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
28 2 Despeckle Filters for Medical Ultrasound Images
Fig. 2.3 Approximation of
singular curves with tensor
wavelet
ˆx
d
k,j
=
⎧
⎨
⎩
0
1
w
d
k,j
ˆy
d
k,j+1
<(qˆσ
d
j
)
2
w
d
k,j
ˆy
d
k,j+1
≥ (qˆσ
d
j
)
2
⎫
⎬
⎭
(2.16)
where y
d
k,j
is the noise-free wavelet coefficients, and ˆσ
d
j
is an estimate of the noise
standard deviation in the high-pass image. q is an adjustable parameter. The empirical
densities ˆp
M
k
|X
k
(m
k
|1), ˆp
M
k
|X
k
(m
k
|0), ˆp
E
k
|X
k
(e
k
|1), and ˆp
E
k
|X
k
(e
k
|0) can be estimated,
where m
k
=
|
w
k
|
and the local spatial activity indicator. e
k
is calculated as the aver-
aged energy of the neighboring coefficients of w
d
k,j
. The data above are obtained by
posteriori estimating, and the estimated values ˆr =
N
k=1
ˆx
k
N −
N
k=1
ˆx
k
,
ˆ
ξ
k
=
ˆp
M
k
|X
k
(m
k
|1)
ˆp
M
k
|X
k
(m
k
|0)
, ˆη
k
=
ˆp
E
k
|X
k
(e
k
|1)
ˆp
E
k
|X
k
(e
k
|0)
can be easily known. The estimated wavelet coefficients ˆy
k
can be calcu-
lated by:
ˆy
k
=
ˆr
ˆ
ξ
k
ˆη
k
1 +ˆr
ˆ
ξ
k
ˆη
k
w
k
(2.17)
where
ˆ
ξ
k
cannot be directly obtained, because the curves of ˆp
M
k
|X
k
(m
k
|1) and
ˆp
M
k
|X
k
(m
k
|0) have significantly trailing portion. In a practical experiment, two seg-
ments function approximates the curve of log (
ˆ
ξ
k
), and an exponent transform is
performed to get
ˆ
ξ
k
.
(2) The Wavelet-Based Filter of Gupta et al. (Gupta)
The wavelet-based filter of Gupta et al. reduces the speckle noise by using Bayesian
maximum a posteriori (MAP) estimated theory in the wavelet domain. The signal
component G
k
(s) of wavelet coefficients is modeled as Gaussian distribution, and
the noise component N
k
(s) is modeled as Rayleigh distribution.
2.2 Despeckle Filters for Ultrasound Images 29
F
k
(s) = G
k
(s) + N
k
(s) (2.18)
where F
k
(s) means the wavelet coefficients of noisy image at k scale and s spatial
position. Then, the optimal estimated value of high-pass wavelet coefficients can be
obtained using Bayesian MAP estimation
ˆ
G = sign(F) · max
⎛
⎝
2
|
F
|
σ
2
G
+ α
2
|
F
|
−
α
4
F
2
+ 4α
4
σ
2
G
+ 4α
4
σ
4
G
2(α
2
+ σ
2
G
)
, 0
⎞
⎠
(2.19)
where sign (·) is the sign function, and ˆσ
G
=
max(σ
2
F
− σ
2
, 0) denotes the esti-
mated standard variance of the signal at a given position. σ
2
F
and σ
2
are the estimated
variance of wavelet coefficients F
k
(s) and N
k
(s) respectively.
(3) The Linear Wavelet Filter of Andria et al. (Andria)
The linear wavelet filter of Andria et al. is a novel despeckle method for ultrasound
medical images. The noisy ultrasound image is decomposed with first level of 2-D
wavelet transform. Comparing the peak signal-to-noise ratio (PSNR) values of the
three reconstructive images (obtained via the inverse wavelet transform with three
orientations, respectively), Andria et al. concluded that horizontal details have more
information, especially for the image suffered with high noise levels. As a result,
Andria filter performs the linear filtering just on the vertical and diagonal details
of the image, keeping the horizontal details. The linear filtering is proposed as a
Gaussian filter with an adjustable kernel size.
