2. Abstract
Ultrasound has been used in a variety of clinical settings, including obstetrics and
gynecology, cardiology and cancer detection. The main advantage of ultrasound is
that, unlike xray imaging, it does not require ionizing radiation, which may
increase the risk of getting cancer. In addition, using ultrasound imaging as a
medical diagnostic tool is more cost effective compared to other radiation free
diagnostic techniques, such as the MRI. However, a difficult and crucial challenge
in ultrasound medical imaging is the necessity of reducing the appearance of
speckles in Bscan images. Speckles are caused by coherent interference of
reflected ultrasound waves by structures smaller than the Rayleigh scattering limit.
Appearance of speckles tend to reduce the perception of small structures, and
ultimately limit diagnostic accuracy of medical imaging system. Since it is difficult
to model speckles using architectural structure of tissues, modeling techniques
involve a more constructive approach: predicting the speckle pattern of an arbitrary
scattering medium for a given transducer geometry; such a model was developed
by Foster et al. (1983).
Investigation of the origin and nature of speckles involve creating a mathematical
model to predict the RF output signal waveform of a particular transducer
geometry, and then extending that model to incorporate the effects of varying sizes
of scatterers on the RF voltage trace. The complete model will then be verified by
comparing theoretical results with experimental results obtained from pulseecho
experiments. For this project, we have focused more on the experimental aspect of
the investigation by producing Bmode images of scatterer phantoms. These
phantoms are standard samples made using the discrete scatterer model, where
scatterers of uniform shape and size are embedded in a homogeneous medium.
Before imaging scatterers smaller than the Rayleigh scattering limit, it is important
to ensure that scatterers much larger than the Rayleigh limit (discrete scatterers)
can be successfully imaged. We conducted pulseecho experiments using a 5 MHz
singleelement focused transducer, and imaged phantoms containing discrete
scatterers of the following sizes: 4.76 mm metal beads, 2.85 mm 3.45mm glass
beads and 1.00 mm 1.03 mm glass beads. The RF voltage traces from the
pulseecho experiments were then converted to Bmode images using a Matlab
program.
2
3. Acknowledgements
First of all, I would like to thank my advisor Dr. Maria Teresa Herd
for giving me the opportunity to do research in her lab. Without her continuous
guidance and support, this thesis would not have been possible. I would also like to
thank her for encouraging and helping me at times when I lost faith in my abilities
over the past year. She has truly been a great role model and mentor.
I would also like to thank the Physics Department for fostering my love for
physics and passion for research. I would like to thank all the professors: Mark
Peterson, Juan Burciaga, Neal Abraham, Spencer Smith, Alexi Arango and Kathy
Aidala for their enthusiasm and inspirational way of teaching.
I am grateful to my lab mates Madeleina, Audrey and Colbie. Thank you,
Audrey and Madeleina for allowing me to observe your experiments, and helping
me learn the fundamentals of handling lab equipments, acquiring and analyzing
data. Thank you Colbie for your assistance during the first half of my project,
when we spent several hours at a stretch pouring over books looking for missing
functions for the RF voltage trace model.
Next, I would like to thank my brother for always pushing me to become a
better physicist than him. I would also like to thank my boyfriend for his constant
encouragement, and for always being available to listen to me whine, complain
and cry over the phone whether he is at home or at an observing run at Mauna Kea
or Coonabarbaran.
Finally, I have to thank my mom for giving me full freedom to explore my
passion and pursue my dreams, no matter how crazy they are. Without her
unconditional support, guidance, and love, I would not have come this far.
Once again, a big thank you to each and everyone!
3
4. Contents
1 Introduction to PulseEcho Ultrasound Imaging 7
1.1 Making Ultrasound Bmode Images …………….. 7
1.2 Speckles in Ultrasound Images ….………………. 9
1.3 Importance of Modeling Speckle ………………... 10
1.4 Current Theoretical Models for Speckles ………... 11
1.4.1 Discrete Scatterer Model ……………... 12
1.4.2 Inhomogeneous Continuum
Model …………………………………. 12
1.5 Modeling Backscattered RF Trace
of Human Tissue ………………………………… 13
1.6 Complex Plane Representation of
Ultrasound Point Spread Function ………………. 14
2 Theory of Wave Propagation: Physics of Ultrasound 15
2.1 Classical Theory of Linear Acoustics …………….. 15
2.1.1 Equation of State ………………………. 16
2.1.2 Equation of Continuity ………………… 18
2.1.3 Euler’s Equation ……………………….. 18
2.2 The Wave Equation ……………………………….. 20
2.3 Wave Equation in the Frequency
Domain ……………………………………………. 21
2.3.1 The Incident Pressure Field ……………... 23
2.3.2 The Scattered Pressure Field …………… 25
2.3.3 The Force on the Receiving Crystal ……. 26
2.3.4 The RF Voltage Trace ………………….. 28
4
7. Chapter 1
Introduction to PulseEcho Ultrasound
Imaging
1.1 Making Ultrasound Bmode Images
Ultrasound has been used in a variety of clinical settings, including obstetrics and
gynecology, cardiology and cancer detection. The main advantage of ultrasound is
that structures can be observed without using radiation. Using ultrasound medical
imaging as a diagnostic tool is also more cost effective compared to other radiation
free diagnostic techniques, such as the MRI. Ultrasound can also be done much
faster than Xray and other radiographic techniques.Ultrasound also provides
multiple modes available for versatile imaging. For instance, Mmode ultrasound
is extremely valuable for accurate evaluation of rapid movements due to its
excellent temporal resolution, and is used for tracking the motion of heart
structures over time.
For the purposes of modeling speckles, we are interested in producing
Bmode images. This is because Bmode images can be easily produced using the
amplitude of reflected ultrasound signals to reflect the intensity or brightness on a
twodimensional gray scale. Owing to a wide gray scale available for ultrasound
imaging, very small differences in echogenicity are possible to visualize.
The essential component of an ultrasound medical imaging system is a
transducer. Transducers are devices that convert electrical signals to mechanical
energy and vice versa. When electrical signals reach a transducer in the form of an
excitation pulse, these signals are converted to sound waves that travel to the
structure being imaged. All of the sound waves are then reflected by the structure,
some of which eventually find their way back to the transducer and the sound
waves are then converted to electrical signals. These electrical signals are collected
7
8. in the form of radio frequency (RF) voltage traces that are processed to produce
Bmode images of the structure.
