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3.1 Introduction of Wavelet 49
Fig. 3.2 The non-stationary signal
Fig. 3.3 Window the non-stationary signal
There is another problem when we use mathematical transform to signal. It always
cost too much energy which is unnecessary and this situation is called high entropy.
Therefore, people started looking for a mathematical transformation method with
low entropy and a new method called sparse representation was proposed.
The concept of wavelet transform was first introduced by J. Morlet in 1974,
an oil signal processing engineer in France. At that time, mathematicians failed to
recognize it. In 1986, the famous mathematician Y. Meyer accidentally constructed a
real wavelet basis, and established a unified method of constructing wavelet base with
50 3 Wavelet and Fast Bilateral Filter Based Despeckling …
Fig. 3.4 Time-frequency diagram
multiscale analysis in cooperation with S. Mallat. From then on, wavelet analysis
began to flourish. Wavelet function is derived from Multi-resolution analysis. The
basic idea of wavelet function is to express the function f(t) as a series of successive
approximation expressions.
All wavelets are obtained by scaling and displacement of basic wavelets. The
basic wavelet is a real valued function with special properties. It is attenuated by
oscillations and usually decays very quickly.
Wavelet analysis is widely applied in many fields, including signal analysis, image
processing, quantum mechanics and theoretical physics.
In order to analyze non-stationary signals, and implementation of sparse repre-
sentation of signals, the starting point of wavelet transform is different from FT and
STFT. Wavelet directly transforms the infinite trigonometric function base of the
Fourier transform into a finite wavelet group whose mean value is 0. It is a special
waveform that exists in a very small area (Fig. 3.5).
The wavelet function is defined as follows:
Fig. 3.5 The basis function of Wavelet transformation
3.1 Introduction of Wavelet 51
WT
(
a, τ
)
=
1
√
a
∞
−∞
f
(
t
)
∗ ψ
t − τ
a
dt (3.1)
In formula (3.1), ψ
t−τ
a
is called mother wavelet. Wavelet transform has two
variables, scale (a) and translation (τ). The variable (a) in the wavelet function con-
trols scaling of wavelet function corresponding to the frequency and is inversely
proportional to frequency. The variable (τ) in the wavelet function controls shift of
wavelet function which corresponds to time domain. Therefore, wavelet transform
can obtain not only the frequency information, but also locate the time ( Fig. 3.6).
All wavelets are obtained by scaling and shift of basic wavelets. The basic wavelet
is a real valued function with special properties. It is attenuated by oscillations and
usually decays very quickly. It is defined as follows:
Definition 1 Suppose that ψ
(
t
)
is an integrable function and satisfies the following
conditions:
∞
−∞
ψ
(
t
)
dt = 0(3.2)
And the Fourier function of ψ
(
t
)
satisfies following conditions:
C
ψ
=
∞
−∞
˜
ψ
(
s
)
2
s
ds < ∞ (3.3)
Fig. 3.6 Wavelet transform of the signal in Fig. 3.2
52 3 Wavelet and Fast Bilateral Filter Based Despeckling …
The ψ
(
t
)
is called the basic wavelet, or the mother wavelet function. Equation (3.3)
is the “allowable condition” of wavelet function. Basic wavelets also have good
attenuation properties in frequency domain. Some basic wavelets are actually zero
outside some interval. This is one of the fastest attenuation wavelets. In the time
domain, it has the characteristics of compact set or approximate compact set, and the
non-zero value domain of the function has a limited range.
The wavelet basis function is generated by the basic wavelet through scaling factor
and shift factor. It is defined as follows:
Definition 2 If the expansion factor is defined as a, and the shift factor is b, the basic
wavelet ψ
(
t
)
is shifted and scaled, and the function of translation is ψ
a,τ
(
t
)
:
ψ
a,τ
(
t
)
=
1
√
a
ψ
t − τ
a
, a > 0;a, τ ∈ R(3.4)
It is called the wavelet basis function.
3.1.2 Continuous Wavelet Transform
Definition 3 If the f unction f(x) in the space of L
2
(
R
)
is expanded under the base
function of the wavelet, the expression is as follows:
CWT
(
a, τ
)
=
1
√
a
∞
−∞
f
(
t
)
∗
ψ
t − τ
a
dt (3.5)
It is called continuous wavelet transform of function f(x), where the parameters
a and τ take the values of continuous variation, and the corresponding wavelet basis
function ψ
a,τ
(
t
)
is also called the continuous wavelet basis function. It has an impor-
tant property that the window area does not change with the parameters a and τ.This
property of wavelet is an important basis for time-frequency analysis. The size of
the window is the product of the time frequency window width t and ω of the
wavelet mother function. The phase plane of wavelet basis function can be obtained
by drawing the time-frequency window on the same graph with different values of a
and τ, as shown in Fig. 3.7.
