
i ∈w
k
p
i
(4.20)
where μ
k
and σ
2
k
are the mean and variance of I in w
k
respectively.
|
w
|
is the number
of pixels in w
k
. ¯p
k
represents the mean of input image p in w
k
. Because of the fact
that a pixel i is involved in all the overlapping windows w
k
that covers i,thevalue
of q
i
in (4.18) is not identical when it is computed in different windows. And thus it
need to average the values of q
i
and obtain following equation:
4.2 Speckle Reduction of Ultrasound Images Based on Wavelet … 83
Fig. 4.2 Input image p
q
i
=
1
|
w
|
k,i ∈w
k
(a
k
I
i
+ b
k
) =¯a
i
I
i
+
¯
b
i
(4.21)
where ¯a
i
=
1
|
w
|
a
k
and
¯
b
i
=
1
|
w
|
b
k
.
All W
ij
can be defined as follows:
W
ij
=
1
|
w
|
2
k,(i, j )∈w
k
(1 +
(I
i
− μ
k
)(I
j
− μ
k
)
σ
2
k
+ ε
) (4.22)
From (4.18) we can ensure that q has an edge only if I has an edge, and we derive
the derivation from both sides of formula (4.18). Thus we can obtain ∇q = a∇I ,
and the guided filter performs better on edge-preserving. The guided filter processing
diagram, and the details for each phase are as Figs. 4.2, 4.3, 4.4 and 4.5.
4.2.2 Improvements on Wavelet Shrinkage Algorithms
For wavelet-basedde-noising methods, we should at first select an appropriate thresh-
olding function, and then could by criteria shrink some wavelet coefficients in wavelet
domains. The core ideas includes: (a) for all wavelet coefficients whose absolute val-
ues are smaller than the thresholding function should be treated as noises, and then
we can set them as zeros; (b) for all wavelet coefficients whose absolute values are
84 4 Despeckle Filtering of Medical Ultrasonic Images …
Fig. 4.3 Guided image I
Fig. 4.4 Smoothed image q
4.2 Speckle Reduction of Ultrasound Images Based on Wavelet … 85
Fig. 4.5 Guided filter processing diagram
greater than the thresholding function should be treated as the useful information,
and then by specific methods we can perform shrinkage processes on them [13].
From most recent researches, the relevant study based on the statistical model
of natural images has enormously developed. Hence statistical method has been
introduced as a new tool in the wavelet-based de-noising theory. Its essential idea is
to construct the statistical model of wavelet coefficients based on the prior probability
distribution. The most common method such as MAP utilizes Bayers’ maximum
posterior estimation to estimate wavelet coefficients of noiseless signals.
(1) MAP Shrinkage Algorithm
In (4.16), by classical MAP estimation method, we have
ˆω(g) = arg max
ω
p
ω|g
(ω|g) (4.23)
Applying Bayers’ theorem to (4.23), we have
ˆω(g) = arg max
ω
p
g
|
ω
(g
|
ω)p
ω
(ω)
= arg max
ω
[
p
n
(g − n)p
ω
(ω)
]
(4.24)
From (4.24), for wavelet-based de-noising methods of the statistical model, we
should at first construct mathematical models of coefficients of noisy signals and
noiseless signals, respectively. From the previous analyses, noisy signals received
by the ultrasonic imaging system subjects to the Rayleigh distribution, and related
86 4 Despeckle Filtering of Medical Ultrasonic Images …
speckle noises in wavelet domains can approximately obey the Gaussian distribution
with zero mean after the logarithmic transform and wavelet transform, namely
p
n
(n) =
1
√
2πσ
n
exp
−
n
2
2σ
2
n
(4.25)
The study of ultrasonic medical images shows that wavelet coefficients of ultra-
sonic images has non-Gaussian statistical features. From Fig. 4.6, we take statistics
of gray histograms for every subband of the image, and then obtain Fig. 4.7. We can
find that the histogram for wavelet coefficients in the high frequency subband has
a long heavy tail, and as well shows sharply spike characteristics in the zero point.
