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8/22/2019 Ultrasound Lecture 2 Post (3)
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Ultrasound Imaging: Lecture 2
Absorption
Reflection
Scatter
Speed of sound
Signal modeling
Signal Processing
Statistics
Interactions of ultrasoundwith tissue
Image formation
Jan 14, 2009
Steering
Focusing
Apodization
Design rules
Beams and Arrays
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Anatomy of an ultrasound beam Near field or Fresnel zone
Far field or Fraunhofer zone Near-to-far field transition,L
2aL
L
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Anatomy of an ultrasound beam
Lateral Resolution (FWHM)
FWHM
numberFR
aFWHM
2
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Anatomy of an ultrasound beam
Depth of Field (DOF)
DOF
2)(7 numberFDOF
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Array Geometries
Schematic of a linear phased array
Definition of azimuth, elevation
Scanning angle shown, q, in negative scan
direction.
ya (elevation)
xa (azimuth)
za (depth)
array pitch
Acoustic beamq
t
trtrp
,
,
N
ii ttrhWtrh 1 ),(,
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Some Basic Geometry
Delay determination:
simple path length difference
reference point: phase center
apply Law of Cosines approximate for ASIC
implementation
In some cases, split delay into 2
parts:
beam steering
dynamic focusing
x
z
x
rqr
0
rx
crr x
rrrxxc
q 22 cos21
fs
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Far field beam steering
For beam steering: far field calculation
particularly easy
often implemented as a fixed
delay
c
xs
qsin
x
z
x
r
0
q
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Beamformation: Focusing Basic focusing type beamformation Symmetrical delays about phase center.
-4 0 -3 0 -2 0 -1 0 0 10 20 30 40
-2 0
-1 0
0
10
20
point
source
wavefront s
befo re correct ion
t ransducer
elements
delay
lines
wavefront s
after correction
summing
stage
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Beamformation: Beam steering
Beam steering with linear phased arrays.
Asymmetrical delays, long delay lines
-4 0 -3 0 -2 0 -1 0 0 10 20 30 40
-2 0
-1 0
0
10
20
po int
source
wavefront s
befo re correction
array
element s
delay
lines
wavefront s
after beam
steering and focusing
summing
stage
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Anatomy of an ultrasound beam
Electronic Focusing
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Grating Lobes
Linear array: 32 element array
3 MHz
pitch l= 0.4 mm
= 0.51 mm
L= N l= 13 mm
How to avoid:
design for horizon-to-
horizon safety275.1
4.
51.)(
)(
g
g
Sin
lSin
q
q
How many elements?
What Spacing?
gql
gq
l
Main Lobe
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Array design
Linear array: 32 element array
3 MHz
pitch l= 0.4 mm
= 0.51 mm
Larray= N l= 13 mm
How to avoid:
design for horizon-to-
horizon safety2
l
How many elements?
What Spacing?
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Apodization
Same array:
32 element array
3 MHz
pitch l= 0.4 mm
= 0.51 mm
Larray = N l= 13 mm
With & w/o Hanning wting.
Sidelobes way down. Mainlobe wider
No effect on grating lobes.
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Summary of Beam Processing
Beam shape is improved by several
processing steps:
Transmit apodization
Multiple transmit focal locations Dynamic focusing
Dynamic receive apodization
Post-beamsum processing
Upper frame: fixed transmit focus
Lower frame: the above steps.
