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18 1 Introduction
6. J. Zhang, G. Lin, L. Wu, Y. Cheng, Speckle filtering of medical ultrasonic images using wavelet
and guided filter, in Ultrasonics (2015.10)
7. J. Zhang, G. Lin, L. Wu, Y. Cheng, Wavelet and fast bilateral filter based de-speckling method
for medical ultrasound images, in Biomedical Signal and Control (2015.3)
8. J. Zhang, C. Wang, Y. Cheng, Comparison of despeckle filters for breast ultrasound images, in
Circuit, System and Signal Process (2015.1)
9. J. Zhang, G. Lin, L. Wu, Y. Cheng, An integrated de-speckling approach for medical ultrasound
images based on wavelet and trilateral filter, in Circuit, System and Signal Process (2017.1)
10. J. Zhang, X. Xiu, J. Zhou, K. Zhao, Z. Tian, Y. Cheng, A novel despeckling method for medical
ultrasound images based on the nonsubsampled shearlet and guided filter, in Circuit, System
and Signal Process (2019.8)
Chapter 2
Despeckle Filters for Medical Ultrasound
Images
Abstract This chapter will introduce several kinds of denosing methods, and speckle
reduction techniques and their applications in the breast ultrasound imaging are dis-
cussed. Five categories of despeckle filters are presented. This chapter focuses on the
comparison of despeckle filters for the breast ultrasound images. Despeckle filters
which are classified into five categories (local adaptive filter, anisotropic diffusion fil-
ter, multi-scale filter, nonlocal means filter, and hybrid filter) are described. The com-
parative experiments of eleven despeckle filters for the two types of simulated images
and clinical ultrasound breast images are presented. Toobjectively and systematically
compare the performance of eleven despeckle filters, several comparison methods
are used, such as the full-reference image quality metrics, the non-reference/blind
image quality metrics, observing the removed noise images, as well as the visual
evaluation of experts.
2.1 Models of Speckle Noise for Medical Ultrasound Images
From the previous chapter, we have a general understanding of the principle of
the generation of ultrasonic images, and the modeling of ultrasonic images is
also based on the knowledge of ultrasonic imaging principles. The models of the
speckle noise for medical ultrasound image is divided into two parts, i.e. models of
envelope-detected echo signal and models of speckle noise. And both of them will be
introduced.
2.1.1 Models of Envelope-Detected Echo Signal
When ultrasonic signals are transmitted to the human body, receiving devices of
ultrasound imaging system detect and store the envelope echo signal. The envelope-
detected echo signal is received, and the main echo signal is reflected back from
the organization of the human body. Assuming that the insonification and result-
ing absorption of the echo signal have been obtained sufficiently with appropriate
© 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_2
19
20 2 Despeckle Filters for Medical Ultrasound Images
dynamic compensation from ultrasound imaging system, the final envelope-detected
echo signal obtained consists of two parts. One part is the reflected signal of the
human body which is a useful signal, and the other is the noise which itself is made
up of two components, i.e. multiplicative noise and additive noise [1].
f (i, j) = g(i, j)n(i, j) + ω(i, j) (2.1)
where n(i, j) and ω(i, j) represent the multiplicative and additive components of the
noise, respectively. (i, j) are the two-dimensional space coordinates. g(i, j) and f(i, j)
denote the original signal and the observed signal respectively.
It is generally accepted that the multiplicative noise is the main component of the
noise, which results from a random scattering phenomenon in imaging resolution
cell. Comparing with the multiplicative noise, the additive noise is very small and
it is composed of the sensor noises. To simplify the model (2.1), the additive noise
ω(i, j ) is generally omitted.
f (i, j) = g(i, j)n(i, j) (2.2)
where the observed signal f(i
, j) is known. g(i, j) and n(i, j) are the signals needed
to be evaluated, especially evaluating the original signal g(i, j).
2.1.2 Models of Speckle Noise
The distributions of speckle noises are different from each other according to the
number of scatterers per resolution cell and the ultrasound imaging system [2]. We
have known the formation principle of the speckle noise. A single frequency ultra-
sonic wave emitted by an ultrasonic probe is scattered by a rough surface of the
body’s tissue to form a series of coherent waves. The noise, which comes from the
fact that these coherent waves interfere with each other, is speckle noise. Suppose
that there are a lot of equal-volume scatterers in the imaging resolution unit [2–5].