2.2.4 Category 4: Nonlocal Means Filter
The idea of nonlocal mean filter is the pixels in the image are not isolated, and there
must be some correlation between the pixels at one point and the pixels elsewhere.
This can be summarized as gray correlation and geometric structure similarity. Sim-
ilar pixels are not confined to a local region, such as long edges, texture and so on.
Natural images contain abundant redundant information, so we can use the image
blocks which can describe the structural features of the images to search for similar
blocks in the whole image [19, 10, 20] (Fig. 2.4).
This algorithm takes a lot of time in calculation. If the image has more pixels, and
the box is larger, the calculation usually takes a few minutes.
(1) The Optimized Bayesian Nonlocal Means (OBNLM) Method
Based on the block-wise nonlocal (NL) means filter and Bayesian nonlocal means
filter, Pierrick et al. proposed the optimized Bayesian NL-means (OBNLM) filter
with block selection for medical ultrasound images. The speckle model is chosen as
the following:
30 2 Despeckle Filters for Medical Ultrasound Images
Fig. 2.4 Schematic diagram
of weights of similar blocks
f (x) = g(x) + g
γ
(x)n(x) (2.20)
where f (x) is the observed noisy image, g(x) is the noise-free image, and the noise
n(x) is modeled as Gaussian noise N(0, σ2). The factor γ depends on ultrasound
devices.
An ultrasound image is divided into several overlapping blocks, and the overlap-
ping blocks mean that a pixel is included in several blocks. The whole pixel values of
a block M are calculated as the weighted average value of the N similar blocks, and
the weight is measured by the similarity between block M and each one of N, respec-
tively. The Pearson distance is proposed for measuring the distance of two blocks to
get their weight of similarity. The whole pixel values of block Bi are estimated by
using Bayesian framework.
ˆg(B
i
) =
|
i
|
j=1
f (B
i
)p(f (B
i
)|f (B
j
))
|
i
|
j=1
p(f (B
i
)|f (B
j
))
(2.21)
where bold letters denote vector, f (B
i
) indicates the vector gathering observed
image in block B
i
, ˆg (B
i
) indicates the estimate of noise-free image in block B
i
,
and p(f (B
i
)|f (B
j
)) is the probability distribution function of f (B
i
) conditionally to
f (B
j
). Since a pixel x
i
is included in several blocks, the final restored value of pixel
x
i
is calculated as the mean of these blocks containing pixel x
i
.
(2) The Probabilistic Patch-based (PPB) Filter
Deledalle et al. recently proposed a probabilistic patch-based (PPB) filter [21], which
performs a weighted average of the values of similar patches. These weights can be
2.2 Despeckle Filters for Ultrasound Images 31
iteratively calculated based on both the similarity of noisy patches and their previous
estimated similarity. The pixel amplitudes f, in the noisy image, are modeled as
Nakagami–Rayleigh distribution
p(f |r) =
2L
L
(L)r
L
f
2L−1
exp
−
Lf
2
r
(2.22)
where r is the underlying reflectivity image and L is the number of looks of SAR
image. Note that L = 1 and ˆr =
t
w(s,t)f
2
t
t
w(s,t)
are chosen for ultrasound images. The
weight w(s, t ), between patch s and patch t at iteration i, is defined as follows:
w(s, t) = exp
⎧
⎪
⎨
⎪
⎩
−
k
⎡
⎢
⎣
2L − 1
h
log
f
s,k
f
t,k
+
f
t,k
f
s,k
+
1
T
ˆ
f
i−1
s,k
−
ˆ
f
i−1
t,k
2
ˆ
f
i−1
s,k
ˆ
f
i−1
t,k
⎤
⎥
⎦
⎫
⎪
⎬
⎪
⎭
(2.23)
where the parameters h and T act as dual parameters to balance the trade-off between
the noise reduction and the fidelity of estimate. f
s,k
and f
t,k
are the kth pixel amplitudes
at the patch s and patch t, respectively. The previous estimates of f
s,k
and f
t,k
are
denoted by
ˆ
f
i−1
s,k
and
ˆ
f
i−1
t,k
. The estimate value
ˆ
f
i
s
(c) of the center pixel c at the patch
s at ith iteration can be obtained by performing a weighted average of all the pixels
in the image,
ˆ
f
i
s
(c) =
t
w(s, t)f
t
(c) (2.24)
where f
t
(c) is the center pixel at the patch t. The iteration is repeated until there is
no more change between two consecutive estimates.