This feature fits nicely to the generalized Laplace distribution, and hence wavelet
coefficients in the high frequency subband can be described by the function of gener-
alized Laplace distribution. With the help of curve fitting toolbox cftool in MATLAB,
Fig. 4.8 is obtained by taking the curve fitting validation on the histogram. We can
also find that the model of generalized Laplace distribution could depict the distribu-
tion of wavelet coefficients of detail subands for noiseless signals, and furthermore
its probability density function is as follows:
p
ω
(ω) =
1
√
2σ
ω
exp
√
2
|
ω
|
σ
ω
(4.26)
Fig. 4.6 Wavelet decompositions for the ultrasonic clinical image (liver)
4.2 Speckle Reduction of Ultrasound Images Based on Wavelet … 87
-200 0 200 400 600 80 0 1000 1200
0
1
2
3
4
5
6
7
8
9
x 10
4
-100 -80 -60 -40 -20 0 20 40 60 80
0
1
2
3
4
5
6
7
8
9
10
x 10
4
-150 -100 -50 0 50 100 150
0
1
2
3
4
5
6
7
8
9
10
x 10
4
-30 -20 - 10 0 10 20 30 40
0
1
2
3
4
5
6
7
8
9
10
x 10
4
-80 -60 -40 -20 0 20 40 60 80
0
0.5
1
1.5
2
2.5
x 10
4
-80 -60 -40 -20 0 20 40 60 80 100
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
x 10
4
-80 -60 -40 -20 0 20 40 60 80 100
0
0.5
1
1.5
2
2.5
x 10
4
HH1
LH1
HL1
Fig. 4.7 Gray histograms corresponding to Fig. 4.6
-30 -20 -10 0 10 20 30
0
0.1
0.2
0.3
0.4
0.5
x vs. y
fit curve
Fig. 4.8 Curve fitting validates the mathematical model
88 4 Despeckle Filtering of Medical Ultrasonic Images …
Based on the preceding analyses, we could assume wavelet coefficients of speckle
noises subject to the Gaussian distribution, and as well assume wavelet coefficients of
noiseless signals obey the Laplace distribution. To study further the MAP estimation,
(4.24) is equivalent to
ˆω(g) = arg max
ω
[log( p
n
(g − n)) + log(p
ω
(ω))] (4.27)
If we define f (ω) = log( p
ω
(ω)), we can obtain from (4.25) and (4.27)
ˆω(g) = arg max
ω
−
(g − ω
2
)
2σ
2
ω
+ f (ω)
(4.28)
If p
ω
(ω) is a strictly convex and differentiable function, the problem is to solve
an equation of ˆω
g −ˆω
σ
2
n
+ f
( ˆω) = 0 (4.29)
From (4.26), since f (ω) =−log(
√
2σ
ω
) −
√
2
|
ω
|
σ
ω
, then we obtain a solution by
solving (4.29)
ˆω(g) = sign(g)
|
g
|
−
2σ
2
n
σ
g
(4.30)
(2) Shrinkage Algorithm Considering Correlations Between Wavelet
Coefficients
The MAP shrinkage algorithm does not concern the correlation from different
wavelet coefficients. However, wavelet coefficients in neighborhood scale and spatial
positions has strong correlations. If wavelet coefficients in some scale (or spatial) are
larger or smaller, wavelet coefficients in neighborhood scale (or spatial) positions are
as well larger or smaller. We mainly consider the correlation of wavelet coefficients
in scale positions in this chapter. Define g
2
as wavelet coefficients corresponding to
the parent scale position g
1
, and then we have
g
1
= ω
1
+ n
1
g
2
= ω
2
+ n
2
(4.31)
Rewrite (4.31) into a compact form, and we have
g = w + n, (4.32)
where g = (g
1
, g
2
), w = (ω
1
,ω
2
), and n = (n
1
, n
2
), respectively. We use the MAP
estimation method in (4.23) to compute w, and have