The total effect of the superposition of the scatterers in the resolution unit can be
described by the random walk in the complex plane, as shown in Fig. 2.1 (left).
The contribution of any scatterer in the diffuse scattering group to the echo signal
is equivalent to one step of increasing in the random walk, and to form the received
complex signal by superposition. As shown in Fig. 2.1 (left), the first vector starts
from the origin, and the second vector starts from the end of the first vector. One by
one in turn, the vector from the origin to the last vector end is the final synthesis vector
(black vector). If each walk can be regarded as an independent random variable, a
great deal of this walk can be explained by using Central Limit Theorem. As shown
in Fig. 2.1 (right), the center point is the 2D complex Gauss distribution, and the
large number of such scattering subgroups follows the Rayleigh distribution. So the
speckle n(i, j) can be modeled by Rayleigh distribution, which is a commonly used
model of speckle noise. On the basis of the signal-to-noise ratio (SNR), the high SNR
2.1 Models of Speckle Noise for Medical Ultrasound Images 21
s
r
Fig. 2.1 (Left) Random walk diagram of scatterers, (right) Isoline diagram of PDF for RF signal r
is modeled by Rician distribution, and the lower SNR is modeled by K-distribution
or Nakagami distribution [6–9].
The distribution of Generalized Gamma is attractive as it contains several com-
monly used distributions, and the probability density distribution (PDF) is as follow-
ing:
f
N
(n; α, υ, γ ) =
γ n
(γ υ −1)
α
γυ
Γ(υ)
exp
−
n
α
γ
, n ≥ 0, α,υ,γ > 0 (2.3)
When parameters υ and γ vary, the PDF of (2.3) shows different distributions
which are frequently used in the processing of speckle reduction. For example, the
PDF changes to the one of Rayleigh distribution (υ = 1, γ = 2), to the one of
Nakagami distribution (γ = 2), and to the one of Weibull distribution (υ = 1). In
practice, we can choose one of these distributions according to the statistic of the real
ultrasound image. Due to the limited dynamic range of commercial display monitors,
most ultrasound imaging systems compress the envelope-detected echo signal with
logarithmic transformation. After logarithmic compression, the envelope-detected
echo signal can be fitted into the limited range of ultrasound imaging system monitors.
So, Eq. (2.2) converts to:
log (f (i, j)) = log (g(i, j)) + log (n(i, j)) (2.4)
And its simplified form is given as follows:
f (x, y) = g(x, y) + n(x, y) (2.5)
Logarithm transformation converts the multiplicative s peckle noise model to an
additive noise model, so that filters developed for additive noise can be applied
to the logarithm transformed images. However, the nonlinear logarithm alters the
statistics of ultrasound images and enhances the weak back scatters. The analysis
22 2 Despeckle Filters for Medical Ultrasound Images
of the log-compressed speckle noise is complicated, and a number of researchers
devote their efforts to this work. Logarithm compression has a stronger effect on the
high intensity tail than on the low intensity part of the PDF curve. Before logarithm
compression, the speckle noise n(i, j) is modeled by Rayleigh distribution, while the
log-compressed speckle noise n(x, y) is approximated as the Gaussian noise.
2.2 Despeckle Filters for Ultrasound Images
The purpose of this section is to describe the eleven despeckle filters which are clas-
sified into five categories including the local adaptive filter, the anisotropic diffusion
filter, the multi-scale filter, the nonlocal means filter, and the hybrid filter [10].
2.2.1 Category 1: Local Adaptive Filter
Adaptive filter is a digital filter which can automatically adjust the performance of
the input signal for digital signal processing. In contrast, non-adaptive filters have
static filter coefficients, which together form a transfer function. The adaptive process
involves an algorithm that uses the cost function to determine how to change the filter
coefficients so as to reduce the cost of the next iteration process. The value function i s
the criterion for determining the optimal performance of the filter, such as the ability
to reduce the noise component in the input signal. The theoretical basis of the block
diagram in Fig. 2.2 is that the variable filter can obtain the estimation of the desired
signal.