The basic components of a typical spherically focused singleelement
transducer are summarized in Figure 1. The active piezoelectric element has a
spherically curved shape to facilitate the focusing of the beam, and its thickness is
usually half the wavelength of the ultrasound produced. It is important to note that
the transducer generates a range of frequencies around a central frequency. This is
in part due to the fact that the piezoelectric element continues to produce
ultrasound for sometime even after the electric signal has been terminated.
There are a number of parameters, which are useful for understanding the
general components of a singleelement transducer. The range of frequencies
generated by the transducer is known as the bandwidth. The quality factor of a
transducer is a dimensionless parameter, which is calculated as the ratio of the
central frequency to the bandwidth; the higher the quality factor, the lower the rate
of energy loss compared to stored energy. Finally, the sensitivity of the transducer
is defined as the ability to detect reflected ultrasound and generate electrical signal.
The sensitivity and bandwidth of the transducer are dictated by the parameters of
the piezoelectric material, transducer backing and electrical and acoustic
impedance matching devices, which are needed to achieve the specific imaging
requirements.
8
9.
Figure 1 : Single Element Ultrasonic Transducer
There is a fundamental relationship between the properties of the generated
ultrasound beam and the quality of the resulting image. The ability to predict beam
characteristics is crucial for medical imaging. There are many factors, which affect
the beam profiles of transmitted and received signals. Most important are the
source geometry, source excitation (sinusoidal or pulsed), target response and the
type of electronic processing applied to signals. The theoretical treatment of beam
profiles is based on Huygen’s Principle, where each radiating source is divided
into infinitesimal elements, each radiating hemispherical wavelets. The resulting
acoustic fields obtained from the superposition of these wavelets obey the classical
theory of linear acoustics described in Chapter 2.
A central aspect of quantitatively characterizing the beam profile is by
calculating the instantaneous pressure field distribution. The function for
calculating the instantaneous pressure field is obtained using the wave equation for
the velocity potential (discussed in detail in Chapter 2). Determining the
instantaneous pressure for incident and scattering field paves the way for the
calculation of pulseecho sensitivity profiles based on the spatiotemporal impulse
response of the transducer (discussed in detail in Chapter 2).
1.2 Speckles in Ultrasound Images
A difficult and crucial challenge in ultrasound medical imaging is the necessity of
reducing the annoying “mottled” or “speckled” appearance of Bscan images.
Speckles are an interference phenomena caused by the interaction of coherent
waves produced by the transducer with multiple scattering structure of the tissue.
Such interactions result in the constructive and destructive coherent summation of
9
10. ultrasound echoes. Speckles tend to reduce the perception of small structures, and
ultimately limit diagnostic accuracy of medical imaging systems.
Generally there is no common enhancement approach for speckle noise
reduction. Different filtering techniques based on statistical methods have been
implemented in an attempt to reduce speckles, such as SignaltoNoise Ratio
(SNR), Peak SignaltoNoise Ratio (PSNR) and Root Mean Square Error (RMSE).
Another important averaging technique used for speckle noise reduction is
angularweighted compounding, which stems from ultrasound elastography.
Elastography is an imaging technique in which local strain in human tissues are
used to measure the axial shifts in tissue due to quasi static compression applied
using ultrasonic transducers; all the measurements are made in terms of RF echo
arrival times. Using elastography measurements, angularweighted factors are
derived from the relationship between axial and lateral strain components of strain
estimated along angular insonification directions. Experimental results using a
uniformly elastic tissue mimicking phantoms demonstrated the improvement in
signaltonoise ratio obtained using angularweighted compounding (Techavipoo
et. al. 2004). However, removing speckle noise from original ultrasound images
still remains a challenge in image processing.
1.3 Importance of Modeling Speckles
Ultrasound imaging is a useful diagnostic tool, especially for biological tissues and
subcellular structures. However, ultrasound medical imaging is limited by two
important factors: spatial resolution and fluctuations in echo amplitude, known as
speckle noise. Speckles are considered one of the major obstacles in image
analysis, which include detection, classification and segmentation problems in
coherent imaging systems.
Since speckle appears in all conventional ultrasound images, numerous
investigations have been carried out to understand the nature of speckles and what
10
11. causes them. Establishing a model for ultrasound imaging would require
developing a mathematical description of the acoustical scattering properties of
tissues; however, this is often hindered due to the lack of knowledge concerning
cellular structures. Kossoff et al. (1976), attempted to relate speckles to the
architectural structure of tissues, while Linzer et al. (1979) had tried to relate
speckle patterns to the different diseased states of soft tissue. But in both studies it
was found that the speckle pattern bore little resemblance to the actual acoustical
tissue microstructure. Rather than directly using speckle patterns to decipher
details of complex cellular structures, a more constructive approach would be to
first predict the speckle pattern of an arbitrary scattering medium for a given
transducer geometry. Therefore developing theoretical models to describe the
scattering medium will help us understand the relationship between the imaging
system and the speckles that arise in the final image.
Another reason for modelling speckles is that they are the primary
mechanism for elastography imaging, an emerging and exciting use of ultrasound
imaging technique. In elastography, it is possible to trace localized elastic
characteristics because local movement is estimated using speckle tracking (Kim et
al. 2011). The application of speckles in elastography confirms that speckles are
not always undesirable. Today, ultrasound elastography has developed into an
image modality suitable for detection and diagnosis of cancers in the breast,
kidneys and thyroid.
1.4 Current Theoretical Models for Speckles
Theoretical models used to describe the scattering medium can be essentially
divided into two categories: the discrete scatterer model and the inhomogeneous
continuum model. For the discrete scatterer model, the scattering medium is
depicted as a collection of points, spheres or cylinders embedded in a
homogeneous medium. On the other hand, the inhomogeneous scattering model
11
12. describes the scattering medium to be one where acoustical inhomogeneities, such
as density, compressibility and velocity, vary in a continuous manner throughout
the medium. Such models of scattering medium are then used in conjunction with
simulation models to investigate both theoretically and experimentally the effect of
speckles on ultrasound images.
1.4.1 Discrete Scatterer Model
Foster et al. (1983), in their study to investigate the effect of transducer geometry
and the position of the point spread function on speckle formation, developed a
mathematical model to simulate ultrasound Bscan images showing speckles. Prior
to the experimental approach adopted by Foster et al., simulation models were
based on two dimensional scattering medium. The dimension perpendicular to the
image plane was ignored for the two dimensional models; thus the number of
scatterers in a given distance offaxis remained constant in those models. Previous
models also excluded transducer geometry and the position dependence of the
point spread function. Foster et al. were able to recognize the limitations of the
twodimensional models, and thus incorporated the threedimensional
characteristics of both the transducer and the scattering medium. The scatterer
phantoms were developed using the discrete scatterer model, and their studies
showed good agreement between theoretical and experimental results.
