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
4.2 Speckle Reduction of Ultrasound Images Based on Wavelet … 89
ˆ
w(g) = arg max
w
p
w|g
(w|g) (4.33)
After a sequence of calculations, (4.33) can be rewritten as
ˆ
w(g) = arg max
w
p
g
|
w
(g
|
w ) p
w
(w)
= arg max
w
p
n
(g − w)p
w
(w)
(4.34)
We at first get the density distribution of noises and prior density distribution of
wavelet coefficients, and then approximate the real value of w by the utility of Bayers’
maximum posterior estimation. From the preceding analysis, we show that speckle
noises of ultrasonic images subject approximately to the Gaussian distribution and
wavelet coefficients of noiseless signals obey the Laplace distribution, namely
p
n
(n) =
1
2πσ
2
n
exp(−
n
2
1
+ n
2
2
2σ
2
n
) (4.35)
p
w
(w) =
1
2σ
2
ω
exp
−
√
2
σ
ω
(
|
ω
1
|
+
|
ω
2
|
)
(4.36)
With the continuity of deductions in (4.34), hence we have (4.37)
ˆ
w(g) = arg max
w
log( p
n
(g − w)) + log( p
w
(w))
. (4.37)
We define f (w) = log( p
w
(w)) by the same manner, and get a description of
distribution of statistical model from (4.35) and (4.36)
ˆ
w(g) = arg max
w
−
(g
1
− ω
1
)
2
2σ
2
n
−
(g
2
− ω
2
)
2
2σ
2
n
+ f
(
w
)
. (4.38)
If p
w
(w) is strictly convex and differentiable, then we need to solve an equation
of
ˆ
w
g
1
−ˆω
1
σ
2
n
+ f
1
(
ˆ
w) = 0, (4.39)
g
2
−ˆω
2
σ
2
n
+ f
2
(
ˆ
w) = 0. (4.40)
The ω
1
and ω
2
are considered as independently identical distributed in (4.36).
From the preceding analysis, wavelet coefficients in neighborhood scales has strong
correlations, and hence the correlation depicted by (4.36) is inaccurate. A new model
derived from the Laplace model is as follows
p
w
(w) =
3
2πσ
2
ω
exp
−
√
3
σ
ω
ω
2
1
+ω
2
2
. (4.41)
90 4 Despeckle Filtering of Medical Ultrasonic Images …
Hence f (w) can be rewritten as
f
(
w
)
= log
3
2πσ
2
ω
−
√
3
σ
ω
ω
2
1
+ ω
2
2
, (4.42)
Proceeding to make deductions, we have the following
f
1
(w) =−
√
3ω
1
σ
ω
ω
2
1
+ ω
2
2
, (4.43)
f
2
(w) =−
√
3ω
2
σ
ω
ω
2
1
+ ω
2
2
. (4.44)
We get the estimation of w by plugging (4.43) and (4.44) into Eqs. (4.39) and
(4.40)
ˆω
1
=
g
2
1
+ g
2
2
−
√
3σ
2
n
σ
ω
+
g
2
1
+ g
2
2
g
1
, (4.45)
where (g)
+
=
0, g < 0
g, g ≥ 0
.Eq.(4.45) is a novel de-correlation shrinkage algorithm
derived from the classical MAP shrinkage algorithm.
From now on, we have obtained the wavelet thresholding function and wavelet
shrinkage algorithm, respectively. Hence the modified wavelet de-noising algorithm
can be written down as follows
ˆω
1
=
⎧
⎪
⎨
⎪
⎩
0, g
1
≤ T
j
√
g
2
1
+g
2
2
−
√
3σ
2
n
σ
ω
+
√
g
2
1
+g
2
2
g
1
, g
1
> T
j
. (4.46)
4.2.3 The Procedures of the Wavelet and Guided Filter Based
Denoising Method
In this section, an improved de-speckling method based on wavelet and guided filter is
proposed, and the schematic diagram of the proposed method is as follows (Fig. 4.9):
(1) Ultrasonic envelope signal received by the ultrasonic imaging system is com-
pressed with logarithmic transformation (Log) to fit in the display range, and
thus the multiplicative noise model is converted to an additive noise model of
4.2 Speckle Reduction of Ultrasound Images Based on Wavelet … 91
Fig. 4.9 The schematic diagram of the proposed algorithm
speckle noise and noise-free signal, which is the commonly medical ultrasound
image. Where the noise-free signal will obey the generalized Gaussian distri-
bution, and the speckle noise will obey Rayleigh distribution.