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I INTERACTIONS OF ULTRASOUND WITH TISSUE
Some essentials of linear propagation
Recall the equation of motion
t
v
x
p
0 (1)
Assume a plane progressive wave in the +x direction that
satisfies the wave equation
ie)(
0
kxtepp (2)
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Substituting 2 into 1 we have
t
vjkep kxtj
0
)(
0
Z
p
c
pv
ep
f
ejk
j
pv
dtejkp
v
kxtj
kxtj
kxtj
0
0
0
)(
0
0
)(
0
0
2
2
Acoustic impedance
(3)
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cZ 0Where
= Characteristic Acoustic Impedance
Define a type of Ohms Law for acoustics
Electrical:
Acoustical:
Extending this analogy to Intensity we have
vZp
IRV
2
0
2
0
2
1
2
1ZvZ
pI
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Propagation at an interface between 2 media
111 cZ 222 cZ
iP
rP
tP
xktj
tt
xktj
rr
xktj
ii
eP
p
ePp
ePp
2
1
1
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Define Reflection/Transmission Coef
i
t
i
r
p
pT
ppR ,
You will show:
212
1212 2
ZZZT
ZZZZR
Example: Fat Bone interface
38.16.7)6.7(2
38.16.738.16.7
TR
70.0 69.1
(4)
(5)
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THE DECIBEL (dB) SCALE
refsBd A
A
LogA 10)( 20
Where A = measured amplitude
Aref = reference amplitude
In the amplitude domain
6 dB is a factor of 2
-6 dB is a factor of .5 (i.e. 6dB down)
20 dB is a factor of 10
-20 dB is a factor of .1 (i.e. 20dB down)
(6)
R fl ti C ffi i t
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0
-10
-20
-30
-40
-50
Reflection Coefficients
Air/solid or liquid
Brass/soft tissue or water
Bone/soft tissue or water
Perspex/soft tissue or water
Tendon/fat
Lens/vitreous or aqueous humour
Fat/non-fatty soft tissuesWater/muscle
Fat/water
Muscle/blood
Muscle/liver
Kidney/liver, spleen/blood
Liver/spleen, blood/brain
Water/soft tissues
R = 1.0
R = .1
R = .01
Reflection
Coef.dB
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3) ULTRASOUND IMAGING AND SIGNAL PROCESSING
Thus far we have been concerned with the ultrasound transducer
and beamformer. Lets now start considering the signal
processing aspects of ultrasound imaging.
Begin by considering the sources of information in an
ultrasound image
a) Large interfaces, let a = structure dimension
a
- specular reflection
-
- reflection coefficient 12
12
ZZ
ZZ
R
where cZ
- strong angle dependance
- refraction effects
density speed of sound
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b) Small interfaces
a-
- Rayleigh scattering
Cos
akD
0
0
0
0
32
2
33
3
Compressibility Density
and Arp , Dr
eikr (7)
Morse and Ingard Theoretical Acoustics
p. 427
*
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SCATTER FROM A RIGID SPHERE
Cosr
a
cDs 31
3
4 32
*
*
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SCATTER FROM A RIGID SPHERE (Mie Scatter)
*
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ATTENUATION
= absorption component + reflectivity component
xepxp 0
The units of are cm-1 for this equation. However attenuation
is usually expressed in dB/cm. A simple conversion is given
by
1686.8 cmcm
dB
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Attenuation in
Various Tissues
15%
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Speed of
Sound in
VariousTissues
0%
5%
10%
15%
-5%
-10%
Assumed speedof sound = 1540
m/s
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SUMMARY ULTRASONIC PROPERTIES
Table 1
Material Speed of Sound Impedance Attenuation Frequency
ms-1 Kg m-2 s-1
X 106
At 1 MHz (dB
cm-1) Dependency
water 1490 @ 23C 1.49 0.002 2
muscle 1585 @ 37C 1.70 1.3-3.3 1.2
fat 1420 @ 37C 1.38 0.63 1.5-2
liver 1560 @ 37C 1.65 0.70 1.2
breast 1500 + 80 @ 37C ------ 0.75 1.5
blood 1570 @ 37C 1.70 0.18 1.2
skull bone 4080 @ 37C 7.60 20.00 1.6air 331 @ STP 0.0004 12.00 2
PZT 4300 @ STP 33.00 ------ --
smc /1540
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2.2 Modeling the signal from a point scatterer
Imagine that we have a transducer radiating into a
medium and we wish to know the received signal due to
a single point scatterer located at position
By modifying the impulse response equation (Lecture 1
Equ. 25 ) we can write:
r
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trhtrhtstgtgt
VktrV rtout ,*,***,*, 2
0
transmit + receive
electromechanical
IRs
scatterer
IR
transmit
IR
receive
IR
pulse (t)
trHtpulse
trhtrhtpulsetrV rtout,*)(
,*,*,
easily
measured
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Now consider a complex distribution of scatterers
Isochronous
volume
rx ri
(1)
(2)
(3)
(4)
At any point in the isochronous volume there exists a transmit
receive path length divided by c for a time, t, such that
c
zt
c
ll
21
zl1
l2
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If we look at the four field points shown on the previous page
we would see the following impulse responses
(1)
(2)
(3)
(4)
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The total signal for a given ray position rx is given by
trHrWtpulsetrVout xiN
iiix ,
1
*)(,
(9)
scatterer
strength
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The resultant signal is the coherent sum of signals resulting
from the group of randomly positioned scatterers that make up
the isochronous volume as a function of time.