1.4.2 Inhomogeneous Continuum Model
In recent years, other approaches were implemented for modeling speckles. One
such approach relies on the surface roughness and the number of elemental
scatterers in the tissue surface. In their study to derive stochastic models for
speckle noise, Daba et al. (2009) outlined two distinct categories for describing
speckle distribution in conjunction with the roughness of scattering surface: fully
12
13. developed, where the surface is comprised of infinite number of scatterers making
the surface very rough, and partially developed, where the surface has finite
number of scatterers. Daba et al. derived the statistical distribution for the partially
developed model using Poisson point process, where each scatterer is assumed to
have statistically independent random variables, which for the purposes of the
theory were amplitude and phase of the backscattered field.
In order to obtain ultrasound images of a particular organ or tissue structure,
a pulse is emitted into the body and is scattered and reflected by density and
propagation velocity perturbations. Therefore developing a model for the received
pulseecho pressure field of a transducer can also help to predict the speckle
pattern from a scattering medium. Using this approach, Jensen (1990) derived an
inhomogeneous wave equation to describe the propagation and scattering of
ultrasound by a collection of point scatterers in a homogeneous medium. The
solution to the wave equation was then combined with the field of a typical
transducer to develop a model for the pulseecho pressure field. This approach
proved to be versatile because it enabled the derivation of scattered field for
different transducer geometries, whose incident pressure fields are known.
1.5 Modeling Backscattered RF Trace of Human
Tissue
Instead of modeling arbitrary scattering medium, like those described in Sections
1.4, Gore and Leeman (1976) developed a realistic approximation of human tissue
inhomogeneities. Their approach involved describing wave propagation in
inhomogeneous media, and then showing that the waves are scattered by
fluctuations in density and compressibility. The model is then used to calculate the
precise form of the backscattered field from tissues, which coincides with actual
signals recorded by diagnostic pulseecho equipment.
From their studies it was found that the usefulness of characterizing human
tissues is dependent on the nature of the structure of the tissue itself. They
13
14. concluded that although two different tissues may have different compressibility
coefficient and density, the backscattered echo sequences are rotationally invariant
to first order. For less restricted systems, without an isotropic distribution of
scatterers, the rotational invariance does not hold; this indicated that there is an
angular dependence of the scattered field—this further confirms why
angularweighted compounding is effective for speckle reduction. Therefore, the
studies concluded that in order to understand the diagnostic capability of grey scale
Bscan images, a knowledge of the scattering processes within the tissues and
quantitative image analysis in conjunction with pattern recognition is important.
1.6 Complex Plane Representation of Ultrasound
Point Spread Function
Conventional ultrasound imaging probes a medium with high frequency
bandlimited acoustic waves, and detects echoes scattered by inhomogeneities
within the medium. A spherical single element transducer placed in the medium is
used for both generation of pulses and reception of these echoes. The site of the
transducer responsible for both transmission and reception is known as the active
aperture, which is a set of piezoelectric crystals. The scattered echoes are summed
up coherently to yield a single RF voltage trace.
In their study, Ng et al. (2005) solve the equation that governs wave
propagation in an inhomogeneous medium to show that the RF ultrasound signal
can be expressed as the result of filtering the tissue reflectivity by a point spread
function and the total pressure field. Ng et al. (2005) further extended their
analysis by establishing a link between the RF signal and the representation of
point scatterers as vectors with random phase in the complex plane. Their analysis
led to an insight into the useful techniques for simulation and analysis of speckle
statistics in the complex plane, in addition to developing a normalized covariance
of the RF signal in terms of the complex envelope of the point spread function.
14
15. The mathematical techniques implemented by Ng et. al. (2005) is relevant to
the modeling speckles because we in the future we aim to carry out impulse
response and instantaneous pressure field calculations in the frequency domain to
obtain the RF output voltage response of the transducer.
15
16. Chapter 2
Theory of Wave Propagation:
Physics of Ultrasound
Acoustic refers to the generation, transmission and reception of energy in the form
of vibrational waves in matter. Acoustic waves are the organized vibrations of the
molecules or atoms of the medium that supports the propagation of these waves.
Acoustic plane waves possess a particular frequency depending on the source of
the waves, and waves whose frequency is beyond the audible range of 20 KHz are
known as ultrasound waves.
2.1 Classical Theory of Linear Acoustics
In general plane acoustic waves through an acoustic medium is a combination of
transverse and longitudinal motion, although longitudinal motion seems to
dominate. The characteristic properties of plane waves is that each acoustic
variable, such as particle displacement, density and pressure, has constant
amplitude in any given plane perpendicular to the direction of propagation. An
expression describing the motion of a particle in a medium in the presence of
acoustic plane waves is given by:
where is the equilibrium displacement of the particle, is the particle
displacement from the equilibrium position, is the angular frequency and φ is
the arbitrary phase factor.
The above equation is one of the many solution of the wave equation. In the
following sections we will derive the wave equation to relate the changes in
16
17. acoustic variables to one another. To derive a wave equation, we utilize the
following:
● Equation of State, which relates pressure to density.
● Equation of Continuity, which incorporates conservation of mass.
● Euler’s Equation, which is derived from Newton’s Second Law.
2.1.1 Equation of State:
The relationships between pressure and density changes in a gas can also be
applied to sound traveling through gases, such as air. In acoustics the equation of
state for an ideal gas is given by:
where γ is a gasdependent constant, and T k is the temperature in Kelvin. However,
for most fluids and tissues, the adiabatic equation of state is either unknown or
highly complex. Thus, this relationship is determined experimentally, and verified
mathematically using the following Taylor Series expansion:
where the partial derivatives are constants determined for adiabatic compression
and expansion of the medium about its equilibrium density, ,and .
2.1.2 Equation of Continuity
In order to relate the motion of the fluid to its compression or dilation, we require a
functional relation between the particle velocity and the instantaneous density. As
the sound waves propagate, mass has to be conserved. Thus, the extra mass that is
carried into a volume due to the propagating wave represents an increase in
17
18. density. We will consider the rate of increase of mass in a fixed (imaginary)
volume V, with the net amount of mass that flows into , per unit time. In other
words, if mass goes into then, because mass is conserved, the amount of mass
inside must be going up. The total mass in is given by the volume
integral . The rate of change of mass in can therefore be written as:
The surface bounding is . The net flow of mass out through a small part of the
surface at point is the dot product where is the outward unit
perpendicular to . The net mass flowing in to is therefore:
To understand better where the come from, consider Figure 2. When the
angle between the beam and the normal to the aperture (call it ) is zero, all the
beam gets through, but as α increases, less and less gets through until, when they
are at rightangles none gets through. In general, the amount getting through is
.