If the wavelet satisfying formula (3.3), it is proved that inverse transform exists
in continuous wavelet transform:
f
(
t
)
=
1
C
ψ
+∞
0
1
a
2
+∞
−∞
CWT
(
a, τ
)
1
√
a
ψ
t − τ
a
dt (3.6)
Continuous wavelet transform has the following important properties:
3.1 Introduction of Wavelet 53
Fig. 3.7 Characteristics of
time-frequency analysis unit
of wavelet basis function
Linearity: The continuous wavelet transform of a function is equal to the sum of the
wavelet transform of the function component.
Time-shift theorem: If the wavelet transform of f(t) is CWT
(
a, τ
)
,thewavelet
transform of f(t − t
0
)isCWT
(
a, τ − t
0
)
.
Time scale theorem: If the wavelet transform of f
(
t
)
is CWT
(
a, τ
)
,thewavelet
transform of f
(
bt
)
is
1
√
b
CWT
(
ba, bτ
)
.
Redundancy: Continuous Wavelet transformation to transform One-Dimensional
signal into Two-dimensional Space. Therefore there is redundant information in the
wavelet transform.
3.1.3 Two-Dimensional Wavelet Transform
In digital image processing, the image can be abstracted as a two-dimensional signal.
So it is necessary to extend wavelet transform from one dimension to two dimensions
when using wavelet transform tool.
The definition is as follows:
Definition 4 Suppose f
(
t
1
, t
2
)
∈ L
2
R
2
is a two-dimensional signal, t
1
and t
2
rep-
resent the transverse coordinates and vertical coordinates of the signal respectively,
and ψ
(
t
1
, t
2
)
represents two-dimensional fundamental wavelets, the expression of
two-dimensional continuous wavelet transform is as follows:
CWT
(
a;τ
1
, τ
2
)
=
1
a
¨
f
(
t
1
, t
2
)
ψ
t
1
− τ
1
a
,
t
2
− τ
2
a
dt
1
dt
2
(3.7)
54 3 Wavelet and Fast Bilateral Filter Based Despeckling …
It will be seen in the later chapter that the two dimensional wavelet transform is
a great method and a very useful tool in image processing.
3.1.4 Discrete Wavelet Transform
Because of the redundancy of continuous wavelet transform, and in order to facilitate
the use of computer-aided analysis and processing in practice, the signal function
f
(
t
)
and its scaling, translation parameters a and τ need to be discretized in numerical
calculation. And it is called discrete wavelet transform (DWT).
In discrete wavelet transform, the process of discretization of parameters a and τ
of wavelet basis function in formula (3.4) will be introduced as follows:
We use the power order discrete value for the scale factor a, that is, let a = a
m
0
,
a
0
> 0, m ∈ Z;
We use uniform discrete values for the translation factor τ. In order to avoid
information loss, the sampling interval must satisfy the Nyquist Sampling Theorem.
Based on this configuration, m increases by 1, the scale factor a doubles, and the
corresponding frequency is reduced by half. When the scale is j, sampling intervals
can be expanded by a
j
0
, because the width of ψ
a
−j
0
t
is a
j
0
times that of ψ
(
t
)
.The
corresponding discrete wavelet basis functions are as follows:
ψ
a
j
0
,kτ
0
(
t
)
= a
−j
2
0
ψ
a
−j
0
t − kτ
0
, j = 1, 2, 3 ··· (3.8)
So the formula of discrete wavelet transform is:
DWT
a
j
0
, kτ
0
=
∞
−∞
f
(
t
)
∗ ψ
a
j
0
,kτ
0
(
t
)
dt (3.9)
Discrete wavelet transform can guarantee the precision of reconstructed signal,
and it is widely used in image processing, signal processing, speech recognition and
other fields.
3.1.5 Multi-resolution Analysis and Mallat Algorithm
The multi-resolution analysis was proposed by S. Mallat in 1988. Based on the con-
cept of space, he explained the multi-resolution characteristic of wavelet analysis
graphically. Mallat proposed a fast algorithm for tower multi-resolution decomposi-
tion and reconstruction of signals in 1989, and that is the classic Mallat algorithm.
3.1 Introduction of Wavelet 55
Multi-resolution analysis means that a signal is composed of a series of signals
with different resolution or progressive space. We can do what we need just by select-
ing the right signal at the right resolution for processing, because the signal is divided
into different resolution spaces. The method of multi-resolution analysis provides
theoretical support for understanding the principle of wavelet transform and con-
structing wavelet. At the same time, it promotes the development of wavelet theory.
