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5.3 Experimental Studies of the Despeckling Method Based … 119
Fig. 5.8 Denoised results of clinical ultrasound hepatic images. a Lee. b SRBF. c Frost. d Trilateral
filter. e DPAD. f OBNLM. g Wavelet-soft shrinkage. h Wavelet-bilateral. i Wavelet-trilateral
• The corresponding parameters of the given image in the above space model are
calculated;
• The NIQE value of a given image can be obtained by comparing this set of param-
eters with the “natural parameter”.
(3) Summary of Experimental Results
A range of representative de-noised ultrasound hepatic images are shown in Fig. 5.8.
Nevertheless, the Lee filter, as shown in Fig. 5.8a, achieves excess smoothness so
that the edge appears blurry. It can be observed that the de-noising image, filtered by
the SRBF, performs not very well in definition but slightly outperforms the Frost. On
the contrary, the Frost has the better structure similarity than the SRBF, as shown in
Fig. 5.8b, c. The trilateral filter shown in Fig. 5.8d performs worse in terms of smooth-
ness but better in edge detection. For the results of DPAD (Fig. 5.8e) and OBNLM
120 5 Despeckling Method for Medical Images Based on Wavelet …
Table 5.4 NIQE and NRSS
of images in Fig. 5.8
Denoising methods NIQE NRSS
Noisy image 7.8147 0.2587
Lee 7.9962
9
0.8009
2
SRBF 9.5426
6
0.6995
5
Frost 9.0833
8
0.7534
3
Trilateral filter 10.2098
5
0.7488
4
DPAD 12.9821
4
0.4862
7
OBNLM 13.3752
3
0.4487
8
Wavelet-soft shrinkage 9.3984
7
0.3951
9
Wavelet-bilateral 14.0081
2
0.5479
6
Wavelet-trilateral 14.9957
1
0.8823
1
(Fig. 5.8f), they both have good performance in suppressing the noise but at the cost
of obscuring the edge details. The wavelet-soft shrinkage method in Fig. 5.8g has
a strong ability in dealing with the small high-frequency noise. However, it shows
poor performance in those particles and speckle noise because of the fact that the
low-frequency part still contains a large amount of noise after the wavelet decompo-
sition. The de-noised effect of the proposed wavelet-trilateral method is apparently
superior to that of wavelet-bilateral in image quality, structure similarity and edge
detection, as shown in Fig. 5.8h, i. As we all know, the noise-free hepatic images do
not exist, and thus, the above four performance indexes cannot be effectively used
in real experiments of clinical ultrasound images. The natural image quality evalu-
ator (NIQE) is applied to evaluate the quality of an image, which is a “completely
blind” image quality analyzer proposed by Mittal et al. The accurate NIQE values of
filtered images are listed in Table 5.4. It can be seen that the wavelet-trilateral filter
achieves the highest NIQE value because it has the best speckle reduction effect and
obtains the smoothest filtered output image (Fig. 5.8i) compared with the others. The
wavelet-bilateral filter with a desirable NIQE value has a better speckle reduction
performance, but the image appears somewhat blurred (Fig. 5.8h). Both the OBNLM
(Fig. 5.8f) and DPAD (Fig. 5.8e) filters get similarly higher values than the other fil-
ters. The trilateral filter (Fig. 5.8d) has a fine speckle reduction performance from
the NIQE value, but the method is not desirable considering the time factor. Then
the SRBF filter (Fig. 5.8b), the Frost filter (Fig. 5.8c) and the wavelet-soft shrinkage
(Fig. 5.8g) have the similar speckle suppression performance with the similar NIQE
values. The Lee filter (Fig. 5.8a) performs very poorly in NIQE metric due to the
blurred edges.
According to the NRSS index in Table 5.4, the algorithm we introduced in this
chapter preserves the best structural similarity for the original image before de-
noising, can obtain the maximum NRSS value, and the value is much larger than
the second largest Lee filter. Not surprisingly, the NRSS value of wavelet-bilateral
is low, but it is proved that the gradient distortion is serious (Figs. 5.7 and 5.8).
5.4 Conclusion 121
5.4 Conclusion
In this chapter a novel wavelet-trilateral method, based on the statistical distribution
of medical ultrasound signal, is proposed, which is the combination of an adap-
tive wavelet shrinkage algorithm and trilateral filter. We conducted simulation and
experimental studies on synthetic images and real clinical medical ultrasonic images,
respectively, and make comparisons with various existing methods with respect to
four criteria. The experimental results are reported by examination of the synthetic
and real clinical ultrasonic images, which shows the visual and quantitative effec-
tiveness of the proposed method. With the proposed method, severe speckle noise
is greatly reduced, and at the same time, the edge and the gradient information are
preserved. For the quantitative performance analysis, four performance indexes are
calculated to validate the better performance of the proposed method.