(2) The 2-D discrete wavelet transformation ( DW T ) is applied for the log-
transformed image, and obtain four frequency domains (LL
1
,LH
1
,HL
1
and
HH
1
). To continue the process of wavelet decomposition for the low frequency
domain LL
1
, four frequency domains (LL
2
,LH
2
,HL
2
and HH
2
) are obtained.
This step is repeated until the maximum decomposition layer J is reached.
(3) According to the statistical properties of speckle noise and noise-free signal in
step 1, an improved wavelet threshold shrinkage algorithm (WT) is designed,
which is used to process the wavelet coefficients of the high frequency subbands
in each layer (LH
j
,HL
j
and HH
j
, j = 1, 2,..., J ). The standard deviation of
speckle noise, image and no-noise signals are estimated respectively.
(4) Considering the fact that the low frequency subband of the last layer (LL
J
)
still exist most large speckle noise and the bilateral filter suffers from two main
disadvantages of high complexity and “gradient distortion”, so the guided filter
is used to filter LL
J
.
(5) Inverse wavelet transformation (IDWT) is used to process the de-noised
wavelet coefficients and obtain the de-noised medical ultrasonic images.
4.3 Experimental Studies of the Wavelet and Guided Filter
Based Method
From the theoretical description of the above sections, an improved de-noising
method is introduced in this chapter based on the characteristics of the speckle
noise in medical ultrasonic images. This method has the superiority in theory com-
pared with the basic de-noising methods. So in order to further verify this method,
more comparison experiments will be carried out in this section. The experiments are
92 4 Despeckle Filtering of Medical Ultrasonic Images …
divided into two branches: The speckle noise simulation experiment and the med-
ical ultrasound image experiment. In experiments, some quantitative performance
measures are used to objectively evaluate the de-noised images.
4.3.1 Simulation Experiments: Synthetic Images
Experiments
In order to quantitatively evaluate the de-noising methods, the performance of the
noise reduction methods is measured using quantitative performance measures such
as peak signal-to-noise ratio (PSNR), structural similarity (SSIM), Pratt’s Figure
of Merit (FoM) and running time [2, 14, 15]. The meanings and their definitions of
PSNR, SSIM and FoM are already introduced in the Sect. 2.3.3 of the Chap. 2.
In the following experiments, the proposed algorithm is compared with other six
de-noising algorithms, namely RBF, DPAD, DTD, Andria, OBNLMand SAR-BM3-
D. These de-noising algorithms can be classified into different categories. Their con-
figuration of parameters is in Table 4.1, and all settings of parameters are determined
by the referenced papers and suggestions from medical experts.
Although simulated experiments would not completely fit with realistic condition,
we could still retain some important information, i.e. de-noising effects, edge pre-
serving condition and efficiency. We could simulate speckle noises spatially related
to ultrasonic images by means of processing the low filtering on complex Gaussian
random fields [16]. Since speckle noises of ultrasonic images are of random grainy
structures, to construct Gaussian random fields, we select different covariance func-
tions in order to get various sorts of speckle noises. Four covariance functions are
selected in this chapter, namely Matern function, Spherical function, Exponential
Table 4.1 Configuration settings for other six de-noising algorithms
Filter Date Category Configuration of parameters
SRBF 2010 Adaptive de-noising method Window size 5 × 5
DPAD 2006 Anisotropic de-noising method Window size 5 × 5, time step
t = 0.2,
Number of iterations t = 100
DTD 2012 Anisotropic de-noising method Window size 3 × 3, time step
t = 0.15
Andria 2012 Multi-scale de-noising method Window size 5 × 5, maximal
decomposition level J = 1
OBNLM 2009 Nonlocal mean de-noising
method
Window size 7 × 7, search region
size 15 × 15, Smooth parameter
h = 0.6
SAR-BM3-D 2012 Hybrid de-noising method Window size 8 × 8, search region
size 39 × 39, Maximal
decomposition level J = 3
4.3 Experimental Studies of the Wavelet and Guided Filter … 93
function and Rational quadratic function, and their expressions and configuration
settings are shown in Table 4.2.
We select Fig. 4.10a as the reference figure, and obtain different noisy images by
constructing different Gaussian random fields. The de-noising results of simulated
experiment are shown in Figs. 4.11 and 4.12 and in Tables 4.3, 4.4, and 4.5
As shown in the Fig. 4.11a, SRBF can filter out most of speckle noises, but often
results in blurring the detail part of images; As shown in Fig. 4.11b and b, DPAD
and DTD have better de-nosing effects, but at the same time will produce apparent
pseudo-traces; As shown in Fig. 4.11e and f, OBNLM and SAR-BM3-D have obvious
advantages of de-nosing effects, but have a long time implementation so that they are
not suitable for real-time medical imaging systems; As shown in Fig. 4.11d, Andria
cannot perform well in de-nosing speckle noises; As shown in Fig. 4.11g, the wavelet-
guided-filters based de-nosing method not only have a relatively well capacity of
de-nosing of speckle noises, but also retain the majority of edge information.
