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Файл:Ординатура / Хирургия / Библиотека им академика М.И. Перельмана / Книга_5837_Библиотеки_им_академика_М_И_Перельмана.pdf
5.2 A Despeckling Approach for Medical Ultrasound Images Based … 109
Table 5.1 Parameter estimation of adaptive wavelet de-noising algorithm
Parameters Descriptions
ˆσ
n
=
m
S
HH
l,k
0.6745
Standard deviation of noise signal m
(
·
)
: Median calculation
ˆσ
s, j
=
1
M
m,n
l,k =1
S
j
l,k
2
M = m × n: the total number of wavelet coefficients in
corresponding wavelet domain
ˆσ
2
r, j
= max
ˆσ
2
s, j
−ˆσ
2
n
, 0
According to Eq. (5.3)
s =
ˆσ
r, j
√
2
Refer to the literature [15]
5.2.5 Trilateral Filter
(1) Weighting Filter
It is well known that there is a correlation between adjacent pixels in a natural
image. That is, the difference between the gray value of a pixel and that of a pixel
in its neighborhood is very small. If the gray value of a pixel is far from that of the
adjacent pixel, it is likely to be noise [12–14].
Based on above, the idea of weighted filter is to use the weighted average of pixels
in the neighborhood of the point to eliminate noise.
The principle of the weighted filter is to select an appropriate window W (x, y)
centered on each pixel f (x, y) in the image, assign a weight w(i, j) to each pixel
in the window by the corresponding algorithm, and typically, the sum of ownership
values in this window is
i, j∈W (x , y)
w(i, j ) = 1.
The output value g(x , y) corresponding to pixel f (x , y) is the weighted average
of all pixels in the window, as follows:
g(x , y) =
i, j∈W (x , y)
f (i, j )w(i, j ) (5.11)
(a) Mean filter
The image is convolution with the kernel with scale (2 K +1) ×(2 L +1), in which
the element value of the core is the reciprocal of the number of elements in the total
core, it is called mean filter. In window (2 K +1) × (2L + 1), the mean filter results
in the average value of all pixels in the window, that is, the weight of each pixel is
the same and the sum of the weights is 1. For example, when K = L = 1:
w(k, l) =
1
9
⎡
⎣
111
111
111
⎤
⎦
(5.12)
Here w(k , l) is the weight matrix of pixels in the core.
110 5 Despeckling Method for Medical Images Based on Wavelet …
0
5
10
15
20
25
0
5
10
15
20
25
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fig. 5.3 Surface diagram of two dimensional Gaussian distribution with window size 23 × 23
(b) Gaussian filter
Gaussian filter, which is discussed here, is a Gaussian low-pass filter in space domain.
The weight matrix w(k, l ) is designed as a two-dimensional Gaussian distribution,
which is used in the Gaussian filter. Figure 5.3 is a surface map of two-dimensional
Gaussian distribution.
It can be seen from the image that the higher the pixel weight is, the smaller
the pixel weight is near the edge point. Therefore, the weight matrix w(k , l) of the
current point in the Gaussian filter template is independent of the size of the pixel
value, and is determined only by the spatial distance to the center corresponding
point. The larger the standard deviation of Gaussian kernel is, the more smooth the
surface map is, for example, when the size of the selected window is 3 × 3 and the
standard deviation of the Gauss kernel is 1, the Gauss kernel is shown as follows:
w(k, l) =
⎡
⎣
0.0751 0.1238 0.0751
0.1238 0.2042 0.1238
0.0751 0.1238 0.0751
⎤
⎦
=
1
4.8976
⎡
⎣
0.3679 0.6065 0.3679
0.6065 1 0.6065
0.3679 0.6065 0.3679
⎤
⎦
(5.13)
(2) Bilateral Filter
We have introduced the principle of bilateral filter in Chap. 3, then we want to
add some knowledge of bilateral filter as a supplement to Chap. 3, moreover, it
paves the way for the trilateral filter that comes out of the back. The bilateral filter
simultaneously smoothens the image and retains the edge of the image. It combines
two distance functions. The one is the spatial filter, which is constituted by the
regional location similarity, and the other is the range filter that is formed by the
gray value (or color) similarity. Thus, the bilateral filter possesses three advantages:
briefness, non-iterative and local adaptive. Since the bilateral filter is implemented
5.2 A Despeckling Approach for Medical Ultrasound Images Based … 111
in a weighted manner in the literature, we, respectively, describe two of its kernels
first in order to better understand the principle of the bilateral filter.
