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6.2 Nonsubsampled Shearlet and Guided Filter Based Despeckle Method 129
coefficients and apply the three-variable shrinkage algorithm to improve denoising
performance [10–14].
Defining ω
2
as the parent coefficient of the original shearlet coefficient ω
1
. ω
3
is
the direction coefficient and opposite of ω
1
. g
1
, g
2,
g
3
represent the noised shearlet
coefficients corresponding to ω
1
,ω
2
,ω
3
, respectively; n
1
, n
2
, n
3
is the noise coeffi-
cients, i.e.
g
1
= ω
1
+ n
1
g
2
= ω
2
+ n
2
g
3
= ω
3
+ n
3
(6.13)
Equations (6.13) can also be written as:
g = ω + n (6.14)
where g =
(
g
1
, g
2
, g
3
)
, ω =
(
ω
1
,ω
2
,ω
3
)
, n =
(
n
1
, n
2
, n
3
)
.
According to the MAP(Maximum A Posteriori) estimation method, the estimated
value
∧
ω
can be obtained as follows:
∧
ω
(g) = arg max
ω
p
ω|g
(ω
|
g)
= arg max
ω
p
g|ω
(
g
|
ω
)
· p
ω
(ω)
= arg max
ω
p
n
(g − ω)p
ω
(ω)
(6.15)
As speckle noise in the ultrasonic image is similar to the Gaussian distribution,
the probability density function p
n
(n) of speckle noise in the ultrasonic image is three
variable Gaussian densities. Thus, producing the following Eq. (6.16):
p
n
(n) =
1
(2πσ
2
n
)
3
2
· exp
−
n
2
1
+ n
2
2
+ n
2
3
2σ
2
n
(6.16)
Since the generalized sphere of high exponential density function at d = 1,α =
1
√
2σ
ω
,β =
√
2 is equivalent to the Laplace density function, the generalized sphere
of high exponential density function with d = 3 is selected as a function p
ω
(ω).
p
ω
(ω) = GSCE(ω, σ
ω
, 3)
= α · exp
−
β
σ
ω
ω
2
1
+ ω
2
2
+ ω
2
3
(6.17)
Let
f (ω) = log(p
ω
(ω)) (6.18)
130 6 Nonsubsampled Shearlet and Guided Filter Based Despeckling …
BasedonEqs.(6.15)–(6.18):
∧
ω
(g) = arg max
ω
log(p
n
(g − ω)) + log
(
p
ω
(ω)
)
= argmax
ω
[−
3
2
log
2πσ
2
n
−
(
g − ω
)
2
2σ
2
n
+ f (ω)]
= argmax
ω
−
3
2
log(2πσ
2
n
) −
(
g
1
− ω
1
)
2
2σ
2
n
−
(
g
2
− ω
2
)
2
2σ
2
n
−
(
g
3
− ω
3
)
2
2σ
2
n
+ f
(
ω
)
(6.19)
where f (ω) is a strictly convex function and differentiable function. Solving the
problem (6.19) is thus equivalent to solving Eqs. (6.20)–(6.22):
g
1
−
∧
ω
1
σ
2
n
+ f
1
(
∧
ω
) = 0 (6.20)
g
2
−
∧
ω
2
σ
2
n
+ f
2
(
∧
ω
) = 0 (6.21)
g
3
−
∧
ω
3
σ
2
n
+ f
3
(
∧
ω
) = 0 (6.22)
f
1
(ω), f
2
(ω), f
3
(ω) are the first derivatives of f (ω) with respect to ω
1
,ω
2
,ω
3
,
respectively. Combined with the formula (6.17), (6.18), we can obtain:
f (ω) = log α −
β
σ
ω
ω
2
1
+ ω
2
2
+ ω
2
3
(6.23)
And then
f
1
(ω) =−
βω
1
σ
ω
ω
2
1
+ ω
2
2
+ ω
2
3
(6.24)
f
2
(ω) =−
βω
2
σ
ω
ω
2
1
+ ω
2
2
+ ω
2
3
(6.25)
f
3
(ω) =−
βω
3
σ
ω
ω
2
1
+ ω
2
2
+ ω
2
3
(6.26)
Substituting (6.20), (6.21), (6.22)into(6.17), (6.18), (6.19), we obtain:
6.2 Nonsubsampled Shearlet and Guided Filter Based Despeckle Method 131
∧
ω
1
·
⎛
⎝
1 +
βσ
2
n
σ
ω
∧
ω
2
1
+
∧
ω
2
2
+
∧
ω
2
3
⎞
⎠
= g
1
(6.27)
∧
ω
2
·
⎛
⎝
1 +
βσ
2
n
σ
ω
∧
ω
2
1
+
∧
ω
2
2
+
∧
ω
2
3
⎞
⎠
= g
2
(6.28)
∧
ω
3
·
⎛
⎝
1 +
βσ
2
n
σ
ω
∧
ω
2
1
+
∧
ω
2
2
+
∧
ω
2
3
⎞
⎠
= g
3
(6.29)
Combined with the formula (6.24–6.26), we obtain:
∧
ω
2
1
+
∧
ω
2
2
+
∧
ω
2
3
+
βσ
2
n
σ
ω
2
= g
2
1
+ g
2
2
+ g
2
3
⇒
∧
ω
2
1
+
∧
ω
2
2
+
∧
ω
2
3
=
g
2
1
+ g
2
2
+ g
2
3
−
βσ
2
n
σ
ω
