Transcript of "On Convolution Model for Ultrasound Echo Signal Processing "
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Proceedings of the IEEE-EMBS International Conference on Biomedical and Health Informatics (BHI 2012)
Hong Kong and Shenzhen, China, 2-7 Jan 2012
On Convolution Model for Ultrasound Echo Signal Processing
Ming Chen and Cishen Zhang
Abstract- Systematic ultrasound signal processing and imag
ing require a model for the ultrasound echo signal so to extract
the tissue information for image construction. Based on careful
studies of the classical ultrasound acoustics and traditional con
volution model and taking into account the echo effects arising
from anatomical tissue properties, this paper proposes a new
convolution model for the ultrasound pulse-echo process, which
presents dominant terms under common practically feasible
conditions of medical ultrasound. The proposed convolution
model can provide an effective means for processing ultrasound
echo signals and constructing images.
Index Terms-Convolution, Modeling, Signal processing, Ul
trasound Imaging.
I. INTRODUCTION
Ultrasound has been an increasingly important imaging
modality in medical diagnosis since late 1930s. It facilitates
the non-invasive observation of human soft tissues as the 2nd
widely-used tool only next to x-ray. While it is applied as
a means of medical imaging without considerably adverse
effect being reported, its real-time capability makes it suit
able in diagnosis, monitoring the anatomical changes during
surgery, chemotherapy, rehabilitation therapy, and many oth
er clinical applications. Unlike the bulky and stationary Com
puter Tomography (CT) and Magnetic Resonance Imaging
(MRI), ultrasound is mobile for medical operators to move
easily and conveniently. Moreover, the cost of purchase and
operating expenses of ultrasound is incomparably lower than
CT and MRI.
One of fundamentals to ameliorate the imaging quality of
ultrasound lies in the full understanding of the ultrasound
echo process. In general, the ultrasound image degradation
is caused by the constructive and destructive interferences of
backscattered signals due to unresolved tissue inhomogeneity
[2]. An accurate mathematical model in view of signal
and system can provide fundamental comprehension of this
complicated echo process.
Since 1980s, convolution models of ultrasound echo pro
cesses have been studied and applied. Attempts were made
to ameliorate the resolution of scanners a posteriori with
digital signal processing [3]. The convolution model based
on linear system theory and ultrasound acoustics was first
systematically formulated in [5], [6], [7]. Using this model,
a number of methods and algorithms have be developed to
deal with ultrasound signal processing and imaging problems
such as despeckling and denoising [8]-[10].
Ming Chen is with School of Electrical and Electronic
Engineering, Nanyang Technological University, Singapore 639798,
anniechenming@gmail.com
Cishen Zhang is with Faculty of Engineering and Industrial Sciences,
Swinburne University of Technology, Hawthorn VIC 3122, Australia,
cishenzhang@swin.edu.au
978-1-4577-2177-9/12/$25 (C) 2012 IEEE 357
Ultrasound
Transducer
Aperture: 5
Tissue of
Interest: V
Fig. l. Coordinate system for calculating the received signal.
This paper proposes a new model of the pulse-echo pro
cess in medical ultrasound scanning. Based on the study of
classical acoustical equations, the proposed model presents
dominant terms under practically feasible conditions of med
ical ultrasound. With the pulse spread function (PSF) of the
ultrasound transducer and the radio frequency (RF) echo
signal as the input and output, respectively, the proposed
model describes the interaction between the PSF and the
scanned tissue, in terms of its acoustical velocity of propa
gation or mass density, as a convolution in the time-spatial
domain. With this model, the objective of the ultrasound
signal processing and imaging becomes to identify the spatial
distribution of the tissue mass density using the measurement
of the RF echo signal. Built upon the acoustical properties of
the ultrasound pulse-tissue interactions, the proposed model
is computationally feasible and can provide an effective
means for medical ultrasound echo signal processing and
imaging.
Section II introduces fundamentals and expressions of
ultrasound acoustics. Section ill provides important assump
tions and approximations for the rest derivations. The new
convolution model of ultrasound echo signal is proposed
in Section IV with a brief guideline of its derivation. And
Section V concludes the paper.
II. ULTRASOUND ACOUSTICS
Under the condition that there is no heat conduction or
conversion of ultrasound to thermal energy and the first-order
Born approximation [5], the two dimensional coordinate
system for representing the ultrasound echo process is shown
in Figure 1, where the left apparatus is the ultrasound
transducer with surface area S and the object on the right
is the tissue of interest. Following from the adiabatic and
hydrodynamic equations as in [l], the ultrasound echo pro-
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cess is modelled as
Pr(rO,t) (1)
where Em is the electrical-magnetic characteristics of med
ical ultrasound machine and ps(ro+ra,t) is the ultrasound
scattered field at the spatial location (ro+ra); ro refers to
the location of transducer while ra the location of transducer
aperture with respect to roo
The ultrasound scattered field can be formulated at the
point denoted by r, by integrating all the spherical waves
from the scattering region:
ps(ro,l)
where
1 _
-V'(p(r,). V'(Pi(r"tI))
Po
2c(r,) (}2Pi(r,,t,)
- � (}t2o
(2)
(3)
V represents the spatial volume of tissue, T denotes the inte
gration interval over time, Po means the average mass density
of tissue; r, denotes the field point in tissue of interest, v
refers to the gradient operation, p(r,) is the perturbation of
mass density r, and calculated as the difference between
the instantaneous mass density and the mean density of the
tissue, c(r,) is the perturbation of propagation velocity of
the tissue at r, and calculated as the difference between the
instantaneous velocity and the mean propagation velocity of
the tissue, Pi(r,,td is the incident pressure at (r,,t,) written
as
(4)
with
(5)
as the the velocity potential and v(t) as the particle velocity
normal to the transducer surface. The integrated functions
Fps(r"ro,t,t,) contains tissue information of interest such as
ultrasound propagation perturbations and gradient of tissue
density perturbations.
The analysis and formulation in this paper are based on
the ultrasound RF echo signals after beamforming. Usually,
commercial ultrasound scanners collect ultrasound radio fre
quency (RF) echo signals line by line in the axial direction,
and then combine the received lines of echo signals into 2-
dimensional echo signals. It is noted that, although the final
ultrasound images are 2-dimensional, still the ultrasound
echo signals are not collected simultaneously in both axial
and lateral direction.
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III. IMPORTANT AND FEASIBLE CONDITIONS
A. Typical acoustical data and properties of tissues
In addition to the heat exchange and Born approxima
tion conditions stated earlier, we will incorporate into the
proposed convolution model typical data of tissue acoustic
properties as given in Table 1. With these data, we exploit the
TABLE I
DENSITIES AND ULTRASOUND VELOCITY FOR SOME COMMON TISSUES
IN NATIONAL PHYSICAL LABORATORY, UK
Tissue Density Velocity of sound
(kg/m3) (m/s)
Liver 1050-1070 1578-1640
Kidney 1050 1560-1580
Muscle 1038-1056 1529-1629
Pancreas 1040-1050 1591
Skin l 1LO-1190 1729+
Spleen 1054 1567-1635
Breast 1020 1430-1570
following anatomical and mechanical properties to facilitate
derivation of the new convolution model.
B. Acoustical parameters
In theoretical acoustics [1], the dependence of the three
acoustical parameters: c(rl)' the propagation velocity of
ultrasound, p(r,), the mass density and K(r,), the compress
ibility, at the spatial location r, within the tissue is given as
1
c(r,) = ---;==:===:====:=
vp(r,)K(r,)
(6)
It follows from [2] that, if the mass density of human
soft tissue varies in a small range and the typical average
propagation velocity is assumed as 1545 mis, the tissue
compressibility can be linearized and represented by its mass
density:
dK
K = KO+- � 1/1012(1545-l.1p) m2/Nt.
dp
Substituting (7) into (6) yields
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c(rt) = ---;==:===:========:==0:=:=
vp(rI)(1545-l.1p(rI))
(7)
(8)
Let c(rl) = Co+c(rd and p(r) =
Po+p(rd where Co and
Po are average propagation velocity of ultrasound waves and
average mass density of the tissue of interest. It is straight
forward to use the first order Tailor series approximation at
Po to simplify the relationship between c(r,) and p(r,) into
where
106 (772.5-l.1po)
[Po(1545-l.1po)J3/2'
106
[Po(1545-l.1PO)]
,
/2·
(9)
It is noted that, both theoretically and practically, the ultra
sound propagation velocity depends on the variation of tissue
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mass density. In another word, if there is no change of tissue
mass density, ultrasound propagation velocity should remain
unchanged, i.e.
c(rl)=O, at p(rl)=O. (10)
This together with (S) results in
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Co=-
yl
r
p=o(;=' IS= 4== S =-=I=.I=po
=;=
)
and
(11)
In fact, when the tissue mass density ranges from 900 kg/m3
to 1200 kg/m3, the difference between a1 and Co is quite
small. Hence, the omitted term (al- co) can be regarded as
the error of linearization.
C. Particle Velocity
The ultrasound scanning for B-mode imaging is a se
quence of repetitive operations and each operation is com
posed of a short duration pulsing time and a long duration
listening or measurement time. During the pulsing time, the
ultrasound probe produces and transmits an ultrasound pulse
to the body tissue controlled by the electronic circuit of
the scanner. In commercial ultrasound scanners, the pulses
produced are characterized by its centre frequency, typically
between 3- 10 MHz. Thus it is practically feasible to model
the particle velocity v(t) normal to the surface of the trans
ducer as a short duration envelop function g(t) modulated by
a trigonometry function of the centre frequency. It is noted
that the particle velocity v(t) is a different physical quantity
from the propagation velocity c(r).
Without loss of generality, the particle velocity function
can be written as
v(t)=g(t)sin(2n fot), (12)
where g(t) is the envelop function of the particle velocity
and fo is the centre frequency of the ultrasound probe.
The envelop function g(t) is of short time duration and
its components in the frequency domain are at significantly
lower frequency than the centre frequency.
It is assumed that the envelop function g(t) satisfies
This assumption has been verified to be feasible by the ve
locities with different shapes of envelope such as sinusoidal
function and Gaussian-shaped sinusoidal function. The en
velop function based formulation of ultrasound echo process
has been recently applied to process clinical ultrasound data
produced excellent imaging result [11].
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IV. NEW CONVOLUTION MODEL OF ULTRASOUND ECHO
SIGNALS
This section provides a brief outline of the derivation of
the proposed convolution model. Incorporating the conditions
and results provided in Section III, the integral solution for
Fps(r],ro,t,t]) in (3) is
S
c(rt)po n2 fJ
(
cot- rl
)--- --g --- x
c6 r] Co
cot- r]
cos(2n fo(--
))
·Co
(14)
This is used to solve the function ps(ro+ra,t) in (1) resulting
in
r r (_ 2c(rJ) d2Pi(rl,t])
) X
iviT c6 d�
8(t- t _ Iro+ra-r11)I co 3 �
---::---,----'-"-.,---dtldr]. (1S)
2nlro+ra- r]1
With the original acoustical model (1)-(S) and all the
derivations before, ultrasound echo signals are mainly de
termined by ultrasound perturbation of propagation velocity
and the effects from the gradient of tissue perturbation of
mass density are negligible. Hence, ultrasound echo signals
can be approximated as
Pr(ro,t)=Em(t) *
rp,(ro+ra,t)d2ra, (16)
tis
where * denotes the convolution over time and
I
r r [_ 2c(rt) d2Pi(rl,tI)
]
iviT c6 dt2
8(t- tl _ Ira+ro-r11)
---'-::--;-----'
c"'-
o
-..,--'- dtld3rl. (17)
2nlra+ro- r]1
As a result, the echo signal can be modeled as the
following convolution model
where * denotes the spatial convolution,
r
PO d3V(t)
vcc(t)=2" ----=;--:
Co ot
is the pulse-echo wavelet of the excitation,
c(rl)
fcc(rl)=---
Co
(1S)
accounts for the inhomogeneities in the tissue due to propa
gation velocity perturbations which give rise to the scattered
signal, and hpe is the modified pulse-echo spatial impulse
response that relates the transducer geometry to the spatial
extent of the scattered field.
In the proposed convolution model (1S), the tissue pa
rameter is represented in terms of the perturbation of the
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propagation velocity c(rl) which can be a distribution of the
tissue characteristic to be estimated for image construction.
In practice, it may be comparatively more convenient to
represent c(rl) in terms of the perturbation of tissue mass
density p(rl) using relation (11) between the two physical
parameters.
We have conducted extensive simulation to verify our
developed convolution model and the results obtained can
verify its feasibility and effectiveness.
Remark: The convolution model (18) is derived based on
the conditions and approximations described in Section III.
This model appears in a similar form as the conventional
model in (ref[5]) which has been popularly known for a
long time since presented. We would point out that our
recent and careful studies have found that there exist sub
stantial mathematical problems in the original derivations
of the convolution model in [5], so this model is not fully
theoretically sound and valid. Consequently, the techniques
and algorithms for ultrasound echo signal processing and
imaging based on such convolution model no longer have
a theoretical foundation. Details of our finding and analysis
will be further reported.
V. CONCLUSION
This paper has presented a new convolution model for
medical ultrasound echo signals. It is based on careful
studies of the classical ultrasound acoustics and traditional
convolution model. The proposed model takes the acoustical
and practical conditions of medical ultrasound into account
and evaluates the echo effects arising from different anatom
ical tissue properties. In reasonable clinical circumstances,
ultrasound echo signals can be characterized in terms of the
perturbation of the ultrasound propagation velocity or mass
density of the tissue. With the new model, the convolution
framework of ultrasound echo signal processes remains valid
despite the mathematical flaw in the derivation of the con
ventional convolution model.
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