Before introducing specific filtering methods, we may start with some of the basic
concepts of MMSE that we will use in this section.
In statistics and signal processing, a minimum mean square error (MMSE) estima-
tor is an estimation method which minimizes the mean square error (MSE), which is
a common measure of estimator quality, of the fitted values of a dependent variable.
In the Bayesian setting, the term MMSE more specifically refers to estimation with
quadratic loss function. In such case, the MMSE estimator is given by the posterior
mean of the parameter to be estimated. Since the posterior mean is cumbersome to
Fig. 2.2 Block diagram of
adaptive filter
2.2 Despeckle Filters for Ultrasound Images 23
calculate, the form of the MMSE estimator is usually constrained to be within a cer-
tain class of functions. Linear MMSE estimators are a popular choice since they are
easy to use, calculate, and very versatile. It has given rise to many popular estimators
such as the Wiener–Kolmogorov filter and Kalman filter.
(1) Lee Filter
In the early days, digital image processing mostly used time-frequency conversion,
but it takes a lot of time to process 2D image signals on the computer. The method
of frequency conversion cannot meet the requirements of real-time image process-
ing. Then people started to work on recursive filtering algorithms. However, when
recursive filtering algorithms were applied to two-dimensional arrays, the algorithm
runs in the spatial domain, where pixels must be processed individually. Therefore,
the process is no longer efficient, and recursive filtering algorithms still not suitable
for the situation of real-time image processing. Ketcham et al. used the entire local
histogram for real-time image enhancement, and Wallis applied local mean and vari-
ance to filter out scanline noise with striking results. Then Lee extends this family of
algorithms to contrast enhancement and noise filtering. Lee filter is one of the typical
methods for speckle filtering of SAR images using local statistical characteristics of
images.
For image denoising, Lee filter has different filtering schemes according to the
different properties of image noise. It uses the minimum mean square error (MMSE)
filtering criterion to carry out the despeckling.
For the model (2.2), the multiplicative Lee filter assumes that the mean and the
variance of observed noisy image are equal to the mean and variance of original noise-
free image, and there is no correlation between the different positions of the noise. The
nonlinear model (2.2) is approximated by a linear model, and the available estimate
g
of the original image can be obtained using minimum mean square error(MMSE)
estimation theory as follows:
g
= E(g) + k[f − E(n)E(g)] (2.6)
where
E(g) =
E(f )
E(n)
k =
E(n)λ
[E(g)]
2
σ
2
n
+[E(n)]
2
λ
λ =
σ
2
f
+[E(f )]
2
σ
2
n
+[E(n)]
2
−[E(g)]
2
E(f) and E(n) are the mean of noisy image and the noise. σ
2
f
and σ
2
n
are the variance
of noisy image and the noise. Lee filter is an adaptive filter which is characterized by a
24 2 Despeckle Filters for Medical Ultrasound Images
direct estimate based on the statistical properties of the acquired image. Since filtering
an image using Lee filter does not need to do the Fourier or other transformations,
Lee filter has very high efficiency [11].
(2) Frost Filter
Frost filter is an adaptive filter proposed by Frost et al. [12], which is designed to
despeckle SAR images based on the local statistics similar to the Lee filter. The
standard technique for enhancing SAR images is to use non-coherent integration.
Enhancement implies increasing the signal-to-noise ratio at the expense of spatial
resolution. Most SAR systems use frequency diversity, i.e., non-overlapping subareas
of the spectrum of the complex image are used to form N independent images.
Continuous scanning of the spectrum is also used. The net effect is to reduce the
bandwidth (degrade the resolution) while improving the signal-to-noise ratio S/N.
However, there are several disadvantages to this approach:
(1) The technique is spatially invariant and thus does not account for the non-
stationarity of the signal;
(2) The technique was developed for coherent optical processing and thus it is
easily implemented with such a processor but is not necessarily optimal for
digital processing. So the frost filtering algorithm was proposed.