+
(6.30)
Substituting (6.27)into(6.24), we obtain:
∧
ω
1
=
ω
2
1
+ ω
2
2
+ ω
2
3
−
βσ
2
n
σ
ω
+
g
2
1
+ g
2
2
+ g
2
3
· g
1
(6.31)
where (g)
+
=
0, g < 0
g, g ≥ 0
, and
∧
ω
1
is obtained by (6.31). It is assumed shearlet coef-
ficients of noisy signals in ultrasound images obey Gaussian distribution, and shearlet
coefficients of real noiseless signals obey Rayleigh distribution. In this circumstance,
∧
ω
1
is the MAP estimate of the shearlet coefficient ω
1
.
∧
ω
1
=
⎧
⎨
⎩
0, g
1
≤ T
j
√
ω
2
1
+ω
2
2
+ω
2
3
−
βσ
2
n
σ
ω
+
√
g
2
1
+g
2
2
+g
2
3
· g
1
, g
1
> T
j
(6.32)
132 6 Nonsubsampled Shearlet and Guided Filter Based Despeckling …
6.2.3 The Overall Steps of Nonsubsampled-Shearlet
and Guided Filter
The schematic diagram of the proposed nonsubsampled shearlet and guided filter
based despeckling method is showed in Fig. 6.3.
(1) The envelope signal of the ultrasonic imaging system is logarithmically trans-
formed to obtain the noisy ultrasound image. And thus the multiplicative noisy
model is changed to an additive noise model of speckle noise and noise-free
signal; this is the common model of medical ultrasound image. The noise-free
signal will obey the generalized Gaussian distribution, whereas the speckle noise
will obey Laplace distribution.
(2) The obtained image is subject to discrete invariant shearlet transform, which is
the multi-scale and multi-directional decomposition. Nonsubsampled Laplacian
pyramid filtering is used to perform multi-scale decomposition. After the k-
level conversion of the image, k high-frequency subbands and 1 low-frequency
subband are obtained. The obtained subband image is subject to shearlet filtering
group multi-directional decomposition.
Medical images contain
speckle noise
Nonsubsampled Laplacian
decomposition
Low frequency image
High frequency image
Guided fi lter
algorithm processing
Nonsubsampled
Laplacian decom position
Shearlet direction filter
decomposition
Low frequency part
High frequency part
Improved Threshold
and Triple Variable
Shrinkage Algorithm
The inverse of the
Nonsubsampled-shearlet
transfor mation
De-speckling
medical images
Fig. 6.3 Schematic diagram of the nonsubsampled shearlet and guided filter based despeckling
method
6.2 Nonsubsampled Shearlet and Guided Filter Based Despeckle Method 133
(3) By using the improved threshold algorithm and the three-variable shrinkage
algorithm (Eqs. 6.7 and 6.32), the shearlet coefficients of the high frequency
part of each scale are subject to threshold shrinkage.
(4) The shearlet coefficients of the low frequency part are filtered by guided filter.
(5) With nonsubsampled shearlet inverse transformation processing denoised med-
ical ultrasound images can be obtained.
6.3 Experiments and Analysis of Results
In order to objectively evaluate the denoising performances of the developed method,
in this section, the peak signal-to-noise ratio (PSNR), the structural similarity (SSIM),
FoM (Pratt ‘s Figure of Merit) and the running time are chosen as evaluation criteria
of the experiments. The meanings and their definitions of PSNR, SSIM and FoM are
already introduced in Sect. 2.3.3 of Chap. 2.