Multi-resolution analysis is essentially composed of a series of successive progres-
sive spaces V
j
which satisfy L
2
(R) condition. These spaces satisfy the following
conditions:
Monotonicity: V
j
⊂ V
j−1
, j ∈ Z;
Approximation:
∞
j=−∞
V
j
∈ L
2
(R),
∞
j=−∞
V
j
=
{
0
}
;
Scalability: For arbitrary j ∈ Z,f (t) ∈ V
j
, there are f (2t) ∈ V
j−1
;
Translational invariance: if f (t) ∈ V
j
, then f (t − 2
j
k) ∈ V
j
, k ∈ Z;
Existence of orthogonal bases: There exists a function φ ∈ V
0
such
that
{
φ(t − n), n ∈ Z
}
is the orthogonal basis of V
0
, then we have V
0
=
span
{
φ(t − n)
}
,
R
φ(t − n)φ (t − m)dt = δ
m,n
;
The set of the space of function is called a multi-resolution analysis which satisfied
all the above conditions. If the function φ(t) generates a multi-resolution analysis,
the function φ(t) is a scale function.
Multi-resolution analysis is a mathematical description for human visual system
to recognize objects. Suppose that we take the space V
j
as the information observed
when the current scale is j. In order to take a closer look at the target object, we need
to increase the scale to j + 1, where the target object information is now located
in the space V
j+1
. Therefore, the details of the information contained in space V
j+1
are r icher than those in space V
j
. Therefore, multi-resolution analysis provides a
powerful method for digital image processing.
3.2 The Limitations of Traditional Wavelet Based
Denoising Method
Wavelet theory has been widely used in image processing due to the advantages of
time–frequency analysis and multi-scale analysis. The wavelet de-noising method
is better than others in the processing of additive noise, and this method has higher
efficiency, which can satisfy general requirements. However, using the wavelet trans-
formation de-noising method to suppress speckle noise of medical ultrasonic images
usually cannot get desirable results, because the low-pass component also contains
some speckle noise. We have introduced the principle of wavelet transform, and the
characteristics of wavelet transform based de-noising method of digital image can
be summarized as follows:
(1) Low entropy: The sparse distribution of wavelet coefficients reduces the entropy
after image transformation.
56 3 Wavelet and Fast Bilateral Filter Based Despeckling …
(2) Multiple resolutions: Because of the multi-resolution method, the non-stationary
features of the signal, such as edges, peaks, breakpoints, etc., can be described
very well.
(3) De-correlation: Because wavelet transform can de-correlate the signal, and the
noise is whitened after the transform, wavelet domain is more favorable than
time domain for de-noising.
(4) Basis selection flexibility: Because wavelet transform can select transform basis
flexibly, different wavelet mother functions can be used for different applications
and different research objects, so as to obtain the best result.
When we conduct an image processing, we usually suppose that the noise is
mainly in the high frequency part. The de-noising method is usually to remove the
noise in the high-frequency part of the image by the threshold shrinkage, and then
obtain the denoised image by removing the high-frequency noise. But this method
of image processing will inevitably leave out some noises which were part of low-
frequency, and thus we still can find speckle noise in images. Now let’s assume that
if we can do some denoising processing on the low-frequency part of the image, it
may be better for the image processing.
3.3 Wavelet and Fast Bilateral Filter Based Despeckling
Method
In recent decades, several image filtering techniques had been proposed to reduce
speckle noise. These despeckling filters are classified into five categories: local adap-
tive filters, anisotropic diffusion filters, multi-scale filters, nonlocal means filters and
hybrid filters. Commonly used adaptive filters (Lee, Frost [1] and SRBF) assume that
speckle noise is essentially multiplicative noise. Anisotropic diffusion filters include
DPAD, SUSAN AD [2] and OSRAD [3]. Nonlocal means filters are novel de-noising
algorithms, such as OBNLM, PPB [4] and Guo [5]. For multi-scale filters (wavelet
soft/hard threshold’s filter and Andria), the wavelet transformation is usually used as
a tool to analyze and process the image. Hybrid filters are the combination of several
methods. For example, the SAR-BM3-D’s filter is the combination of a multi-scale
filter with a nonlocal means filter. Wavelet theory has been widely used in image
processing due to the advantages of time–frequency analysis and multi-scale analy-
sis. The wavelet de-noising method is better than others in the processing of additive
noise, and this method has higher efficiency, which can satisfy general requirements.
In this section, a novel de-speckling method based on wavelet transformation and
bilateral filter is proposed for medical ultra-sound images. The bilateral filter not
only has better speckle reduction performances but also can preserve image edge
details. To improve efficiency and to shorten running time, a fast bilateral filter based
on FFT is adopted in this section. Therefore, advantages of the wavelet denoising
method and the fast bilateral filter will be combined. The main idea of this method
is as follows: On the basis of the traditional wavelet denoising method, an improved
3.3 Wavelet and Fast Bilateral Filter Based Despeckling Method 57
wavelet threshold function and a novel shrinkage algorithm are developed according
to the statistical properties of speckle noise of the medical ultrasound images in
wavelet domain. This method can effectively suppress the speckle noise in high-pass
component. Speckle noise in the low-pass approximation component of medical
ultrasound images in the wavelet domain is filtered by the fast bilateral filter. The
proposed combined approach not only guarantees speckle reduction but also can
greatly shorten running time.
