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52 S. Balocco et al.
Fig. 2 (a) Synthetic Ultrasound envelope-detected images B(x, y) (128 × 128 pixels size), simu-
lated using a 5 MHz central frequency, and
and Rayleigh curve fitting in B
R1
and B
σ
= 2and
x
R2
σ
= 1.5 PSF parameters. (b) Histogram
y
I0(x,y) be an echogenicity model (an image with different intensities corresponding
to different tissues of the object being imaged) in which the variables x and y are
the lateral and axial coordinates, respectively. Firstly, subresolution variations in
object impedance should be introduced by multiplying the echogenicity model by a
Gaussian white noise with zero mean and unitary variance:
T (x,y)=I
(x,y) ×G(x, y) (2)
0
T(x, y), accounts for acoustic impedance inhomogeneities in the object due to
density and acoustic speed perturbation that generates the scattering.
The V(x, y) ultrasonic radiofrequency (RF) echo data can then be obtained by a
convolution:
V (x,y)=h (x,y) ∗T (x,y) (3)
where h(x, y) is the PSF or impulse response of a hypothetical US imaging system.
It is assumed [16,17]thath(x,y) is separable, i.e., h(x,y)=h
(x) h2(y) where h1is
1
a Gaussian-weighted sinusoidal function determined by:
h
(x,
σ
)=sin(k0x)exp−x2/2
x
1
where k
σ
x
for the transmitting and receiving aperture h
= 2πf0/v, v is the speed of sound in tissue, f0is the center frequency, and
0
represents the pulse-width of transmitting ultrasonic wave. The spatial response
(x) is determined by:
2
(y)=expy2/2
h
2
σ
2
σ
x
2
y
(4)
(5)
where
σ
represents the beam-width of transmitting ultrasonic wave.
y
The image of the envelope-detected amplitude, B(x,y) shown in Fig. 2a, is given
by:
B(x, y)=
V (x,y)+iˆV (x,y)
whereˆV (x,y) is the Hilbert transform of V (x, y) and i is the imaginary unit.
(6)
Ultrasound Despeckle Methods 53
3 Ultrasound Speckle Reduction Methods
In literature, the problem of speckle reduction has been tackled by means ofdifferent
techniques. In this section will be presented the main categories of speckle reduction
filters along with a brief description of each filter principle.
3.1 Linear Filters
Linear filtering, e.g., Wiener filter [18], was the first adaptative approach used to
deal with multiplicative noise [19]. Although linear filtering reduces the amount
of noise in the image it over-smooths the transitions and anatomical details that
should be preserved [20]. This difficulty arises because images corrupted with this
type of noise often present low signal to noise ratio (SNR) and the non-linear
and multiplicative nature of the corrupting process makes the linear approach not
appropriate [21]. Thisdifficulty can be overcome by adopting specific multiplicative
models to describe the speckle noise [15, 22–24]. So in the last two decades, non-
linear filtering methods have been successfully used to deal with the multiplicative
noise using several approaches [25], e.g., median filtering, Bayesian and wavelet
based methods and anisotropic Diffusion (AD) [16].
3.2 Median Filters
Median filter has been extensively used to process multiplicative and impulsive
noise [24, 26] because it is a very simple technique and the visual results are good
with a limited tuning. Several variants of the basic median filter were proposed,e.g.,
adaptive median filters. An example is the Weighted Median Filter (WMF) proposed
in [27] for despeckling of ultrasound images. The filter considers a small window
centered at each pixel as usual but each element of the input image is considered
several times according to the local statistics of the image. More recently [28]
proposes a novel stochastically driven filtering method with constant and variable
size windows. A simpler version of this filter with constant window, is proposed in
[29]. In [28] a noveldespeckling method iteratively removes outliers by determining
the local mean and standard deviation from an adaptively varying window. By removing outliers(local extrema) at each iteration, this method produces a convergent
sequence of images by squeezing the stochastically distributed pixels values to a
limiting value. It was experimentally shown by the authors that the proposed filter
outperforms all the median filters considered in the experiments. Another technique