By combining the spatial filter with the range filter, the bilateral filter is obtained
as in Eq. (5.14):
h(x) = k
−1
(x )
∞
−∞
∞
−∞
f (ξ )c(ξ , x)s( f (ξ ), f (x ))dξ (5.14)
with the normalization:
k(x) =
∞
−∞
∞
−∞
c(ξ, x)s( f (ξ ), f (x ))d ξ (5.15)
where c(ξ , x ) and s ( f (ξ), f (x )) measure the photometric similarity and the geo-
metric closeness between the pixel at the neighboring center x and that of a nearby
point ξ , respectively. They are expressed as follows:
c(ξ, x) = exp
−
(
ξ − x
)
2
2σ
2
c
(5.16)
s( f (ξ ), f (x )) = exp
−
(
f (ξ ) − f (x)
)
2
2σ
2
s
(5.17)
(3) Trilateral Filter
Although the bilateral filter has the advantage of edge-preserving characteristic, it
usually leads to gradient distortion and heavy time-consumption. What’s more, it
is ineffective in dealing with the impulse noise. So we propose the novel idea that
adds the “pulse” weight into the original bilateral filter except for the original Gaus-
sian distance weight and the tightness weight. In addition, the problem of gradient
distortion is simultaneously solved by the trilateral filter, since the impulse weight
expresses the gradient information of an image to some extent.
Begin with predigesting the principle Eq. (5.14) of the bilateral filter into the
below: Reformulate Eq. (5.14)asEq.(5.18).
h
(
x
)
=
ξ ∈Ω
w
(
x,ξ
)
f
(
ξ
)
ξ ∈Ω
w
(
x,ξ
)
(5.18)
where Ω
x
(
N
)
:=
{
x +
(
i, j
)
:−N ≤ i, j ≤ N
}
denotes the adjacency which is cen-
tered at point x, and N is the width of a half filter. We actually choose Ω = Ω
x
(
1
)
for practical effect. And the weighted function is replaced by:
w
(
x,ξ
)
= w
S
(
x,ξ
)
w
R
(
x,ξ
)
(5.19)
112 5 Despeckling Method for Medical Images Based on Wavelet …
where w
S
(
x,ξ
)
and w
R
(
x,ξ
)
denote the spatial filter and the range filter, respectively.
According to the trilateral filter, the weighted function should be formulated to figure
out the noisy point in the image.
Firstly, the function f
m
(
x
)
is introduced to estimate whether the pixel x is the edge
or the noisy point. d
(
x,ξ
)
is defined as the absolute difference of the intensities of
the pixels between x and ξ :
d
(
x,ξ
)
=
|
f
(
x
)
− f
(
ξ
)
|
(5.20)
Thus, f
m
(
x
)
can be set:
f
m
(
x
)
=
m
i =1
g
i
(
x
)
(5.21)
where g
i
(
x
)
represents the ith smallest value except for d
(
x,ξ
)
for ξ ∈ Ω
x
(
1
)
.And
m ∈
[
2, 7
]
, in this manuscript m = 4 generally. This modification roots in the idea
that when the pixel x is proved as the edge of the image, the gray scales of half
pixels in its neighborhood are the same as the gray scale at x, such that f
m
(
x
)
obtains
a relatively small value. On the contrary, if the pixel x is the noisy point, f
m
(
x
)
is
endowed with a relatively large value. Then, the weighted function (5.19) is modified
to Eq. (5.22) by adding an impulse weight:
w
(
x,ξ
)
= w
S
(
x,ξ
)
w
R
(
x,ξ
)
1−H
(
x,ξ
)
w
I
(
ξ
)
H
(
x,ξ
)
(5.22)
where w
I
(
x,ξ
)
= exp
−
f
2
m
(
x
)
2σ
2
I
means pulse weight function. And H
(
x,ξ
)
= 1 −
exp
−
f
m
(
x
)
+ f
m
(
ξ
)
2
2
2σ
2
H
assumes that H
(
x,ξ
)
≈ 0asx is an edge point, and H
(
x,ξ
)
≈
1asx is a corrupted point. The parameter σ
I
determines the approximate threshold to
penalize the high f
m
(
x
)
values. The parameter σ
H
controls the shape of the function
H
(
x,ξ
)
. Furthermore, by means of the trilateral filter, the renovated image pixel
h
(
x
)
is denoted by:
h
(
x
)
=
ξ ∈Ω
w
(
x,ξ
)
f
(
ξ
)
ξ ∈Ω
w
(
x,ξ
)
(5.23)
5.3 Experimental Studies of the Despeckling Method Based … 113
5.3 Experimental Studies of the Despeckling Method Based
on Wavelet and Trilateral Filter
In this section, experimental studies of the despeckling method based on wavelet and
trilateral filter are conducted. In order to evaluate the performance of each algorithm
comprehensively and objectively, four quantitative performance indexes including
Pratt’s figure of merit (FOM), peak signal-to-noise ratio (PSNR), running time and
structural similarity (SSIM) are adopted. The meanings and their definitions of PSNR,
SSIM and FoM are already introduced in Sect. 2.3.3 of Chap. 2.
