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3.3 Wavelet and Fast Bilateral Filter Based Despeckling Method 59
Fig. 3.8 Gaussian distribution
where σ is standard deviation, and μ is the mean.
3.3.2 The Bilateral Filter and the Fast Bilateral Filter
C. Tomasi and R. Manduchi in 1998 proposed a de-noising algorithm of bilateral
filtering which is a kind of nonlinear de-noising algorithms [10, 11]. The bilateral
filter smooths image while preserving the edges. It combines gray levels or colors
based on both their geometric closeness and their photometric similarity, and prefers
near values to distant values in both domain and range [12].
(1) Structure of Image
Digital images can be divided into flat and edge regions from the whole structure
representing the low frequency part and high frequency part of the image in the
frequency domain respectively. The low-frequency part of the image refers to the
signal component in which the gray value of the pixel changes slowly in a certain
region. In general, the background of the image is the low-frequency part. The high
frequency part of the image refers to the signal component in which the gray value
of the pixel changes greatly in a certain region, and the high frequency reflects the
edge feature and detail information of the image. The edge of the image has two
characteristics: direction and amplitude. The change of gray amplitude along edge
tangential direction is relativelygentle, while in vertical edge direction is more severe.
But in the process of improving the SNR of image by traditional filtering, the high
60 3 Wavelet and Fast Bilateral Filter Based Despeckling …
frequency component of the image is inevitably removed, resulting in the edge blur.
Therefore, it is very meaningful to preserve image details and useful high frequency
information in the process of noise removal.
(2) Gaussian Filter
Gaussian filter is essentially a signal filter, which is used to smooth the signal pro-
cessing. Its formula is as follows:
G(x, y) =
1
√
2πσ
e
x
2
+y
2
2σ
2
(3.17)
where σ is the scale of Gaussian filter. The greater the value of σ , the more obvious
the smoothing effect, and the better the denoising effect, but at the same time, the
greater the loss of high-frequency edge details [9, 13, 6, 12] (Fig. 3.9).
When the image is filtered by Gaussian filter, the data in the current window is
convoluted with Gaussian kernel to calculate the pixel value of the current position.
(3) The Bilateral Filter and the Fast Bilateral Filter
Bilateral filtering combines range filtering with domain filtering, and it can be applied
to color images just as easily as they are applied to black-and-white ones. In the
smooth area of the image, the pixel value varies very little and the corresponding
pixel range weight is close to 1, where the spatial domain weight plays a major role,
which is equivalent to Gaussian ambiguity. In the edge region of the image, the pixel
values vary greatly, and the weight of the pixel range becomes larger, thus preserving
the edge information.
Fig. 3.9 Gaussian kernel function
3.3 Wavelet and Fast Bilateral Filter Based Despeckling Method 61
The main idea of wavelet denoising method is to retain the wavelet coefficients
of the low-pass component (LL), while wavelet coefficients of high-pass component
(LH, HL, HH) are shrunk with the wavelet threshold function. However, only using
the wavelet transformation based de-noising algorithm to suppress speckle noise in
medical ultrasonic images does not produce optimal results. In the experiment, it
can be found that the wavelet coefficients in the low-pass component still contain
some speckle noise. In order to effectively eliminate speckle noise in the low-pass
component, the bilateral filter is selected to filter the speckle noise in the low-pass
parts. The bilateral filter is divided into traditional bilateral filter and fast bilateral
filter. We will introduce both of them in follow.
The output of bilateral filter at each pixel is a weighted average of its neighbors.
The weight is assigned to each neighbor, which decreases with both the distance in
the image plane (the spatial domain S) and the distance on the intensity axis (the
range domain R). When a Gaussian G
σ
is used as a decreasing function, the structure
of traditional bilateral filter is given as following
I
b
p
=
1
W
b
p
q∈S
G
σ
s
(
p − q
)G
σ
r
(
I
p
− I
q
)I
q
(3.18)
where I is the input image, and I
b
is the result of the bilateral filter. W
b
p
=
q∈S
G
σ
s
(
p − q
)G
σ
r
(
I
p
− I
q
) normalizes the sum of the weights. The param-
eter σ
s
defines the size of the spatial neighborhood to filter a pixel, and σ
r
controls
the weight with its adjacent pixels according to the intensity difference.