The noisy image in this theory is modeled as follows:
f (x, y) =[g(x, y) · n(x, y)]∗h(x, y) (2.7)
where f (x, y), g(x, y), and n(x, y) denote noisy image, ideal noise-free image, and
noise in spatial coordinate (x, y), respectively. h(x, y) is the system impulse response.
The noise-free image g(x, y) is estimated from f (x, y) using MMSE estimator. The
pixel value is calculated by a weighted average of pixel values using the neighborhood
pixels in the spatial domain. The weights ω are determined by the local statistics of
the data
ω = kαe
−α
|
t
|
(2.8)
where α ∝ σ
f
/f , k is a constant normalization factor, and |t| is the spatial geometric
distance between the center pixel and the neighborhood pixels.
(3) Speckle Reducing Bilateral Filter (SRBF)
Balocco et al. [5] proposed an adaptive speckle reducing bilateral filter (SRBF) [13],
which is tailored to ultrasound images. SRBF is derived from a general bilateral filter.
The bilateral filter is a local adaptive filter [14], and a non-linear, edge-preserving,
and noise-reducing smoothing filter for images. The feature of the filter is that it
preserves the shape edge of the image. It replaces the intensity of each pixel with a
weighted average of intensity values from nearby pixels. This weight can be based
on a Gaussian distribution. The SRBF is given as
2.2 Despeckle Filters for Ultrasound Images 25
h(x) = k
−1
(x)
ξ∈(x)
f (ξ ) · c(ξ , x)s(f (ξ ), f (x))d ξ (2.9)
and the constant normalization factor k(x)is:
k(x) =
ξ∈(x)
c(ξ, x)s (f (ξ), f (x))dξ (2.10)
where f and h denote the noisy input image and output image. Function c measures
the geometric distance between the center pixel x and the neighborhood pixels ξ.
Function s measures the similarity of the pixel values between x and ξ . The speckle
noise is modeled by a Rayleigh distribution. SRBF modifies both c function and s
function according to the speckle noise statistics of ultrasound images. The existence
of the function k allows the filter to retain the energy of the image. However, SRBF
still have the same shortcoming as bilateral filter, it can lead to staircase effect,
intensity plateaus that lead to images appearing like cartoons, and gradient reversal,
introduction of false edges in the image.
2.2.2 Category 2: Anisotropic Diffusion Filter
Anisotropic diffusion, which is firstly proposed by Perona and Malik [15], has caused
a great impact and been widely used in image de-noising. Anisotropic diffusion also
called Perona–Malik diffusion is a technique aiming at reducing image noise without
removing significant parts of the image content, typically edges, lines or other details
that are important for the interpretation of the image. Anisotropic diffusion resembles
the process that creates a scale space, where an image generates a parameterized
family of successively more and more blurred images based on a diffusion process.
Each of the resulting images in this family is given as a convolution between the
image and a 2D isotropic Gaussian filter, where the width of the filter increases with
the parameter. This diffusion process is a linear and space-invariant transformation
of the original image. Anisotropic diffusion is a generalization of this diffusion
process: it produces a family of parameterized images, but each resulting image is a
combination between the original image and a filter that depends on the local content
of the original image. As a consequence, anisotropic diffusion is a non-linear and
space-variant transformation of the original image. The four divergence formulas are
deviated from the current pixel in four directions. The general continuum model is
as follows:
∂I
∂t
= div[c(
|
∇I
|
) ·∇I ]
I(t = 0) = I
0
(2.11)
26 2 Despeckle Filters for Medical Ultrasound Images
where ∇←denotes the gradient operator, div represents the divergence operator,
| • | indicates the magnitude, c· indicates the diffusion coefficient, and I
0
is the
obtained initial image. Anisotropic diffusion filtering is mainly used to smooth the
image, which overcomes the defect of Gaussian blur. Anisotropic diffusion preserves
the edge of the image when smoothing the image, which is very similar to the bilateral
filtering. Anisotropic filtering treats the image as a heat field. Each pixel is considered
as a heat flux, and the relationship between the current pixel and the surrounding
pixel is used to determine whether to diffuse to the surrounding pixel.