In this section, the experiments are divided into two branches: One is speckle noise
simulation studies, and the other is the experiments on the real medical ultrasound
image (hepatic image). The experimental schematic diagram is given in Fig. 5.4.
5.3.1 Experimental Results on Synthetic Images
In this section, the proposed approach is compared with other existing 8 kinds of de-
noising algorithms, namely Lee, SRBF, the Frost, trilateral filter, DPAD, OBNLM,
wavelet-soft shrinkage and wavelet-bilateral. The images, as shown in Fig. 5.5,are
used to intuitively estimate the de-noising effects of the proposed algorithm, and the
parameters of every method are listed in Table 5.2. The simulated images, as shown
in Fig. 5.5, are filtered by 9 de-noising methods, and the de-noising results are shown
in Fig. 5.6 and Table 5.3. Here all the algorithms are executed at MATLAB R2013a in
a computer with Windows7, an Intel Pentium dual-core processor, 2.1 GHz dominant
frequency, 2 GB internal storage and 320 GB hard disk.
According to the simulation performances in Table 5.3 and the de-noised images
shown in Fig. 5.6, it can visually be concluded that the proposed method obtains
the highest comprehensive performance. The proposed method not only achieves
the considerable effect, but also preserves the visible edge and undistorted gradient
information. To describe the performance of the proposed wavelet-trilateralalgorithm
in detail and more accurately, its results are compared with other 8 methods with
each evaluation criterion, detail by detail.
Fig. 5.4 Experimental schematic diagram
114 5 Despeckling Method for Medical Images Based on Wavelet …
(a) noise-free image with 400 × 400.
(b) The image with speckle noise
(σ
2
= 0.1) size.
Fig. 5.5 Synthetic images
Table 5.2 Parameters of algorithms
Algorithms Categories Categories experiment parameters
Lee [16] Local adaptive Window size 3 × 3
SRBF [17] Local adaptive Window size 3 × 3
Frost [14] Local adaptive Window size 5 × 5
Trilateral filter [9] Local adaptive Window size 5 × 5, σ
c
= 4, σ
s
= 0.3,
σ
I
= 2, σ
H
= 3
DPAD [18] Anisotropic
diffusion
Window size 5 × 5,
Time step t = 0.2,
Iteration times t = 100
OBNLM [11] Nonlocal means Window size 7 × 7
Search area size 15 × 15
Smoothing parameter h = 0.6
Wavelet-soft shrinkage [5] Multi-scale Window size 3 × 3, wavelet = ‘coif1’
“coif1” decomposition level J = 4
Wavelet-bilateral Hybrid Window size 5 × 5, σ
c
= 4, σ
s
= 0.3
Wavelet-trilateral Hybrid Window size 5 × 5, σ
c
= 4, σ
s
= 0.3
Notice that the above robust de-noising methods belong to different categories,
respectively. But this distinction does not incur any contradiction.