A useful model of the signal is:
ttCostatytVout 2
Envelope Modulated
carrier
Phase
Grayscale informationfor B-scan Image
How do we calculate a(t) and (t)?
Velocity informationfor Doppler
(10)
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3.3 Hilbert Transform
The Hilbert transform is an unusual form of filtration in which thespectral magnitude of a signal is left unchanged but its phase
is altered by for negative frequencies and for
positive frequencies2
2
Definition
)(*1
1
xfx
xdxx
xfxFH
(11)
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In the frequency domain
sH FsjxF )sgn(
Consider the Hilbert transform of Cos x
RE RE
(12)
IMIM
xCos sjSgn
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The application of two successive Hilbert transforms results
in the inversion of the signal we have 2 successiverotations in the negative frequency range and 2
rotations in the positive frequency range. Thus the total
shift in each direction is .
2
2
1
II
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xfxx
xF
xH
111
sFsjsj sgnsgn
xfF
sF
s
1
1
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The Hilbert transform is interesting but what good is it?
ANALYTIC SIGNAL THEORY
Consider a real function . Associate with this function
another function called the analytic signal defined by:
where = Hilbert Transform
The real part of the analytic signal is the function itself whereas
the imaginary part is the Hilbert transform of the function.
Note that the real and imaginary components of the analytic
signal are often called the in phase, I, and quadrature, Q,
components.
tjztytf tz(13)
ty
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Just as complex phasors simplify many problems in AC
circuit analysis the analytic signal simplifies many signal
processing problems.
The Fourier transform of the analytic signal has an interesting
property.
0,2
0,0
][
sY
s
YsSgnY
YsSgnjjYtjzty
s
s
ss
s sy
0
2
s
Ys
(14)
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Equation 14 gives us an easy way to calculate the analytic
signal of a function:
1) Fourier transform function
2) Truncate negative frequencies to zero
3) Multiply positive frequencies by 2
4) Inverse Fourier Transform
Recall that our resultant ultrasound signal can be expressed
as:
ttCostaty 2
Its analytic signal is then
ttetatf 2(15)
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which on the complex plane looks like:
IM
RE
ty
t
ta tz
Where
and the phase is given by
tztyta22
ty
tzTant )(1
(16)
(17)
a(t) envelope
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Demodulation: estimate using
1) Analytic signal method using FFT (slow)
2) Analytic signal using baseband quadrature approach
3) Sampled quadrature
)(),( tta QI,
Baseband Quadrature Demodulation
X
X
Low
Pass
Low
Pass
tCos 2
tQtSin
2 ty
tIt )Re(
Baseband
Inphase Signal
)()Im( tQt
Baseband
Quadrature Signal
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tCostCosa
tCostCosaI
tt
ttt
22
2
22
Use shift and convolution theorems to calculate spectra
ttnote :
2
2 tjt eAI
(slowly varying)
tjeA 2 221
tjt eAI 21
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tjtj
t eAeAI
2
1
2
1
tt CostaI 2
1 tj
t etas)(
2
1
Similarly
tt SintaQ )(2
1 Baseband
Analytic
SignalNo carrier
Phase preserved
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ttt
t
ttt
QIa
a
CosSinaQI
22
2
22222
2
41
)(4
1
Thus
)(
)(
)()(
)(
tI
tQArcTan
I
Q
Tan
t
t
tt
and
Sampled Quadrature
Begin with the signal of the ultrasound waveform
ttt Cosay 2
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Sample with period 1T
* **
)(nTI
T
tIIIy t
)(nTQ
T
tIIIy t
Recall that the quadrature signal is the Hilbert Transform of the
inphase component of the analytic signal i.e. for a cos wave it
is a negative sine wave. Thus we see that . . .