3.3.1 The Model of Medical Ultrasound Image
The reasonableness of speckle noise models not only accurately describes the process
of speckle noise formation, but also determines the reliability and precision of de-
noising algorithms, and hence is a prior condition for proposing effective de-noising
methods. Assuming that insonification and the resulting echo signal absorption have
sufficiently obtained appropriate dynamic compensation from the ultrasound imag-
ing system, the final ultrasonic envelope signal obtained consists of two parts: the
reflected signal of the human body, which is a useful signal, and the noise itself,
which is made up of two components, multiplicative noise and additive noise. Mul-
tiplicative noise is associated with the principle of ultrasonic signal imaging, which
results from a random scattering phenomenon in imaging cell resolution. Additive
noise can be considered system noise, such as sensor noise [6–9]. The ultrasonic
envelope signal f
pre
(i, j) is modeled as follows:
f
pre
(i, j) = g
pre
(i, j)n
pre
(i, j) + w
pre
(i, j) (3.10)
where (i, j) ∈ Z
2
are the two-dimensional spatial coordinates, and the superscript pre
is the preliminary signal obtained by system. g
pre
(i, j) and f
pre
(i, j) denote the original
signal and the observed signal respectively. n
pre
(i, j) and w
pre
(i, j) represent the
multiplicative and additive components of the noise respectively, where the n
pre
(i, j)
is the main component of noise.
The effect of additive noise w
pre
(i, j) on the qualities of the medical ultrasound
images is less significant than the multiplicative noise n
pre
(i, j), and in order to
simplify the model (3.10), the additive noise w
pre
(i, j) is generally omitted and the
following model is obtained:
f
pre
(i, j) = g
pre
(i, j)n
pre
(i, j), (i, j) ∈ Z
2
(3.11)
Due to the limited dynamic range of commercial display monitors, ultrasound
imaging systems compress the envelope signal with logarithmic transformation to fit
in the display range which we have mentioned in last chapter. Logarithmic amplifi-
cation converts the model (3.11) into the classical additive noise model as follows:
log(f
pre
(i, j)) = log(g
pre
(i, j)) + log(n
pre
(i, j)) (3.12)
58 3 Wavelet and Fast Bilateral Filter Based Despeckling …
where the signal log(f
pre
(i, j)) is the common medical ultrasonic image.
Since wavelet transformation is a linear transformation, the following model is
obtained after two-dimensional discrete wavelet transformation for model (3.12):
W
j
l,k
(f ) = W
j
l,k
(g) +W
j
l,k
(n) (3.13)
where j = 1, 2,...,J , l, k ∈ Z
2
. W
j
l,k
(f ), W
j
l,k
(g) and W
j
l,k
(n) represent the wavelet
coefficients of noisy images, noise-free images and the speckle noise respectively.
The superscript j is the decomposition layers of wavelet transformation, and the
subscript (l, k ) is wavelet domain coordinates. J denotes the largest decomposition
layers. In order to facilitate the representation, we can rewrite (3.13)as
F
j
l,k
= G
j
l,k
+ N
j
l,k
(3.14)
Since the Bayesian maximum a posteriori estimation will be used to develop a
new wavelet shrinkage algorithm in this chapter, and since the prior probability of
noise-free signal and speckle noise is the premise of using the Bayesian maximum a
posteriori estimation, we consider that the wavelet coefficients of noise-free signal
will obey the generalized Laplacian distribution, and the wavelet coefficients of
speckle noise will obey zero mean Gaussian distribution.
The wavelet coefficients of noise-free signal G
j
l,k
obey the generalized Laplacian
distribution, and the probability distribution is:
p
G
(g) =
v
2s(
1
v
)
exp
−
g
s
v
, s, v > 0 (3.15)
where (a) =
∞
0
x
a−1
exp(−x)dx represents Gamma function. s is scale parameter,
and v is shape parameter. When v is selected as 1, formula (3.15) becomes the
Laplacian distribution, which is a special model of generalized Laplacian distribution.
The wavelet coefficients of speckle noise N
j
l,k
obey zero mean Gaussian distribu-
tion
p
N
(n) =
1
√
2πσ
N
exp(−
n
2
2σ
2
N
) (3.16)
where σ
N
denotes the standard deviation of noise in wavelet domain.
Gaussian distribution is as follows (Fig. 3.8):