Intuitively, i t can be observed that the original noisy image is improved by every
method to some extent. The Lee filter, as shown in Fig. 5.6a, obviously has the min-
imum time cost and obtains the maximal SSIM value that represents the best edge
detection. But Lee filter weakly restrains the noise. The de-noising effect of SRBF,
viewed as Fig. 5.6b, achieves better than Lee but is still limited, because it appears
blurry without any outstanding virtue. Except that the edge details can be recognized,
5.3 Experimental Studies of the Despeckling Method Based … 115
Fig. 5.6 Denoised images. a Lee. b SRBF.c Frost. d Trilateral filter. e DPAD. f OBNLM. g Wavelet-
soft shrinkage. h Wavelet-bilateral. i Wavelet-trilateral
Frost, as shown in Fig. 5.6c, works poorly in suppressing the noise. Among the men-
tioned local adaptive de-noising methods, the trilateral filter, as shown in Fig. 5.6d,
not only performs best on noise reduction, but also keeps the most unambiguous
edge. The DPAD, as shown in Fig. 5.6e, performs not so well in noise reduction and
edge preservation. With the OBNLM filter, as shown in Fig. 5.6f, the clear outline
at the edge can be noticed obviously, but on the other hand the entire image appears
blurry. Our idea is proved by the slightly prominent results of the trilateral filter
and the wavelet-soft shrinkage. The latter, as shown in Fig. 5.6g, emerges apparently
vague and inferior smoothness. According to the effects of smoothness or edge detec-
tion, the hybrid de-noising methods, such as wavelet-bilateral (Fig. 5.6h), obviously
116 5 Despeckling Method for Medical Images Based on Wavelet …
Table 5.3 Performances of 9 denoising algorithms
Algorithms FOM PSNR/dB SSIM Running time/s
Noisy image 0.3170 18.7605 0.4708 –
Lee 0.7969
5
22.2006
9
0.8559
1
3.5123
1
SRBF 0.6465
7
24.9425
5
0.7525
6
11.0831
6
Frost 0.8146
4
23.5470
7
0.8419
3
11.3765
7
Trilateral filter 0.8216
3
25.7178
2
0.8247
4
9.2761
3
DPAD 0.7054
6
25.3775
4
0.7234
7
32.7633
8
OBNLM 0.8762
2
22.0271
8
0.7098
9
49.8702
9
Wavelet-soft shrinkage 0.3320
9
24.2765
6
0.7162
8
7.3319
2
Wavelet-bilateral 0.4482
8
25.5692
3
0.8102
5
9.4418
4
Wavelet-trilateral 0.8825
1
26.3174
1
0.8479
2
9.5162
5
outperform any one of the combined filters. Compared with the other 8 methods, the
proposed algorithm, the wavelet-trilateral (Fig. 5.6i), not only has a relatively similar
structure with the noise-free image but also smoothes the original images, and at the
same time the edge is sharpened well. In order to objectively evaluate the de-noising
performance, the comparison of performance indexes is listed in Table 5.3, which
can better reflect the de-noising effects. The FOM value measures the similarity of
the image edges between the noise-free image and the filtered output image (where
the largest value one represents the ideal case). The proposed algorithm with the
highest FOM value provides the supreme edge preserving effect, and the OBNLM
filter outperforms all other methods except for the proposed wavelet-trilateral. The
FOM does not have much difference between the trilateral filter and the Frost. The
former demonstrates slightly better than the latter, which means they have simi-
lar edge-preserving characteristics. The Lee does not obtain a satisfied value in the
absence of good edge. The DPAD, with an inferior FOM value, performs poorly
on the edge. Among the aforesaid local adaptive methods, the SRBF obtains the
minimum value, which means the lowest ability to keep edge details. The speckle
noise is hardly optimized by the wavelet-bilateral from its lower FOM value. The
wavelet-soft shrinkage possesses the close value with the noisy image, which means
it lacks the capacity to detect edge. The PSNR value in Table 5.3 demonstrates the
quality of the results after executing the de-noising operation. It is noted that the
PSNR value of the wavelet-trilateral has the peak value among the all mentioned
methods. Additionally, it outperforms the trilateral filter, which is better at the opti-
mization process. Although the wavelet-bilateral performed feebly in edge detection
as mentioned above, it obtained a desirable PSNR value and slightly exceeded the
DPAD of high quality. Filtered by the SRBF and the wavelet-soft shrinkage meth-
ods, the gradual smoothness and elimination of the de-noised results are noticeable.