2.2.5 Category 5: Hybrid Filter (Nonlocal Means
and Multi-scale Hybrid Filter)
Kostadin Dabov et al. proposed a denoising algorithm called Block matching 3D
denoising (BM3D) [22], which is using redundant information of image to finish
denoising. There is a connection between adjacent pixels in the image, and BM3D
takes advantage of this connection. It groups similar blocks, establishes a 3D array,
and converts the 3D array into sparse representation in transform domain by using
collaborative filtering method to restore the image. The combination of transform
domain method and nonlocal image modeling technique enable BM3D to obtain
excellent PSNR values, at the same time, it also can preserves some essential unique
features of image. These abilities make it worthy of being called state-of-the art
techniques [23] (Fig. 2.5).
32 2 Despeckle Filters for Medical Ultrasound Images
Fig. 2.5 Flowchart of the BM3D [23]
(1) The SAR-Oriented Version of Block-Matching 3-D (SAR-BM3-D) Filter
SAR-BM3-D filter, based on the block-matching 3-D (BM3-D) filter, combines the
nonlocal means filter with the wavelet transform method for ultrasound images. To
simplify the model f (x,y) = g(x,y)n(x,y), (x, y) is replaced with s, and Eq. f (x,y)
= g(x,y)n(x,y) is expressed as f (s) = g(s)n(s), and then its additive form can be
rewritten as
f (s) = g(s) +[n(s) − 1]g(s) = g(s) + v(s) (2.25)
where both signal g(s) and noise v(s ) are assumed spatially uncorrelated. Referencing
BM3-D filter, an image is divided into many small blocks. For each reference block,
the similar blocks are arranged in the same group, thus the image is divided into
several groups of 3-D form. Performing collaborative filtering on the groups, all
filtered blocks are returned to their original location and contribute with the suitable
weights to a basic estimate of the image. Performing undecimated discrete wavelet
transform on each 3-D group, Eq. (2.25) is converted to
F = G + V (2.26)
In the first step, the block similarity measure d
1
of block B
s
and B
t
is defined as:
d
1
[a(B
S
), a(B
t
)]=
k
log
a(s + k)
a(t + k )
+
a(t + k )
a(s + k)
(2.27)
where a(B
s
) and a(B
t
) denote the vector of observed values drawn from block B
s
and
B
t
, respectively. k is used to scan the whole block and the subscript 1 indicates that
this measurement is used in the first step. The estimation
ˆ
G
1
is a coarse estimate and
it can be calculated based on the transformed coefficients of each group
ˆ
G
1
(i) = max
⎛
⎝
0,
#
F
2
$
SB(i)
−
σ
2
u
(1+σ
2
u
)
#
f
2
$
θ
#
F
2
$
SB(i)
⎞
⎠
F(i) (2.28)
2.2 Despeckle Filters for Ultrasound Images 33
where
·
SB(i)
denotes the average over the sub-band comprising the ith coefficient.
·
θ
indicates the average over the whole group and σ
2
u
is a known parameter depend-
ing on the speckle format.
In the second step, the similarity measure d
2
between block B
s
and B
t
takes into
account the additional information provided by the first step:
d
2
[a(B
s
), a(B
t
)]=
k
log
%
a(s + k)
a(t + k )
+
a(t + k )
a(s + k)
+ γ
ˆx(s + k) −ˆx(t + k)
2
ˆx(s + k)ˆx(t + k )
&
(2.29)
where ˆx indicates a coarse estimate of noise-free image, and γ is a weigh factor.
After wavelet transform on each group, the empirical Wiener filtering is performed
and the estimate
ˆ
G
2
of noise-free image can be calculated as:
ˆ
G
2
(i) =
ˆ
G
2
1
(i)
ˆ
G
2
1
(i) +
#
V
2
$
θ
F(i) (2.30)
where
#
V
2
$
θ
=
1
|
θ
|
i∈θ
[F(i) −
ˆ
G
1
]
2
. Then, inverse wavelet transform on image
estimate
ˆ
G
2
(i) is performed.