is proposed in [30] where a new efficient 3D median despeckling algorithm is
described and implemented in VHDL. Experimental tests performed by the authors
lead to a processing time of 0.03 s to filter a 128 ×128 ×128 3D volume using a
50 MHz processor.
54 S. Balocco et al.
3.3 Wavelet-Based Filters
Wavelet-based despeckling algorithm has also been extensively used in medical
imaging since the seminal works of Donoho on soft-thresholding were published
[31]. This method is based on the multi-scale decomposition of the noisy image and
in the processing of the image coefficients at coarser scales [25, 32–36]. For example, in [33] the authors compute the multi-scale decomposition of the logarithmic of
the ultrasound image and model the coefficients at each scale by the alpha-stable
heavy-tailed distribution. A Bayesian-based non-linear operator is then used to
remove the speckle noise at each scale. A similar approachis described in [35]using
generalized Gaussian distribution (GGD).In[34] the same multi-scale decomposi-
tion is used to identify wavelet coefficients at different scales with high correlation.
It is assumed that these coefficients describe anatomical details of the image which
should be preserved. This allows the elimination of the noise without removing
anatomical information. Comprehensive surveys on Wavelets in medical imaging
can be found in [37, 38]. The wavelet denoising algorithms for images corrupted
with multiplicative noise usually use a logarithmic version of the noisy original
image, by assuming that multiplicative noise is converted into additive Gaussian
one as suggested in [24] although, this assumption is not true, as discussed in [20].
3.4 Bayesian Filters
A different group of algorithms, called Bayesian methods, formulates the denoising task as an estimation problem, where the likelihood function and a prior
distribution are jointly maximized. Let X be a N × M unknown image to be
estimated/reconstructed from a noisy image, Y.Themaximum a posteriori (MAP)
estimate of X is the solution of the following optimization problem
ˆ
X = arg min
where
E(X,Y )= E
Data fidelity term
E
(X,Y )=−log p(Y|X ), is called the data fidelity term and attracts the solution
Y
toward the data while E
(X)=−log p(X), is called the prior or internal energy and
X
regularizes the solution removing the noise.
The computation of X , based on the minimization of the data fidelity term
E
(X,Y),isthemaximum likelihood (ML) estimation problem and it is usually
Y
an ill-posed problem because the solution is not unique and it may not depend
continuously on the data [39–41]. To overcome this difficulty a regularization term
E(X,Y ) (7)
X
(X,Y)
Y
+ EX(X)
Prior term
(8)
Ultrasound Despeckle Methods 55
is added, turning the problem into a well-posed problem. The distribution p(X)
introduces prior knowledge about the image to be estimated, and it usually favors
smooth solutions. The MAP energy function, E(X,Y ) has a global minimum, called
MAP solution, which is, very often, difficult to find because the MAP optimization
function (8) may not be convex.
It is usually assumed that there are no natural priors for medical images. The
common assumption about these images is that they are band-limited, changing
slowly in space except near the organs boundaries where abrupt transitions are
expected. This is a difficult criterion because the location of the transitions are
unknown and must be estimated. This can be done by modeling X as Markov
Random Field (MRF) where neighboring pixels should have similar intensities,
except if they are located at a transition. Three issues must be chosen in this
framework: (1) the statistical observation model, (2) the prior distribution and
(3) the optimization method [41]. A unifying Bayesian framework, based on the
Sylvester–Lyapunov equation able to deal with Rayleigh and Poisson noise and
with several prior distributions was recently proposed by Sanches [42]. Bayesian
methods are often combined with the wavelet decomposition leading to efficient
and fast algorithms [43–45].