The bilateral filter has been proven effective to filter speckle noise in the low-pass
component, but it has a high computational cost, and is difficult to be applied in real-
time systems. Especially for the medical ultrasound images, the image resolutions
are usually very large, which limit the application of the bilateral filter to a great
extent, and thus the fast bilateral filter is proposed.
The fast bilateral filter is also known as higher-dimensional bilateral filter. In the
fast bilateral filter, the gray levels of pixels are combined to form a 3D space from a
2-D image domain. Thus, the original complex nonlinear component has been shifted
to a linear convolution of a three-dimensional Gaussian kernel function and a three-
dimensional image function. Using this higher dimensional space, the convolution
computation can be downsampled without significant influences on the resulting
accuracy. Fast Fourier transformation is applied to calculate linear convolution. This
approximation technique enables a speed-up of several orders of magnitude and
controls the induced error. The structure of fast bilateral filter is as follows:
BI(x, y) =
IY (x, y)
EY (x, y)
=
int erp(G ⊗ IX,
x
s
s
,
y
s
s
,
I(x,y)
s
r
)
int erp(G ⊗ EX,
x
s
s
,
y
s
s
,
I(x,y)
s
r
)
(3.19)
62 3 Wavelet and Fast Bilateral Filter Based Despeckling …
Fig. 3.10 Processing diagram of bilateral filtering
where IX (x, y, z) =
z, z = I(x, y)
0, z = I (x, y)
represents the three-dimensional image matrix,
and EX (x, y, z) =
1, z = I(x, y)
0, z = I (x, y)
represents the three-dimensional weight matrix.
int erp is the interpolation function. G is a Gaussian kernel function after linearization.
s
s
and s
r
denote the sample rate of spatial domain and range domain respectively.
In 3D space, sampling, convolution and interpolation realize the acceleration for
several orders of magnitude. We would like to show more detail of bilateral filter in
Chap. 5 (Fig. 3.10).
3.3.3 A Despeckling Method Based on Wavelet and Fast
Bilateral Filter
On the basis of the traditional wavelet denoising method, an improved despeckling
method based on wavelet shrinkage algorithm [13] and fast bilateral filter is proposed.
(1) According to the statistical properties of medical ultrasound images in the
wavelet domain, an improved threshold function based on the universal thresh-
old function is proposed.
3.3 Wavelet and Fast Bilateral Filter Based Despeckling Method 63
(2) According to the statistical models of noise and signal in the wavelet domain,
Bayesian maximum a posteriori estimation is applied to design a shrinkage
algorithm.
(3) The speckle noise in the low-pass approximation component is filtered by the
fast bilateral filter to ensure that this denoising algorithm has a stronger ability
to suppress speckle noise in the low-pass approximation component.
(1) Bayesian Maximum a Posteriori Estimation
In Bayesian statistical theory, the maximum posterior estimator (MAP) is a com-
monly used method, that is, under the condition of the observed data y, the variable
w maximizes the posterior probability density P
w|y
(w|y):
ˆw(y) = arg max
w
P
w|y
(w|y)
(3.20)
Bayesian maximum a posteriori estimation is to obtain the maximum of likelihood
function on the premise of adding prior information. It requires not only the maximum
of likelihood function, but also in the situation of maximum of prior probability.
Formula (3.16) and (3.15) are the prior information of wavelet coefficient distribution.
So what is likelihood function? The likelihood function represents the possibility
that the parameter w
1
(compare to the other parameter w
2
) is true under a given data
sample y.
(2) An Improved Wavelet Threshold Function
As can be seen from experiments, most of the wavelet coefficients corresponding to
noise signals in the noisy image after wavelet transform are small. The denoising
method in wavelet domain is to filter the wavelet coefficients below the threshold
value, so that the noise in the image can be filtered.