(1) Detail Preserving Anisotropic Diffusion (DPAD)
The DPAD filter [3] has made reasonable improvements to correctly estimate the
speckle statistical properties of ultrasound images based on the Speckle Reducing
Anisotropic Diffusion (SRAD) filter [16]. The discrete form of Eq. (2.11)is:
I
t+t
P
= I
t
P
+
t
| η
P
|
div[c (C
P
, t)∇ I
t
P
] (2.12)
where I
t
P
indicates the discrete image at p coordinate position. t represents the
time step, and |ηp| is the number of pixels in the window. ∇I
t
P
indicates the gradient
value, and Cp,t represents the ratio between the local standard deviation and the
local mean. The DPAD filter focuses on the estimation of the coefficient of signal
and noise statistics and it is an iterative algorithm. Assuming that the noise’s mean
and variance are constants throughout the image, the diffusion coefficient can be
calculated as follows:
c(R
P
) =
1 + 1/C
2
P
1 + 1/C
2
μ
(2.13)
where C
2
P
=
Var(f
P
)
¯
f
2
P
·
|
η
P
|
|
η
P
|
−1
and C
2
μ
can be estimated as the median value of C
2
P
.The
denoising effect of DPAD is closely related to the estimation of noise.
(2) Oriented Speckle Reducing Anisotropic Diffusion (OSRAD)
The OSRAD filter, based on the SRBF filter, takes advantage of the relationship
between the local variance and local geometry of an image [17]. The diffusion coef-
ficient c(R
P
) is calculated using Eq. (2.13)ofDPADfilter.
By combining the SRAD filter with a matrix anisotropic diffusion, OSRAD filter
has a characteristic of filtering an ultrasound image along the structures. For 2-
Dultrasound images, Eq. (2.11) is extended to a matrix expression:
∂I
∂t
= div
c(R
P
) 0
0 c
ta
·∇I
(2.14)
where c
ta
denotes an adjustable constant. The relationship between the local variance
in the direction of a principal curvature and the corresponding curvature of the image
2.2 Despeckle Filters for Ultrasound Images 27
is as follows:
Var(I) = γ
|
∇I
|
2
κ
2
(2.15)
where Var (I ) indicates a local variance of the i mage, γ = σ
4
/ 2 is an adjustable
constant, and κ is the associated curvature value.
2.2.3 Category 3: Multi-scale Filter
The typical method for the multi-scale analysis method, the wavelet theory, is pro-
posed by Meyer and Mallat in the late 1980s. Wavelet theory has both spatial and
frequency resolution, thus it is widely used to compress and de-noise signal. One of
the key reasons for the great success of wavelet analysis in multidisciplinary fields is
that it can represent one-dimensional piecewise smooth or bounded variation func-
tions more sparsely than Fourier analysis. Unfortunately, the excellent properties of
the wavelet analysis in one dimension cannot be simply extended to two-dimensional
or higher dimensions. This is because the separable Wavelets (Separable wavelet)
of one-dimensional wavelets have only a limited direction and cannot “optimally”
represent a high dimensional function containing line or surface singularity, but in
fact it has a wired or surface singular function. Numbers are very common in high
dimensional space. The smooth boundary of natural object makes the discontinuity
of natural image often reflect singularity on smooth curve, not just point singularity.
In other words, in the case of high dimension, wavelet analysis cannot make full
use of the geometric characteristics of the data itself, and is not the best or “sparse”
function representation method. The purpose and motive force of the development of
multi-scale geometric analysis (Multiscale Geometric analysis) after wavelet analy-
sis is to develop a new optimal representation method of high-dimensional function,
like Shearlets (Fig. 2.3).
(1) The Generalized Likelihood Method (GLM)
In the process of GLM filtering, the non-decimated wavelet transform is proposed in
order to keep the wavelet transform with shift invariant features [18]. After J–level
wavelet transform, the wavelet coefficients at scale 2
j
(1 ≤ j ≤ J) and spatial position
k are indicated by w
d
k,j
. The superscript d denotes the detail part of signal in wavelet
domain.
The highlight of GLM filter is the preliminary classification of wavelet coefficients
without accurate prior knowledge of the noise distribution in wavelet domain. The
aim of this classification is to judge wavelet coefficients of interest and to estimate
the conditional probability density function of the image features and the background
noise. The classification is given by: