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RF Ultrasound Estimation from B-Mode Images 21
Fig. 9 Application of the decompression method to BUS image of the liver (a) and carotid plaque
(b). The plaque contour is marked for ease of visualization
Rayleigh distribution provides a good description of the data in a very significant
part of the images, essentially where strong scattering phenomena do not occur.
Moreover, when the local comparison between the Gamma and Rayleigh distributions is carried out, it is observed that in most regions of the studied images, the
Rayleigh distribution closely approaches the Gamma distribution. The only exceptions occur in regions of substantial echogenicity, where the Gamma distribution is
more suitable to describe the data. These results validate the adopted decision and
does not include in the proposed decompressionmethod the interpolation operation.
This operation is the source of the Gamma distribution, but as it was confirmed in
22 J. Seabra and J.M. Sanches
a
Rayleigh, avg. ccoeff: 0.695
Goodness of Fit Test with...
Gamma , avg. ccoeff: 0.9 13
0.5
0
−0.5
0.8
0.6
0.4
0.2
0
Gamma vs. Rayleigh,
avg. ccoeff: 0.835
0.5
0
−0.5
Liver
b
Rayleigh, avg. ccoeff: 0.507
Gamma , avg. ccoeff: 0.7 61
0.5
0
−0.5
0.5
0
−0.5
Gamma vs. Rayleigh,
avg. ccoeff: 0.765
0.8
0.6
0.4
0.2
0
Carotid plaque
Fig. 10 Color-scaled maps of the GoF test when the data is locally compared with ML Rayleigh
distribution (left), Gamma distribution (middle). GoF map associated with the local comparison
between ML Rayleigh and Gamma local distributions (right)
this last comparison study, the simpler Rayleigh distribution is able to describe the
data in almost all regions of the images but at the transitions.
6 Conclusions
Standard ultrasound equipment performs nonlinear compression of the envelope
data thus changing some of its attractive statistical properties.
This chapter proposes a statistical model for log-compressed BUS data which
allows to parameterize the most significant operating settings of ultrasound equipments and revert the nonlinear compression, providing an estimate of ERF data.
The estimated envelope intensity can be used by a variety of algorithms that rely
on the statistics of the ultrasound signal. These include segmentation and speckle
tracking algorithms, speckle reduction methods (proposed in the next chapter),
tissue classification methods, etc.
The method here presented relies on statistics of the compressed signal, which
follows a double-exponential distribution and makes use of a realistic mapping
function, designated as LCL, first proposed in [20] which is able to provide an
estimate of the ERF imagegiven thatparameters related to dynamicrange and linear
gain are known. The decompression method makes use of this prior knowledge to
accurately estimate such parameters and recover the ERF image.
RF Ultrasound Estimation from B-Mode Images 23
Experiments performed in synthetic and real data show the accuracy of the
estimates obtained for the decompression parameters. Moreover, this method is
robust because it is able to provide similar outcomes for images acquired with
different operating settings. On the other hand, similar decompression parameters
were obtained for different images acquired with fixed operating settings.
The Rayleigh distribution has shown to correctly describe the ERF estimated
data which has important consequences in the assumptions made for designing the
decompression method presented in this chapter.
Finally, a study recently presented in [29] compared the compression parameter
estimation of the well-established method proposed in [7, 8] with the approach
described in this chapter, observing that the later provides better results in terms
of parameter estimation accuracy. As pointed out in [29] this could be explained
as the decompression method proposed in this chapter is based on the statistics
for the compressed signal, while the approach presented in [7, 8] uses statistics for
the uncompressed signal, and attempts to match theoretically calculated normalized
moments with those determined directly from the image. The process of fitting
the moments calculated in the image with theoretical moments of the exponential
distribution (cf. [8]) is extremely sensitive to the order of the moment n, and this
could create uncertainty on the decompression parameter to be estimated.