References
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IEEE Trans. Image Process. 14, 1747–1754 (2005)
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Ultrason. Ferroelectr. Freq. Control 58, 82–101 (2011)
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median filter, in Pattern Recognition (CCPR), 2010 Chinese Conference on (IEEE, 2010),
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425–455 (1994)
10. MathWorks. Wavelet Toolbox (2013)
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(2010)
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Genom. 11, 1471–1480 (2010)
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Chapter 6
Nonsubsampled Shearlet and Guided
Filter Based Despeckling Method
for Medical Ultrasound Images
Abstract In this chapter, a novel despeckling method based on nonsubsampled
shearlet transformation and a guided filter is presented. First, a nonsubsampled Lapla-
cian pyramid filter is used to decompose the noisy image thus decomposing the image
into high-frequency and low-frequency subbands. Under the direction of the non-
sampling filter bank, a highfrequency subband multi-directional decomposition is
obtained. Next, based on the threshold function and the correlation of the shearlet
coefficients in the transformation domain, an improved threshold s hrinkage algorithm
is proposed to perform the threshold shrinkage processing on the shearlet coefficients
of the high-frequency subbands. Finally, the low-frequency subbands in the transfor-
mation domain are processed by the guided filter, and a denoised ultrasonic image
is obtained by the inverse transformation of the shearlet. So as to verify the effec-
tiveness of the proposed method, experiments were conducted, and the results were
compared to those of other existing denoising filters. These showed the proposed
method performs more effectively at denoising and delivers clearer image detail.
6.1 Introduction of Shearlet
Wavelet transform (WT) has been used for despeckling; this can hunt the one-
dimensional point singularity effectively, but it cannot show the changes in curves and
straight lines. To compensate for the shortage of WT, ridgelet transform can be used
to capture the line’s singularity in an image, but it cannot effectively capture the curve
singularity.Toovercome this problem, Donoho et al. have used the curvelet transform
(CVT) to represent a two-dimensional singularity of the smooth curve, however CVT
does not support multi-resolution representations of the geometry, nor can it structure
the curve segment directly in the discrete domain. The contourlet transform improves
greatly the effectiveness of image processing, but it shows the lack of shift invari-
ance in the Gibbs phenomenon. The nonsubsampled contourlet transform (NSCT) is
a good solution to the problem of higher dimension singularity, and solves the prob-
lem of shift invariance, but its disadvantages are its limited directionality and large
computations. The shearlet transform [1–7] has the advantages of multi-scale, local-
ization, anisotropy and directionality, but cannot solve the shift invariance problem.
© Springer Nature Singapore Pte Ltd. 2020
J. Zhang and Y. Cheng, Despeckling Methods for Medical Ultrasound Images,
https://doi.org/10.1007/978-981- 15-0516-4_6
123
124 6 Nonsubsampled Shearlet and Guided Filter Based Despeckling …
Easley et al. proposed a nonsubsampled shearlet transform (NSST), which provides
better multi-directional and shift invariance problems through the Laplacian pyramid
and several shearlet filters. In recent years, the despeckling technology of medical
images processing has experienced a breakthrough by using the non-subsampling
shearlet transform. This said, though the high-frequency part of the image can be
handled by NSST, the denoising effectiveness in the low-frequency part of the image
is not satisfactory because the low-pass component can still contain some speckle
noise after the NSST.
A guided filter is adopted in this chapter for the low frequency part of the image.
Not only does the guided filter perform better at speckle reduction, it can also pre-
serve details of the image edge. Thus, a method of combining guided filtering with
NSST is proposed in this paper to address the despeckling problem in ultrasound
images. Shearlet coefficients have a certain correlation with their parent coefficients,
so correlations between the two are considered. To estimate the threshold the three
variable non-Gaussian model was applied and a three-variable shrinkage algorithm
developed to improve despeckling performance. The effectiveness and superiority of
the method proposed is apparent in the experimental and comparison studies shown.
