Jan 14, 2009
Ultrasound Imaging: Lecture 2
Beams and Arrays
•
•
•
•
Steering
Focusing
Apodization
Design rules
Image formation
• Signal modeling
• Signal Processing
• Statistics
Interactions of ultrasound
with tissue
•
•
•
•
Absorption
Reflection
Scatter
Speed of sound
Anatomy of an ultrasound beam
•
•
•
Near field or Fresnel zone
Far field or Fraunhofer zone
Near-to-far field transition, L
L
L
a2

Anatomy of an ultrasound beam
•
Lateral Resolution (FWHM)
2a
FWHM 
 F  number
R
FWHM
Anatomy of an ultrasound beam
•
DOF  7 ( F  number) 2
Depth of Field (DOF)
DOF
Array Geometries
•
•
•
ya (elevation)
Schematic of a linear phased
array
Definition of azimuth, elevation
Scanning angle shown, q, in
negative scan direction.
xa (azimuth)
q
za (depth)
array pitch
hr , t  

N
1
Wi hi (r , t  t )
 
 r , t
pr , t    
t
Acoustic beam
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
x
0
1

c

rx
r,q
r
r  rx

c
z
x 2  2rx cosq  r 2  r
  s   f

Far field beam steering
x sin q
s 
c
• For beam steering:
– far field calculation
particularly easy
– often implemented as a fixed
delay
x
x
r
q
0
z
Beamformation: Focusing
•
•
Basic focusing type beamformation
Symmetrical delays about phase center.
20
delay
lines
s umming
s tage
wav efron ts
before co rrection
10

0
po int
s ource
-10
wav efron ts
after correctio n
-20
-40
-30
-20
transd ucer
elements
-10
0
10
20
30
40
Beamformation: Beam steering
•
•
Beam steering with linear phased arrays.
Asymmetrical delays, long delay lines
20
delay
lines
s umming
s tage
wav efron ts
before co rrection
10

0
-10
array
wav efron ts
elements
after b eam
s teering and focusing
-20
-40
-30
-20
-10
0
po int
s ource
10
20
30
40
Anatomy of an ultrasound beam
• Electronic Focusing
Grating Lobes
How many elements?
qg
What Spacing?
l
•
•
Linear array:
– 32 element array
– 3 MHz
– ‘pitch’ l = 0.4 mm
–  = 0.51 mm
– L= N l = 13 mm

l
qg
Sin (q g ) 
How to avoid:
– design for horizon-tohorizon safety
Main Lobe

l
.51
Sin (q g ) 
 1.275
.4
Array design
How many elements?
What Spacing?
•
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-tohorizon safety

l
2
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.
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.
I INTERACTIONS OF ULTRASOUND WITH TISSUE
Some essentials of linear propagation
Recall the equation of motion
 p
v
 0
x
t
(1)
Assume a plane progressive wave in the +x direction that
satisfies the wave equation
ie
p  p0 e ( t  kx )
(2)
Substituting 2 into 1 we have
p0 jke j (t  kx )   0
v
p0
0
v
t
jk  e j (t  kx ) dt
p0 jk j (t  kx )
e
v
0 j 
2
p0 e j t  kx 

 0  2 f
v
p
p

 0c Z
(3)
Acoustic impedance
Where
Z  0c
= Characteristic Acoustic Impedance
Define a type of Ohm’s Law for acoustics
Electrical:
Acoustical:
V  IR
p  vZ
Extending this analogy to Intensity we have
2
1 p0
1
2
I
 Zv0
2 Z
2
Propagation at an interface between 2 media
Z1  1c1
Pi
Z2  2c2
Pt
Pr
pi  Pi e j t  k1 x 
pr  Pr e j t  k1 x 
pt  P e j t  k 2 x 
t
Define Reflection/Transmission Coef
R
pr
,
pi
T
pt
pi
(4)
You will show:
Z 2  Z1
R
Z 2  Z1
2Z 2
T
Z1  Z 2
Example: Fat – Bone interface
R
7.6  1.38
7.6  1.38
 0.70
T
2(7.6)
7.6  1.38
 1.69
(5)
THE DECIBEL (dB) SCALE
A( dBs )
Where A
Aref
=
=
 A
 20 Log10 
A
 ref