3.5 Anisotropic Diffusion Filters
AD algorithms were introduced by Perona and Malik [46] in 1990. The general
∂
equation is
I
= div((g|∇I|) ∇I) where ∇ is the gradient operator,Ω(p) is the
∂Ω
(p)
spatial neighborhoodof the coordinate of a generic pixel p in the noisy image I and
div is the divergence operator.
This filter removes the noise by computing a local average of the central pixel
intensity with the ones of its neighbors. The iterative process achieves a balance
between averaging (in homogeneous regions) and the identity filter (where edges
exist) according to a coefficient proportional to the directional gradient. This
balance depends on the coefficient of variation (g|∇I|) inside the neighborhood
Ω
If (g|∇I|) → 0 the algorithm behaves as an all-pass filter while when (g|∇I|) →1an
isotropic diffusion is achieved (Gaussian filtering).The general anisotropic equation
was successively improved by several authors [16, 47] by proposing a coefficient
of variation able to better estimate the edge orientations. In particular the Speckle
Reducing Anisotropic Diffusion(SRAD) algorithm [16], was based on a coefficient
able to enhance the boundaries. The filler preserves the object shapes by inhibiting
diffusion across boundaries and, at the same time, enhance the shape frontiers by
promoting the AD on either side of the edge. Successively, Krissian [48] proposed
an extension of SRAD called Oriented Speckle Reduction Anisotropic Diffusion
(OSRAD), that allows different levels of filtering along the image contours and their
principal curvature directions.
.
56 S. Balocco et al.
In 1998, Tomasi [49] introduced the Bilateral Filter (BF) frameworkin which the
output pixel’s value is a Gaussian-weighted average of its neighbors in both space
and intensity range. The general BF functional can be expressed as:
h(p)=
−1
Γ
(p)
f (ξ)c (ξ, p) s ( f (ξ), f (p)) d
Ω
(p)
ξ
(9)
with the normalization factor:
Γ
(p)=
c(ξ, p) s ( f (ξ), f (p)) d
Ω
(p)
ξ
(10)
where f is the input image, h is the output image,
of apixel p in the image,
in
Ω
. The classical BF framework [49] defines both c and s functions as unbiased
ξ
is the integration variable representing pixels coordinates
Ω
(p) is the spatial neighborhood
isotropic Gaussian functions.
Comaniciu [50] proposed the mean shift approach based on a statistical local
modes analysis of the image distribution in the joint spatial-range domain. The
relationship between AD, adaptive smoothing, bilateral filtering and mean shift
procedure was finally established by Barash and Comaniciu [51] in 2004 demonstrating that such families of filters, under specific conditions, are formally identical.
More recently, adaptive filters based on local noise statistics were proposed by Guo
and Thakur [52, 53]. Dantas [54] introduced a filter based on a set of modified
Gabor filters (MGFs). However, most filters are developed independently of the
image nature and its noise statistical model. In contrast, Aysal [1] embedded in a
basic filter framework the Rayleigh noise statistics. Finally Balocco [55] proposed
a fully automatic speckle reducing bilateral filter tailored to US Images called
SRBF. In [55], the edge-preserving feature for US images is obtained by embedding
noise statistics in the filter framework by modifying both c and s functions of the
classical BF framework. As a consequence, the filter is able to tackle the noise
multiplicative behavior modulating selectively the smoothing strength with respect
to local statistics.
4 Filter Comparison
This last category (anisotropic filters, described in Sect. 3) has been provedto obtain
the strongest reduction of speckle noise, while maintaining a low computational
cost. For this reason the next section will present various experiments, extracted
by the article [55], focusing on a comparison of such filter categories. In each subsection several state-of-the-art filters are compared through experiments performed
on both in silico and in vivo data. In particular, an experiment has been designed
for illustrating the interest of using an accurate filtering method as pre-processing
stage, in order to improve the performance of the segmentation stages.
Ultrasound Despeckle Methods 57
4.1 Anisotropic Filters Comparison
This section presents two experiments comparing the performance of several stateof-the-art anisotropic filters. Specific details about the noise distributions can be
found in [55].