nTIT
tIIIy t
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If the inphase and quadrature signals are slowly varying
we can get the quadrature signal simply by sampling the
inphase signal 90 or period later
Sampling t = nT for I samples
t = nT+T/4 for Q sample
1
42)()(
)2()()(
T
nTTnTCosnTanTQnTrTCosnTanTI
let
nTSinntanTnCosnTanTQ
TnCosnTanTnCosnTanTI
)()2
2()()(
)()()2()()(
(18)
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1
Overall Imager Block Diagram
Doppler
Beamformer
Digital
Receive
Beamformer
Beamformer
Central
Control
Digital
Transmit
Beamformer
Transmit
Demux
Receive
Mux
Transducer
Connectors
System
Control
Image
Proces-
sing
2 3 4 5 6
Imaging System Signals
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Imaging System Signals
Doppler
Beamformer
Digital
Receive
Beamformer
Beamformer
Central
Control
Digital
Transmit
Beamformer
Transmit
Demux
Receive
Mux
Transducer
Connectors
System
Control
Image
Proces-
sing
23 4 5
6
1
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Coarse and Fine Beamforming Delays
Ho()e-j/4
Ho()e-j/2
Ho()e-j3/4
Ho()
MUXFIFO
Input from
ADC at 20to 40 MHz,
8 to 12 bits
Output withdelay accuracy
up to 160 MHz
To apodization
and further
processing
Coarse
Delay
Control
Fine
Delay
Control
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SIGNAL STATISTICS
Recall that the ultrasound signal is the sum of harmoniccomponents with random phase and amplitude. It can be shown
that the probability density function for such a situation is
Gaussian with zero mean i.e.
2
2
221)(
y
eyp
(19)
The quadrature signal will also be Gaussian with the
same standard deviation
2
2
22
1)(
z
ezp
(20)
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Since p(y) and p(z) are independent random variables the joint
probability density function is given by
2
22
2
2
2
2
22
22
2
1
2
1
2
1),(
zy
zy
e
eezyp
(21)
The probability of a joint event (corresponding to a particularamplitude of the envelope) is the probability that:
)(zp
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)(zp
)(yp
total area = ada2
The probability thata lies between
a and a + da is
222 zya
dae
a
daap
a2
2
222
2
)(
adad
adad
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So that the probability density function for the radio
frequency signal is given by
22
22
a
ea
ap
Rayleigh Prob.
Density function
aa
)(ap
few white pixels
many gray pixels
few
blackpixels
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The speckle in an ultrasound image is described by this
probability density function. Lets define the signal as
and the noise as the deviation from this value
arms
2122
12aaaN Thus
daea
daapaa
a
o
o
2
2
22
2
Recall
2a
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Thus:
21
21
22
22
2
22
2
N
aSNR
SNR = 1.91 and is invariant (25)
Note that the SNR in ultrasound imaging is independent of
signal level. This is in contrast to x-ray imaging where the
noise is proportional to the square root of the number of
photons.
S kl N i i Ult d I
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Speckle Noise in an Ultrasound Image
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a
ias
as
i
00
x
Lets make several independent measurements of
so and si
These measurements will form distributions
i 0
is 0s
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The parameter used to define image quality includes both
the observed contrast and the noise due to speckle in the
following fashion:
Define Contrast:
Define Normalized
speckle noise as:
and finally, define our quality factor as the contrast to
speckle noise ratio (CSR)
0
0
s
ss i
0
2122
0
si
22
0
0
i
issCSR
(26)
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Suggested Ultrasound Book References:
General Biomedical Ultrasound (and physical/mathematical foundations):
Foundations of Biomedical Ultrasound, RSC Cobbold, Oxford Press 2007.
General Biomedical Ultrasound (bit more applied): Diagnostic Ultrasound Imaging: inside out
TL Szabo Academic Press 2004.
Ultrasound Blood flow detection/imaging: Estimation of blood velocities with ultrasound
JA Jensen Cambridge university press 1996
Basic acoustics: Theoretical Acoustics PM Morse and KU Ingard, Princeton University Press(many editions).
Bubble behaviour: The Acoustic bubble TG Leighton Academic Press 1997.
Nonlinear Acoustics: Nonlinear Acoustics Hamilton and Blackstock, Academic Press 1998.