6.1.1 The Basic Principle of Shearlet Transformation
Shearlet system [8, 9] in the situation of 2D is defined as follows:
M
AS
(ϕ) =
ϕ
j,l,k
(x) =
|
det A
|
j/2
ϕ(S
l
A
j
x − k):j, l ∈ z, k ∈ z
2
(6.1)
where ϕ ∈ L
2
(R
2
). A is a multi-scale anisotropic matrix. S is a multi-directional
clipping matrix. A and S are 2×2 reversible matrices.
|
det S
|
= 1 and j, l , k represent
scaling, direction, and shift parameter respectively. A
j
is associated with the scaling
transformation and S
l
is associated with the geometric transformation to keep the
area constant.
Let
N
a,s
=
1 s
01
a 0
0
√
a
=
a
√
as
0
√
a
(6.2)
where (a, s) ∈ R
+
× R, the affine system
M
N
a,s
(ϕ) = M
a,s,t
(ϕ) =
ϕ
a,s,t
(
x
)
= a
−3/4
ϕ(N
−1
a,s
(x − t)): a ∈ R
+
, s ∈ R, t ∈ R
2
(6.3)
is called a continuous shearlet system [10, 11].
6.1 Introduction of Shearlet 125
(a)
(b)
ξ
ξ
Fig. 6.1 a The segmentation of frequency plane induced by shearlet. b The frequency support of
a shearlet
∧
ϕ
j,l,k
Let
A =
40
02
, S =
11
01
(6.4)
for any ζ =
(
ζ
1
,ζ
2
)
∈ R
2
, ζ
1
= 0, let φ
(
x
)
be:
ˆ
φ(ζ) =
ˆ
φ(ζ
1
,ζ
2
) =
ˆ
φ
1
(ζ
1
)
ˆ
φ
2
ζ
2
ζ
1
(6.5)
ˆ
φ
(
ζ
)
denotes the Fourier transformation of φ
(
ξ
)
.
ˆ
φ
1
,
ˆ
φ
2
are continuous wavelets,
ˆ
φ
1
∈ C
∞
(R),
ˆ
φ
2
∈ C
∞
(R). The support of
ˆ
φ
1
is
[
−1/2, −1/16
]
∪
[
1/16, 1/2
]
, and
the support of
ˆ
φ
2
is [−1, 1]. According to the support conditions of
ˆ
φ
1
,
ˆ
φ
2
,itiseasy
to know that the function
ˆ
φ
j,l,k
has support as follows:
(
ζ
1
,ζ
2
)
: ζ
1
∈
−2
2j−1
, −2
2j−4
∪
2
2j−4
, 2
2j−1
,
ζ
1
ζ
2
+ l2
−j
≤ 2
−j
(6.6)
Figure 6.1a illustrates the segmentation of frequency plane which is induced by
the shearlet transform. Figure 6.1b shows the frequency support of a shearlet
ˆ
φ
j,l,k
on a pair of trapezoids of approximate size 2
2j
× 2
j
.
6.1.2 Nonsubsampled-Shearlet Transformation and Its
Inverse Transformation
This section presents nonsubsampled shearlet (NSST) transformation and its inverse
processing. The schematic diagram of nonsubsampled shearlet transformation is
shown in Fig. 6.2. The NSST is not only shift-invariant, but also expansive multi-
directionally and multi-scale.
126 6 Nonsubsampled Shearlet and Guided Filter Based Despeckling …
Fig. 6.2 The schematic diagram of nonsubsampled shearlet transformation
For image f, f will be decomposed into L layers and the details of nonsubsampled
shearlet transformation are as following 5 steps:
(1) The Laplacian pyramid method is used to decompose f
j−1
a
into a low-pass filtered
image f
j
a
and a high-pass filtered image f
j
d
;
(2) The
ˆ
f
j
d
is calculated in the pseudo-polar coordinate network, and through pseudo-
polar discrete Fourier transform (PDFT) obtained the matrix Pf
j
d
;
(3) Band pass filtering is performed on the matrix Pf
j
d
;
(4) Redefine the sampling values of Cartesian coordinate. Then, obtain the shearlet
coefficients by using the inverse two-dimensional FFT, or by the inverse pseudo-
PDFT .
(5) Repeat the above steps until j = L.
The inverse of the nonsubsampled shearlet transformation is exactly the reverse
of the above five steps, the details as follows:
(1) The pseudo-PDFT operation is performed on the shearlet coefficients of j layers;
(2) Adopt the inverse filter to handle the data from the previous step to obtain the
matrix Pf
j
d
;
(3) Perform an inverse PDFT on Pf
j
d
to obtain the image f
j
d
;
(4) Apply the Laplace pyramid reconstruction algorithm to obtain image f
j−1
a
;
(5) Repeat the above four steps until j = 0, and obtain the image f.