The first set of experiments (Sect. 4.1.1) assesses the performance and the
robustness of all the denoising filters when applied to in silico images corrupted with
Rayleigh noise. Then, a second experiment (Sect. 4.1.2), compares qualitatively and
quantitatively the best two algorithm of the first experiment on in vivo Ultrasound
images This second study aims at evaluating how each denoising filter improves the
performance of a segmentation algorithm.
4.1.1 In Silico Experiments
A set of in silico experiments was designed to compare the performance of the
anisotropic filters on images characterized by a controlled amount of noise. The
edge-preserving performance in this case has been tested by evaluating the Sum
of Square Differences (SSD) criterion between the ground truth and the denoised
images which provides local comparison of the filtering results with the groundtruth data.
Experimental Image Set
A set of US synthetic images, consisting of two regions, of low (R2) and high
(R1) scatterer intensityrespectively,was generatedusing the convolutional approach
[16, 17] detailed in the Sect. 2.2. The echogenicity model amplitude in R1 (I
unitary, while in R2 (I
R2
) can be tuned using a parameterγ= I
0
R1
0
/I
R2
0
R1
)is
0
representing
the brightness contrast between the two areas. Since the noise is multiplicative,
indirectly indicates the SNR. The higher is the contrast between the two areas the
higher is the SNR. The use of the
γ
variable does not relate with the amplitude of
US equipment contrast.
In the whole set of images, a 5 MHz central frequency probe is simulated, and an
anisotropic noise distribution is obtained using a PSF with parameters
and
σ
= 1.5 ×k,wherek will be used as noise scaling factor.
y
σ
x
= 1 ×k
Fig. 1 illustrates a 128 ×128 pixel image of the set with echogenicity model
amplitudes I
R1
= 1andI
0
of the image exhibits two Rayleigh distributions (Fig. 2b), with
μ
= 0.154 (in the central region) and
R2
R2
= 0.2. The histogram computed in two uniform regions
0
ˆ
σ
= 1.55 and
R1
μ
ˆ
σ
= 0.272 and
R2
= 1.55 (in the upper
R1
bright region) respectively. The ratio between the Rayleigh parameter estimators in
the bright (R1) and the dark (R2) areas of the envelope image (Fig. 2a) are equal to
the ratio of I
R1
0
over I
R2
since speckle noise is purely multiplicative.
0
γ
58 S. Balocco et al.
Performance Comparison
In this section, various state-of-the-art speckle reducing techniques based on the
AD framework are compared. Classic Bilateral Filter (CLASSIC BF)[49], AD
[47], Rayleigh-Maximum-Likelihood Filter (RMLF) [1], Adaptive filter based on
second order statistics filter (AF) [53], MGF [54], Guo filter (GUO) [52], meanshift (MEAN-SHIFT)
[16], Oriented speckle reduction AD
1
[51], Speckle reduction anisotropic diffusion2(SRAD)
3
(OSRAD) [48] and SRBF [55] are applied
to a set of 50 synthetic images generated with the same contrast and speckle size
(
γ
= 5andk = 1) in order to qualitatively (Fig. 3) and quantitatively (Table 1)
evaluate the respective performances.
4
Particularly Table 1, for each method, lists
the parameters, compares the features (number of iterations, computational time),
and reports the similarity scores.
As it can be observed in Fig. 3a, b, the SRBF [55] presents a superior edgepreserving behavior, thus outperforming filtering method based on zero mean noise
assumptions. RMLF, AF, MGF, and GUO and MEAN-SHIFT methods are unable
to fully remove the speckle noise in the uniform area although they exhibit border
preservation properties (see Fig. 3c–g). On the other hand, SRAD and OSRAD
better smooth the noise close to the lower edge of the image, thanks to their edgeenhancing feature. Even though SRAD and OSRAD use a fixed spatial support,
the latter achieves better performances since it emphasizes the smoothing behavior
along the tangential direction to the object boundary (Fig. 3h, i).