Considering every pixel can be included into several groups, a pixel value must
be calculated using the weighted average:
ˆx(s) =
1
T
w
θ
ˆx
θ
(s), θ ∈ (s) (2.31)
where ˆx is the final estimate of noise-free image, ˆx
θ
indicates the estimate of 3-D
groups after inverse wavelet transform, and w
θ
represents the corresponding weight.
The set of all group (s) is comprised of x(s) and T =
θ∈(s)
w
θ
is the normalizing
factor.
2.3 Experiments for Simulation and Ultrasound Images
To better compare the denoising performance of different filters, simulation and
clinical ultrasound images are adopted in this chapter to conduct the experimental
studies. Some full-reference image quality metrics (PSNR, SSIM, and FoM) and
the natural image quality evaluator (NIQE) are applied to compare the despeckling
performance of the eleven filters [24–26].
34 2 Despeckle Filters for Medical Ultrasound Images
2.3.1 Field II Simulation
Field II is an ultrasonic system designed by Danish ultrasonic expert J. A. Jensen
et al. using acoustic principles. The Field program system uses the concept of spatial
impulse responses as developed by Tupholme and Stepanishen in a series of papers.
The program can be used to calculate the acoustic field of emission and pulse echo
for a large number of different ultrasonic sensors under the condition of pulse and
continuous wave, and to simulate any kind of linear imaging and lifelike image of
human body tissue.
2.3.2 Configuration of Experimental Parameters
The parameters of eleven filters in this experiment are listed in Table 2.1. To get better
filtered result, these parameters are proposed by both the cited papers and medical
experts.
The first simulation method is based on Field II simulation. In Fig. 2.6a, the
“Cyst” phantom is a result of the Field II. Figure 2.6b is the corresponding noise-
free image obtained by the approach proposed in the paper which was published by
Peter C. Tay. They propose an iterative filtering method in which the local extremum
of the b-mode image is regarded as the outlier and only these outliers are locally
smoothed. The second simulation result is obtained by contaminating the noise-free
image with “speckle” noise (by using the function “imnoise” in MATLAB). Based
on the noise-free image Fig. 2.6b, c is the resulting image calculated by the second
Table 2.1 The parameters of eleven filters for experiments
Method Experiment parameters
Lee [11] Window size 3 × 3
Frost [12] Window size 5 × 5
SRBF [5] Window size 5 × 5
DPA1D [3] Window size 5 × 5, time step_t = 0.2, iteration times t = 100
OSRAD [17] dt = 0.05, T = 2, σ = 1, Neighborhood = 1, implicit iteration times t =
200
GLM [18] Window size 3 × 3, decomposition level J = 4, threshold factor K = 4
Gupta [27] Window size 5 × 5, decomposition level J = 4
Andria [4] Window size 5 × 5, decomposition level J = 1
OBNLM [10] Block size 7 × 7, search area size 15 × 15, smoothing parameter h = 0.7
PPB [21] Number of looks L = 1, search area size 23 × 23, path size 7 × 7, α =
0.92, T = 0.2, iteration times t = 4
SAR-BM3-D [22] Number of looks L = 1, decomposition level J = 3, block size 8 × 8,
search area size 39 × 39
2.3 Experiments for Simulation and Ultrasound Images 35
Fig. 2.6 Simulation images. a Image produced by field II simulation, b corresponding noise-free
image, c image produced by “imnoise” in MATLAB with variance 0.4
simulation method with variance 0.4. The resolution of these simulation images is
350 × 500 pixels, representing approximately 29 × 41 mm in Fig. 2.6. Notice that
the white dashed [28] (Table 2.1).
Line region at Fig. 2.6a, c is the selected area to show the performance of different
filters later.
2.3.3 Full-Reference Image Quality Metrics
Since noise-free images exist in simulation images experiments, the performance of
eleven filters for simulating images assessment is measured by using these full refer-
ence image quality metrics, such as peak signal-to-noise ratio (PSNR), the structural
similarity (SSIM) index, and Pratt’s figure of merit (FoM) [29].