Figure 2: Vector dot product
18
19. The divergence theorem, sometimes called Gauss’ divergence theorem, equates the
outward flux of a vector field over a closed surface with the volume integral of the
divergence of the vector field over the region enclosed by the surface:
where is a vector field. This can be used to convert Equation (6) to the
following form:
Equating Equations (7) and (4), we get:
As this equation is zero for any arbitrary volume , the integrand must also be
zero hence:
2.1.3 Euler’s Equation:
Let us consider a medium of density and a particle with an incremental volume,
whose size is smaller than the wavelength of the ultrasound. The length of this
small volume in the longitudinal direction is , where is the area of the cross
section perpendicular to the longitudinal axis and the mass of the incremental
volume is given by . Along the longitudinal direction, there exists an
excess pressure denoted by . In conjunction with the pressure field, and
coupled to its longitudinal velocity field represents the motion of the particle.
19
20. If the particle moves with the velocity in the presence of an acoustic field, a
minute force acts on the particle due to the change in pressure. Newton’s force
equation can be applied to the volume element, where the net force on the volume
can be written in partial differential form:
where . Notice the extra term in the force equation; this term takes
into account the fact that acceleration is not constant within the fluid for the
infinitesimal displacement traveled in a time . Thus acceleration is a
function of both time and displacement.
Now , Equation (11) becomes:
Since pressure is force per unit area and :
Taking the limit as , Equation (12):
Rearranging Equation (13), we get Euler’s Equation in one dimension:
Following from Equation (14), Euler’s Equation in three dimensions is:
20
21.
2.2 The Wave Equation
The wave equation in an ideal fluid can be derived from hydrodynamics and the
adiabatic relation between pressure and density. The equation for conservation of
mass, Euler’s equation (Newton’s second law), and the adiabatic equation of state
are respectively:
and for convenience, we will define the quantity:
where c will turn out to be the speed of sound in an ideal fluid. In the above
equations, is the density, the particle velocity, the pressure. The ambient
quantities of the quiescent (time independent) medium are identified by the subscript
0. We use small perturbations for the pressure and density, and note that is also a
small quantity; that is, the particle velocity which results from density and pressure
perturbations is much smaller than the speed of sound.
Retaining higherorder terms in Equation ( 3 ) yields a nonlinear wave equation. The
nonlinear effects we include are contained in the quadratic density term in the
equation of state, ( 3 ). We first multiply Equation ( 15 ) by and take its divergence;
21
22. next, we take the partial derivative of ( 9) with respect to time. Substituting one into
the other yields :
Here, the indices i, j = 1, 2, 3 indicate x, y and z components, respectively. Tensor
notation is used; repeated indices signify a summation (e.g., ).
The first term on the righthand side of ( 16) can be rewritten using ( 3) and (15) as:
2.3 Wave Equation in the Frequency Domain
Conventional ultrasound imaging interrogates a medium with high frequency, band
limited waves and detects echoes scattered by inhomogeneities within the medium,
where scatterers represent such inhomogeneities. A singleelement transducer
immersed in the medium both generates ultrasound waves and receives the echoes.
The single element transducer consists of two types of piezoelectric crystals, the
transmitting and the receiving crystals. The transmitting crystal is excited coherently
to produce a focused beam, whereas the receiving crystal detects the scattered echoes,
which are then summed up coherently to yield an RF voltage trace. At each
transmission, the emitted wave propagating through the medium gives rise to an
incident pressure field, and the scattered wave gives rise to a scattered pressure field.
Multiple RF traces are formed by moving the centre of the crystals and repeating the
process; by lining up these individual traces next to each other in an image space, an
RF image is formed by summing the RF traces resulting from the scattered pressure
field over the piezoelectric surface and filtering the sum by the electromechanical
impulse response of the crystals.
22
23. With the physical description of the transducer mechanism in mind, we will consider
the mathematical expressions for the incident and scattered pressure fields in the
following sections. The wave equation (17) that we derived in the time domain can
also be expressed in the frequency domain as follows:
where
and are density and compressibility terms respectively. The presence of
scatterers in the medium may be modelled by adding spatially dependent variables
and to the density and compressibility terms. The term is the scattering
operator defined as follows:
and the scattering terms and are:
Since Equation (18) is linear partial differential equation (PDE) we can write its
general solution as the sum of the solution to the corresponding homogeneous
equation (i.e. with the RHS set to zero) and any particular solution. Denoting
as the solution to the homogeneous equation as and the particular solution
as , we can therefore write the total field as:
23
24.
In order to assign a physical interpretation to when we set RHS of Equation
(18) to zero we have effectively set . Thus we see that
then is the pressure field that develops in the absence of any scatterers, which by
definition is the incident pressure field. We also know that the scattered pressure field
must obey Equation (18), and so we assign our particular solution to be the scattered
pressure field . With these physical interpretations in mind, we see that the
total pressure field is indeed the sum of the incident and scattered pressure fields.
2.3.1 The Incident Pressure Field
In order to obtain an expression for the incident pressure field, we need to consider the
mechanics and geometry of the transducer. The generalized threedimensional
coordinate system is shown in Figure 3.
Figure 3: Coordinate system for describing scattering in an inhomogeneous medium
where represents the area over the transmitter crystal, is the location of the
center of , is an arbitrary point on , is the volume within which the
24
25. scatterers being considered are contained and is an arbitrary point in . The
surface may be considered to consist of infinitesimally small area element,
each of which behaves as a simple point source. The HuygenFresnel principle states
that each area element contributes a spherically expanding wave to the incident
pressure field. The incident pressure field ca then be obtained by summing the
spherical wave contribution from each area element.
If we assume that the radius of curvature of is large enough, then is considered
to be effectively flat. As a result, the incident pressure field can be expressed in terms
of the Rayleigh integral:
where is the temporal Fourier transform of the normal velocity on the
transmitter crystal’s surface; this normal velocity is not constant and varies from point
to point on . We note that term corresponds to normal acceleration in
the time domain, since the factor corresponds to time differentiation. We have
written on the lefthand side of Equation (24) to explicitly indicate the dependence
of the incident pressure field on the location of the center of .
Although we have not shown the proof for Equation (24), we can intuitively see that it
is indeed the HuygenFresnel principle expressed mathematically: the integral on RHS
describes the summation of complexvalued spherically expanding waves (represented
by the complex exponential term), each weighted by the normal acceleration at its
source and decaying in amplitude with increasing distance from its source.