Since the above analysis of the de-nosing effects strongly depend on human eye’s
intuitions, the judgement from human eye sometimes does not fit well with realistic
image quality, and hence utilities of quantitative assessment indexes are necessary
to analyze the de-nosing effects; As shown in Table 4.3, Andria alike with classical
wavelet de-nosing methods has the minimum PSNR since it does not process wavelet
coefficients in low frequency domains, and results in the preservation of large grainy
speckle noises. SRBF, DPAD and DTD all do not obtain relatively ideal PSNR values;
OBNLM and SAR-BM3-D have a relatively strong capability of de-nosing speckle
noises, and hence obtain a nice PSNR value. The proposed algorithm has the highest
PSNR value.
Compared with PSNR, SSIM is better suitable to the judgement on images. As
shown in Table 4.4, Andria has the minimum SSIM. Since Andria does not discard
enough speckle noises, it means that the de-noised image has the least similarity
Table 4.2 Expression of covariance functions and related parameters
Covariance function Expression Parameters
Matern function r
θ
(l) =
⎧
⎨
⎩
1
2
θ
2
−1
(θ
2
)
l
θ
1
θ
2
κ
θ
2
l
θ
1
l = 0
1 l = 0
,
θ
1
,θ
2
> 0
θ
1
= 1,θ
2
= 1.2
Spherical function
r
θ
(l) =
1 −
3
2
l
θ
+
1
2
l
θ
3
l ≤ θ
0 ot h er w i s e
,θ > 0
θ
1
= 1,θ
2
= 1.2
Exponential function r
θ
(l) = θ
lθ
2
1
,θ
1
∈ (0, 1), θ
2
∈ (0, 2] θ
1
= 0.5,θ
2
= 1.2
Rational quadratic function
r
θ
(l) = (1 +
1
θ
2
1
)
−θ
2
,θ
1
,θ
2
> 0
θ
1
= 1,θ
2
= 1.2
94 4 Despeckle Filtering of Medical Ultrasonic Images …
(a) reference figure (b) Matern (c) Spherical
(d) Exponential (e) Rational quadratic
Fig. 4.10 Synthetic image and simulated image with associated speckle noises
structure with the reference image. As shown in Table 4.4, SRBF, DPAD and DTD
do not perform well by means of the i mage quality index SSIM. OBNLM, SAR-
BM3-D and the proposed algorithm have obtained relatively ideal SSIM values. This
indicates that the output image could better preserve the information of structures,
and hence have more similarity structure with the reference image.
To denoise ultrasonic medical images, it should have a relatively strong denoising
capability, and could sufficiently preserve the detail information of edges as well,
and the index of FoM can be used to measure the capacity of edge preservation.
From Table 4.5, compared with other de-noising algorithms, the proposed algorithm
has the highest value of FoM, and this reveals that the algorithm could sufficiently
preserve the tissue structures and edge information of diseased tissues during the
process of de-noising.
In the consideration of running time, DPAD and DTD all belong to iterative algo-
rithms, and spend a relatively long iterative time in order to obtain better de-noising
results. However, OBNLM and SAR-BM3-D would spend a lot of time search-
ing similar regions in the process of de-noising, and hence could not be appropri-
ately applied to real-time imaging systems. Therefore, the wavelet-guided-filters de-
noising algorithm could outperform other de-noising algorithms in several aspects,
namely de-noising effects, edge preservation and efficiency.