PSNR is the most common and the most widely used measurement of the quality
of an evaluation image. In order to measure the quality of a processed image, we
usually use t he PSNR value to determine whether a processing is satisfactory or not.
The PSNR is given by:
PSNR(f ,
ˆ
f ) = 10 log
10
255
2
MSE
(2.32)
where f denotes the original noise-free image and
ˆ
f is the de-noised image. MSE
represents the mean-squared error, which is defined as:
MSE =
1
MN
M
x=1
N
y=1
(f
xy
−
ˆ
f
xy
)
2
(2.33)
36 2 Despeckle Filters for Medical Ultrasound Images
where Mand N are the width and height of the image, respectively.
Although PSNR and MSE have very direct mathematical meanings and are cal-
culated easily, they are not very well matched to perceived visual quality [28]. A
great deal of effort has gone into the development of quality assessment methods
that take advantage of known characteristics of the human visual system (HVS).
The majority of the proposed perceptual quality assessment models have followed
a strategy of modifying the MSE measure so that errors are penalized in accordance
with their visibility. Zhou Wang et al. developed a measure of structural similarity
(SSIM) that compares local patterns of pixel intensities that have been normalized
for luminance and contrast. SSIM is an objective method to assess the perceptual
image quality, which is very close to visual perception of human. As SSIM has this
superior character, it is used to compare the performance of different filters in this
chapter. The formula of SSIM is as follows:
SSIM (f ,
ˆ
f ) =
(2μ
f
μ
ˆ
f
+ c
1
)(2σ
f ,
ˆ
f
+ c
2
)
(μ
2
f
+ μ
2
ˆ
f
+ c
1
)(σ
2
f
+ σ
2
ˆ
f
+ c
2
)
(2.34)
where μ
f
, μ
ˆ
f
, σ
2
f
, and σ
2
ˆ
f
are the means and variances of reference image f and image
ˆ
f , σ
f ,
ˆ
f
is the covariance between image f and
ˆ
f . c
1
and c
2
are constants. Notice that
the reference image f is noise-free image and
ˆ
f is the filtered image of simulating
noisy image in this chapter. Because both c
1
and c
2
are chosen positive constants,
the value of SSIM ranges from 0 to 1. SSIM = 1 is the perfect result representing
that two images have same structure.
To quantitatively compare the quality of the detected edge pixels, Pratt’s figure of
merit is adopted, which is defined as follows:
FoM (f ,
ˆ
f ) =
1
max(N
f
, N
ˆ
f
)
N
f
i=1
1
1+αd
2
(2.35)
where N
f
and N
ˆ
f
represent the number of ideal and actual edge pixels. α is a constant,
and α = 19. d is the separation distance of an actual edge pixels normal to a line
of ideal edge pixels. The ideal edge pixels and actual edge pixels are binary maps,
which are calculated from the noise-free image and the de-noised image with Canny
edge detector separately in this chapter. To extract the ideal edge map, the Canny
detector with the standard deviation of the Gaussian filter σ = 4 is applied to noise-
free image. Considering the fairness of detecting edge for the filtered images, the
same Canny detector is used to extract the edge of these filtered images. Note that
FoM is normalized where FoM = 1 is the perfect result. The perfect result means
that the actual edge map is the same with ideal edge map.
2.3 Experiments for Simulation and Ultrasound Images 37
2.3.4 Performances of Different Filters
In the experiment of this chapter, the speckle reducing filters are applied to the
simulated images, and some examples are shown in Fig. 2.6a, c. Only the regions
inside the white lines in Fig. 2.6a, c are presented in Fig. 2.7. The values of image
quality metrics in the results are averaged over a set of twenty images with a different
level noise [8, 29].
(a) noisy image (b)Lee (c) Frost (d)SRBF
(e) DPAD (f) OSRAD (g) GLM (h) Gupta
(i) Andria (j) OBNLM (k) PPB (l) SAR-BM3-D
Fig. 2.7 Filtered output from simulation images