For a typical transducer, we can consider the nominal normal velocity weighted by a
spatially variable term to account for apodization, a filtering technique that
removes discontinuities at the beginning and end of the sampled time record. We can
also model focusing by considering the normal velocity at each to be delayed by
.
Substituting into Equation (24), we obtain:
25
26.
For convenience we define a new quantity that we define to be the spatial
transfer function:
Equation (25) can then expressed more compactly as:
This compact expression allows us to view the incident pressure field as the result of
temporally filtering the nominal normal velocity by the spatial transfer
function . The spatial transfer function incorporates
the effects of apodization, and accounts for the spatial distribution of the incident
pressure field.
2.3.2 The Scattered Pressure Field
In order to find an expression for the scattered pressure field, we will consider the
Green’s function method. In our case, we consider spherical waves scattered from a
volume to be propagating into an effectively unbounded medium, in which case the
Green’s function takes the following form :
The particular solution to Equation (23) is then the product of the RHS and the
Green’s function integrated over the volume. Furthermore if we define and to be
26
27. zero outside of , then we can perform the integration over all of the three dimensional
space, and the scattered pressure field can be expressed as the convolution integral:
Since we consider weak scattering, we assume that .
in Equation (23) becomes negligible, and .
Rewriting Equation (29) with substituted by yields:
This approximation is referred to as the first Born approximation, and the equation (30)
states that the scattered pressure field is, to a first approximation, the spherically
expanding wave convolved with the scattering term. If we regard scatterers to be
idealised points in , then it is the equivalent of saying that each point scatterers
contribute spherically expanding wave independent of each other. Thus in making the
Born approximation, it is assumed that multiple scattering (i.e. waves scattered off a
particle that are then scattered off other particles ) is negligible.
By substituting the expression for , we can rewrite the scattering
pressure field entirely in terms of the transducer’s characteristics and the scattering
operator:
2.3.3 The Force on the Receiving Crystal
Recall that the received RF voltage trace is obtained by summing the scattered pressure
field over the area of the receiving crystal, and filtering the sum using the
electromechanical impulse response of the transducer. In this section we will derive the
27
28. summation formula for the scattered pressure field over the receiving crystal, and the
formula will be denoted by , which is essentially the force on the receiving
crystal.
Before we develop an expression for , we will introduce two lemmas. These
lemmas are essential in the derivation of the formula for the force on the receiving
crystal.
Lemma 1 (Refer to Appendix A for proof): For any vector valued function A(x) and
scalar function b(x), if A(x) is zero outside some volume V’ then:
Lemma 2 (Refer to Appendix B for proof): At locations that are far away from the
receiving crystal:
Returning now to , if we assume the same apodization and focusing at
reception as at transmission, then:
where,
We recognize that the integral is equal to ,
and so:
From the definition of the scattering operator in Equation (20), we can rewrite Equation
(34) as follows:
28
29.
To simplify Equation (35), we note that has finite limit, which allows us to use
Lemma 1 to simplify the integral on RHS of Equation (35):
At sufficiently large distances away from the receiving crystal, we can use Lemma 2 to
further simplify the force equation:
2.3.4 The RF Voltage Trace
In this section, we will discuss the conversion of the force on the receiving crystal into
a voltage RF trace. If we model the electromechanical transfer function of the
transducer to be and the voltage trace to be , we have:
Substituting in the expression for , we have:
For convenience, we can group together the characteristics of the medium together, and
the electromechanical properties of the transducer together. We adopt definitions similar
to those in Equation (27):
29
30.
where represents convolution in the space domain.
In keeping with the terminology introduced in Equation (27), we refer to the quantities
and respectively as the pulseecho wavelet and the tissue reflectivity or
scatterer field.
We can also express the voltage trace in the time domain as follows:
where and represent convolution in the space and time domain respectively.
If we regard the quantity as the input signal and the quantity as output
signal, then Equations (40) and (43) indicate that the imaging system has a
spatiotemporal transfer function and the impulse response function
. Thus, the definition of the transfer function this way neatly
distinguishes between the electromechanical characteristics of the transducer (represented
30
31. by the pulseecho wavelet) and the geometry of the transducer, represented by
.
31
32. Chapter 3
PulseEcho Experiments Using a Single
Element Transducer
The main goal of this project is to obtain Bmode images of scatterers small enough
to cause speckles, and then compare experimental results with theoretical
predictions from a mathematical model that would simulate Bmode images of
those scatterers showing speckles. Developing this model will rely on much of the
theoretical background described in section 2.3 of Chapter 2, which incorporates
transducer geometry and its corresponding pressurefield calculations for predicting
the RF voltage trace for that transducer geometry. However before an experimental
prototype and its corresponding theoretical model can be developed for describing
speckles, the first step would be to model and gather experimental data for discrete
scatterers, which are larger than speckles and the scatterers that cause speckles. This
chapter describes the experimental procedures for obtaining Bmode images of
discrete scatterers.
The process of obtaining Bmode images for scatterers within a
homogeneous medium is very challenging. This is because measurements and data
collection are very sensitive to external factors, for example transducer alignment
and sample material. For instance, a comparison of backscatter coefficient
measurements performed at eight different ultrasound physics laboratories found
variations in the final results of almost two orders of magnitude (Wear, et. al, 2005).
However, some of this variation also arises due to differences in experimental
techniques.
Techniques for measuring acoustic properties of phantoms may vary in
several ways, for instance bandwidth of the technique and the type of transducer
used. For our experiments, we have used a broadband, singleelement focused 5
MHz transducer. The advantage of using a broadband transducer lies in the fact that
32
33. it allows for simultaneous RF data acquisition over a range of frequencies; this also
means that the transducer can be excited with a short pulse rather than a
quasicontinuous wave excitation. Also using a singleelement transducer instead of
an array of transducers reduces the complexity of data analysis of the signals
received.
3.1 Making Scatterer Phantoms:
Scatterer phantoms are a uniform distribution of symmetric structures in a
homogeneous medium, based on the discrete scatterer model mentioned in Chapter
1. Scatterer phantoms were made by embedding beads in an agarose base, enclosed
by plastic petri dishes of about 3.6 cm in diameter and 0.8 cm in height.
Using the speed of sound in agarose at 24 °C, 1500 m/s, and the peak
frequency of the transducer, which for the purposes of our experiment was 5 MHz,
the minimum size of the scatterers were calculated:
The minimum size of scatterers were calculated assuming Rayleigh scattering takes
place, where scatterers are assumed to be smaller than the wavelength of ultrasound
waves. This value is significant for the following reasons: first, it provides an
estimate of the size of scatterers, which may cause speckles (these scatterers are
expected to be of the order of several hundred microns); further more it enables us
to determine how large discrete scatterers should be compared to clusters of
indistinguishable scatterers that cause speckles.