References
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study. Serbian J Electr Eng 6(3):419–425
A Rayleigh Mixture Model for IVUS Imaging
Jos´e Seabra, Francesco Ciompi, Petia Radeva, and Jo˜ao Miguel Sanches
Abstract Carotid and coronary vascular problems, such as heart attack or stroke,
are often originated in vulnerable plaques. Hence, the accurate characterization of
plaque echogenic contents could help in diagnosing such lesions.
The Rayleigh distribution is widely accepted as an appropriate model to describe
plaque morphology although it is known that other more complex distributions
depending on multiple parameters are usually needed whenever the tissues show
significant heterogeneity.
In this chapter a new model to describe the tissue echo-morphology by using a
mixture of Rayleigh distribution is described. This model, called Rayleigh Mixture
Model (RMM), combines the robustness of a mixture model with the mathematical
simplicity and adequacy of the Rayleigh distributions to deal with the speckle
multiplicative noise that corrupts the ultrasound images.
The method for the automatic estimation of the RMM mixture parameters by
using the Expectation Maximization (EM) algorithm is described.
The performance of the proposed model is evaluated with a database of in-vitro
IVUS samples. We show that the mixture coefficients and Rayleigh parameters
explicitly derived from the mixture model are able to accurately describe different
plaque types and to significantly improve the characterization performance of an
already existing methodology.
J. Seabra () • J.M. Sanches
Institute for Systems and Robotics, Department of Bioengineering from the Instituto
Superior T´ecnico/Technical University of Lisbon, Portugal
e-mail: mail2jseabra@gmail.com; jmrs@ist.utl.pt
F. Ciompi • P. Radeva
Computer Vision Center, Campus UAB, Edifici O, Bellaterra, Spain
University of Barcelona, Gran Via de Les Cortes Catalanes, 585, 08007 Barcelona, Spain
e-mail: fciompi@maia.ub.es; petia@cvc.uab.es
J.M. Sanches et al. (eds.), Ultrasound Imaging: Advances and Applications,
DOI 10.1007/978-1-4614-1180-2
2, © Springer Science+Business Media, LLC 2012
25
26 J. Seabra et al.
1 Introduction
Atherosclerotic plaques may eventually present high risk of rupture, consequently
leading to brain stroke or heart attack [1]. Albeit vulnerable plaque is a concept
well accepted as a clinical entity with potential harmful consequences, its echomorphology, and pathological evolution it is not yet well understood. Hence, it is
important to objectively characterize the plaque echo-morphology to identify this
kind of lesions and develop or refine methods for risk prediction.
Ultrasound images are corrupted by a characteristic granular pattern, called
speckle [2], that depends on the number of scatterers (reflectors) as well as their
size. This speckle signal is usually considered noise and there is a lot of work in
the literature proposing methods to its removal [3–6]. However, speckle encodes
information about tissue acoustic properties [7] that can be used for diagnostic
purposes.
As pointed out in [8], features extracted from these noisy images can be considered as tissue histological descriptors. Moreover, IVUS is an imaging technique
which enables to clearly assess the arterial wall internal echo-morphology. The
technical procedure of acquiring IVUS data consists in introducing a catheter, carrying a rotating ultrasound emitter inside the vessel. During rotation, a piezoelectric
transducer transmits US waves and collects the reflected components which are
afterward converted into electrical signals (A-lines) and sampled by an Analog
to Digital Converter (see Fig. 1b). The IVUS image is obtained by processing
the received echoes is a 360-degree tomographic view of the inner arterial walls
(Fig. 1a). The proximity of the ultrasound probe from the inner arterial walls makes
Fig. 1 (a) IVUS image represented in cartesian coordinates and (b) its corresponding polar
representation;
the probe
ρ
represents the depth in the tissue andθthe position (angle) in the rotation of
A Rayleigh Mixture Model for IVUS Imaging 27
tissue sample
Mixture of
single
Rayleigh
Distribution
Fig. 2 Hypothetical acoustic tissue model, including different scattering phenomena which points
out the need for using a mixture model of distributions
Rayleigh
Distribution
it possible to use high frequency US probes and therefore obtain high quality
US images. Consequently, IVUS is commonly considered a suitable technique for
accurate in-vivo characterization of the coronary plaques composition [9].