6.3.1 Experimental Verification
This section validates the algorithm from the following three aspects:
(1) Experimental studies of Wavelet transform, Shearlet transform and NSST based
methods for 4 different clinical ultrasound images are conducted.
(2) Comparison and experimental studies of NSST combined with different low-
frequency filters are conducted.
(3) Comparison and experimental studies of Nonsubsampled Shearlet + Guided
with other existing Filters are conducted.
In the experimental comparison studies, all the algorithms are programmed in
MATLAB 2011a (Win32), and the computer configuration are: Windows7 sp1 sys-
tem, Core i5-2400, CPU3.10 GHz, 320G hard drive, 2G memory.
The clinical ultrasound images of the kidney, arteries, liver and mammary gland
are used for comparisons. The clinical images were obtained from Vivid7 ultrasound
imaging equipment of General Electric Medical System Co., Ltd. The clinical images
were as follows (Fig. 6.4):
(1) Experimental studies for 4 different clinical ultrasound images
Experimental studies of Wavelet transform, Shearlet transform and NSST based
methods for 4 different clinical ultrasound images are conducted. Comparisons and
experimental results are presented in Figs. 6.5, 6.6, 6.7 and 6.8 and Table 6.1.
We can see from the above visual results that the shearlet transform based de-
noising has a better effect than that of the wavelet transform. We present further the
experimental results of three algorithms in Table 6.1.
134 6 Nonsubsampled Shearlet and Guided Filter Based Despeckling …
(a) female breast tissue (b) arterial tissue
(c) kidney tissue (d) liver tissue
Fig. 6.4 Clinical ultrasound images
Noisy images Wavelet transform Shearlet transform NSST
Fig. 6.5 Results of experimental studies for ultrasound Female breast tissue image with σ = 0.05
From the above table, it can be seen that NSST based de-noising has the best
performance effect. It is shown that the shearlet transform algorithm based on trans-
lation invariance is superior to the ordinary shearlet transform, and the performance
of de-noising is better.
(2) Comparison experimental studies of NSST combined with different low-
frequency filters
6.3 Experiments and Analysis of Results 135
Noisy images Wavelet transform Shearlet transform NSST
Fig. 6.6 Results of experimental studies for ultrasound Liver tissue image with σ = 0.05
Noisy images Wavelet transform Shearlet transform NSST
Fig. 6.7 Results of experimental studies for ultrasound Kidney tissue image with σ = 0.05
Noisy images Wavelet transform Shearlet transform NSST
Fig. 6.8 Results of experimental studies for ultrasound Arterial tissue image with σ = 0.05
According to the principle of image decomposition, the noise presents not only in
the high frequency, also exists in low frequency. So in this section, experiments of
NSST combined with low-frequency filters for de-noising are conducted (Figs. 6.9,
6.10, 6.11 and 6.12).
We also use the denoising index to quantify and show the de-noising effect. For
NSST +bilateral, NSST +triangular, NSST + guided, the experimental comparison
results are presented in the Table 6.2.
Analysis of the above results, it can be seen that while NSST are employed to pro-
cess high-frequency in all cases, different filters are adopted in the low-frequency pro-
cessing de-noising. And the results show that for guide filtering in the low-frequency,
better and satisfied performances are achieved.