Beads of the following size ranges (from largest to smallest) were selected:
4.76 mm, 2.85 mm 3.45 mm and 1.0 mm 1.3mm. Each petri dish was first filled
with a relatively thin layer of liquid agarose. This layer was allowed to congeal, and
then a second layer of liquid agarose was poured on top of the first layer. As soon
33
34. as the second layer was poured, the scatterers were embedded into this layer. The
reason why scatterers were not embedded in the first agarose layer is because when
scatterers are offset from the base of the petri dish, it is much easier to differentiate
RF signals of the base from those of the scatterers when the RF data is converted to
Bmode images (since the scatterers are quite small, RF signals from the base have
a much higher intensity, and appear “brighter” in the grey scale Bmode image).
Figure 4 shows the scatterer phantoms. The beads were placed in a
symmetric pattern in order to obtain an approximate value for the spacing between
each bead, and knowing this spacing in conjunction with scatterer size enables
better interpretation of Bmode images. The 4.76 mm beads and the 2.85 mm 3.45
mm beads were arranged in a rectangular array using a tweezer, with an average
spacing of about 6 mm and 3mm respectively. Since the 1.0 mm 1.03 mm beads
were too small to be arranged linearly, they were arranged radially outward from
the center using the tip of a thin metal strip, and arranged radially with respect to a
2.85 mm 3.45 mm glass bead placed at the center of the agar base; the average
spacing was about 2 mm.
(a) (b) (c)
Figure 4: Scatterer Phantoms (a) Metal beads: 4.76 mm, spacing between beads about
6 mm, (b) 2.85 mm 3.45 mm glass beads, spacing between beads about 3 mm,
(c) 1.0 mm 1.03 mm glass beads, spacing between beads about 2 mm.
34
35. 3.2 Experimental Setup:
As with all acoustic property measurements, the results of the pulseecho
measurement can depend on temperature. Therefore, every attempt should be made
to achieve a uniform temperature in your sample, which is typically room
temperature (22.0 °C). For typical size phantom samples, the samples should be in
the water tank at this temperature for 1 to 2 hours prior to starting measurements.
This should allow enough time for the sample to uniformly be at 24 °C. For our
experiments, we used a 5 MHz focused transducer with a usable bandwidth of
about 60% of its central frequency (from about 3.5 MHz to 6.5 MHz).
The initial setup of the experiments are illustrated in Figure 5:
Figure 5: Experimental setup for pulseecho experiment
The stage of the xyz translator is held in position with screws, such that the
transducer will face downwards when mounted onto the stage. After the stage is
securely stationed, the transducer is carefully mounted on the stage and held firmly
in position by tightening a pair of screws.
For each phantom, the top of the petri dish is taken off and then the bottom
surface is securely attached to the base of a large, glass container using double
35
36. sided tape. The phantoms should be fixed rather firmly to prevent them from being
displaced as they are being submerged in water. The container is then filled with
deionized water until the meniscus of the water level above the phantoms is about
5 cm, which is approximately equal to the focal length of the transducer (52.2
mm). The phantoms should be roughly at the focal length so that the distance
between each phantom and the transducer can be later adjusted to observe
scatterers at the correct time delay, specifically at the focus.
3.3 Electronic Setup:
The pulseecho measurements require the use of an oscilloscope (Tektronix TDS
3014C Digital Phosphor Oscilloscope with 1.25 GS/s digitization), translator
motor for positioning the transducer, and pulsereceiver (Olympus
PanametricsNDT 5800). A pulsereceiver is a device that does just as its name
suggest: it sends out a voltage spike or pulse, and then turns itself into a broadband
receiver amplifier. The transducer is connected to the pulsereceiver using a BNC
cable, where the transducer is attached to the mounting bracket of the male UHF
adaptor.
In order to provide a trigger to the oscilloscope, a cable is connected from the
"Ext Trig/+Sync" jack of the pulsereceiver to the "Ext" jack of the oscilloscope to
provide external trigger to the oscilloscope. Then "Gated RF" output of the
pulsereceiver is connected to channel 1 of the oscilloscope. In order to provide
some lowpass filtering, a lowpass filter is then inserted between the Gated RF
jack and the oscilloscope in order to ensure that signals associated with the central
frequency of the transducer, and its bandwidth are detected. Note that the filters
have an input and an output side. A schematic of the setup is shown in Figure 6.
36
37.
Figure 6: Transducer, Oscilloscope and PulseReceiver connections
The mode of the pulsereceiver is set to "PulseEcho". The settings for
"Gain", "Attenuation", and "Energy" are determined, and for our experiments they
are usually set to 20 dB, 0, 50 J respectively. After the pulse/receiver is turned
on, the gate settings and oscilloscope timevoltage divisions are adjusted to locate
the high amplitude echo signals from the front and back surfaces of the sample.
3.4 Aligning the Transducer:
The front surface of the sample defines one geometric plane in space. The
combination of the x and yaxes of the transducer translation system define another
geometric plane. It is important that the surface of the sample be as close as
possible to parallel to the plane of motion of the transducer. If this is not the case,
37
38. then the distance the beam travels in water will vary as the transducer moves to
obtain independent power spectrum estimates. Unfortunately, there is no automatic
alignment system with this setup. However, you can use the computer to determine
if the two planes are parallel.
The motor of the xyz translator is controlled using a program called
COSMOS 3.6.1. The program has two different settings, which control the motors:
the “Quick MoveSingle Axis” and “Virtual Jog”. The “Quick Move” setting is
most useful for carrying out raster scans because it moves the motor a specific
distance from a point of reference, defined by the stepsize associated with each
click. However, the “Virtual Jog” setting is used for fine tuning the position of the
transducer; it is useful for centering the transducer, and locating the edge of the
sample.
The translator motors are labelled 1 through 3, where 1 corresponds to the
yaxis, 2 corresponds to the xaxis and 3 corresponds to the zaxis. In order to
lower the transducer into the glass container, the “Quick Move” setting is used to
control motor 3. The transducer is vertically lowered until it is sufficiently below
the surface of the water at an appropriate distance from the phantom, and this is
determined using the time delay. For the time delay calculation the variables (in
s) and are considered, where:
is the speed of sound in the agar base of the phantom and water,.