6.2 Nonsubsampled Shearlet and Guided Filter Based Despeckle Method 127
6.2 Nonsubsampled Shearlet and Guided Filter Based
Despeckle Method
In this chapter, a novel de-speckle method for medical ultrasound imaging, which
is based on nonsubsampled shearlet transform (NSST) and guided filter is intro-
duced. Firstly, the nonsubsampled Laplacian pyramid filtering is used to decompose
the noisy image, and thus the image is decomposed into high frequency subbands
and low frequency subbands. Under the direction of non-sampling filter bank, we
obtain a high-frequency subband multi-directional decomposition; Secondly, based
on the threshold function and the correlation of the shearlet coefficients in the trans-
form domain, an improved threshold shrinkage algorithm is proposed to perform
the threshold shrinkage processing on the shearlet coefficients of the high frequency
subbands. Finally, the low f requency subbands in the transform domain are processed
by the guided filter, and the de-noised ultrasonic image is obtained by the inverse
transformation of the shearlet. In order to verify the effectiveness of the proposed
method, experiments are conducted, and the results were compared with those of
other existing de-noising filter. Showing that the method we be introduced in this
chapter as a strong de-noising performances and maintain image details.
6.2.1 An Improved Shearlet Threshold Function
The threshold effect of shearlet de-noising method is similar to that of wavelet de-
noising method. In this method, the noise information in shearlet domain is filtered
by threshold processing.
Comparing several threshold methods shows each has certain advantages and
disadvantages [10, 11]. The Level Shrink threshold function solves the problem of
uniform threshold, thereby providing a basis for improving the threshold algorithm.
The Bayes Shrink threshold function is proposed to solve the problem of adaptive
parameter selection on the basis of signal statistics. Given the advantages of the
aforementioned methods, a new threshold algorithm is proposed in Eq. (6.7) better
suited for medical ultrasound imaging.
T = a
j
(k
1
σ
n
2ln(M ) +k
2
σ
2
n
/σ
ω,j
) (6.7)
where j(= 1, 2,...,J ) are the decomposition layers of shearlet transformation. J is
the largest number of decomposed layers. a
j
belongs to the adaptive parameter of
j layers. M is the number of the shearlet coefficients in the corresponding shearlet
domain. By setting a
j
= 1/ ln(j + 1), k
1
= 0.5 k
2
= 0.5inEq.(6.7), which can
weaken the effect of M on the threshold, and meanwhile make the threshold function
more stable, Eq. (6.7) can be simplified as follows:
128 6 Nonsubsampled Shearlet and Guided Filter Based Despeckling …
T =
1
ln(j + 1)
0.5σ
n
2ln(M ) +0.5σ
2
n
/σ
ω,j
(6.8)
In the threshold function (6.8), the standard deviation σ
ω,j
of the noise and the
standard deviation σ
n
of the noise-free image can be obtained as follows:
σ
n
can be obtained from the median absolute value of G
1,k
of the shearlet coef-
ficient in the first layers of the high frequency subbands in the shearlet transform
decomposition, i.e.
∧
σ
n
=
median(
G
1,k
)
0.6745
(6.9)
Since the wavelet transform is a linear transformation, and the shearlet transform
is a high-dimensional expansion of wavelet transform, thus it also belongs to the
linear transformation.
As σ
w
and σ
n
operate independently, we obtain:
σ
2
g,j
= σ
2
ω,j
+ σ
2
n,j
(6.10)
Since the coefficients of the noise-free signal and the coefficients of the speckle
noise obey the zero-mean model distribution, the standard deviation σ
g,j
of the shear-
let coefficients in the j layer can be obtained from the shearlet coefficients of this
layer:
σ
g,j
=
1
M
G
j,k
2
(6.11)
where j, k represent the number of layers and the direction of the layer, respectively.
According to the Laplacian distribution of shearlet coefficient, Eq. (6.10) can be
formulated as Eq. (6.12).
ˆσ
ω,j
=
max( ˆσ
2
g,j
−ˆσ
2
n
, 0) (6.12)
6.2.2 Three-Variable Shrinkage Algorithm
Commonly used shrinkage algorithms are the soft threshold shrinkage and the hard
threshold shrinkage. These algorithms ignore the correlations between coefficients.
When the soft threshold shrink algorithm is applied to the practical application, the
denoised image tends to smooth. The hard t hreshold shrinkage algorithm, however,
greatly affects the image details. Since shearlet coefficients have a certain correlation
with their parent coefficients, this section will consider the correlations between the