Other filters with the similar PSNR value, namely the Frost, OBNLM and Lee, do
not have the comparable performances with the above methods. The SSIM value
in Table 5.3 quantities the discrepancy of the structures between the original image
5.3 Experimental Studies of the Despeckling Method Based … 117
and de-noised result. A larger value means the better performance (where the perfect
value is one). The Lee, with the largest value among the nine methods, achieves the
best structure similarity. However, it should be noted that although our algorithm
comes off second best, compared with Lee, the difference value between them is
quite small as SSI M = 0.008 that stands for the excellent efficiency in struc-
ture similarity. Similarly, it is readily observed that the high SSIM value of Frost
is close to that of the proposed algorithm. Frost also has a satisfactory ability to
hold the structure. The part of our method, named trilateral filter, outperforms the
hybrid wavelet-bilateral method, with SS I M = 0.0145. Among the local adaptive
optimization algorithms, the SRBF performs the worst. But it surpasses the DPAD,
wavelet-soft shrinkage and OBNLM of rather lower SSIM, which make the struc-
ture of the noisy image distorted. The real-time property increasingly arouses much
attention in medical fields, and the optimization speed as a vital factor should be
considered. The running time in Table 5.3 represents the time of the de-speckle pro-
cess, which measures the efficiency quantification of speckle reduction. It is easy to
find that the Lee filtering has the minimum value. In this manuscript, for real-time
property, we focus on the two component parts of our method and the two kinds
of hybrid methods. It can be seen that the wavelet-soft shrinkage has the less time
expenditure compared with the other three filters. Although our method expends the
maximum time, it should be noted that the difference values with wavelet-bilateral
and trilateral filter are Time = 0.0744s and Time = 0.2401s, respectively. The
speeds between the SRBF and Frost have slight differences and far surpass those
of the DPAD and OBNLM, which are inadequate in real-time systems due to the
time-consuming characteristics. According to the above interpretations, the superi-
ority of the proposed wavelet trilateral method is obviously observed. The proposed
method not only effectively smoothens medical ultrasound images contaminated by
speckle and impulse noise, but also keeps the gradient information. At the same
time, it maintains the advantages of the original bilateral filter which is achieved by
suppressing the noise and sharpening the edges simultaneously.
5.3.2 Experiments on Real Clinical Ultrasonic Images
In this section, the experiments are conducted on real clinical ultrasound hepatic
images with the resolution 460 ×380, and 256 gray scales, captured via Vivid 7 ultra-
sound unit (GE Medical System) with an operating frequency range of 2.3–10.0 MHz
after logarithmic compression and produced by ultrasound medical experts. The orig-
inal image and the de-noised results are shown in Figs. 5.7 and 5.8, respectively.
(1) Sharpness Algorithm Based on Similarity of Gradient Structure (NRSS)
The objective evaluation index of NRSS is a relatively new technology, and it
is mainly used in the embedded system. It has not been used to evaluate the results of
ultrasonic image de-noising in previous literatures, and thus using NRSS to evaluate
the ultrasonic image is a novel attempt. The idea of image structure similarity (SSIM)
118 5 Despeckling Method for Medical Images Based on Wavelet …
is that the distortion of the perceptual image can be obtained as long as the structural
information of the target can be calculated. Based on this idea, Yang Chunling et al.
proposed a novel non-parametric image quality evaluation index based on NRSS
without reference structure. The advantages of mathematical model and structural
similarity evaluation method of imaging system are well combined in NRSS. The
main idea of NRSS is to construct a virtual reference image. The evaluation steps
are as follows:
(a) constructing the reference image of a given image through a low-pass filter;
(b) The structural similarity between a given image and a reference image is calcu-
lated to evaluate the quality of a given image.
A large number of experiments show that the results obtained with the process of
performance evaluation of NRSS are consistent with subjective evaluation methods
and other fully parametric and non-parametric performance indicators. Therefore,
this chapter applies NRSS to the evaluation of the effect of medical ultrasound image
de-noising.
(2) No Reference Image Quality Assessment (NIQE)
The natural image quality evaluator (NIQE) is applied to evaluate the quality of an
image, which is a “completely blind” image quality analyzer proposed by Mittal et al.
Different from the previous full reference image quality assessment PSNR, NIQE
does not need any reference image and only evaluates the quality according to the
given image itself.
The steps to evaluate a given image using NIQE are as follow:
• Calculating the parameters of the input general image in this model, which are
regarded as the “natural parameters” for the evaluation of the image;
Fig. 5.7 Clinical ultrasound
hepatic images