(6)
measured amplitude
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)
Reflection Coefficients
0
R = .1
Reflection Coef. dB
R = 1.0
-10
-20
-30
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 tissues
Water/muscle
Water/soft tissues
Fat/water
Muscle/blood
Muscle/liver
R = .01
-40
-50
Kidney/liver, spleen/blood
Liver/spleen, blood/brain
3) ULTRASOUND IMAGING AND SIGNAL PROCESSING
Thus far we have been concerned with the ultrasound transducer
and beamformer. Let’s 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
where
-
Z  c
Z 2  Z1
R
Z 2  Z1
density
speed of sound
strong angle dependance
refraction effects
b)
Small interfaces
-
a  
- Rayleigh scattering

k 2 a 3     0 3   3 0

D  

Cos 
3  0
2  0

Compressibility
ikr
e
and p r,   A
D  
r
Density
(7)
Morse and Ingard Theoretical Acoustics
p. 427
SCATTER FROM A RIGID SPHERE
*
*
4   2 a 3
1  3Cos 
Ds     
3 c  r
SCATTER FROM A RIGID SPHERE (Mie Scatter)
*
ATTENUATION
  = absorption component + reflectivity component
p  x   p0e  x
The units of   are cm-1 for this equation. However attenuation
is usually expressed in dB/cm. A simple conversion is given
by


dB
 8.686  cm 1
cm

Attenuation in
Various Tissues
15%
Speed of
Sound in
Various
Tissues
10%
5%
Assumed speed
of sound = 1540
m/s
0%
-5%
-10%
SUMMARY ULTRASONIC PROPERTIES
Table 1
Material
Speed of Sound
ms-1
Impedance
Kg
m-2 s-1
X 106
Attenuation
Frequency
At 1 MHz (dB
cm-1)
Dependency 
water
1490 @ 23ºC
1.49
0.002
2
muscle
1585 @ 37ºC
1.70
1.3-3.3
1.2
fat
1420 @ 37ºC
1.38
0.63
1.5-2
liver
1560 @ 37ºC
1.65
0.70
1.2
breast
1500 + 80 @ 37ºC
------
0.75
1.5
blood
1570 @ 37ºC
1.70
0.18
1.2
skull bone
4080 @ 37ºC
7.60
20.00
1.6
0.0004
12.00
2
------
--
air
PZT
331 @ STP
4300 @ STP
c  1540 m / s
33.00
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 r
By modifying the impulse response equation (Lecture 1
Equ. 25 ) we can write:


V
Vout r , t    k 0 * g , t * g 2 t * st  * ht r , t * hr r , t 
t


transmit + receive
electromechanical
IR’s
scatterer
IR
transmit
IR
pulse (t)
Vout r , t   pulse t  *h t r , t  * hr r , t 
 pulse (t ) * H r , t 
easily
measured
receive
IR
Now consider a complex distribution of scatterers
Isochronous
volume
(4)
(1)
z
l1
rx
l2
ri
(2)
(3)
At any point in the isochronous volume there exists a transmit –
receive path length divided by c for a time, t, such that
l1  l2
z
t 
c
c
If we look at the four field points shown on the previous page
we would see the following impulse responses
(1)
(2)
(3)
(4)
The total signal for a given ray position rx is given by

N
Voutrx , t   pulse(t ) * Wi ri  H rxi,t
i 1
scatterer
strength

(9)
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:
Vout t   y t   a t Cos 2 t   t 
Envelope
Modulated
carrier
Grayscale information
for B-scan Image
How do we calculate a(t) and  (t)?
(10)
Phase
Velocity information
for Doppler
3.3
Hilbert Transform
The Hilbert transform is an unusual form of filtration in which the
spectral magnitude of a signal is left unchanged but its phase
is altered by   2 for negative frequencies and   2 for
positive frequencies
Definition
1  f  x 
FH  x  
dx 

  x  x
1

* f ( x)
x
(11)
In the frequency domain
FH  x   j sgn( s)  Fs 
(12)
Consider the Hilbert transform of Cos  x 
RE
RE
IM
IM

Cos  x 
jSgn s 
II
  1
The application of two successive Hilbert transforms results
in the inversion of the signal – we have 2 successive   2

rotations in the negative frequency range and 2  2
rotations in the positive frequency range. Thus the total
shift in each direction is  .

1
1
1
 FH  x   

 f x 
x
x x

  j sgn s   j sgn s   F s 
 1   F s 
1 Fs    f  x 
The Hilbert transform is interesting but what good is it?
ANALYTIC SIGNAL THEORY
Consider a real function y t . Associate with this function
another function called the analytic signal defined by:
f t   y t   jz t 
where
z t  = Hilbert Transform
(13)
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.
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.
[ y t   jzt ]  Y s   j j Sgns   Y s 
y s 
 Y s   Sgns  Y  s 
 0, s  0

 2Y s  , s  0
2Ys 
s0
(14)
Equation 14 gives us an easy way to calculate the analytic
signal of a function:
1)
2)
3)
4)
Fourier transform function
Truncate negative frequencies to zero
Multiply positive frequencies by 2
Inverse Fourier Transform
Recall that our resultant ultrasound signal can be expressed
as:
y t   at Cos 2  t  t 
Its analytic signal is then