136 6 Nonsubsampled Shearlet and Guided Filter Based Despeckling …
Table 6.1 Experimental results of three algorithms. NSST, Shearlet and Wavelet. PSNR/dB
Human tissue
structure
Algorithms σ = 0.01 σ = 0.02 σ = 0.03 σ = 0.04 σ = 0.05
Liver NSST 31.88 31.55 31.38 30.80 28.97
Shearlet 31.30 31.02 30.99 30.19 29.50
Wav e l et 30.32 30.12 29.92 29.45 29.11
Kidney NSST 30.90 30.71 29.38 28.28 28.84
Shearlet 29.97 29.64 28.57 27.68 29.31
Wav e l et 29.14 28.12 27.60 26.18 29.02
Breast NSST 31.18 30.97 29.63 29.38 28.34
Shearlet 30.55 30.19 29.12 28.84 29.28
Wav e l et 30.15 29.82 28.43 27.13 28.75
Artery NSST 31.80 30.81 29.88 28.23 27.71
Shearlet 30.66 30.13 29.51 28.02 28.80
Wav e l et 30.15 29.22 28.56 27.13 27.91
Noisy image NSST + bilateral filter NSST + triangular filter NSST + guided filter
Fig. 6.9 Results of experimental studies for ultrasound Female breast tissue image with σ = 0.05
Noisy image NSST + bilateral filter NSST + triangular filter NSST + guided filter
Fig. 6.10 Results of experimental studies for ultrasound Liver tissue image with σ = 0.05
6.3 Experiments and Analysis of Results 137
Noisy image NSST + bilateral filter NSST + triangular filter NSST + guided filter
Fig. 6.11 Results of experimental studies for ultrasound Kidney tissue image with σ = 0.05
Noisy image NSST + bilateral filter NSST + triangular filter NSST + guided filter
Fig. 6.12 Results of experimental studies for ultrasound Arterial tissue image with σ = 0.05
(3) Comparative experimental studies of Nonsubsampled Shearlet + Guided
with other existing Filters.
In order to further verify the proposed algorithm in this chapter, SRBF, DTD, Andria,
OBNLM, wavelet + guided, NSST + bilateral, NSST + triangular are chosen to
conduct the comparison studies. Algorithm parameters are shown in Table 6.3.
For ultrasound Breast tissue image, experimental de-noising results are shown in
Fig. 6.13.
We also use the denoising index to quantify and show the de-noising effect. The
experimental comparison results are presented in the Table 6.4 (Fig. 6.14).
It can be recognized that the SRBF filter (Fig. 6.13b) has a better speckle reduc-
tion performance but the edges appear somewhat blurred. In Fig. 6.13c, the DPAD
filter shows a strong speckle reduction but introduces small artifacts. In terms of
the multi-scale filters, the Andria filter (Fig. 6.13d) only reduces a little of speckle.
The OBNLM filter (Fig. 6.13e) has excellent speckle suppression performance, but
it is not desirable considering the time factor. The comparison between the proposed
method and Wavelet + guided (Fig. 6.13f) shown that the anisotropic properties of
the shearlet are superior to the isotropic properties of the wavelet. From the compar-
ison of the algorithm of this paper with NSST + bilateral (Fig. 6.13g) and NSST +
triangular (Fig. 6.13i), it can be comprehended that guided filter has better perfor-
mance in low-frequency denoising. Only the proposed method has the strong speckle
reduction and keeps retaining the edge information at the same time.
138 6 Nonsubsampled Shearlet and Guided Filter Based Despeckling …
Table 6.2 Experimental comparison results for different low frequency filter (PSNR, dB)
Human tissue
structure
Algorithms σ = 0.01 σ = 0.02 σ = 0.03 σ = 0.04 σ = 0.05
Breast NSST +
bilateral
31.98 31.65 31.48 31.10 30.20
NSST +
triangular
32.30 31.92 31.79 31.19 30.44
NSST +
guided
32.42 32.12 31.92 31.45 30.76
Liver NSST +
bilateral
31.10 30.91 29.68 28.58 27.85
NSST +
triangular
32.17 31.04 30.17 29.68 28.14
NSST +
guided
32.74 32.12 31.16 29.98 28.95
Kidney NSST +
bilateral
31.28 30.98 29.83 29.48 28.65
NSST +
triangular
31.55 31.19 30.12 29.84 28.80
NSST +
guided
32.15 31.82 31.43 30.13 29.54
Arterial NSST +
bilateral
31.90 30.91 30.48 28.73 28.15
NSST +
triangular
32.16 31.13 30.71 29.02 28.98
NSST +
guided
32.85 32.22 31.56 30.13 29.76
Table 6.3 Parameter settings
Algorithm Category Experimental parameters
SRBF [15] Local adaptive algorithm Window size 55
DPAD [16] Anisotropic diffusion de-noising
algorithm
Window size 5 × 5
Andria [17] Multi-scale de-noising algorithm Window size 5 × 5
OBNLM [18] Nonlocal mean algorithm Window size 7 × 7, Smoothing
parameters 0.6
Wav e l et + guided Mixed de-noising algorithm Window size 5 × 5
NSST + bilateral Mixed de-noising algorithm Window size 5 × 5
NSST + triangular Mixed de-noising algorithm Window size 5 × 5
NSST + guided Mixed de-noising algorithm Window size 5 × 5