Studies which rely on tissue appearance [10, 11] were pursued to qualitatively
and subjectively characterize plaque echo-morphology as soft (echolucent), fibrous
(intermediate echogenicity), mixed (several acoustical subtypes), and calcified
(strongly echogenic). Given the high variability in tissue appearance, the IVUS
imaging parameters (such as brightness and contrast) are often tuned to improve
visualization. This pre-processing operation modifies the IVUS signal properties,
hinders the comparison of tissues on different images and prevents the application
of appearance-based methods. Thus, analysis from RF data is needed to obtain
discrimination of plaques. Recently, automatic quantitative methods for plaque
characterization have been proposed, based either on high-order statistical texture
analysis [12–14] and on spectral featuresextracted from the raw RF signals acquired
by the IVUS equipment [15–17].
The work presented in this chapter aims to model the atherosclerotic plaque
through the analysis of the envelope backscattered IVUS data. For this purpose,
an hypothetical model is considered where a scanned tissue sample suffers from a
certain number of scattering phenomena, as depicted in Fig.2.
28 J. Seabra et al.
The commonest modelfor speckle formationis known as fully [18] and considers
a tissue orregion composedby a large number ofscatterers, actingas echo reflectors.
These scatterers arise from structural inhomogeneities with size approximately
equal or smaller than the wavelength of the ultrasound, such as in the parenchyma,
where there are changes in acoustic impedance on a microscopic level within the
tissue. Under fully developed speckle, pixel intensities in envelope images are well
modeled by Rayleigh PDFs [2, 19]. When this condition does not hold, other more
complex parametric models, such as K [20], Rician [21], homodyned-K [22], and
Nakagami [23] are suitable to describe the data.
The motivation to use the single parameter Rayleigh distribution comes from
the fact that the regions defining atherosclerotic tissue are piecewise homogeneous
and do not present strong scatterers nor edges, as it happens across the rest of the
image, where other speckle conditions are verified and other statistical models are
more convenient. These other models, such as Rice, K or Nakagami distributions,
depend on a large number of parameters which makes the estimation of tissue echomorphology a hard task.
Plaque echo-morphology may result from different types of components, spatial
organization, and complexity which determine different scattering phenomena
where the Rayleigh distribution would be a reasonable approximation but a compound statistical model would be more appropriate. Hence, the description of tissue
echo-morphology may be tackled with complex distributions depending on multiple
parameters or with a mixture of simple distributions. In this chapter a combination
of Rayleigh distributions, called Rayleigh Mixture Model (RMM), is proposed to
describe the tissue echo-morphology in atherosclerotic plaques from IVUS images.
The coefficients of the mixture are estimated with the Expectation Maximization
(EM) algorithm [24] adapted to this kind of mixture.
The RMM consists of a technique to describe a particular data distribution by
linearly combining different Rayleigh PDFs. Up to our knowledge, the RMM was
never used for tissue characterization in ultrasound, although these models have
been successfully employed in other fields, such as in underwater acoustics and
speech processing problems [25,26].
This chapter is organized as follows. First, in Sect. 2.1 a comprehensive mathematical formulation of the mixture model is provided, using the EM algorithm
for estimating the coefficients and Rayleigh parameters of the mixture. Second,
the adequacy of the proposed model to describe the envelope ultrasound data is
evaluatedon validatedIVUS data of different plaquetypes (Sect. 3.3). Moreover,the
RMM is applied for modeling plaques as monolithic objects, i.e., by considering all
the pixels enclosed in the plaque. The features explicitly obtained from the mixture
model (cf. Sect. 2.1) are used to investigate the discriminative power of the model
for identifying different tissue types, namely fibrotic, lipidic, and calcified ones. In
Sect. 3.4 the ability of the RMM for pixel-wise classification of plaque composition
is evaluated when using only the proposed features and when combining them
with textural and spectral features recently proposed [27]. Finally, we investigate
the significance of the obtained classification improvement when using the RMM
features (cf. Sects.3.5 and 3.6).