For our calculations we set the offset to zero, assuming that sound waves are not
significantly reflected from the top surface of the sample. Since all the RF data is
collected in a window of 10 s, the “window” term in the expression for is set to
10 and multiplied by 0.000001 to convert into microseconds. After obtaining the
38
39. values of and , and given that the sample is roughly located at the focal length
of the transducer, the time delay is calculated as follows:
where , is the speed of sound in water. For our experiments,
time delay was calculated to be 64.8 s. On the oscilloscope, the signal appears as
an envelope, which outlines the variation in amplitude of the scattered signal
against a time scale. Usually, the first high amplitude signals correspond to the
interface between the surrounding water and top surface of the phantom. The next
set of signals are recognized as scatterers distributed in the agar phantom because
the signal amplitude is much smaller than that of the interface, and each signal
corresponds to a scatterer. However, the amplitude of the scatterer signals is
largely dependent on the size of the scatterers. The last set of signals correspond to
the base of the phantom petri dish, and look very similar to the initial set of signals
due to the interface. Next, the voltage time scale is adjusted to locate the signal
from the top surface of the phantom. After the top surface is located, the time delay
of the oscilloscope is set to 64.8 s. Next, in order to find the correct distance
between the transducer and phantom, motor 3 is used to further adjust the vertical
height of the transducer until the signal from the top surface is at the origin of the
voltagetime axis. Once the appropriate distance is determined, the time delay is
offset by 5 s to 69.8 s so that the signal inside the phantom fills the oscilloscope
screen, which we use to window our RF signal.
Before centering the transducer, the axis of the stage is manually adjusted
using positioning knobs on the translation mount; this ensures that the transducer is
parallel to the surface of the sample. Once the transducer’s vertical orientation is
adjusted, the “Quick Move” setting of the program is used to move motors 1 and 2
to align the transducer to the center of the phantom.
39
40. Once we have centered the transducer, we then located the edge of the
sample in the x and y direction by observing changes in the signal amplitude. The
“Quick Move” setting is used to move motors 1 and 2 slowly along the x and y
direction. The edges usually appear as a subenvelope of signals, whose amplitude
increases and then decreases rapidly. This subenvelope noticeably diminishes as
we move just off the edge of the sample until there is no signal at all. Figure 7
shows an RF data at time delay for 4.76 mm scatterers.
Figure 7: RF data at time delay for 4.76 mm scatterers
3.5 Collecting RF PulseEcho Signal Data:
After the edge is located, the next step would be to figure out exactly how many
steps away from the center, and this is done using the “Virtual Jog” setting.
“Virtual Jog” uses step sizes, which are defined as the smallest move of the
motorized xyz stage. For instance, when the edge is located in the x direction,
motor 2 is moved away from the edge using a step size of 330 (any step size less
than or equal to 400 should be reasonable). As the transducer is moved towards the
40
41. edge of the phantom, it is important to keep track of the number of clicks it takes
for the edge signals to appear and eventually diminish again. The same procedure
is repeated in the y direction using motor 1.
In order to obtain RF voltage trace data, a raster scan is performed. But
before the scan can be performed it is important to know the beam width. The
beam width of the sound waves is calculated, assuming limited diffraction through
a circular aperture for the concave transducer surface:
where is the beamwidth, is the focal length (0.0522 m), is the diameter
of the transducer element (0.5 inches) is the speed of sound in the phantom and
is the frequency of the transducer (5 MHz).
For the 5 MHz focused transducer, the beam width is 0.015 mm. The beam
width is converted to motor steps, and the corresponding value (238 motor steps) is
set as the step size for the raster scan in order to obtain independent beam lines.
Figure 8 shows the rectangular pattern of signal reception, where the transducer is
moved in a square with five steps in the x and y direction. When the raster scan is
complete, all the data acquired from the oscilloscope is saved in a USB and loaded
into a MATLAB program, the raw data is the RF voltage trace. The program is
also designed to produce Bmode images of the horizontal or vertical
crosssections of the phantoms by mapping the vector arrays (associated with the
RF data) on twodimensional space domain, that is the xy plane in our case. Refer
to Appendix C for a detailed description of how the program works.
41
43. Chapter 4
Obtaining Bmode Images
Before we add complexity to our experiments and make phantoms consisting of
scatterers in the Rayleigh scattering limit, it is important to verify phantom
preparation methods and reasonably image discrete scatterers. In this chapter, we
will look at Bmode images of the discrete scatterer phantoms discussed in Chapter
3. Having Bmode images of discrete scatterers will not only test accuracy of
preparation methods, but it will also enable comparison of experimental results with
the theoretical results once we establish our own mathematical model for obtaining
RF voltage trace and Bmode images for a singleelement transducer geometry.
4.1 Bmode Image of 4.76 mm Metal Beads:
Figure 9: RF data for 4.76 mm scatterers.
43
44. Figure 10 : Bmode image of 4.76 mm scatterers
For the 4.76 mm metal beads, the brightest bands indicated in Figure 10 represent
the top and bottom surface of a single scatterer. The distance between the respective
bright bands is roughly the size of a scatterer, thus confirming that the bright bands
arise due to sound waves striking the scatterer head on. Looking at the RF data for
these scatterers, we are also able to identify the top and bottom surface of the
scatterer at about 70 s and 73.5 s. Thus, Bmode imaging of 4.76 mm scatterers,
was successful.
44
46. Figure 12: Bmode image of 2.85 mm 3.45 mm glass beads
Since the spacing between theses scatterers (about 3 mm) was approximately the
same as the average size of the scatterers (about 3.15 mm), we observed a series of
bright bands across the plane of the phantom. This is because as the scatterers were
relatively clustered, sound waves struck the edges of most of the scatterers giving
rise to a linear array of bright bands. However, some of the sound waves managed
to strike a single scatterer head on, and again we observe a much brighter set of
bands as indicated in Figure 12. The distance between the brightest bands is about
2.75 mm, which is roughly equal to the average diameter of a single scatterer.
From Figure 11 we observe the front and back surface of the scatterer to be around
71 s and 72.8 s respectively, and this further confirms that the 2.85 mm 3.45
mm scatterers were successfully imaged.
4.3 Bmode Image of 1.0 mm 1.03 mm Glass
Beads:
Figure 13: RF data for 1.0 mm 1.03 mm glass beads
46
47.
Figure 14: Bmode image of 1.0 mm 1.03 mm glass beads
Spacing between the 1.0 mm 1.03 mm scatterers (about 2mm) was not too large
compared to the average size of the scatterers (about 1.02 mm), and we again
observe a series of relatively bright bands like those in the Bmode image for the
2.85 mm 3.45 mm scatterers. Among the series of bright bands, we observe
brighter bands corresponding to the front and back surface of a single scatterer as
seen in Figure 14. We estimate the distance between the bright bands to be around
1mm, which again is roughly the same size as the diameter of a single scatterer.