These results are in agreement with the average SSD similarity score, computed
over the 50 synthetic images, reported in Table 1 (last column). In this experiment
the reconstruction accuracy of SRBF and OSRAD are similar; however, it is worth
noting that SRBF is faster (about 20 times) and fully automatic, as summarized in
the feature description listed in Table 1.
Filter Robustness
The previous section illustrated that among the considered anisotropic approaches,
the OSRAD and SRBF filters provide the best denoising outcome. In this section we
designed a set of experiment to investigate the robustness of such filters when the
size of the speckle changes. Both filters are applied to synthetic images generated
with different speckle sizes (by varying k while keeping
γ
= 5.7) and the results are
illustrated in Fig. 4.
1
EDISON: Code for the Edge Detection and Image SegmentatiON system, http://www.caip.
rutgers.edu/riul/research/code/EDISON/index.html
2
http://viva.ee.virginia.edu/research ultrasounddenoising.html
3
http://serdis.dis.ulpgc.es/∼krissian/HomePage/
4
Implementation in Matlab software and computed on a Pentium IV dual core Intel processor.
Ultrasound Despeckle Methods 59
. Pairwise filtering comparison of (a)CLASSICBF,
0
(b) AD, (c)RMLF,(d)AF,(e)MGF,(f) GUO, (g) MEAN-SHIFT, (h)SRADand(i) OSRAD versus SRBF algorithm.
Fig. 3 Normalized Intensity profile of denoised images, superimposed on the ground-truth shape I
60 S. Balocco et al.
1,000 1.57 [s] 1,913 ±153
1,000 6.305[s] 1,635 ±168
= 0.2
Experimental Computational time SSD
=1.3 300 1.411 [s] 3,692 ±139
σ
)
σ
= 0.5 1 0.028 [s] 3,138 ±114
1
) k
1,2
= 0.05
) k
τ
1 coefficient of variation (
= 0.05
asm
as described in [53] D = 2, CT = 3.2,
,E
H
t
asm
asm
= 0.8, H
= 0.4, L
= 1
inv
t
inv
H
L
E
= 0.04 1 47 [s] 2,810±148
0
α
= 5
= 10
= 4 1 0.055 [s] 2,384 ±190
= 0.05
ax
2
τ
ax
0
σ
σ
W = 2,
lat
, k
σ
0
,
ax
σ
asm
,L
inv
,H
inv
Filter orientations
Lateral and axial Gabor
,D,CT,L
0
α
= 1 10 3.312 [s] 2,246 ± 129
σ
) N = 6,
σ
R2
R2
neighbors
Support User interaction parameters Iterations n. (128×128 pixels) (mean ±std)
–– – –– 7,104±180
0
OSRAD [48] Four neighbors Choice of uniform region Top stripe of I
GUO [52] Twenty four Fully automatic None 1 3.117 [s] 2,622 ±129
RMLF [1] Eight neighbors 2 smoothing coefficients (k
CLASSIC BF [49]UserdefinedN Fully automatic 8 6 0.364 [s] 2,736±130
Tabl e 1 Features and performance comparison of speckle reducing filters
AD [47] Four neighbors Noise variance size (
Noised image I
MGF [54] Frequency filter Probe central frequency k
AF[53] Adaptive W
MEAN-SHIFT [51]UserdefinedN Noise variance (
SRBF [55] Adaptive Fully automatic None 6 0.364 [s] 1,288 ±127
SRAD [16] Four neighbors Choice of uniform region Top stripe of I
Ultrasound Despeckle Methods 61
Fig. 4 Average SSD (a) between the echogenicity image I0and the image filtered using OSRAD
and SRBF when the speckle noise scaling factor k ranges from 0.5 to 3.5. Intensity profile of I
(b–c) superimposed on the OSRAD and SRBF filtering result when the noise scaling is equal to 1
and to 2.5, respectively.
0