2  t  t  
f t   a t  e
(15)
which on the complex plane looks like:
2
Where at   yt   z t 
IM
a t 
2
t
y t 
z t 
(16)
and the phase is given by
RE
  z (t ) 

 y t  
 t   Tan1 
(17)
a(t) envelope
Demodulation: estimate a(t ), (t ) using I , Q
1) Analytic signal method using FFT (slow)
2) Analytic signal using baseband quadrature approach
3) Sampled quadrature
Baseband Quadrature Demodulation
Low
Pass
X
Cos 2 t
yt 
X
Sin2 t
Q t 
Low
Pass
Re(t )  I t 
Baseband
Inphase Signal
Im( t )  Q(t )
Baseband
Quadrature Signal
note :  t    t
I t   a t Cos2  t  t   Cos2  t

t

 at   Cos 2  t 
 2

(slowly varying)

  Cos2 t

Use shift and convolution theorems to calculate spectra
I t   A   e
j 2 t
2





1
 A   e j t 
I t   A  e

2
 2
2
1
jt

2
1
1
j t
I t    A   e      A       e jt
2
2
1
 I t   a t   Cos t
2
1
st   a (t )e jt
2
Similarly
1
Qt   a (t )  Sin t
2
Baseband
Analytic
Signal
No carrier
Phase preserved
1 2
I t   Q t   a t  ( Sin 2  Cos 2 )
4
1 2
 a t 
4
2
Thus
and
2
a t   2 I 2 t  Q 2  t 
Tan( t ) 
 Q(t )
I (t )
 t  ArcTan
 Q (t )
I (t )
Sampled Quadrature
Begin with the signal of the ultrasound waveform
yt   at  Cos2  t 
Sample with period T  1
t
yt   III    I nT 
T 
t
yt   III  
T 
 I (nT )
t
yt   III  
T 
 Q (nT )
*
*
*
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 . . .
If the inphase and quadrature signals are slowly varying
we can get the quadrature signal simply by sampling the
inphase signal 90º or ¼ period later
Sampling
t=
t=
nT for I samples
nT+T/4 for Q sample
I ( nT )  a ( nT )  Cos( 2 rT  nT )
 
Q ( nT )  a ( nT )  Cos 2 nT  T
let
T
 nT 
4
1

I ( nT )  a ( nT )  Cos( 2  n   nT )  a ( nT ) Cos ( n T )
Q ( nT )  a ( nT )  Cos( 2  n    nT )  a ( nt ) Sin nT
2
(18)
Overall Imager Block Diagram
Doppler
Beamformer
Receive
Mux
2
Transducer
Connectors
Digital
Receive
Beamformer
6
3 4 5
Beamformer
Central
Control
Transmit
Demux
Image
Processing
Digital
Transmit
Beamformer
1
System
Control
Imaging System Signals
Doppler
Beamformer
Receive
Mux
2
Transducer
Connectors
Digital
Receive
Beamformer
3
4 5
Beamformer
Central
Control
Transmit
Demux
Image
Processing
Digital
Transmit
Beamformer
6
1
System
Control
Coarse and Fine Beamforming Delays
Coarse
Delay
Control
Fine
Delay
Control
Ho()
Ho()e-j/4
FIFO
MUX
Ho()e-j/2
Input from
ADC at 20
to 40 MHz,
8 to 12 bits
Ho()e-j3/4
To apodization
and further
processing
Output with
delay accuracy
up to 160 MHz
SIGNAL STATISTICS
Recall that the ultrasound signal is the sum of harmonic
components 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.
1
p( y ) 
e
2
 y2
2 2
(19)
The quadrature signal will also be Gaussian with the
same standard deviation
1
p( z ) 
e
2
 z2
2 2
(20)
Since p(y) and p(z) are independent random variables the joint
probability density function is given by
1
p( y, z ) 
e
2


2 2
 y2  z2
1
2
 y2
2
e

1

e
2
z2
2 2
(21)
2 2
The probability of a joint event (corresponding to a particular
amplitude of the envelope) is the probability that:
p(z )
adad
total area = 2 ada
d
da
a
p( y )
The probability that
a lies between
a and a + da is
p ( a )da 
2 a
2
2
a 2
e
2 2 da
a2  y2  z2
So that the probability density function for the radio
frequency signal is given by
p a  
a
2
a 2
e
Rayleigh Prob.
Density function
2 2
many gray pixels
p(a )
few
black
pixels
few white pixels
a
a
The speckle in an ultrasound image is described by this
probability density function. Let’s define the signal as a
and the noise as the rms deviation from this value
Thus
2

Recall

N  a  a    a  a
1
2
a   a pa  da
o


o
a
2

2
a 2
e
2
2
da
2
2

1
2
Thus:

a
SNR  
N


2
2

 
2
2
2
2
1
 2

1
 2  2 2
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.
Speckle Noise in an Ultrasound Image
s0  a0
a
si  a i
x
Let’s make several independent measurements of
so and si
These measurements will form distributions
i
si
0
s0
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:
s0  si
s0

2
0
i
s0
2

1
2
and finally, define our quality factor as the contrast to
speckle noise ratio (CSR)
CSR 
s0  si
0 i
2
2
(26)
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.