4.3 Experimental Studies of the Wavelet and Guided Filter … 95
(a) SRBF
(b) DPAD
(c) DTD
(d) Andria
(e) OBNLM
(f) SAR-BM3-D
(g) Wavelet+guided
Fig. 4.11 De-noising results of simulated experiment
96 4 Despeckle Filtering of Medical Ultrasonic Images …
Fig. 4.12 Performance comparison of de-noising results for simulated experiment (Average Time)
Table 4.3 Performance comparison of de-noising results for simulated experiment (PSNR)
De-noising methods Category of speckle noises
Matern Spherical Exponential Rational quadratic
Noisy image 18.1473 18.4704 19.3003 19.6245
SRBF 21.2280 21.7657 22.2021 24.3563
DPAD 21.7363 22.6797 23.9577 25.1396
DTD 22.4295 22.3275 23.6643 25.5842
Andria 20.2312 20.1870 21.4657 23.1408
OBNLM 23.5083 23.4033 24.7755 26.8010
SAR-BM3-D 24.4116 24.1130 25.5658 27.1266
Wav e l et + guided 26.6388 27.8089 30.5196 33.4731
Table 4.4 Performance comparison of de-noising results for simulated experiment (SSIM)
De-noising methods Category of speckle noises
Matern Spherical Exponential Rational quadratic
Noisy image 0.1886 0.2156 0.2184 0.1883
SRBF 0.7309 0.8451 0.8452 0.8226
DPAD 0.8488 0.8754 0.8898 0.8247
DTD 0.8663 0.8833 0.8899 0.8714
Andria 0.6751 0.7196 0.7227 0.6712
OBNLM 0.9129 0.9317 0.9409 0.9250
SAR-BM3-D 0.9590 0.9621 0.9684 0.9711
Wav e l et + guided 0.9290 0.9472 0.9498 0.9333
4.3 Experimental Studies of the Wavelet and Guided Filter … 97
Table 4.5 Performance comparison of de-noising results for simulated experiment (FoM)
De-noising methods Category of speckle noises
Matern Spherical Exponential Rational quadratic
Noisy image 0.1618 0.1380 0.1499 0.1871
SRBF 0.1581 0.2075 0.2244 0.1843
DPAD 0.1786 0.2032 0.2284 0.2355
DTD 0.1535 0.1868 0.2110 0.1590
Andria 0.1725 0.1405 0.1495 0.1288
OBNLM 0.1783 0.2646 0.2798 0.1892
SAR-BM3-D 0.2471 0.4283 0.3481 0.3029
Wav e l et + guided 0.2827 0.5984 0.5444 0.3657
4.3.2 Experiments on Clinical Medical Ultrasound Images
The analysis method for speckle noise simulation experiment is different from that of
the experiment for medical ultrasound images, and thus the two experiments should
be studied separately [17, 18]. The simulation experiments have been presented in
Sect. 4.3.1, and the experiments for medical ultrasound images will be carried out in
this section. In order to comprehensively analyze the de-noising effect of the proposed
method for medical ultrasound images, a clinical medical ultrasound image with
lesions (a breast ultrasound image of women, 340 × 350 pixels) will be introduced
for experiment, which is shown in Fig. 4.13a. The clinical ultrasound breast images
are acquired using Vivid 7 ultrasound unit (GE Medical System) with an operating
frequency range of 2.3–10.0 MHz. The ultrasound breast images are recorded as
displayed on a Vivid 7 monitor after logarithmic compression. The image resolution
is 350 × 340 pixels with 256 gray levels, approximately representing 36 × 35 mm.
There are 200 asymptomatic and 200 symptomatic ultrasound breast images obtained
for the clinical experiment, which are provided by ultrasound medical experts.
Because of the fact that the medical ultrasound images without noise do not
exist, PSNR, SSIM and other quantitative performance measures which need noise-
free images as reference images will not work for real medical ultrasound images.
Thus, some different methods will be used to analyze and evaluate the de-noise
results. Firstly, analyzing the de-noised images by observing the characteristics of
speckle suppression and feature preservation. Secondly, NIQE (no-reference image
quality assessment) [19] will be used for evaluate de-noised images. Finally, with
the removed noise image, which is the difference between clinical ultrasound image
and the corresponding filtered image, to judge if the artifacts appear in the filtered
image.
The de-noising performance will be compared with other de-noising methods,
and the parameter settings of other 6 de-noising methods are same with simulation
experiment. The de-noised results are shown in Figs. 4.13, 4.14 and Table 4.6.
98 4 Despeckle Filtering of Medical Ultrasonic Images …
(a) Original image (b) SRBF (c) DPAD
(d) Andria (e) OBNLM (f) SAR-BM3-D
(g) Wavelet-Soft Shrinkage (h) Wavelet+bilateral (i) Wavelet+guided
Fig. 4.13 De-noised results of clinical ultrasound image
4.3.3 Discussion of the Experimental Results on Clinical
Medical Ultrasound Images
According to the experimental results of the simulation and medical ultrasound
images presented in the previous 2 sections, we can analyze the de-noising per-
formance based on the quantitative indicators.
The noisy breast medical ultrasound image is filtered by different denoising meth-
ods and the denoised images are shown in Fig. 4.13. Figure 4.13a is a breast ultra-
sound image of a woman. From the Fig. 4.13b, it can be seen that the SRBF filter
has a good speckle suppression performance at the expense of blurred image. With