Donoho and Johnstone designed a universal wavelet threshold function, i.e. T =
σ
N
√
2logM , where M is the number of the wavelet coefficients in the corresponding
wavelet domain. However, when M becomes large, the big threshold could smooth
out some useful information, and thus this threshold function maybe ineffective on the
noise removal in medical ultrasonic images. The following improvement is designed
in order to obtain the desirable effect:
T
j
= a
j
σ
N
2logM (3.21)
where j(= 1, 2, ···, J ) are the decomposition layers of wavelet transformation, J
denotes the largest decomposition layers. In this chapter, a
j
represents the adaptive
parameter of j layer, and is determined experimentally. After wavelet decomposition,
the wavelet coefficients in different decomposition layers have different distribution,
and thus the selection of a
j
should be based on the j, and a
j
is selected as
1
ln(j+1)
.
But this choice of a
j
is not the optimal one, and if a
j
is selected appropriately, the
proposed method will reflect more advantageously.
64 3 Wavelet and Fast Bilateral Filter Based Despeckling …
(3) An Improved Wavelet Shrinkage Algorithm
The wavelet coefficients of noise-free signal G
j
l,k
and speckle noise N
j
l,k
obey the
generalized Laplasse distribution and Gaussian distribution respectively. In order to
simplify the calculation, v is selected as 1, so Eq. (3.15) can be simplified as follows
p
G
(g) =
1
2s
exp(−
|
g
|
s
), s > 0 (3.22)
Bayesian maximum a posteriori estimation is used to get the estimation of signal
in wavelet domain. In the process of calculating posterior probability, the following
Bayesian equation is used
p
G|F
(g|f ) =
1
p
F
(f )
p
F|G
(f |g) ·p
G
(g) =
1
p
F
(f )
p
N
(f − g) · p
G
(g) (3.23)
With Eq. (3.16) and Eqs. (3.22) and (3.23) can be restated as
p
G|F
(g|f ) =
1
p
F
(f )
·
1
2
√
2πsσ
N
× exp
2σ
2
N
|g|−s(f − g)
2
2sσ
2
N
(3.24)
In order to get the maximum posteriori probability, the first-order derivative equal
to zero of ln(p
G
|
F
(g
|
f )) with respect to g leads to Eq. (3.25)
ˆg = sign(f ) · max(|f |−
σ
2
N
s
, 0) (3.25)
where ˆg is the estimation of g, f is assumed in phase with noise-free signal g.
Finally,according to the above equation, a new wavelet shrinkage algorithm based
on the improved threshold function and shrinkage algorithm is proposed:
ˆg =
0, f ≤ T
j
sign(f ) · max
|f |−
σ
2
N
s
, 0
, f = T
j
(3.26)
(4) Parameter Estimation
The standard deviation of speckle noise σ
n
is obtained from the wavelet coefficients
of highest frequency in wavelet domain (F
HH
1
i,k
)
ˆσ
N
=
median(
F
HH
1
i,k
)
0.6745
(3.27)
3.3 Wavelet and Fast Bilateral Filter Based Despeckling Method 65
where ˆσ
N
is the estimation of σ
N
, this equation had been proven by Donoho insoft-
threshold. With Eq. (3.14), it can be obtained as follows:
σ
2
F,j
= σ
2
G,j
+ σ
2
N
(3.28)
Because G
j
l,k
and N
j
l,k
were modeled as zero mean models (Eqs. 3.15 and 3.16),
the standard deviation of wavelet coefficients in j layer σ
F,j
can be calculated by the
standard deviation of wavelet coefficients F
j
l,k
ˆσ
F,j
=
1
N
m,n
l,k =1
(F
j
l,k
− mean)
2
=
1
N
m,n
l,k =1
(F
j
l,k
)
2
(3.29)
where N = m×n is the total number of wavelet coefficients in corresponding wavelet
domain, m and n are the rows and columns of wavelet coefficients in corresponding
wavelet domain respectively.
BasedonEqs.(3.27)–(3.29)
ˆσ
2
G,j
= max(ˆσ
2
F,j
−ˆσ
2
N
, 0) (3.30)
In this chapter, scale parameter s in Eq. (3.25) can be estimated as the following
equation:
s =[0.5(σ
2
F,j
− σ
2
N
)]
0.5
=
ˆσ
G,j
√
2
(3.31)
(5) Steps for Despeckling Method Based on Wavelet and Fast Bilateral Filter
• The ultrasonic envelope signal received by ultrasonic imaging system is com-
pressed with logarithmic transformation, and thus the multiplicative noise model
is converted to an additive noise model. If the image is the common medical ultra-
sonic image which has been processed with the logarithmic transformation, this
step can be omitted.