Figure 13 also indicates the front and back surface of the scatterer as seen on the RF
data; the front surface is located at around 71.8 s, whereas the back surface is
around the 72.4 s mark.
47
48. Chapter 5
Conclusion and Future Work
From the results of our experiments, we can conclude that we have been able to
successfully produce Bmode images of discrete scatterers, which are larger than
scatterers that cause speckles. The results also indicate the phantom preparation
method complies with the discrete scatterer model, where the scatterer phantom is
depicted as a collection of points, spheres or cylinders embedded in a homogeneous
medium. However, the results presented in this thesis are the first steps towards
achieving the main goal of the experiment: producing Bmode images of scatterers
in the Rayleigh scattering limit.
Now that we have a mathematical model for predicting the RF voltage trace,
as discussed in Chapter 2, the next step would be to extend that model to describe
scatterers of various sizes (higher or lower than the Rayleigh scattering limit), and
simulate ultrasound Bscan images of scatterer phantoms. We started some
preliminary modeling of the incident and scattered pressure fields, and both these
models are discussed in detail in Appendix D and E. However, these models are
still in the early stages of development and require addition of further complexity,
such as developing subfunctions for apodization and scattering terms, in order for
them to be more realistic.
Thus in order to effectively model speckles, the process of writing up a
complete program model will involve pressurefield calculations similar to those
discussed in Chapter 2. These calculations will help model the RF voltage trace of a
given transducer geometry (a single element focused transducer in our case), and
then convert the RF data to Bmode images for comparison between experimental
and theoretical results. After a working model is established, keeping in mind the
discrete scatterer model for scatterer phantoms, the theoretical Bmode images will
be compared with experimental results (including the ones described in Chapter 4)
48
49. to verify the accuracy of the mathematical model for describing speckles.
Achieving a good agreement between experimental and theoretical results will
enable a better understanding of the nature of speckles and what causes them.
Developing a model that accurately describes speckles will help us develop an
ultrasonic microscope to map scatterers at high frequency locations within
biological systems, which we are unable to detect using current ultrasound imaging
techniques.
49
50.
Appendix:
A Proof of Lemma 1 (Chapter 2, 2.3.3)
We begin with the identity:
which can be verified by expanding the RHS and simplifying. Integrating both
sides over :
By the divergence theorem, the second integral on the RHS is equal to the surface
integral:
where is some reference surface enclosing and is a unit vector normal
to . Since is zero outside , the surface integral reduces to zero and
Equation follows. Q.E.D.
50
51.
B Proof of Lemma 2 (Chapter 2, 2.3.3)
We define the wave vector , where , in other words
is a unit vector parallel to and is a vector also parallel to
but with a magnitude . We can then rewrite Equation for the spatial transfer
function as,
Taking the gradient,
We apply the condition , which is equivalent to
since . The above equation then becomes:
We also assume that the direction of does not vary very much over the receiving
crystal, which allows the term on the RHS to be factored out of the integral,
51
52. Lemma 2 follows from this immediately. Q.E.D.
In practice, is satisfied for virtually the entire imaged region
since the wavelength from medical ultrasonic transducers are usually very short. In
the context of our experiments described in Chapter 3, the singleelement focused
transducer is transmitting at 5 MHz with an average speed of sound of 1490 m/s in
agar and water. For this transducer , which is small enough to
be considered negligible.
The condition that the direction of not vary very much over the receiving
crystal is much stricter and is only satisfied in regions far away from the receiving
crystal. To quantify, what is exactly meant by “far away” consider the following
figure:
Figure A: Coordinate system for calculating an approximation to
For a typical focused transducer, we expect the majority of the transmitted
acoustic energy to be concentrated in a small region enclosing the axial axis, and
we have chosen the scenario where lies along this axis. We can restate our
requirement that can be uniform over the surface of the receiving crystal by
requiring to be approximately parallel to .
This requirement is most difficult to satisfy when we consider the point on
farthest from the center of the receiving crystal, i.e. when .
52
53. Our requirement that be approximately parallel to can be stated in terms of
the dot product ; in the case where (as shown in
the figure above), we require . If we allow a 10% error in the
approximation of , we effectively impose the constraint where
. From the figure , and so our requirement that
translates to requiring .
What we have demonstrated in this brief discussion is that the approximation
considered as a part of lemma 2 is satisfied well at axial depths that are greater
than the diameter of the receiving crystal. For noncircular apertures, this diameter
is the smallest circle within which the receiving crystal can be ascribed.
53
55. Figure C1: Matlab program for converting RF data to Bmode images
Figure C1 shows the Matlab program that was written to convert the RF data
to Bmode images. The RF data is loaded on to the program using the defined
variables for file path and name of the designated folder, where all the data is
saved. Since the RF data file is essentially a threedimensional array vector,
variables describing the RF vector array and the Bmode array are defined. The
vector array can be visualized as shown in Figure C2:
Figure C2: Threedimensional vector array
(Source:http://www.mathworks.com/help/matlab/math/multidimensionalarrays.html)
55
56.
In the context of our pulseecho experiments, ‘row’ is RF amplitude the along the
xaxis of the sample plane, ‘column’ is the RF amplitude along the sample in the y
direction and ‘page’ is the time interval between between transmitted and received
signals. Both the x and y components of the vector array have two sets of signal
associated with them: the input RF signal from the transmitting crystal, and the
output RF signal from the receiving crystal.
The main job of the program is to convert the threedimensional vector array
into a 2by2 vector matrix, which is then converted to a twodimensional Bmode
image. This process is initiated by the iteration loop spanning lines 26 through 32
Currently this forloop has a limitation that it can do a maximum of nine iterations,
which means that it can not handle more than nine sets of data from the raster scan;
we use five sets of data to produce the Bmode image. The final Bmode image in
our case is a horizontal cross section along the sample plane, therefore only the
output RF signal is relevant along the x axis. Thus in line 29 of Figure C1
“sampleData (:, 2, :)” indicates that we are only considering the output RF signal
in the x direction. The operation “squeeze” then removes singleton dimensions and
converts the threedimensional vector array into a 2by2 matrix in the space
domain.
The twodimensional vector array is then converted to a Bmode array by
taking the natural logarithm of the RF amplitude data vector array, which
graphically corresponds to the intensity or brightness level that we see in the final
Bmode image. Next, the x and y distances along the plane of the sample are
defined (line 4354) using stepsize (the smallest distance covered by the translator
motor), window size in microseconds and the number of raster scan data sets. Then
the image layout is specified as shown in Figure C1 between lines 57 to 67.
56