A Rayleigh Mixture Model for IVUS Imaging 29
2 Methods
In this section a mathematical formulation of the problem is provided and the
estimation algorithm for the coefficients of the mixture (weights) and Rayleigh
parameters of each component, by using the EM algorithm, is described.
2.1 Rayleigh Mixture Model
Let Y = {yi},1 ≤ i ≤ N, be a set of pixel intensities of a given region of interest,
particularly a plaque, from an ultrasound image. Pixel intensities are considered
random variables which are described by the following mixture of L distributions
L
p(y
|Ψ)=
i
∑
j=1
θjφ
), (1)
j(yi
Ψ=(θ
where
coefficients of the mixture and
φ
)=p(yi|
j(yi
The condition
1
σ
,...,
),
j
θ
,
σ
,...,
σ
L
1
L
∑
j=1
) is the vector of parameters to estimate.
L
σ
are the parameters of each Rayleigh component,
j
y
)=
j
i
2
σ
j
exp−
p(y
|
σ
i
θ
= 1 must hold to guarantee that p(yi|Ψ) is a true
j
2
y
i
, (2)
2
2
σ
j
θ
j
are the
distribution function.
The parameters
properties of the tissue at the ith location [28]. The effect of changing
σ
associated with the pixel intensity yi, characterize the acoustic
j
σ
in the
shape of the distribution and thus in the image intensity is illustrated in Fig. 3.The
joint distribution of the pixel intensities, considered independent and identically
distributed (i.i.d.), is given by,
N
p(Y|
Ψ
The goal is to estimate
where
L (Y,
Ψ
by maximizing the likelihood function,
ˆ
Ψ
= arg maxΨL (Y,Ψ), (4)
ML
Ψ
)=log p(Y|Ψ)=
)=
p(yi|Ψ). (3)
∏
i
N
∑
i=1
log
L
∑
j=1
θ
jpj(yi
|
σ
). (5)
j
30 J. Seabra et al.
0.06
0.05
0.04
0.03
p(y)
0.02
0.01
0
0 50 100 150 200 250
Fig. 3 Rayleigh PDFs generated with parameter 102<σ< 103(from darker to lighter curves)
y
The maximization of (5) is a difficult task because it consists of a logarithmic
function of a sum of terms. To overcome this difficulty the EM [24] method is
used where a set of hidden variables are introduced, K = {k
The value of k
pixel intensity, y
= j informs us about the mixture component j that generated the ith
i
, with probability p(yi|
i
σ
) defined in (2).
k
i
} with ki∈{1, ...,L}.
i
Each nth iteration of the EM method is composed of two steps
• Estep:Where the expectation of the new likelihood function, L (Y,K,
computed with respect to K,
Q(Y,
n
Ψ
,Ψ)=EK[L (Y, K(
n
Ψ
),Ψ)] (6)
and
• Mstep:Where a new estimate of
Ψ,Ψ
, is obtained by maximizing the
n+1
function Q,
n+1
Ψ
= arg maxΨQ(Y,
n
Ψ
,Ψ). (7)
These two steps alternate until convergence is achieved, which happens when
n+1
|
Ψ
n
−
Ψ
| <ξ, e.g.,ξ= 10−3.
The likelihood function involving all unknowns, visible and hidden, is
N
L (Y, K,
Ψ
)=log p(Y, K|Ψ)=
log p(yi,ki|Ψ)
∑
i=1
N
log p(yi|
=
∑
i=1
σ
)+log p(ki|
k
i
σ
k
θ
k
i
)
, (8)
i
Ψ
),is