• The 2-D discrete wavelet transformation (DWT) is applied for the log-transformed
image, and four frequency domains (LL
1
,LH
1
,HL
1
and HH
1
) are obtained. To
proceed 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.
• According to Eq. (3.22), the fast bilateral filter is used to filter the low-pass com-
ponent of the last layer (LL
J
).
• Based on Eq. (3.26), the new threshold shrinkage algorithm is used to process the
wavelet coefficients of high-pass components 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 obtained by Eqs. (3.28), (3.29) and (3.30) respectively. The threshold of
each layer is obtained from Eq. (3.24), and parameter s is determined by Eq. (3.31).
66 3 Wavelet and Fast Bilateral Filter Based Despeckling …
Fig. 3.11 The schematic
diagram of the despeckling
method based on wavelet and
fast bilateral filter
• The inverse wavelet transformation is used to obtain the de-noised medical ultra-
sonic images.
The schematic diagram of the despeckling method based on wavelet and fast
bilateral filter is as follows (Fig. 3.11).
3.4 Experimental Studies of the Despeckling Method
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)
[14], FoM (Pratt ‘s Figure of Merit) [15] 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 the Sect. 2.3.3 of the Chap. 2.
The images in Fig. 3.12 are used for experiments. Figure 3.12a is designed for
simulation (350 ×350 pixels), and Fig. 3.12b is a clinical medical ultrasound image
with lesions (a breast ultrasound image of women, 340 × 350 pixels).
In this section, the experiments are divided into two branches (Fig. 3.13). One is
speckle noise simulation experiment, and the other one is the clinical medical ultra-
sound image experiment. The experimental schematic diagram is given in Fig. 3.13.
3.4.1 Experiments for Simulated Ultrasound Images
In order to quantitatively estimate the de-noising effect of the method proposed in this
section, two speckle noise simulation experiments are conducted [16]. The low-pass
filter is used for the Gaussian random field of complex domain, and its amplitude is
3.4 Experimental Studies of the Despeckling Method 67
(a) Simulated image (b) woman’s breast ultrasound
image
Fig. 3.12 Images used for experiments
Fig. 3.13 Experimental schematic diagram
simulated as speckle noise. The simulated image is shown in Fig. 3.14a. The param-
eters of the Gaussian random field are: the mean μ = 0, and the rational quadratic
covariance function r
θ
(l) = (1+
1
θ
2
1
)
−θ
2
, where two non-negative parameters selected
as θ
1
= 1,θ
2
= 1.2 respectively.
68 3 Wavelet and Fast Bilateral Filter Based Despeckling …
(a) Noisy image (b) Traditional bilateral
filter
(c) Fast bilateral filter
Fig. 3.14 Denoised results of simulation experiment 1
Table 3.1 Performance comparison of simulation experiment 1
PSNR SSIM FoM Time (s)
Noisy image 24.7141 0.4593 0.1763 –
Traditional bilateral filter 27.8670 0.6823 0.2432 7.7813
Fast bilateral filter 27.0225 0.8497 0.2367 1.1079
3.4.2 Simulation Experiment 1
Since our method achieves acceleration through approximation, we have to measure
the numerical accuracy by comparing the outputs of our technique with traditional
bilateral filter. Figure 3.14a is selected as noise-free image, the result is shown in
Fig. 3.14 and Table 3.1. Where the Gaussian parameters (space and rang) of two
filters are chosen as σ
s
= 2.5, σ
r
= 0.045 respectively.
3.4.3 Simulation Experiment 2
In order to evaluatethe accuracy and performance of the proposed de-noising method,
in this experiment, the results obtained with the proposed de-noising method are
compared with those of other six kinds of de-noising methods (SRBF, DPAD, Andria,
OBNLM, SAR-BM3-D and the Wavelet-Soft Shrinkage). The parameters of six
filters in this experiment are listed in Table 3.2. To get better filtered result, these
parameters are proposed by both the cited papers and medical experts. Figure 3.14a
is chosen as noise-free image, and the simulation experimental results are shown in
Fig. 3.15 and Table 3.3. Where all the algorithms ran at Matlab R2010a.